Research ArticleComparison Study on the Exact Dynamic StiffnessMethod for Free Vibration of Thin and Moderately ThickCircular Cylindrical Shells

Xudong Chen1 and Kangsheng Ye2

1School of Naval Architecture and Civil Engineering Jiangsu University of Science and Technology Zhangjiagang 215600 China2Department of Civil Engineering Tsinghua University Beijing 100084 China

Correspondence should be addressed to Kangsheng Ye yekstsinghuaeducn

Received 3 September 2016 Revised 21 November 2016 Accepted 24 November 2016

Academic Editor Marcello Vanali

Copyright copy 2016 X Chen and K Ye This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Comparison study on free vibration of circular cylindrical shells between thin and moderately thick shell theories when usingthe exact dynamic stiffness method (DSM) formulation is presented Firstly both the thin and moderately thick dynamic stiffnessformulations are examined Based on the strain and kinetic energy the vibration governing equations are expressed in theHamiltonform for both thin and moderately thick circular cylindrical shells The dynamic stiffness is assembled in a similar way as that inclassic skeletal theory With the employment of the Wittrick-Williams algorithm natural frequencies of circular cylindrical shellscan be obtained A FORTRAN code is written and used to compute the modal characteristics Numerical examples are presentedverifying the proposed computational framework Since the DSM is an exact approach the advantages of high accuracy no-missingfrequencies and good adaptability to various geometries and boundary conditions are demonstrated Comprehensive parametricstudies on the thickness to radius ratio (ℎ119903) and the length to radius ratio (119871119903) are performed Applicable ranges of ℎ119903 are foundfor both thin and moderately thick DSM formulations and influences of 119871119903 on frequencies are also investigated The followingconclusions are reached frequencies of moderately thick shells can be considered as alternatives to those of thin shells with highaccuracy where ℎ119903 is small and 119871119903 is large without any observation of shear locking

1 Introduction

Free vibration analysis of shell structures is important forfurther dynamic analysis in engineering Among all shelltypes circular cylindrical shells are the most widely used in agood variety of applications for example tubes containmentvessels and aircraft fuselagesTherefore exact modal charac-teristic analysis of circular cylindrical shells is of considerablescientific and engineering significance

It is well known that shell theories can be categorised intothin moderately thick and thick forms Since the Kirchhoff-Love hypothesis for thin elastic shell theory was established[1] much effort was spent on its vibration developmentArnold and Warburton [2] analysed the flexural vibrationof thin cylindrical shells with general boundary conditionsusing Rayleighrsquos principle Leissa [3] provided a systematic

review and summary on the vibration of different shelltheories and shell types Bardell et al [4] investigated thefree vibration of isotropic open cylindrical shell panels byapplying an ℎ-119901 version finite element method Tan [5]introduced a substructuring method for predicting the nat-ural frequencies of circular cylindrical shells with arbitraryboundary conditions

On the other hand moderately thick shell theory whichderives from the Reissner-Mindlin hypothesis is less fre-quently used in analysing the free vibration behaviour ofshells due to its complexity Naghdi and Cooper [6] con-ducted vibration analysis on circular cylindrical shells withthe consideration of shear deformation and rotary inertia byemploying elastic wave method Mirsky and Herrmann [7]studied the nonaxisymmetric vibration of moderately thickcircular cylindrical shells Tornabene and Viola [8] proposed

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 9748135 14 pageshttpdxdoiorg10115520169748135

2 Shock and Vibration

an approach to analyse free vibration of parabolic shells usingGeneralised Differential Quadrature method (GDQ) basedon the first-order shear deformation theory Mantari et al[9] applied a new accurate higher-order shear deformationtheory to perform free vibration analysis of multilayeredshells

Although the free vibration of shells has been extensivelyexamined there are continuing research efforts made withnew methods and shell types for example 119901-version mixedfinite element by Kim et al [10] GDQ method for doublycurved shells of revolution by Tornabene et al [11 12] higher-order theory for thin-walled beams by Pagani et al [13] andthree-dimensional analysis method by Ye et al [14 15]

The dynamic stiffness method (DSM) is an exact methodsince the dynamic stiffness is computed directly from theexact vibration governing equations It is suitable for com-putational analysis of free vibration of continuous systemswith infinite degrees of freedom In conjunction with theWittrick-Williams (W-W) algorithm [16 17] and the recursiveNewtonrsquos method [18 19] the DSM has been employed infree vibration of skeletal structures for example nonuniformTimoshenko beams [20] three-layered sandwich beams [21]and functionally graded beams [22] Pagani et al [23]employed the exact dynamic stiffness elements to investigatethe free vibration of thin-walled structures A recent paper[24] applied the DSM to the free vibration analysis ofthin circular cylindrical shells The present authors used toemploy the DSM to investigate the free vibration of shells ofrevolution [25] and the complete theory was given in [26]based on the thin shell assumption Further extension of theDSM to moderately thick circular cylindrical shells [27] wasmade with the consideration of transverse shear deformationand rotary inertia

As an exactmethod the dynamic stiffnessmethod has thefollowing advantages

(1) Themethod is exact and theoretically solutions can beobtained with any desired accuracy

(2) The original eigenvalue problems of two-dimensionalpartial differential equations (PDEs) are degraded tothe eigenvalue problems of a set of one-dimensionalordinary differential equations (ODEs) by introduc-ing the circumferential vibration modes Since themethod is exact frequencies and vibration modes ofany order can be calculated with a small number ofelements

(3) The Wittrick-William algorithm is mathematicallystrict and ensures that no eigenvalue will be omitted

(4) TheDSM formulation is general and versatilemakingit capable of solving the free vibration of circularcylindrical shells with a variety of geometries andarbitrary boundary conditions

Though the free vibration formulation using the DSM hasbeen established for both thin and moderately thick circularcylindrical shells there is no comprehensive comparisonstudy available currently to address the applicability of DSMformulations between the two different shell theories Thispaper attempts to show the compatibility on the free vibration

h

rO

L

u

vw

x

120579

Figure 1 Coordinates of a circular cylindrical shell

of circular cylindrical shells between thin and moderatelythick DSM formulationsTheDSM formulations of both thinand moderately thick circular cylindrical shells are well com-pared in Sections 2 and 3 Afterwards the Wittrick-Williamsalgorithm is presented providing a solution to computingthe number of clamped-end natural frequencies 1198690 Fournumerical examples are given in Section 5 demonstrating thecapability and reliability of the proposed method Extensiveparametric studies are performed in Section 6 on the ratiosof thickness to radius (ℎ119903) and length to radius (119871119903) tofind out the applicable ranges of each theory Differencesbetween frequencies from thin and moderately thick DSMformulations are analysed The following conclusions arereached in the final the compatibility of DSM between thinand moderately thick formulations is proven without shearlocking

2 Vibration Equations

This section theoretically presents a comparison study onDSM formulations for free vibration of both thin andmoderately thick circular cylindrical shells Figure 1 showsthe coordinate system of a homogeneous isotropic andcircumferentially closed circular cylindrical shell The axialand circumferential directions of the shell are termed as119909 and120579 respectivelyThe length of the shell is 119871 radius is 119903 and thethickness is ℎ Youngrsquos modulus is 119864 Poissonrsquos ratio is ] andthe density is 120588

For thin shells the vibration can be represented by thetranslational displacements of the middle surface Δthin =119906 V 119908T along 119909 120579 and 119903 directions while for moderatelythick shells the angular displacements 120595119909 120595120579T of themiddle surface along 119909 and 120579 directions are independentwhich results in an expansion of the displacement vector toΔthick = 119906 V 119908 120595119909 120595120579T

The strain-displacement relationship at the middle sur-face of a circular cylindrical shell can be written in the formof

120576 = [L] Δ (1)

where 120576 is the strain vector and Δ is the displacementvector which can be Δthin or Δthick as mentioned aboveThe expressions of the differential matrix [L] for both thinand moderately thick circular cylindrical shells can be foundin [3]

Shock and Vibration 3

dx

h

rd120579

Figure 2 The diagram of an infinitesimal portion of a circularcylindrical shell

Similarly the internal force-strain relationship at themiddle surface of a circular cylindrical shell can also beexpressed as

N = [D] 120576 (2)

where N is the internal force vector and [D] is the stiffnessmatrix which can also be found in [3]

Denote 119889119860 as the area of an infinitesimal portion in themiddle surface of a circular cylindrical shell (see Figure 2)and the area of this infinitesimal portion is 119889119860 = 119903119889120579119889119909The total strain energy 119880 for free vibration of the circularcylindrical shell can be integrated along the whole middlesurface 119878 as

119880 = ∬119878

12 120576T [D] 120576 119889119860 (3)

and the total free vibration kinetic energy is obtained as

119879 = 1205882 ∬119878ℎ [(120597119906120597119905 )

2 + (120597V120597119905 )2 + (120597119908120597119905 )

2]119889119860 (4)

for thin shell or

119879 = 1205882 ∬119878ℎ[(120597119906120597119905 )

2 + (120597V120597119905 )2 + (120597119908120597119905 )

2]+ ℎ312 [(120597120595119909120597119905 )2 + (120597120595120579120597119905 )2]119889119860

(5)

for moderately thick circular cylindrical shellsThe dynamic displacement functions Δthin and Δthick

are expressed differently as

Δthin = 119906119899 (119909) cos 119899120579 sin120596119905V119899 (119909) sin 119899120579 sin120596119905119908119899 (119909) cos 119899120579 sin120596119905

Δthick =

119906 = 119906119899 (119909) cos 119899120579 sin120596119905V = V119899 (119909) sin 119899120579 sin120596119905119908 = 119908119899 (119909) cos 119899120579 sin120596119905120595119909 = 120595119909119899 (119909) cos 119899120579 sin120596119905120595120579 = 120595120579119899 (119909) sin 119899120579 sin120596119905

(6)

(e)

0 Lxa xb

Figure 3 The mesh division along the axis

119899 is the circumferential wave number 120596 is a circular fre-quency 119906119899(119909) V119899(119909) 119908119899(119909) 120595119909119899(119909) and 120595120579119899(119909) are axialvibration functions

The Hamilton principle suggests

120575int11990521199051

(119880 minus 119879) 119889119905 = 0 (7)

where (1199051 1199052) is an arbitrary time interval during vibrationThe equations ofmotion can be derived from (7) Substitutingthe dynamic displacement functions (6) into equations ofmotion the free vibration governing differential equationscan be obtained By rewriting the governing differentialequations into the Hamilton form generalised displacementsu and forces k are obtained where u and k are given inthe Appendix

The vibration governing equations can be written in theHamilton form of

[J] z1015840 = [S] z (8)

where ()1015840 denotes the differentiation with respect to 119909 andz = u kT [J] = [ 0 minusII 0 ] and I is an identity matrixwith the order of four (or five) for thin (or moderately thick)shells The coefficient matrix [S] is a symmetric eighth- (ortenth-) order matrix for thin (or moderately thick) circularcylindrical shells Details of [S] can also be found in theAppendix

3 Dynamic Stiffness Matrix

Should the circumferential wave number 119899 be a specific value(8) is one-dimensional with respect to 119909 only Thereforea circular cylindrical shell can be analysed as a generalone-dimensional skeleton-like structure with four (or five)degrees of freedom 1199061 1199062 1199063 1199064 (or 1199061 1199062 1199063 1199064 1199065)for thin (ormoderately thick) shells Accordingly the internalforce vectors for both thin andmoderately thick shell theoriesare V1 V2 V3 V4T and V1 V2 V3 V4 V5T respectivelyThedynamic stiffness of free vibration of a circular cylindricalshell can now be set up in a similar way as that in classic one-dimensional skeletal theory

Figure 3 shows that a circular cylinder is divided into sev-eral shell segments along its axial direction Taking element(119890) for example the 119909 coordinates at its two ends are 119909119886 and

4 Shock and Vibration

119909119887 respectivelyThe segment-end displacement vector d119890 isdefined as

d119890thin = 1199061 (119909119886) 1199062 (119909119886) 1199063 (119909119886) 1199064 (119909119886) 1199061 (119909119887) 1199062 (119909119887) 1199063 (119909119887) 1199064 (119909119887)T

d119890thick = 1199061 (119909119886) 1199062 (119909119886) 1199063 (119909119886) 1199064 (119909119886) 1199065 (119909119886) 1199061 (119909119887) 1199062 (119909119887) 1199063 (119909119887) 1199064 (119909119887) 1199065 (119909119887)T

(9)

and the force vector F119890 isF119890thin = minusV1 (119909119886) minusV2 (119909119886) minusV3 (119909119886)

minus V4 (119909119886) V1 (119909119887) V2 (119909119887) V3 (119909119887) V4 (119909119887)T

F119890thick = minusV1 (119909119886) minusV2 (119909119886) minusV3 (119909119886) minusV4 (119909119886) minus V5 (119909119886) V1 (119909119887) V2 (119909119887) V3 (119909119887) V4 (119909119887) V5 (119909119887)T

(10)

Mathematically apply the boundary conditions in (11) or (12)to (8) in turn

d119890thin = 119890119895 119895 = 1 8 (11)

d119890thick = 119890119895 119895 = 1 10 (12)

where 119890119895 is a unit vector with the 119895th element equal to oneand the corresponding element dynamic stiffness matrix [k]119890is formulated by

[k]119890thin = [F119890119895=1 F119890119895=2 F119890119895=3 F119890119895=4 F119890119895=5 F119890119895=6 F119890119895=7 F119890119895=8]thin [k]119890thick = [F119890119895=1 F119890119895=2 F119890119895=3 F119890119895=4 F119890119895=5 F119890119895=6 F119890119895=7 F119890119895=8 F119890119895=9 F119890119895=10]thick

(13)

Due to the complexity (8) is solved computationally usingadaptive ordinary differential equations (ODEs) solver COL-SYS [33 34] The global dynamic stiffness matrix K of acircular cylindrical shell is assembled in a regular FEMway

4 Wittrick-Williams Algorithm

The dynamic stiffness method is used in conjunction withtheWittrick-Williams (W-W) algorithmTheW-Walgorithmmerely gives the number of frequencies below 120596lowast with theexpression of

119869 = 1198690 + 119869119870 (14)

where 119869 is the total number of frequencies exceeded bythe trial frequency 120596lowast 119869119870 = 119904K(120596lowast) is the sign countof the global dynamic stiffness matrix K and equal to thenumber of negative elements on the diagonal of an uppertriangular matrix obtained from K by applying standardGaussian eliminationwithout row exchange 1198690 is the numberof clamped-end frequencies exceeded by 120596lowast and can beaccumulated from 1198691198900 of each shell segment element over thewhole meridian as is expressed in

1198690 = sum1198691198900 (15)

To solve 1198691198900 a substructuring method is employed by takingadvantage of the self-adaptability of COLSYSWhen comput-ing the dynamic stiffness matrix of a shell segment elementfor example element (119890) a submesh will be generated byCOLSYS adaptively as is shown in Figure 4

Applying the Wittrick-Williams algorithm again on thesubmesh we have

1198691198900 = sum119869()0 + 119904 K (120596lowast) (16)

where notation ldquordquo stands for submesh Since COLSYS iscapable of controlling the error tolerance adaptively thenumber of clamped-end natural frequencies exceeded by 120596lowaston the submesh has to be zero suggesting 119869()0 = 0 Otherwisethe clamped-end modes can be enlarged infinitely resultingin the unsatisfaction of the given error tolerance

The following will introduce the formulation of K(120596lowast) in(16) Taking a subelement () with the coordinates (119909119894 119909119894+1)on element (119890) for instance its corresponding dynamicstiffness matrix [k]() can be obtained by linearly combiningthe eight (or ten) solutions of (8) and (11) or (12) Note thatthe enddisplacement and force vectors for this subelement ()are d() and F() respectively and the subelement stiffnessmatrix [k]() satisfies

[k]() d() = F() (17)

By applying the boundary conditions in (11) or (12) in turnagain for thin or moderately thick circular cylindrical shells(18) is obtained

[k]() [B]() = [C]() (18)

where

[B]()= [d()119895=1 d()119895=2 d()119895=3 d()119895=4 d()119895=5 d()119895=6 d()119895=7 d()119895=8]thin [C]()= [F()119895=1 F()119895=2 F()119895=3 F()119895=4 F()119895=5 F()119895=6 F()119895=7 F()119895=8]thin

(19)

Shock and Vibration 5

xa xi xi+1 xb

(e)

Figure 4 Submesh on element (119890)

for thin shells or

[B]() = [d()119895=1 d()119895=2 d()119895=3 d()119895=4 d()119895=5 d()119895=6 d()119895=7 d()119895=8 d()119895=9 d()119895=10]thick [C]() = [F()119895=1 F()119895=2 F()119895=3 F()119895=4 F()119895=5 F()119895=6 F()119895=7 F()119895=8 F()119895=9 F()119895=10]thick

(20)

for moderately thick shellsSince 119869()0 = 0 [B]() is nonsingular (otherwise120596lowast is one of

clamped-end frequencies which is conflicted with 119869()0 = 0)and (21) is obtained

[k]() = [C]() ([B]())minus1 (21)

Thus K(120596lowast) can be assembled regularly from [k]() Based onthe above theories a FORTRAN code was written and usedto solve the free vibration of both thin and moderately thickcircular cylindrical shells in this paper

5 Validation

In this section four examples on free vibration of circularcylindrical shells are presented to show the reliability andaccuracy of the proposed method in the above sectionsExamples in Sections 51 and 52 are calculated with thin shelltheory in Section 53 both thin and moderately thick shelltheories are employed to study the free vibration of the sameshell and results are given opposite for comparison examplesin Section 54 are performed with the moderately thick shelltheory

51 Free Vibration of a Thin Clamped-Clamped Shell Zhanget al [28] and Xuebin [29] analysed the natural frequenciesof a clamped-clamped (C-C) circular cylindrical shell basedon the Flugge thin shell theory by employing the FEM codeMSCNASTRAN and wave propagation approach respec-tively The clamped boundary condition is considered as

119906 = V = 119908 = 120597119908120597119909 = 0 (22)

The length of the shell is 119871 = 20m radius 119903 = 1m thicknessℎ = 001m Density 120588 = 7850 kgm3 Youngrsquos modulus119864 = 210GPa Poissonrsquos ratio ] = 03 In this example thinshell theory is used in formulating the dynamic stiffness andthe first nine frequencies are tabulated in Table 1 where 119898denotes the number of half-waves along the axial direction

It can be observed from Table 1 that the frequenciesobtained using the present method agree very well withthat from the FEM (MSCNASTRAN) and wave propagation

approach Further investigation into Table 1 shows that thelowest natural frequency of a circular cylindrical usuallycomes with a relatively larger 119899 that is 119899 = 2 in this case

52 Free Vibration of a Thin Clamped-Free Shell In Sharma[30] the free vibration of a clamped-free (C-F) circularcylindrical shell is investigated based on the Rayleigh-Ritzmethod and Flugge shell theory The geometry parametersof the shell are length 119871 = 0502m radius 119903 = 00635mand thickness ℎ = 000163m and the material propertiesare Youngrsquos modulus 119864 = 210GPa Poissonrsquos ratio ] = 028and density 120588 = 7800 kgm3 A thin shell formulation is usedin the present dynamic stiffness approach Table 2 presentsthe lowest four frequencies of the shell with a relatively largercircumferential wave number 119899 = 5 and 6

It can be concluded that the results agree very well with[30] and the errors are no larger than 04 By comparingthe results from the present method and the Rayleigh-Ritzmethod used in Sharma [30] the reliability of the presentmethod is verified

53 Free Vibration of a Shell with Different Boundary Condi-tions Pagani et al [23] applied the one-dimensional higher-order beam theory and exact dynamic stiffness elements tostudy the free vibration of solid and thin-walled structuresin which a thin-walled cylinder with a circular cross-section(Figure 5) is analysed In a recent paper [13] they employedboth the radial basis functions (RBFs) method and CarreraUnified Formulation (CUF) to examine the free vibrationbehaviour of both open and closed thin-walled structures

The length of the examined cylinder is 119871 = 20mouter diameter is 119889 = 2m and thickness is ℎ = 002mThe material properties are Youngrsquos modulus 119864 = 75GPaPoissonrsquos ratio ] = 033 and density 120588 = 2700 kgm3 Table 3shows the natural frequencies of this thin-walled cylindricalbeam with different boundaries namely both ends simplysupported (S-S) both ends clamped (C-C) clamped-free(C-F) and free-free (F-F) The first and second flexural(bending) shell-like and torsional natural frequencies arepresented Solutions from the present thin and moderatelythick DSM 2D MSCNASTRAN and [13 23] are given forcomparison

6 Shock and Vibration

Table 1 The lowest nine frequencies of the C-C circular cylindricalshell (Hz)

Order (119898 119899) Present Reference [28] Reference [29]1 (1 2) 1200 1225 12132 (1 3) 1956 1964 19613 (2 3) 2310 2318 23284 (2 2) 2716 2769 28065 (1 1) 2829 mdash 30096 (3 3) 3148 316 31977 (1 4) 3642 367 36488 (2 4) 3728 3755 37389 (3 4) 3960 3987 3977mdash not given

Table 2The lowest four frequencies of the C-F shell with 119899 = 5 and6 (Hz)

119899 = 5 119899 = 6Present Sharma [30] Present Sharma [30]

1 23619 23666 34628 346972 24021 24064 34990 350503 25050 25091 35768 358074 27062 27160 37120 37166

It can be observed that the natural frequencies computedfrom the present dynamic stiffness method agree very wellwith that from 2D MSCNASTRAN code for both thin andmoderately thick shell theories Comparing with the datafrom the higher-order beam theories [13 23] the presentmethod provides an excellent agreement for flexural andtorsional modes The differences of frequencies for the firstand second shell-like modes are larger and this can belargely attributed that the formulations established in [1323] are based on higher-order beam theories Obviouslybetter agreement can be achieved if exact shell theories areemployed by [13 23] Further investigation by the presentDSM approach shows that 119899 = 1 corresponds to flexural(bending) mode while 119899 = 0 and 2 correspond to tor-sional and shell-like vibration modes respectively whichdemonstrate that the current DSM approach is capable ofshowing the correct vibration modes as presented in existingliterature

54 Free Vibration of a Moderately Thick Shell Bhimaraddi[32] analysed the free vibration of a simply supported (S-S) moderately thick circular cylindrical shell by using thehigher-order displacement model with the consideration ofin-plane inertia rotary inertia and shear effect In [31]natural frequencies of the circular cylindrical shell subjectsto different ℎ119903 and 119871119903 are computed and compared withthe solutions from 3D elasticity theory Results from themoderately thick DSM approach are also presented and listedin Table 4 In this example Poissonrsquos ratio is 03 and 1205960 =(120587ℎ)radic119866120588 where 119866 is the shear modulus

According to Table 4 results from the present methodagree very well with those from [31 32] demonstrating the

h d

Figure 5 The cross-section of a thin-walled cylinder

high accuracy and adaptability of the present method forcircular cylindrical shells with different ℎ119903 and 1198711199036 Parametric Study

61 Ratio of Thickness to Radius (ℎ119903) An important param-eter to determine whether a shell is thin or moderatelythick is the thickness to radius ratio (ℎ119903) According tothe classic shell theory a critical value for ℎ119903 between thinand moderately thick shell theories is 005 In this sectioninfluences of ℎ119903 on natural frequencies are investigatedextensively from 0005 to 01 for both thin and moderatelythick DSM formulations

611 Low Order Frequencies Consider a circular cylindricalshell with 119871119903 = 12 and Poissonrsquos ratio ] = 03 Toinvestigate the differences of natural frequencies between thinand moderately thick DSM formulations a notation 120578 in (23)is introduced as

120578 = 10038161003816100381610038161003816100381610038161003816ΩthinΩthick

minus 110038161003816100381610038161003816100381610038161003816 (23)

where Ωthin and Ωthick are the nondimensional frequencyparameters defined as

Ω = 120596119903radic120588 (1 minus ]2)119864 (24)

Consequently log10(120578) is the index of differenceFigure 6 shows the log10(120578) for the first frequencies (119898 =1) under 119899 = 0 1 2 and 3 between thin and moderately thick

DSM formulations with a variety of boundary conditionsfor example clamped-clamped (C-C) clamped-free (C-F)clamped-simply supported (C-S) and both ends simplysupported (S-S) In these cases ℎ119903 is investigated from 0025to 01

If log10(120578) le minus2 (which suggests the difference is nolarger than 1) it can be considered that results from the thinand moderately thick dynamic stiffness formulations are ingood agreement with each other Furthermore frequenciescomputed from themoderately thick shell theory can be usedas good alternatives to those computed from the thin shelltheory Referring to Figure 6 the following conclusions canbe reached

Shock and Vibration 7

Table 3 Natural frequencies of the thin-walled cylinder with different boundary conditions (Hz)

BCs Model I flexural II flexural I shell-like II shell-like I torsional II torsional

S-S

Present thin 14025 51506 14851 22931 80790 161579Present thick 14021 51502 14828 22877 80792 161585NAS2D [23] 13978 51366 14913 22917 80415 160810Reference [23] 14022 51503 18405 25460 80786 161573Reference [13] 14294 51567 18608 25574 80639 162551

C-C

Present thin 28577 69110 17350 30239 80790 161579Present thick 28574 69105 17327 30193 80792 161585NAS2D [23] 28498 68960 17396 30225 80415 160810Reference [23] 28576 69110 20484 32222 80786 161573Reference [13] 28354 69096 20463 31974 80838 161579

C-F

Present thin 5081 29092 14153 17427 40395 121184Present thick 5076 29086 14140 17370 40396 121188NAS2D [23] 5059 29001 14235 17435 40209 120620Reference [23] 5077 29090 23069 25239 40394 121181Reference [13] 5047 29002 23003 24979 40431 121203

F-F

Present thin 30939 77045 14036 14105 80788 161576Present thick 30931 77038 14032 14076 80792 161585NAS2D [23] 30829 76806 14129 14171 80415 160810Reference [23] 30932 77043 22987 23053 80789 161577Reference [13] 30945 77052 22864 23048 80787 161592

Table 4 Lowest natural frequencies (times1205960) of a moderately thick circular cylindrical shell with different circumferential wave numbers 119899119871119903 ℎ119903 = 012 ℎ119903 = 018119899 = 1 119899 = 2 119899 = 3 119899 = 4 119899 = 1 119899 = 2 119899 = 3 119899 = 42

Present 003724 002349 002449 003664 005632 003893 004939 007726Reference [31] 003730 002359 002462 003686 005652 003929 004996 007821Reference [32] 003730 002365 002470 003687 005653 003944 005009 007833

1Present 005861 004969 004765 005508 009420 008501 008995 011056

Reference [31] 005853 004978 004789 005545 009402 008545 009093 011205Reference [32] 005856 004986 004799 005550 009409 008562 009109 011202

05Present 010087 010238 010658 011463 018915 019406 020451 022180

Reference [31] 010057 010234 010688 011528 018894 019467 020616 022450Reference [32] 010047 010224 010674 011508 018832 019403 020544 022361

025Present 027310 027650 028219 029020 049751 050299 051216 052506

Reference [31] 027491 027849 028447 029287 050338 050937 051934 053325Reference [32] 027286 027641 028233 029064 049818 050418 051416 052808

(1) For low order frequencies good agreement is reachedbetween thin and moderately thick dynamic stiffnessformulations with ℎ119903 varying from 0025 to 01 Thesmaller the ℎ119903 the smaller the difference

(2) The difference between frequencies from thin andmoderately thick DSM formulations increases withthe increase of 119899 as log10(120578) always has a smallermagnitude under 119899 = 0 than those under 119899 = 1 2and 3

(3) It can be argued that the difference is insensitiveto boundary conditions The differences are roughlywithin 10minus5sim10minus2 in this investigation and the trends

of curves are similar for different boundary condi-tions

612 High Order Frequencies Investigation on the differ-ences between high order frequencies computed from thethin and moderately thick DSM formulations is examined inthis subsection Since it is known that the DSM formulationis not sensitive to boundary conditions [26] a C-S circu-lar cylindrical shell is studied The geometry and materialproperties are the same to the example in Section 611 if notspecifically stated

For high order frequencies ℎ119903 is investigated from verysmall value 0005 to a relatively large value 01 and the first

8 Shock and Vibration

005 0075 010025ℎr

minus1

minus2

minus3

minus4

minus5log 1

0(120578

)

n = 0

n = 1

n = 2

n = 3

(a) C-C

005 0075 010025ℎr

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)n = 0

n = 1

n = 2

n = 3

(b) C-F

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

n = 0

n = 1

n = 2

n = 3

005 0075 010025ℎr

(c) C-S

005 0075 010025ℎr

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

n = 0

n = 1

n = 2

n = 3

(d) S-S

Figure 6 The differences of low order frequencies between thin and moderately thick DSM formulations

ten frequencies under 119899 ranging from 0 to 10 from boththin and moderately thick DSM formulations are computedand compared The differences are shown in Figure 7 andconcluded as follows

(1) When ℎ119903 le 0025 log10(120578) is smaller than 10minus2most of the time which suggests that frequenciescomputed frommoderately thick shell theory are well

compatible with those from thin shell theory Noshear locking is observed

(2) When ℎ119903 ge 005 only frequencies with a small 119899(eg 119899 le 2) produce an acceptable difference (sim10minus2)With the increase of n the difference becomes quitenoticeable The higher the 119899 the more obvious thedifference

Shock and Vibration 9

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

(a) ℎ119903 = 0005

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

(b) ℎ119903 = 001

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

2 4 6 8 100m

(c) ℎ119903 = 0025

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

(d) ℎ119903 = 005

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

(e) ℎ119903 = 0075

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

(f) ℎ119903 = 01

Figure 7 The differences for frequencies between thin and moderately thick C-S circular cylindrical shell theories

(3) The order 119898 under a specific circumferential wavenumber 119899 does not have a significant impact on thedifferences as most curves in Figure 7 are largely flatand stable with the increase of119898

62 Ratio of Length to Radius (119871119903) Parameter 119871119903 controlsthe relative length of a circular cylindrical shell A larger119871119903 indicates a long shell and vice versa In this subsectionthe frequency trends and the compatibility between thin andmoderately thick DSM formulations with respect to 119871119903 arestudied In this subsection a C-C circular cylindrical shell

with ] = 03 is examined Figure 8 plots the trends of the firstfrequency parameterΩ under each 119899 with 119871119903 from 02 to 10

According to Figure 8 for a typical long circular cylin-drical shell (eg 119871119903 = 10) the lowest natural frequency isobtained with 119899 = 2 with the decrease of 119871119903 the circumfer-ential wave number 119899 corresponding to the lowest natural fre-quency slightly increases to 119899 = 3 or 4 when 119871119903 is very small(eg119871119903 = 02 and the shell is actually degraded to a ring) thefirst natural frequency under each 119899 increases monotonicallyand the smallest is reached at 119899 = 0The trends shown by Fig-ure 8 are also in good agreementwith other researchersrsquo work

10 Shock and Vibration

Ω

0

2

4

6

8

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(a) ℎ119903 = 0025 moderately thick shell

Ω

0

2

4

6

8

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(b) ℎ119903 = 0025 thin shell

0

3

6

9

12

Ω

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(c) ℎ119903 = 005 moderately thick shell

0

3

6

9

12

Ω

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(d) ℎ119903 = 005 thin shell

Figure 8 The trends of the first natural frequency parameters Ω under different 119899for example Markus [35] These obtained trends are applica-ble to both thin and moderately thick DSM formulations

Figure 9 shows the differences of the first frequenciesunder each 119899 computed from the thin and moderately thickDSM formulations of the circular cylindrical shell with ℎ119903 =0025 and 005 It can be understood fromFigure 9 that (1) thesmaller the ℎ119903 the smaller the difference (2) the larger the119871119903 the smaller the difference (3) conditions of ℎ119903 le 0025and 119899 le 10 need to be satisfied and a minor difference(log10(120578) le 10minus2) is required for a normal length shell thatis 119871119903 ge 2

7 Conclusions

This paper conducts comparison study on the frequenciesfrom thin and moderately thick circular cylindrical shellscomputed using the DSM formulations and comprehensiveparametric studies on ℎ119903 and 119871119903 are performed Based onthese investigations conclusions are outlined as follows

(1) Natural frequencies computed from the moder-ately thick DSM formulations can be consideredas satisfactory alternatives to those from thin DSMformulations when ℎ119903 le 0025 and 119871119903 ge 2

Shock and Vibration 11

0

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(a) ℎ119903 = 0025

0

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(b) ℎ119903 = 005

Figure 9 Differences between the first frequencies under each 119899 with different 119871119903

(differences are around or smaller than 10minus2) Sincethe DSM is exact no shear locking is observed

(2) Differences between frequencies from thin and mod-erately thick DSM formulations change with a varietyof factors The smaller ℎ119903 the smaller the differencethe smaller119871119903 the larger the difference the larger the119899 the larger the difference

It is proved that moderately thick DSM formulations canbe used to analyse the free vibration of normal length thin

circular cylindrical shells without shear locking demon-strating that the applicable range of moderately thick DSMformulations is expanded These merits should be attributedto that the dynamic stiffness method is an exact method

Though this comparison study is performed on circularcylindrical shells similar conclusions can be extended to thefree vibration of other shaped shells of revolution

Appendix

uthin =

1199061 = 1199061198991199062 = V119899

1199063 = 1199081198991199064 = 1199081015840119899

kthin =

V1 = 119903120587119864ℎ1 minus ]2[]119899119903 V119899 + ]

1119903119908119899 + 1199061015840119899]V2 = 119903120587119864ℎ2 (1 + ]) [minus119899119903 119906119899 + (1 + ℎ2121199032) V1015840119899 + 119899ℎ261199032 1199081015840119899]V3 = 119903120587119864ℎ2 (1 minus ]2) [119899ℎ

2

61199032 V1015840119899 + (2 minus ]) 1198992ℎ261199032 1199081015840119899 minus ℎ26 119908101584010158401015840119899 ]V4 = 119903120587119864ℎ12 (1 minus ]2) [minus]119899ℎ

2

1199032 V119899 minus ]1198992ℎ21199032 119908119899 + ℎ211990810158401015840119899 ]

12 Shock and Vibration

uthick =

1199061 = 1199061198991199062 = V119899

1199063 = 1199081198991199064 = 1205951199091198991199065 = 120595120579119899

kthick =

V1 = 119903120587119864ℎ1 minus ]2[1199061015840119899 + ]

119899119903 V119899 + ]119903119908119899]V2 = 12 119903120587119864ℎ1 + ]

[minus119899119903 119906119899 + (1 + ℎ2121199032) V1015840119899 minus 119899ℎ2121199032120595119909119899 + ℎ21211990321205951015840120579119899]V3 = 119903120587120581119866ℎ [1199081015840119899 + 120595119909119899]V4 = ℎ212 119903120587119864ℎ1 minus ]2

[1205951015840119909119899 + ]119899119903120595120579119899]

V5 = ℎ224 119903120587119864ℎ1 + ][V1015840119899119903 minus 119899119903120595119909119899 + 1205951015840120579119899]

[S]thin =

[[[[[[[[[[[[[[[[[[

11990411 0 0 11990414 0 11990416 0 011990422 11990423 0 11990425 0 0 1199042811990433 0 11990435 0 0 1199043811990444 0 11990446 11990447 0

11990455 0 0 0sym 11990466 0 0

0 011990488

]]]]]]]]]]]]]]]]]]

(A1)

where nonzero elements in the upper triangular domain are

11990411 = 120588120587119903ℎ1205962 minus 120587119864ℎ11989922119903 (1 minus ]) + 61198992119903120587119864ℎ(121199032 + ℎ2) (1 + ]) 11990414 = minus 1198992ℎ3120587119864(121199032 + ℎ2) (1 + ]) 11990416 = 12119899119903121199032 + ℎ2 11990422 = 120588120587119903ℎ1205962 minus 1198992120587119864ℎ (121199032 + ℎ2)

121199033 11990423 = minus120587119864ℎ121198991199032 + 1198993ℎ2121199033 11990425 = minus]119899119903 11990428 = ]

1198991199032 11990433 = 120588ℎ1199031205871205962 minus 120587119864ℎ119903 minus 1198994ℎ3120587119864121199033

11990435 = minus]119903 11990438 = ]11989921199032 11990444 = minus 21198992ℎ3119903120587119864(121199032 + ℎ2) (1 + ])

11990446 = minus 2119899ℎ2121199032 + ℎ2 11990447 = 111990455 = 1 minus ]2119903120587119864ℎ 11990466 = 24119903 (1 + ])120587119864ℎ (121199032 + ℎ2)

11990488 = 12 (1 minus ]2)119903120587119864ℎ3

[S]thick

Shock and Vibration 13

=

[[[[[[[[[[[[[[[[[[[[[[[

11990411 0 0 0 0 0 11990417 0 0 1199041lowast1011990422 11990423 0 11990425 11990426 0 0 0 011990433 0 11990435 11990436 0 0 0 0

11990444 0 0 0 11990448 0 1199044lowast1011990455 0 0 0 11990459 011990466 0 0 0 0

11990477 0 0 1199047lowast10sym 11990488 0 0

11990499 011990410lowast10

]]]]]]]]]]]]]]]]]]]]]]]

(A2)

where nonzero elements in the upper triangular domain are

11990411 = 119903120587120588ℎ120596211990417 = 119899119903

1199041lowast10 = minus 1198991199032 11990422 = 120587(119903120588ℎ1205962 minus 120581119866ℎ119903 minus 1198992119864ℎ119903 ) 11990423 = minus119899120587119903 (119864ℎ + 120581119866ℎ) 11990425 = 120587120581119866ℎ11990426 = minus]119899119903 11990433 = 120587(119903120588ℎ1205962 minus 119864ℎ119903 minus 1198992120581119866ℎ119903 ) 11990435 = 119899120587120581119866ℎ11990436 = minus]119903 11990444 = 1199031205871205881205962 ℎ312 11990448 = minus1

1199044lowast10 = 119899119903 11990455 = 120587(1199031205881205962 ℎ312 minus 1198992119864ℎ312119903 minus 119903120581119866ℎ) 11990459 = minus]119899119903 11990466 = 1 minus ]2119903120587119864ℎ 11990477 = 2 (1 + ])119903120587119864ℎ

1199047lowast10 = minus2 (1 + ])1199032120587119864ℎ 11990488 = 1119903120587120581119866ℎ 11990499 = 12 (1 minus ]2)

119903120587119864ℎ3 11990410lowast10 = 2 (1 + ])119903120587119864ℎ (12ℎ2 + 11199032 )

120581 = 120587212 (A3)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully appreciate the financial support fromNational Natural Science Foundation of China (51078198)Tsinghua University (2011THZ03) Jiangsu University ofScience and Technology (1732921402) and Shuang-ChuangProgram of Jiangsu Province for this research

References

[1] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover England UK 1927

[2] R N Arnold and G B Warburton ldquoThe flexural vibrationsof thin cylindersrdquo Proceedings of the Institution of MechanicalEngineers vol 167 no 1 pp 62ndash80 1953

[3] A W Leissa ldquoVibration of shellsrdquo Report 288 National Aero-nautics and Space Administration (NASA) Washington DCUSA 1973

[4] N S Bardell J M Dunsdon and R S Langley ldquoOn thefree vibration of completely free open cylindrically curvedisotropic shell panelsrdquo Journal of Sound and Vibration vol 207no 5 pp 647ndash669 1997

[5] D-Y Tan ldquoFree vibration analysis of shells of revolutionrdquoJournal of Sound and Vibration vol 213 no 1 pp 15ndash33 1998

[6] P M Naghdi and R M Cooper ldquoPropagation of elastic wavesin cylindrical shells including the effects of transverse shearand rotatory inertiardquo The Journal of the Acoustical Society ofAmerica vol 28 no 1 pp 56ndash63 1956

[7] I Mirsky and G Herrmann ldquoNonaxially symmetric motionsof cylindrical shellsrdquo The Journal of the Acoustical Society ofAmerica vol 29 pp 1116ndash1123 1957

[8] F Tornabene and E Viola ldquo2-D solution for free vibrationsof parabolic shells using generalized differential quadraturemethodrdquo European Journal of Mechanics ASolids vol 27 no6 pp 1001ndash1025 2008

[9] J L Mantari A S Oktem and C Guedes Soares ldquoBending andfree vibration analysis of isotropic and multilayered plates andshells by using a new accurate higher-order shear deformationtheoryrdquo Composites Part B Engineering vol 43 no 8 pp 3348ndash3360 2012

14 Shock and Vibration

[10] J G Kim J K Lee and H J Yoon ldquoFree vibration analysisfor shells of revolution based on p-version mixed finite elementformulationrdquo Finite Elements in Analysis and Design vol 95 pp12ndash19 2015

[11] F Tornabene ldquoFree vibrations of laminated composite doubly-curved shells and panels of revolution via the GDQ methodrdquoComputer Methods in Applied Mechanics and Engineering vol200 no 9ndash12 pp 931ndash952 2011

[12] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe local GDQmethod applied to general higher-order theories of doubly-curved laminated composite shells and panels the free vibrationanalysisrdquo Composite Structures vol 116 pp 637ndash660 2014

[13] A Pagani E Carrera and A Ferreira ldquoHigher-order theoriesand radial basis functions applied to free vibration analysisof thin-walled beamsrdquo Mechanics of Advanced Materials andStructures vol 23 no 9 pp 1080ndash1091 2016

[14] T Ye G Jin Y Chen and S Shi ldquoA unified formulationfor vibration analysis of open shells with arbitrary boundaryconditionsrdquo International Journal ofMechanical Sciences vol 81pp 42ndash59 2014

[15] T Ye G Jin S Shi and X Ma ldquoThree-dimensional freevibration analysis of thick cylindrical shells with general endconditions and resting on elastic foundationsrdquo InternationalJournal of Mechanical Sciences vol 84 pp 120ndash137 2014

[16] F W Williams and W H Wittrick ldquoAn automatic computa-tional procedure for calculating natural frequencies of skeletalstructuresrdquo International Journal of Mechanical Sciences vol 12no 9 pp 781ndash791 1970

[17] W H Wittrick and F W Williams ldquoA general algorithmfor computing natural frequencies of elastic structuresrdquo TheQuarterly Journal of Mechanics and Applied Mathematics vol24 pp 263ndash284 1971

[18] S Yuan K Ye F W Williams and D Kennedy ldquoRecursivesecond order convergence method for natural frequencies andmodes when using dynamic stiffness matricesrdquo InternationalJournal for Numerical Methods in Engineering vol 56 no 12pp 1795ndash1814 2003

[19] S Yuan K Ye and FWWilliams ldquoSecond order mode-findingmethod in dynamic stiffness matrix methodsrdquo Journal of Soundand Vibration vol 269 no 3ndash5 pp 689ndash708 2004

[20] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007

[21] J R Banerjee C W Cheung R Morishima M Pereraand J Njuguna ldquoFree vibration of a three-layered sandwichbeam using the dynamic stiffness method and experimentrdquoInternational Journal of Solids and Structures vol 44 no 22-23pp 7543ndash7563 2007

[22] H Su J R Banerjee and C W Cheung ldquoDynamic stiffnessformulation and free vibration analysis of functionally gradedbeamsrdquo Composite Structures vol 106 pp 854ndash862 2013

[23] A Pagani M Boscolo J R Banerjee and E Carrera ldquoExactdynamic stiffness elements based on one-dimensional higher-order theories for free vibration analysis of solid and thin-walled structuresrdquo Journal of Sound and Vibration vol 332 no23 pp 6104ndash6127 2013

[24] N El-Kaabazi and D Kennedy ldquoCalculation of natural fre-quencies and vibration modes of variable thickness cylindricalshells using the Wittrick-Williams algorithmrdquo Computers andStructures vol 104-105 pp 4ndash12 2012

[25] X Chen Research on dynamic stiffness method for free vibrationof shells of revolution [MS thesis] Tsinghua University 2009(Chinese)

[26] XChen andKYe ldquoFreeVibrationAnalysis for Shells of Revolu-tion Using an Exact Dynamic Stiffness Methodrdquo MathematicalProblems in Engineering vol 2016 Article ID 4513520 12 pages2016

[27] X Chen and K Ye ldquoAnalysis of free vibration of moderatelythick circular cylindrical shells using the dynamic stiffnessmethodrdquo Engineering Mechanics vol 33 no 9 pp 40ndash48 2016(Chinese)

[28] X M Zhang G R Liu and K Y Lam ldquoVibration analysisof thin cylindrical shells using wave propagation approachrdquoJournal of Sound and Vibration vol 239 no 3 pp 397ndash4032001

[29] L Xuebin ldquoStudy on free vibration analysis of circular cylin-drical shells using wave propagationrdquo Journal of Sound andVibration vol 311 no 3-5 pp 667ndash682 2008

[30] C B Sharma ldquoCalculation of natural frequencies of fixed-freecircular cylindrical shellsrdquo Journal of Sound and Vibration vol35 no 1 pp 55ndash76 1974

[31] A E ArmenakasDCGazis andGHerrmannFreeVibrationsof Circular Cylindrical Shells PergamonPressOxfordUK 1969

[32] A Bhimaraddi ldquoA higher order theory for free vibrationanalysis of circular cylindrical shellsrdquo International Journal ofSolids and Structures vol 20 no 7 pp 623ndash630 1984

[33] U Ascher J Christiansen and R D Russell ldquoAlgorithm 569COLSYS collocation software for boundary value ODEs [D2]rdquoACM Transactions on Mathematical Software vol 7 no 2 pp223ndash229 1981

[34] U Ascher J Christiansen and R D Russell ldquoCollocationsoftware for boundary-value ODEsrdquo ACM Transactions onMathematical Software vol 7 no 2 pp 209ndash222 1981

[35] S Markus The Mechanics of Vibrations of Cylindrical ShellsElsevier Science Limited 1988

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2 Shock and Vibration

an approach to analyse free vibration of parabolic shells usingGeneralised Differential Quadrature method (GDQ) basedon the first-order shear deformation theory Mantari et al[9] applied a new accurate higher-order shear deformationtheory to perform free vibration analysis of multilayeredshells

Although the free vibration of shells has been extensivelyexamined there are continuing research efforts made withnew methods and shell types for example 119901-version mixedfinite element by Kim et al [10] GDQ method for doublycurved shells of revolution by Tornabene et al [11 12] higher-order theory for thin-walled beams by Pagani et al [13] andthree-dimensional analysis method by Ye et al [14 15]

The dynamic stiffness method (DSM) is an exact methodsince the dynamic stiffness is computed directly from theexact vibration governing equations It is suitable for com-putational analysis of free vibration of continuous systemswith infinite degrees of freedom In conjunction with theWittrick-Williams (W-W) algorithm [16 17] and the recursiveNewtonrsquos method [18 19] the DSM has been employed infree vibration of skeletal structures for example nonuniformTimoshenko beams [20] three-layered sandwich beams [21]and functionally graded beams [22] Pagani et al [23]employed the exact dynamic stiffness elements to investigatethe free vibration of thin-walled structures A recent paper[24] applied the DSM to the free vibration analysis ofthin circular cylindrical shells The present authors used toemploy the DSM to investigate the free vibration of shells ofrevolution [25] and the complete theory was given in [26]based on the thin shell assumption Further extension of theDSM to moderately thick circular cylindrical shells [27] wasmade with the consideration of transverse shear deformationand rotary inertia

As an exactmethod the dynamic stiffnessmethod has thefollowing advantages

(1) Themethod is exact and theoretically solutions can beobtained with any desired accuracy

(2) The original eigenvalue problems of two-dimensionalpartial differential equations (PDEs) are degraded tothe eigenvalue problems of a set of one-dimensionalordinary differential equations (ODEs) by introduc-ing the circumferential vibration modes Since themethod is exact frequencies and vibration modes ofany order can be calculated with a small number ofelements

(3) The Wittrick-William algorithm is mathematicallystrict and ensures that no eigenvalue will be omitted

(4) TheDSM formulation is general and versatilemakingit capable of solving the free vibration of circularcylindrical shells with a variety of geometries andarbitrary boundary conditions

Though the free vibration formulation using the DSM hasbeen established for both thin and moderately thick circularcylindrical shells there is no comprehensive comparisonstudy available currently to address the applicability of DSMformulations between the two different shell theories Thispaper attempts to show the compatibility on the free vibration

h

rO

L

u

vw

x

120579

Figure 1 Coordinates of a circular cylindrical shell

of circular cylindrical shells between thin and moderatelythick DSM formulationsTheDSM formulations of both thinand moderately thick circular cylindrical shells are well com-pared in Sections 2 and 3 Afterwards the Wittrick-Williamsalgorithm is presented providing a solution to computingthe number of clamped-end natural frequencies 1198690 Fournumerical examples are given in Section 5 demonstrating thecapability and reliability of the proposed method Extensiveparametric studies are performed in Section 6 on the ratiosof thickness to radius (ℎ119903) and length to radius (119871119903) tofind out the applicable ranges of each theory Differencesbetween frequencies from thin and moderately thick DSMformulations are analysed The following conclusions arereached in the final the compatibility of DSM between thinand moderately thick formulations is proven without shearlocking

2 Vibration Equations

This section theoretically presents a comparison study onDSM formulations for free vibration of both thin andmoderately thick circular cylindrical shells Figure 1 showsthe coordinate system of a homogeneous isotropic andcircumferentially closed circular cylindrical shell The axialand circumferential directions of the shell are termed as119909 and120579 respectivelyThe length of the shell is 119871 radius is 119903 and thethickness is ℎ Youngrsquos modulus is 119864 Poissonrsquos ratio is ] andthe density is 120588

For thin shells the vibration can be represented by thetranslational displacements of the middle surface Δthin =119906 V 119908T along 119909 120579 and 119903 directions while for moderatelythick shells the angular displacements 120595119909 120595120579T of themiddle surface along 119909 and 120579 directions are independentwhich results in an expansion of the displacement vector toΔthick = 119906 V 119908 120595119909 120595120579T

The strain-displacement relationship at the middle sur-face of a circular cylindrical shell can be written in the formof

120576 = [L] Δ (1)

where 120576 is the strain vector and Δ is the displacementvector which can be Δthin or Δthick as mentioned aboveThe expressions of the differential matrix [L] for both thinand moderately thick circular cylindrical shells can be foundin [3]

Shock and Vibration 3

dx

h

rd120579

Figure 2 The diagram of an infinitesimal portion of a circularcylindrical shell

Similarly the internal force-strain relationship at themiddle surface of a circular cylindrical shell can also beexpressed as

N = [D] 120576 (2)

where N is the internal force vector and [D] is the stiffnessmatrix which can also be found in [3]

Denote 119889119860 as the area of an infinitesimal portion in themiddle surface of a circular cylindrical shell (see Figure 2)and the area of this infinitesimal portion is 119889119860 = 119903119889120579119889119909The total strain energy 119880 for free vibration of the circularcylindrical shell can be integrated along the whole middlesurface 119878 as

119880 = ∬119878

12 120576T [D] 120576 119889119860 (3)

and the total free vibration kinetic energy is obtained as

119879 = 1205882 ∬119878ℎ [(120597119906120597119905 )

2 + (120597V120597119905 )2 + (120597119908120597119905 )

2]119889119860 (4)

for thin shell or

119879 = 1205882 ∬119878ℎ[(120597119906120597119905 )

2 + (120597V120597119905 )2 + (120597119908120597119905 )

2]+ ℎ312 [(120597120595119909120597119905 )2 + (120597120595120579120597119905 )2]119889119860

(5)

for moderately thick circular cylindrical shellsThe dynamic displacement functions Δthin and Δthick

are expressed differently as

Δthin = 119906119899 (119909) cos 119899120579 sin120596119905V119899 (119909) sin 119899120579 sin120596119905119908119899 (119909) cos 119899120579 sin120596119905

Δthick =

119906 = 119906119899 (119909) cos 119899120579 sin120596119905V = V119899 (119909) sin 119899120579 sin120596119905119908 = 119908119899 (119909) cos 119899120579 sin120596119905120595119909 = 120595119909119899 (119909) cos 119899120579 sin120596119905120595120579 = 120595120579119899 (119909) sin 119899120579 sin120596119905

(6)

(e)

0 Lxa xb

Figure 3 The mesh division along the axis

119899 is the circumferential wave number 120596 is a circular fre-quency 119906119899(119909) V119899(119909) 119908119899(119909) 120595119909119899(119909) and 120595120579119899(119909) are axialvibration functions

The Hamilton principle suggests

120575int11990521199051

(119880 minus 119879) 119889119905 = 0 (7)

where (1199051 1199052) is an arbitrary time interval during vibrationThe equations ofmotion can be derived from (7) Substitutingthe dynamic displacement functions (6) into equations ofmotion the free vibration governing differential equationscan be obtained By rewriting the governing differentialequations into the Hamilton form generalised displacementsu and forces k are obtained where u and k are given inthe Appendix

The vibration governing equations can be written in theHamilton form of

[J] z1015840 = [S] z (8)

where ()1015840 denotes the differentiation with respect to 119909 andz = u kT [J] = [ 0 minusII 0 ] and I is an identity matrixwith the order of four (or five) for thin (or moderately thick)shells The coefficient matrix [S] is a symmetric eighth- (ortenth-) order matrix for thin (or moderately thick) circularcylindrical shells Details of [S] can also be found in theAppendix

3 Dynamic Stiffness Matrix

Should the circumferential wave number 119899 be a specific value(8) is one-dimensional with respect to 119909 only Thereforea circular cylindrical shell can be analysed as a generalone-dimensional skeleton-like structure with four (or five)degrees of freedom 1199061 1199062 1199063 1199064 (or 1199061 1199062 1199063 1199064 1199065)for thin (ormoderately thick) shells Accordingly the internalforce vectors for both thin andmoderately thick shell theoriesare V1 V2 V3 V4T and V1 V2 V3 V4 V5T respectivelyThedynamic stiffness of free vibration of a circular cylindricalshell can now be set up in a similar way as that in classic one-dimensional skeletal theory

Figure 3 shows that a circular cylinder is divided into sev-eral shell segments along its axial direction Taking element(119890) for example the 119909 coordinates at its two ends are 119909119886 and

4 Shock and Vibration

119909119887 respectivelyThe segment-end displacement vector d119890 isdefined as

d119890thin = 1199061 (119909119886) 1199062 (119909119886) 1199063 (119909119886) 1199064 (119909119886) 1199061 (119909119887) 1199062 (119909119887) 1199063 (119909119887) 1199064 (119909119887)T

d119890thick = 1199061 (119909119886) 1199062 (119909119886) 1199063 (119909119886) 1199064 (119909119886) 1199065 (119909119886) 1199061 (119909119887) 1199062 (119909119887) 1199063 (119909119887) 1199064 (119909119887) 1199065 (119909119887)T

(9)

and the force vector F119890 isF119890thin = minusV1 (119909119886) minusV2 (119909119886) minusV3 (119909119886)

minus V4 (119909119886) V1 (119909119887) V2 (119909119887) V3 (119909119887) V4 (119909119887)T

F119890thick = minusV1 (119909119886) minusV2 (119909119886) minusV3 (119909119886) minusV4 (119909119886) minus V5 (119909119886) V1 (119909119887) V2 (119909119887) V3 (119909119887) V4 (119909119887) V5 (119909119887)T

(10)

Mathematically apply the boundary conditions in (11) or (12)to (8) in turn

d119890thin = 119890119895 119895 = 1 8 (11)

d119890thick = 119890119895 119895 = 1 10 (12)

where 119890119895 is a unit vector with the 119895th element equal to oneand the corresponding element dynamic stiffness matrix [k]119890is formulated by

[k]119890thin = [F119890119895=1 F119890119895=2 F119890119895=3 F119890119895=4 F119890119895=5 F119890119895=6 F119890119895=7 F119890119895=8]thin [k]119890thick = [F119890119895=1 F119890119895=2 F119890119895=3 F119890119895=4 F119890119895=5 F119890119895=6 F119890119895=7 F119890119895=8 F119890119895=9 F119890119895=10]thick

(13)

Due to the complexity (8) is solved computationally usingadaptive ordinary differential equations (ODEs) solver COL-SYS [33 34] The global dynamic stiffness matrix K of acircular cylindrical shell is assembled in a regular FEMway

4 Wittrick-Williams Algorithm

The dynamic stiffness method is used in conjunction withtheWittrick-Williams (W-W) algorithmTheW-Walgorithmmerely gives the number of frequencies below 120596lowast with theexpression of

119869 = 1198690 + 119869119870 (14)

where 119869 is the total number of frequencies exceeded bythe trial frequency 120596lowast 119869119870 = 119904K(120596lowast) is the sign countof the global dynamic stiffness matrix K and equal to thenumber of negative elements on the diagonal of an uppertriangular matrix obtained from K by applying standardGaussian eliminationwithout row exchange 1198690 is the numberof clamped-end frequencies exceeded by 120596lowast and can beaccumulated from 1198691198900 of each shell segment element over thewhole meridian as is expressed in

1198690 = sum1198691198900 (15)

To solve 1198691198900 a substructuring method is employed by takingadvantage of the self-adaptability of COLSYSWhen comput-ing the dynamic stiffness matrix of a shell segment elementfor example element (119890) a submesh will be generated byCOLSYS adaptively as is shown in Figure 4

Applying the Wittrick-Williams algorithm again on thesubmesh we have

1198691198900 = sum119869()0 + 119904 K (120596lowast) (16)

where notation ldquordquo stands for submesh Since COLSYS iscapable of controlling the error tolerance adaptively thenumber of clamped-end natural frequencies exceeded by 120596lowaston the submesh has to be zero suggesting 119869()0 = 0 Otherwisethe clamped-end modes can be enlarged infinitely resultingin the unsatisfaction of the given error tolerance

The following will introduce the formulation of K(120596lowast) in(16) Taking a subelement () with the coordinates (119909119894 119909119894+1)on element (119890) for instance its corresponding dynamicstiffness matrix [k]() can be obtained by linearly combiningthe eight (or ten) solutions of (8) and (11) or (12) Note thatthe enddisplacement and force vectors for this subelement ()are d() and F() respectively and the subelement stiffnessmatrix [k]() satisfies

[k]() d() = F() (17)

By applying the boundary conditions in (11) or (12) in turnagain for thin or moderately thick circular cylindrical shells(18) is obtained

[k]() [B]() = [C]() (18)

where

[B]()= [d()119895=1 d()119895=2 d()119895=3 d()119895=4 d()119895=5 d()119895=6 d()119895=7 d()119895=8]thin [C]()= [F()119895=1 F()119895=2 F()119895=3 F()119895=4 F()119895=5 F()119895=6 F()119895=7 F()119895=8]thin

(19)

Shock and Vibration 5

xa xi xi+1 xb

(e)

Figure 4 Submesh on element (119890)

for thin shells or

[B]() = [d()119895=1 d()119895=2 d()119895=3 d()119895=4 d()119895=5 d()119895=6 d()119895=7 d()119895=8 d()119895=9 d()119895=10]thick [C]() = [F()119895=1 F()119895=2 F()119895=3 F()119895=4 F()119895=5 F()119895=6 F()119895=7 F()119895=8 F()119895=9 F()119895=10]thick

(20)

for moderately thick shellsSince 119869()0 = 0 [B]() is nonsingular (otherwise120596lowast is one of

clamped-end frequencies which is conflicted with 119869()0 = 0)and (21) is obtained

[k]() = [C]() ([B]())minus1 (21)

Thus K(120596lowast) can be assembled regularly from [k]() Based onthe above theories a FORTRAN code was written and usedto solve the free vibration of both thin and moderately thickcircular cylindrical shells in this paper

5 Validation

In this section four examples on free vibration of circularcylindrical shells are presented to show the reliability andaccuracy of the proposed method in the above sectionsExamples in Sections 51 and 52 are calculated with thin shelltheory in Section 53 both thin and moderately thick shelltheories are employed to study the free vibration of the sameshell and results are given opposite for comparison examplesin Section 54 are performed with the moderately thick shelltheory

51 Free Vibration of a Thin Clamped-Clamped Shell Zhanget al [28] and Xuebin [29] analysed the natural frequenciesof a clamped-clamped (C-C) circular cylindrical shell basedon the Flugge thin shell theory by employing the FEM codeMSCNASTRAN and wave propagation approach respec-tively The clamped boundary condition is considered as

119906 = V = 119908 = 120597119908120597119909 = 0 (22)

The length of the shell is 119871 = 20m radius 119903 = 1m thicknessℎ = 001m Density 120588 = 7850 kgm3 Youngrsquos modulus119864 = 210GPa Poissonrsquos ratio ] = 03 In this example thinshell theory is used in formulating the dynamic stiffness andthe first nine frequencies are tabulated in Table 1 where 119898denotes the number of half-waves along the axial direction

It can be observed from Table 1 that the frequenciesobtained using the present method agree very well withthat from the FEM (MSCNASTRAN) and wave propagation

approach Further investigation into Table 1 shows that thelowest natural frequency of a circular cylindrical usuallycomes with a relatively larger 119899 that is 119899 = 2 in this case

52 Free Vibration of a Thin Clamped-Free Shell In Sharma[30] the free vibration of a clamped-free (C-F) circularcylindrical shell is investigated based on the Rayleigh-Ritzmethod and Flugge shell theory The geometry parametersof the shell are length 119871 = 0502m radius 119903 = 00635mand thickness ℎ = 000163m and the material propertiesare Youngrsquos modulus 119864 = 210GPa Poissonrsquos ratio ] = 028and density 120588 = 7800 kgm3 A thin shell formulation is usedin the present dynamic stiffness approach Table 2 presentsthe lowest four frequencies of the shell with a relatively largercircumferential wave number 119899 = 5 and 6

It can be concluded that the results agree very well with[30] and the errors are no larger than 04 By comparingthe results from the present method and the Rayleigh-Ritzmethod used in Sharma [30] the reliability of the presentmethod is verified

53 Free Vibration of a Shell with Different Boundary Condi-tions Pagani et al [23] applied the one-dimensional higher-order beam theory and exact dynamic stiffness elements tostudy the free vibration of solid and thin-walled structuresin which a thin-walled cylinder with a circular cross-section(Figure 5) is analysed In a recent paper [13] they employedboth the radial basis functions (RBFs) method and CarreraUnified Formulation (CUF) to examine the free vibrationbehaviour of both open and closed thin-walled structures

The length of the examined cylinder is 119871 = 20mouter diameter is 119889 = 2m and thickness is ℎ = 002mThe material properties are Youngrsquos modulus 119864 = 75GPaPoissonrsquos ratio ] = 033 and density 120588 = 2700 kgm3 Table 3shows the natural frequencies of this thin-walled cylindricalbeam with different boundaries namely both ends simplysupported (S-S) both ends clamped (C-C) clamped-free(C-F) and free-free (F-F) The first and second flexural(bending) shell-like and torsional natural frequencies arepresented Solutions from the present thin and moderatelythick DSM 2D MSCNASTRAN and [13 23] are given forcomparison

6 Shock and Vibration

Table 1 The lowest nine frequencies of the C-C circular cylindricalshell (Hz)

Order (119898 119899) Present Reference [28] Reference [29]1 (1 2) 1200 1225 12132 (1 3) 1956 1964 19613 (2 3) 2310 2318 23284 (2 2) 2716 2769 28065 (1 1) 2829 mdash 30096 (3 3) 3148 316 31977 (1 4) 3642 367 36488 (2 4) 3728 3755 37389 (3 4) 3960 3987 3977mdash not given

Table 2The lowest four frequencies of the C-F shell with 119899 = 5 and6 (Hz)

119899 = 5 119899 = 6Present Sharma [30] Present Sharma [30]

1 23619 23666 34628 346972 24021 24064 34990 350503 25050 25091 35768 358074 27062 27160 37120 37166

It can be observed that the natural frequencies computedfrom the present dynamic stiffness method agree very wellwith that from 2D MSCNASTRAN code for both thin andmoderately thick shell theories Comparing with the datafrom the higher-order beam theories [13 23] the presentmethod provides an excellent agreement for flexural andtorsional modes The differences of frequencies for the firstand second shell-like modes are larger and this can belargely attributed that the formulations established in [1323] are based on higher-order beam theories Obviouslybetter agreement can be achieved if exact shell theories areemployed by [13 23] Further investigation by the presentDSM approach shows that 119899 = 1 corresponds to flexural(bending) mode while 119899 = 0 and 2 correspond to tor-sional and shell-like vibration modes respectively whichdemonstrate that the current DSM approach is capable ofshowing the correct vibration modes as presented in existingliterature

54 Free Vibration of a Moderately Thick Shell Bhimaraddi[32] analysed the free vibration of a simply supported (S-S) moderately thick circular cylindrical shell by using thehigher-order displacement model with the consideration ofin-plane inertia rotary inertia and shear effect In [31]natural frequencies of the circular cylindrical shell subjectsto different ℎ119903 and 119871119903 are computed and compared withthe solutions from 3D elasticity theory Results from themoderately thick DSM approach are also presented and listedin Table 4 In this example Poissonrsquos ratio is 03 and 1205960 =(120587ℎ)radic119866120588 where 119866 is the shear modulus

According to Table 4 results from the present methodagree very well with those from [31 32] demonstrating the

h d

Figure 5 The cross-section of a thin-walled cylinder

high accuracy and adaptability of the present method forcircular cylindrical shells with different ℎ119903 and 1198711199036 Parametric Study

61 Ratio of Thickness to Radius (ℎ119903) An important param-eter to determine whether a shell is thin or moderatelythick is the thickness to radius ratio (ℎ119903) According tothe classic shell theory a critical value for ℎ119903 between thinand moderately thick shell theories is 005 In this sectioninfluences of ℎ119903 on natural frequencies are investigatedextensively from 0005 to 01 for both thin and moderatelythick DSM formulations

611 Low Order Frequencies Consider a circular cylindricalshell with 119871119903 = 12 and Poissonrsquos ratio ] = 03 Toinvestigate the differences of natural frequencies between thinand moderately thick DSM formulations a notation 120578 in (23)is introduced as

120578 = 10038161003816100381610038161003816100381610038161003816ΩthinΩthick

minus 110038161003816100381610038161003816100381610038161003816 (23)

where Ωthin and Ωthick are the nondimensional frequencyparameters defined as

Ω = 120596119903radic120588 (1 minus ]2)119864 (24)

Consequently log10(120578) is the index of differenceFigure 6 shows the log10(120578) for the first frequencies (119898 =1) under 119899 = 0 1 2 and 3 between thin and moderately thick

DSM formulations with a variety of boundary conditionsfor example clamped-clamped (C-C) clamped-free (C-F)clamped-simply supported (C-S) and both ends simplysupported (S-S) In these cases ℎ119903 is investigated from 0025to 01

If log10(120578) le minus2 (which suggests the difference is nolarger than 1) it can be considered that results from the thinand moderately thick dynamic stiffness formulations are ingood agreement with each other Furthermore frequenciescomputed from themoderately thick shell theory can be usedas good alternatives to those computed from the thin shelltheory Referring to Figure 6 the following conclusions canbe reached

Shock and Vibration 7

Table 3 Natural frequencies of the thin-walled cylinder with different boundary conditions (Hz)

BCs Model I flexural II flexural I shell-like II shell-like I torsional II torsional

S-S

Present thin 14025 51506 14851 22931 80790 161579Present thick 14021 51502 14828 22877 80792 161585NAS2D [23] 13978 51366 14913 22917 80415 160810Reference [23] 14022 51503 18405 25460 80786 161573Reference [13] 14294 51567 18608 25574 80639 162551

C-C

Present thin 28577 69110 17350 30239 80790 161579Present thick 28574 69105 17327 30193 80792 161585NAS2D [23] 28498 68960 17396 30225 80415 160810Reference [23] 28576 69110 20484 32222 80786 161573Reference [13] 28354 69096 20463 31974 80838 161579

C-F

Present thin 5081 29092 14153 17427 40395 121184Present thick 5076 29086 14140 17370 40396 121188NAS2D [23] 5059 29001 14235 17435 40209 120620Reference [23] 5077 29090 23069 25239 40394 121181Reference [13] 5047 29002 23003 24979 40431 121203

F-F

Present thin 30939 77045 14036 14105 80788 161576Present thick 30931 77038 14032 14076 80792 161585NAS2D [23] 30829 76806 14129 14171 80415 160810Reference [23] 30932 77043 22987 23053 80789 161577Reference [13] 30945 77052 22864 23048 80787 161592

Table 4 Lowest natural frequencies (times1205960) of a moderately thick circular cylindrical shell with different circumferential wave numbers 119899119871119903 ℎ119903 = 012 ℎ119903 = 018119899 = 1 119899 = 2 119899 = 3 119899 = 4 119899 = 1 119899 = 2 119899 = 3 119899 = 42

Present 003724 002349 002449 003664 005632 003893 004939 007726Reference [31] 003730 002359 002462 003686 005652 003929 004996 007821Reference [32] 003730 002365 002470 003687 005653 003944 005009 007833

1Present 005861 004969 004765 005508 009420 008501 008995 011056

Reference [31] 005853 004978 004789 005545 009402 008545 009093 011205Reference [32] 005856 004986 004799 005550 009409 008562 009109 011202

05Present 010087 010238 010658 011463 018915 019406 020451 022180

Reference [31] 010057 010234 010688 011528 018894 019467 020616 022450Reference [32] 010047 010224 010674 011508 018832 019403 020544 022361

025Present 027310 027650 028219 029020 049751 050299 051216 052506

Reference [31] 027491 027849 028447 029287 050338 050937 051934 053325Reference [32] 027286 027641 028233 029064 049818 050418 051416 052808

(1) For low order frequencies good agreement is reachedbetween thin and moderately thick dynamic stiffnessformulations with ℎ119903 varying from 0025 to 01 Thesmaller the ℎ119903 the smaller the difference

(2) The difference between frequencies from thin andmoderately thick DSM formulations increases withthe increase of 119899 as log10(120578) always has a smallermagnitude under 119899 = 0 than those under 119899 = 1 2and 3

(3) It can be argued that the difference is insensitiveto boundary conditions The differences are roughlywithin 10minus5sim10minus2 in this investigation and the trends

of curves are similar for different boundary condi-tions

612 High Order Frequencies Investigation on the differ-ences between high order frequencies computed from thethin and moderately thick DSM formulations is examined inthis subsection Since it is known that the DSM formulationis not sensitive to boundary conditions [26] a C-S circu-lar cylindrical shell is studied The geometry and materialproperties are the same to the example in Section 611 if notspecifically stated

For high order frequencies ℎ119903 is investigated from verysmall value 0005 to a relatively large value 01 and the first

8 Shock and Vibration

005 0075 010025ℎr

minus1

minus2

minus3

minus4

minus5log 1

0(120578

)

n = 0

n = 1

n = 2

n = 3

(a) C-C

005 0075 010025ℎr

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)n = 0

n = 1

n = 2

n = 3

(b) C-F

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

n = 0

n = 1

n = 2

n = 3

005 0075 010025ℎr

(c) C-S

005 0075 010025ℎr

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

n = 0

n = 1

n = 2

n = 3

(d) S-S

Figure 6 The differences of low order frequencies between thin and moderately thick DSM formulations

ten frequencies under 119899 ranging from 0 to 10 from boththin and moderately thick DSM formulations are computedand compared The differences are shown in Figure 7 andconcluded as follows

(1) When ℎ119903 le 0025 log10(120578) is smaller than 10minus2most of the time which suggests that frequenciescomputed frommoderately thick shell theory are well

compatible with those from thin shell theory Noshear locking is observed

(2) When ℎ119903 ge 005 only frequencies with a small 119899(eg 119899 le 2) produce an acceptable difference (sim10minus2)With the increase of n the difference becomes quitenoticeable The higher the 119899 the more obvious thedifference

Shock and Vibration 9

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

(a) ℎ119903 = 0005

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

(b) ℎ119903 = 001

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

2 4 6 8 100m

(c) ℎ119903 = 0025

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

(d) ℎ119903 = 005

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

(e) ℎ119903 = 0075

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

(f) ℎ119903 = 01

Figure 7 The differences for frequencies between thin and moderately thick C-S circular cylindrical shell theories

(3) The order 119898 under a specific circumferential wavenumber 119899 does not have a significant impact on thedifferences as most curves in Figure 7 are largely flatand stable with the increase of119898

62 Ratio of Length to Radius (119871119903) Parameter 119871119903 controlsthe relative length of a circular cylindrical shell A larger119871119903 indicates a long shell and vice versa In this subsectionthe frequency trends and the compatibility between thin andmoderately thick DSM formulations with respect to 119871119903 arestudied In this subsection a C-C circular cylindrical shell

with ] = 03 is examined Figure 8 plots the trends of the firstfrequency parameterΩ under each 119899 with 119871119903 from 02 to 10

According to Figure 8 for a typical long circular cylin-drical shell (eg 119871119903 = 10) the lowest natural frequency isobtained with 119899 = 2 with the decrease of 119871119903 the circumfer-ential wave number 119899 corresponding to the lowest natural fre-quency slightly increases to 119899 = 3 or 4 when 119871119903 is very small(eg119871119903 = 02 and the shell is actually degraded to a ring) thefirst natural frequency under each 119899 increases monotonicallyand the smallest is reached at 119899 = 0The trends shown by Fig-ure 8 are also in good agreementwith other researchersrsquo work

10 Shock and Vibration

Ω

0

2

4

6

8

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(a) ℎ119903 = 0025 moderately thick shell

Ω

0

2

4

6

8

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(b) ℎ119903 = 0025 thin shell

0

3

6

9

12

Ω

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(c) ℎ119903 = 005 moderately thick shell

0

3

6

9

12

Ω

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(d) ℎ119903 = 005 thin shell

Figure 8 The trends of the first natural frequency parameters Ω under different 119899for example Markus [35] These obtained trends are applica-ble to both thin and moderately thick DSM formulations

Figure 9 shows the differences of the first frequenciesunder each 119899 computed from the thin and moderately thickDSM formulations of the circular cylindrical shell with ℎ119903 =0025 and 005 It can be understood fromFigure 9 that (1) thesmaller the ℎ119903 the smaller the difference (2) the larger the119871119903 the smaller the difference (3) conditions of ℎ119903 le 0025and 119899 le 10 need to be satisfied and a minor difference(log10(120578) le 10minus2) is required for a normal length shell thatis 119871119903 ge 2

7 Conclusions

This paper conducts comparison study on the frequenciesfrom thin and moderately thick circular cylindrical shellscomputed using the DSM formulations and comprehensiveparametric studies on ℎ119903 and 119871119903 are performed Based onthese investigations conclusions are outlined as follows

(1) Natural frequencies computed from the moder-ately thick DSM formulations can be consideredas satisfactory alternatives to those from thin DSMformulations when ℎ119903 le 0025 and 119871119903 ge 2

Shock and Vibration 11

0

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(a) ℎ119903 = 0025

0

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(b) ℎ119903 = 005

Figure 9 Differences between the first frequencies under each 119899 with different 119871119903

(differences are around or smaller than 10minus2) Sincethe DSM is exact no shear locking is observed

(2) Differences between frequencies from thin and mod-erately thick DSM formulations change with a varietyof factors The smaller ℎ119903 the smaller the differencethe smaller119871119903 the larger the difference the larger the119899 the larger the difference

It is proved that moderately thick DSM formulations canbe used to analyse the free vibration of normal length thin

circular cylindrical shells without shear locking demon-strating that the applicable range of moderately thick DSMformulations is expanded These merits should be attributedto that the dynamic stiffness method is an exact method

Though this comparison study is performed on circularcylindrical shells similar conclusions can be extended to thefree vibration of other shaped shells of revolution

Appendix

uthin =

1199061 = 1199061198991199062 = V119899

1199063 = 1199081198991199064 = 1199081015840119899

kthin =

V1 = 119903120587119864ℎ1 minus ]2[]119899119903 V119899 + ]

1119903119908119899 + 1199061015840119899]V2 = 119903120587119864ℎ2 (1 + ]) [minus119899119903 119906119899 + (1 + ℎ2121199032) V1015840119899 + 119899ℎ261199032 1199081015840119899]V3 = 119903120587119864ℎ2 (1 minus ]2) [119899ℎ

2

61199032 V1015840119899 + (2 minus ]) 1198992ℎ261199032 1199081015840119899 minus ℎ26 119908101584010158401015840119899 ]V4 = 119903120587119864ℎ12 (1 minus ]2) [minus]119899ℎ

2

1199032 V119899 minus ]1198992ℎ21199032 119908119899 + ℎ211990810158401015840119899 ]

12 Shock and Vibration

uthick =

1199061 = 1199061198991199062 = V119899

1199063 = 1199081198991199064 = 1205951199091198991199065 = 120595120579119899

kthick =

V1 = 119903120587119864ℎ1 minus ]2[1199061015840119899 + ]

119899119903 V119899 + ]119903119908119899]V2 = 12 119903120587119864ℎ1 + ]

[minus119899119903 119906119899 + (1 + ℎ2121199032) V1015840119899 minus 119899ℎ2121199032120595119909119899 + ℎ21211990321205951015840120579119899]V3 = 119903120587120581119866ℎ [1199081015840119899 + 120595119909119899]V4 = ℎ212 119903120587119864ℎ1 minus ]2

[1205951015840119909119899 + ]119899119903120595120579119899]

V5 = ℎ224 119903120587119864ℎ1 + ][V1015840119899119903 minus 119899119903120595119909119899 + 1205951015840120579119899]

[S]thin =

[[[[[[[[[[[[[[[[[[

11990411 0 0 11990414 0 11990416 0 011990422 11990423 0 11990425 0 0 1199042811990433 0 11990435 0 0 1199043811990444 0 11990446 11990447 0

11990455 0 0 0sym 11990466 0 0

0 011990488

]]]]]]]]]]]]]]]]]]

(A1)

where nonzero elements in the upper triangular domain are

11990411 = 120588120587119903ℎ1205962 minus 120587119864ℎ11989922119903 (1 minus ]) + 61198992119903120587119864ℎ(121199032 + ℎ2) (1 + ]) 11990414 = minus 1198992ℎ3120587119864(121199032 + ℎ2) (1 + ]) 11990416 = 12119899119903121199032 + ℎ2 11990422 = 120588120587119903ℎ1205962 minus 1198992120587119864ℎ (121199032 + ℎ2)

121199033 11990423 = minus120587119864ℎ121198991199032 + 1198993ℎ2121199033 11990425 = minus]119899119903 11990428 = ]

1198991199032 11990433 = 120588ℎ1199031205871205962 minus 120587119864ℎ119903 minus 1198994ℎ3120587119864121199033

11990435 = minus]119903 11990438 = ]11989921199032 11990444 = minus 21198992ℎ3119903120587119864(121199032 + ℎ2) (1 + ])

11990446 = minus 2119899ℎ2121199032 + ℎ2 11990447 = 111990455 = 1 minus ]2119903120587119864ℎ 11990466 = 24119903 (1 + ])120587119864ℎ (121199032 + ℎ2)

11990488 = 12 (1 minus ]2)119903120587119864ℎ3

[S]thick

Shock and Vibration 13

=

[[[[[[[[[[[[[[[[[[[[[[[

11990411 0 0 0 0 0 11990417 0 0 1199041lowast1011990422 11990423 0 11990425 11990426 0 0 0 011990433 0 11990435 11990436 0 0 0 0

11990444 0 0 0 11990448 0 1199044lowast1011990455 0 0 0 11990459 011990466 0 0 0 0

11990477 0 0 1199047lowast10sym 11990488 0 0

11990499 011990410lowast10

]]]]]]]]]]]]]]]]]]]]]]]

(A2)

where nonzero elements in the upper triangular domain are

11990411 = 119903120587120588ℎ120596211990417 = 119899119903

1199041lowast10 = minus 1198991199032 11990422 = 120587(119903120588ℎ1205962 minus 120581119866ℎ119903 minus 1198992119864ℎ119903 ) 11990423 = minus119899120587119903 (119864ℎ + 120581119866ℎ) 11990425 = 120587120581119866ℎ11990426 = minus]119899119903 11990433 = 120587(119903120588ℎ1205962 minus 119864ℎ119903 minus 1198992120581119866ℎ119903 ) 11990435 = 119899120587120581119866ℎ11990436 = minus]119903 11990444 = 1199031205871205881205962 ℎ312 11990448 = minus1

1199044lowast10 = 119899119903 11990455 = 120587(1199031205881205962 ℎ312 minus 1198992119864ℎ312119903 minus 119903120581119866ℎ) 11990459 = minus]119899119903 11990466 = 1 minus ]2119903120587119864ℎ 11990477 = 2 (1 + ])119903120587119864ℎ

1199047lowast10 = minus2 (1 + ])1199032120587119864ℎ 11990488 = 1119903120587120581119866ℎ 11990499 = 12 (1 minus ]2)

119903120587119864ℎ3 11990410lowast10 = 2 (1 + ])119903120587119864ℎ (12ℎ2 + 11199032 )

120581 = 120587212 (A3)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully appreciate the financial support fromNational Natural Science Foundation of China (51078198)Tsinghua University (2011THZ03) Jiangsu University ofScience and Technology (1732921402) and Shuang-ChuangProgram of Jiangsu Province for this research

References

[1] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover England UK 1927

[2] R N Arnold and G B Warburton ldquoThe flexural vibrationsof thin cylindersrdquo Proceedings of the Institution of MechanicalEngineers vol 167 no 1 pp 62ndash80 1953

[3] A W Leissa ldquoVibration of shellsrdquo Report 288 National Aero-nautics and Space Administration (NASA) Washington DCUSA 1973

[4] N S Bardell J M Dunsdon and R S Langley ldquoOn thefree vibration of completely free open cylindrically curvedisotropic shell panelsrdquo Journal of Sound and Vibration vol 207no 5 pp 647ndash669 1997

[5] D-Y Tan ldquoFree vibration analysis of shells of revolutionrdquoJournal of Sound and Vibration vol 213 no 1 pp 15ndash33 1998

[6] P M Naghdi and R M Cooper ldquoPropagation of elastic wavesin cylindrical shells including the effects of transverse shearand rotatory inertiardquo The Journal of the Acoustical Society ofAmerica vol 28 no 1 pp 56ndash63 1956

[7] I Mirsky and G Herrmann ldquoNonaxially symmetric motionsof cylindrical shellsrdquo The Journal of the Acoustical Society ofAmerica vol 29 pp 1116ndash1123 1957

[8] F Tornabene and E Viola ldquo2-D solution for free vibrationsof parabolic shells using generalized differential quadraturemethodrdquo European Journal of Mechanics ASolids vol 27 no6 pp 1001ndash1025 2008

[9] J L Mantari A S Oktem and C Guedes Soares ldquoBending andfree vibration analysis of isotropic and multilayered plates andshells by using a new accurate higher-order shear deformationtheoryrdquo Composites Part B Engineering vol 43 no 8 pp 3348ndash3360 2012

14 Shock and Vibration

[10] J G Kim J K Lee and H J Yoon ldquoFree vibration analysisfor shells of revolution based on p-version mixed finite elementformulationrdquo Finite Elements in Analysis and Design vol 95 pp12ndash19 2015

[11] F Tornabene ldquoFree vibrations of laminated composite doubly-curved shells and panels of revolution via the GDQ methodrdquoComputer Methods in Applied Mechanics and Engineering vol200 no 9ndash12 pp 931ndash952 2011

[12] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe local GDQmethod applied to general higher-order theories of doubly-curved laminated composite shells and panels the free vibrationanalysisrdquo Composite Structures vol 116 pp 637ndash660 2014

[13] A Pagani E Carrera and A Ferreira ldquoHigher-order theoriesand radial basis functions applied to free vibration analysisof thin-walled beamsrdquo Mechanics of Advanced Materials andStructures vol 23 no 9 pp 1080ndash1091 2016

[14] T Ye G Jin Y Chen and S Shi ldquoA unified formulationfor vibration analysis of open shells with arbitrary boundaryconditionsrdquo International Journal ofMechanical Sciences vol 81pp 42ndash59 2014

[15] T Ye G Jin S Shi and X Ma ldquoThree-dimensional freevibration analysis of thick cylindrical shells with general endconditions and resting on elastic foundationsrdquo InternationalJournal of Mechanical Sciences vol 84 pp 120ndash137 2014

[16] F W Williams and W H Wittrick ldquoAn automatic computa-tional procedure for calculating natural frequencies of skeletalstructuresrdquo International Journal of Mechanical Sciences vol 12no 9 pp 781ndash791 1970

[17] W H Wittrick and F W Williams ldquoA general algorithmfor computing natural frequencies of elastic structuresrdquo TheQuarterly Journal of Mechanics and Applied Mathematics vol24 pp 263ndash284 1971

[18] S Yuan K Ye F W Williams and D Kennedy ldquoRecursivesecond order convergence method for natural frequencies andmodes when using dynamic stiffness matricesrdquo InternationalJournal for Numerical Methods in Engineering vol 56 no 12pp 1795ndash1814 2003

[19] S Yuan K Ye and FWWilliams ldquoSecond order mode-findingmethod in dynamic stiffness matrix methodsrdquo Journal of Soundand Vibration vol 269 no 3ndash5 pp 689ndash708 2004

[20] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007

[21] J R Banerjee C W Cheung R Morishima M Pereraand J Njuguna ldquoFree vibration of a three-layered sandwichbeam using the dynamic stiffness method and experimentrdquoInternational Journal of Solids and Structures vol 44 no 22-23pp 7543ndash7563 2007

[22] H Su J R Banerjee and C W Cheung ldquoDynamic stiffnessformulation and free vibration analysis of functionally gradedbeamsrdquo Composite Structures vol 106 pp 854ndash862 2013

[23] A Pagani M Boscolo J R Banerjee and E Carrera ldquoExactdynamic stiffness elements based on one-dimensional higher-order theories for free vibration analysis of solid and thin-walled structuresrdquo Journal of Sound and Vibration vol 332 no23 pp 6104ndash6127 2013

[24] N El-Kaabazi and D Kennedy ldquoCalculation of natural fre-quencies and vibration modes of variable thickness cylindricalshells using the Wittrick-Williams algorithmrdquo Computers andStructures vol 104-105 pp 4ndash12 2012

[25] X Chen Research on dynamic stiffness method for free vibrationof shells of revolution [MS thesis] Tsinghua University 2009(Chinese)

[26] XChen andKYe ldquoFreeVibrationAnalysis for Shells of Revolu-tion Using an Exact Dynamic Stiffness Methodrdquo MathematicalProblems in Engineering vol 2016 Article ID 4513520 12 pages2016

[27] X Chen and K Ye ldquoAnalysis of free vibration of moderatelythick circular cylindrical shells using the dynamic stiffnessmethodrdquo Engineering Mechanics vol 33 no 9 pp 40ndash48 2016(Chinese)

[28] X M Zhang G R Liu and K Y Lam ldquoVibration analysisof thin cylindrical shells using wave propagation approachrdquoJournal of Sound and Vibration vol 239 no 3 pp 397ndash4032001

[29] L Xuebin ldquoStudy on free vibration analysis of circular cylin-drical shells using wave propagationrdquo Journal of Sound andVibration vol 311 no 3-5 pp 667ndash682 2008

[30] C B Sharma ldquoCalculation of natural frequencies of fixed-freecircular cylindrical shellsrdquo Journal of Sound and Vibration vol35 no 1 pp 55ndash76 1974

[31] A E ArmenakasDCGazis andGHerrmannFreeVibrationsof Circular Cylindrical Shells PergamonPressOxfordUK 1969

[32] A Bhimaraddi ldquoA higher order theory for free vibrationanalysis of circular cylindrical shellsrdquo International Journal ofSolids and Structures vol 20 no 7 pp 623ndash630 1984

[33] U Ascher J Christiansen and R D Russell ldquoAlgorithm 569COLSYS collocation software for boundary value ODEs [D2]rdquoACM Transactions on Mathematical Software vol 7 no 2 pp223ndash229 1981

[34] U Ascher J Christiansen and R D Russell ldquoCollocationsoftware for boundary-value ODEsrdquo ACM Transactions onMathematical Software vol 7 no 2 pp 209ndash222 1981

[35] S Markus The Mechanics of Vibrations of Cylindrical ShellsElsevier Science Limited 1988

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Shock and Vibration 3

dx

h

rd120579

Figure 2 The diagram of an infinitesimal portion of a circularcylindrical shell

Similarly the internal force-strain relationship at themiddle surface of a circular cylindrical shell can also beexpressed as

N = [D] 120576 (2)

where N is the internal force vector and [D] is the stiffnessmatrix which can also be found in [3]

Denote 119889119860 as the area of an infinitesimal portion in themiddle surface of a circular cylindrical shell (see Figure 2)and the area of this infinitesimal portion is 119889119860 = 119903119889120579119889119909The total strain energy 119880 for free vibration of the circularcylindrical shell can be integrated along the whole middlesurface 119878 as

119880 = ∬119878

12 120576T [D] 120576 119889119860 (3)

and the total free vibration kinetic energy is obtained as

119879 = 1205882 ∬119878ℎ [(120597119906120597119905 )

2 + (120597V120597119905 )2 + (120597119908120597119905 )

2]119889119860 (4)

for thin shell or

119879 = 1205882 ∬119878ℎ[(120597119906120597119905 )

2 + (120597V120597119905 )2 + (120597119908120597119905 )

2]+ ℎ312 [(120597120595119909120597119905 )2 + (120597120595120579120597119905 )2]119889119860

(5)

for moderately thick circular cylindrical shellsThe dynamic displacement functions Δthin and Δthick

are expressed differently as

Δthin = 119906119899 (119909) cos 119899120579 sin120596119905V119899 (119909) sin 119899120579 sin120596119905119908119899 (119909) cos 119899120579 sin120596119905

Δthick =

119906 = 119906119899 (119909) cos 119899120579 sin120596119905V = V119899 (119909) sin 119899120579 sin120596119905119908 = 119908119899 (119909) cos 119899120579 sin120596119905120595119909 = 120595119909119899 (119909) cos 119899120579 sin120596119905120595120579 = 120595120579119899 (119909) sin 119899120579 sin120596119905

(6)

(e)

0 Lxa xb

Figure 3 The mesh division along the axis

119899 is the circumferential wave number 120596 is a circular fre-quency 119906119899(119909) V119899(119909) 119908119899(119909) 120595119909119899(119909) and 120595120579119899(119909) are axialvibration functions

The Hamilton principle suggests

120575int11990521199051

(119880 minus 119879) 119889119905 = 0 (7)

where (1199051 1199052) is an arbitrary time interval during vibrationThe equations ofmotion can be derived from (7) Substitutingthe dynamic displacement functions (6) into equations ofmotion the free vibration governing differential equationscan be obtained By rewriting the governing differentialequations into the Hamilton form generalised displacementsu and forces k are obtained where u and k are given inthe Appendix

The vibration governing equations can be written in theHamilton form of

[J] z1015840 = [S] z (8)

where ()1015840 denotes the differentiation with respect to 119909 andz = u kT [J] = [ 0 minusII 0 ] and I is an identity matrixwith the order of four (or five) for thin (or moderately thick)shells The coefficient matrix [S] is a symmetric eighth- (ortenth-) order matrix for thin (or moderately thick) circularcylindrical shells Details of [S] can also be found in theAppendix

3 Dynamic Stiffness Matrix

Should the circumferential wave number 119899 be a specific value(8) is one-dimensional with respect to 119909 only Thereforea circular cylindrical shell can be analysed as a generalone-dimensional skeleton-like structure with four (or five)degrees of freedom 1199061 1199062 1199063 1199064 (or 1199061 1199062 1199063 1199064 1199065)for thin (ormoderately thick) shells Accordingly the internalforce vectors for both thin andmoderately thick shell theoriesare V1 V2 V3 V4T and V1 V2 V3 V4 V5T respectivelyThedynamic stiffness of free vibration of a circular cylindricalshell can now be set up in a similar way as that in classic one-dimensional skeletal theory

Figure 3 shows that a circular cylinder is divided into sev-eral shell segments along its axial direction Taking element(119890) for example the 119909 coordinates at its two ends are 119909119886 and

4 Shock and Vibration

119909119887 respectivelyThe segment-end displacement vector d119890 isdefined as

d119890thin = 1199061 (119909119886) 1199062 (119909119886) 1199063 (119909119886) 1199064 (119909119886) 1199061 (119909119887) 1199062 (119909119887) 1199063 (119909119887) 1199064 (119909119887)T

d119890thick = 1199061 (119909119886) 1199062 (119909119886) 1199063 (119909119886) 1199064 (119909119886) 1199065 (119909119886) 1199061 (119909119887) 1199062 (119909119887) 1199063 (119909119887) 1199064 (119909119887) 1199065 (119909119887)T

(9)

and the force vector F119890 isF119890thin = minusV1 (119909119886) minusV2 (119909119886) minusV3 (119909119886)

minus V4 (119909119886) V1 (119909119887) V2 (119909119887) V3 (119909119887) V4 (119909119887)T

F119890thick = minusV1 (119909119886) minusV2 (119909119886) minusV3 (119909119886) minusV4 (119909119886) minus V5 (119909119886) V1 (119909119887) V2 (119909119887) V3 (119909119887) V4 (119909119887) V5 (119909119887)T

(10)

Mathematically apply the boundary conditions in (11) or (12)to (8) in turn

d119890thin = 119890119895 119895 = 1 8 (11)

d119890thick = 119890119895 119895 = 1 10 (12)

where 119890119895 is a unit vector with the 119895th element equal to oneand the corresponding element dynamic stiffness matrix [k]119890is formulated by

[k]119890thin = [F119890119895=1 F119890119895=2 F119890119895=3 F119890119895=4 F119890119895=5 F119890119895=6 F119890119895=7 F119890119895=8]thin [k]119890thick = [F119890119895=1 F119890119895=2 F119890119895=3 F119890119895=4 F119890119895=5 F119890119895=6 F119890119895=7 F119890119895=8 F119890119895=9 F119890119895=10]thick

(13)

Due to the complexity (8) is solved computationally usingadaptive ordinary differential equations (ODEs) solver COL-SYS [33 34] The global dynamic stiffness matrix K of acircular cylindrical shell is assembled in a regular FEMway

4 Wittrick-Williams Algorithm

The dynamic stiffness method is used in conjunction withtheWittrick-Williams (W-W) algorithmTheW-Walgorithmmerely gives the number of frequencies below 120596lowast with theexpression of

119869 = 1198690 + 119869119870 (14)

where 119869 is the total number of frequencies exceeded bythe trial frequency 120596lowast 119869119870 = 119904K(120596lowast) is the sign countof the global dynamic stiffness matrix K and equal to thenumber of negative elements on the diagonal of an uppertriangular matrix obtained from K by applying standardGaussian eliminationwithout row exchange 1198690 is the numberof clamped-end frequencies exceeded by 120596lowast and can beaccumulated from 1198691198900 of each shell segment element over thewhole meridian as is expressed in

1198690 = sum1198691198900 (15)

To solve 1198691198900 a substructuring method is employed by takingadvantage of the self-adaptability of COLSYSWhen comput-ing the dynamic stiffness matrix of a shell segment elementfor example element (119890) a submesh will be generated byCOLSYS adaptively as is shown in Figure 4

Applying the Wittrick-Williams algorithm again on thesubmesh we have

1198691198900 = sum119869()0 + 119904 K (120596lowast) (16)

where notation ldquordquo stands for submesh Since COLSYS iscapable of controlling the error tolerance adaptively thenumber of clamped-end natural frequencies exceeded by 120596lowaston the submesh has to be zero suggesting 119869()0 = 0 Otherwisethe clamped-end modes can be enlarged infinitely resultingin the unsatisfaction of the given error tolerance

The following will introduce the formulation of K(120596lowast) in(16) Taking a subelement () with the coordinates (119909119894 119909119894+1)on element (119890) for instance its corresponding dynamicstiffness matrix [k]() can be obtained by linearly combiningthe eight (or ten) solutions of (8) and (11) or (12) Note thatthe enddisplacement and force vectors for this subelement ()are d() and F() respectively and the subelement stiffnessmatrix [k]() satisfies

[k]() d() = F() (17)

By applying the boundary conditions in (11) or (12) in turnagain for thin or moderately thick circular cylindrical shells(18) is obtained

[k]() [B]() = [C]() (18)

where

[B]()= [d()119895=1 d()119895=2 d()119895=3 d()119895=4 d()119895=5 d()119895=6 d()119895=7 d()119895=8]thin [C]()= [F()119895=1 F()119895=2 F()119895=3 F()119895=4 F()119895=5 F()119895=6 F()119895=7 F()119895=8]thin

(19)

Shock and Vibration 5

xa xi xi+1 xb

(e)

Figure 4 Submesh on element (119890)

for thin shells or

[B]() = [d()119895=1 d()119895=2 d()119895=3 d()119895=4 d()119895=5 d()119895=6 d()119895=7 d()119895=8 d()119895=9 d()119895=10]thick [C]() = [F()119895=1 F()119895=2 F()119895=3 F()119895=4 F()119895=5 F()119895=6 F()119895=7 F()119895=8 F()119895=9 F()119895=10]thick

(20)

for moderately thick shellsSince 119869()0 = 0 [B]() is nonsingular (otherwise120596lowast is one of

clamped-end frequencies which is conflicted with 119869()0 = 0)and (21) is obtained

[k]() = [C]() ([B]())minus1 (21)

Thus K(120596lowast) can be assembled regularly from [k]() Based onthe above theories a FORTRAN code was written and usedto solve the free vibration of both thin and moderately thickcircular cylindrical shells in this paper

5 Validation

In this section four examples on free vibration of circularcylindrical shells are presented to show the reliability andaccuracy of the proposed method in the above sectionsExamples in Sections 51 and 52 are calculated with thin shelltheory in Section 53 both thin and moderately thick shelltheories are employed to study the free vibration of the sameshell and results are given opposite for comparison examplesin Section 54 are performed with the moderately thick shelltheory

51 Free Vibration of a Thin Clamped-Clamped Shell Zhanget al [28] and Xuebin [29] analysed the natural frequenciesof a clamped-clamped (C-C) circular cylindrical shell basedon the Flugge thin shell theory by employing the FEM codeMSCNASTRAN and wave propagation approach respec-tively The clamped boundary condition is considered as

119906 = V = 119908 = 120597119908120597119909 = 0 (22)

The length of the shell is 119871 = 20m radius 119903 = 1m thicknessℎ = 001m Density 120588 = 7850 kgm3 Youngrsquos modulus119864 = 210GPa Poissonrsquos ratio ] = 03 In this example thinshell theory is used in formulating the dynamic stiffness andthe first nine frequencies are tabulated in Table 1 where 119898denotes the number of half-waves along the axial direction

It can be observed from Table 1 that the frequenciesobtained using the present method agree very well withthat from the FEM (MSCNASTRAN) and wave propagation

approach Further investigation into Table 1 shows that thelowest natural frequency of a circular cylindrical usuallycomes with a relatively larger 119899 that is 119899 = 2 in this case

52 Free Vibration of a Thin Clamped-Free Shell In Sharma[30] the free vibration of a clamped-free (C-F) circularcylindrical shell is investigated based on the Rayleigh-Ritzmethod and Flugge shell theory The geometry parametersof the shell are length 119871 = 0502m radius 119903 = 00635mand thickness ℎ = 000163m and the material propertiesare Youngrsquos modulus 119864 = 210GPa Poissonrsquos ratio ] = 028and density 120588 = 7800 kgm3 A thin shell formulation is usedin the present dynamic stiffness approach Table 2 presentsthe lowest four frequencies of the shell with a relatively largercircumferential wave number 119899 = 5 and 6

It can be concluded that the results agree very well with[30] and the errors are no larger than 04 By comparingthe results from the present method and the Rayleigh-Ritzmethod used in Sharma [30] the reliability of the presentmethod is verified

53 Free Vibration of a Shell with Different Boundary Condi-tions Pagani et al [23] applied the one-dimensional higher-order beam theory and exact dynamic stiffness elements tostudy the free vibration of solid and thin-walled structuresin which a thin-walled cylinder with a circular cross-section(Figure 5) is analysed In a recent paper [13] they employedboth the radial basis functions (RBFs) method and CarreraUnified Formulation (CUF) to examine the free vibrationbehaviour of both open and closed thin-walled structures

The length of the examined cylinder is 119871 = 20mouter diameter is 119889 = 2m and thickness is ℎ = 002mThe material properties are Youngrsquos modulus 119864 = 75GPaPoissonrsquos ratio ] = 033 and density 120588 = 2700 kgm3 Table 3shows the natural frequencies of this thin-walled cylindricalbeam with different boundaries namely both ends simplysupported (S-S) both ends clamped (C-C) clamped-free(C-F) and free-free (F-F) The first and second flexural(bending) shell-like and torsional natural frequencies arepresented Solutions from the present thin and moderatelythick DSM 2D MSCNASTRAN and [13 23] are given forcomparison

6 Shock and Vibration

Table 1 The lowest nine frequencies of the C-C circular cylindricalshell (Hz)

Order (119898 119899) Present Reference [28] Reference [29]1 (1 2) 1200 1225 12132 (1 3) 1956 1964 19613 (2 3) 2310 2318 23284 (2 2) 2716 2769 28065 (1 1) 2829 mdash 30096 (3 3) 3148 316 31977 (1 4) 3642 367 36488 (2 4) 3728 3755 37389 (3 4) 3960 3987 3977mdash not given

Table 2The lowest four frequencies of the C-F shell with 119899 = 5 and6 (Hz)

119899 = 5 119899 = 6Present Sharma [30] Present Sharma [30]

1 23619 23666 34628 346972 24021 24064 34990 350503 25050 25091 35768 358074 27062 27160 37120 37166

It can be observed that the natural frequencies computedfrom the present dynamic stiffness method agree very wellwith that from 2D MSCNASTRAN code for both thin andmoderately thick shell theories Comparing with the datafrom the higher-order beam theories [13 23] the presentmethod provides an excellent agreement for flexural andtorsional modes The differences of frequencies for the firstand second shell-like modes are larger and this can belargely attributed that the formulations established in [1323] are based on higher-order beam theories Obviouslybetter agreement can be achieved if exact shell theories areemployed by [13 23] Further investigation by the presentDSM approach shows that 119899 = 1 corresponds to flexural(bending) mode while 119899 = 0 and 2 correspond to tor-sional and shell-like vibration modes respectively whichdemonstrate that the current DSM approach is capable ofshowing the correct vibration modes as presented in existingliterature

54 Free Vibration of a Moderately Thick Shell Bhimaraddi[32] analysed the free vibration of a simply supported (S-S) moderately thick circular cylindrical shell by using thehigher-order displacement model with the consideration ofin-plane inertia rotary inertia and shear effect In [31]natural frequencies of the circular cylindrical shell subjectsto different ℎ119903 and 119871119903 are computed and compared withthe solutions from 3D elasticity theory Results from themoderately thick DSM approach are also presented and listedin Table 4 In this example Poissonrsquos ratio is 03 and 1205960 =(120587ℎ)radic119866120588 where 119866 is the shear modulus

According to Table 4 results from the present methodagree very well with those from [31 32] demonstrating the

h d

Figure 5 The cross-section of a thin-walled cylinder

high accuracy and adaptability of the present method forcircular cylindrical shells with different ℎ119903 and 1198711199036 Parametric Study

61 Ratio of Thickness to Radius (ℎ119903) An important param-eter to determine whether a shell is thin or moderatelythick is the thickness to radius ratio (ℎ119903) According tothe classic shell theory a critical value for ℎ119903 between thinand moderately thick shell theories is 005 In this sectioninfluences of ℎ119903 on natural frequencies are investigatedextensively from 0005 to 01 for both thin and moderatelythick DSM formulations

611 Low Order Frequencies Consider a circular cylindricalshell with 119871119903 = 12 and Poissonrsquos ratio ] = 03 Toinvestigate the differences of natural frequencies between thinand moderately thick DSM formulations a notation 120578 in (23)is introduced as

120578 = 10038161003816100381610038161003816100381610038161003816ΩthinΩthick

minus 110038161003816100381610038161003816100381610038161003816 (23)

where Ωthin and Ωthick are the nondimensional frequencyparameters defined as

Ω = 120596119903radic120588 (1 minus ]2)119864 (24)

Consequently log10(120578) is the index of differenceFigure 6 shows the log10(120578) for the first frequencies (119898 =1) under 119899 = 0 1 2 and 3 between thin and moderately thick

DSM formulations with a variety of boundary conditionsfor example clamped-clamped (C-C) clamped-free (C-F)clamped-simply supported (C-S) and both ends simplysupported (S-S) In these cases ℎ119903 is investigated from 0025to 01

If log10(120578) le minus2 (which suggests the difference is nolarger than 1) it can be considered that results from the thinand moderately thick dynamic stiffness formulations are ingood agreement with each other Furthermore frequenciescomputed from themoderately thick shell theory can be usedas good alternatives to those computed from the thin shelltheory Referring to Figure 6 the following conclusions canbe reached

Shock and Vibration 7

Table 3 Natural frequencies of the thin-walled cylinder with different boundary conditions (Hz)

BCs Model I flexural II flexural I shell-like II shell-like I torsional II torsional

S-S

Present thin 14025 51506 14851 22931 80790 161579Present thick 14021 51502 14828 22877 80792 161585NAS2D [23] 13978 51366 14913 22917 80415 160810Reference [23] 14022 51503 18405 25460 80786 161573Reference [13] 14294 51567 18608 25574 80639 162551

C-C

Present thin 28577 69110 17350 30239 80790 161579Present thick 28574 69105 17327 30193 80792 161585NAS2D [23] 28498 68960 17396 30225 80415 160810Reference [23] 28576 69110 20484 32222 80786 161573Reference [13] 28354 69096 20463 31974 80838 161579

C-F

Present thin 5081 29092 14153 17427 40395 121184Present thick 5076 29086 14140 17370 40396 121188NAS2D [23] 5059 29001 14235 17435 40209 120620Reference [23] 5077 29090 23069 25239 40394 121181Reference [13] 5047 29002 23003 24979 40431 121203

F-F

Present thin 30939 77045 14036 14105 80788 161576Present thick 30931 77038 14032 14076 80792 161585NAS2D [23] 30829 76806 14129 14171 80415 160810Reference [23] 30932 77043 22987 23053 80789 161577Reference [13] 30945 77052 22864 23048 80787 161592

Table 4 Lowest natural frequencies (times1205960) of a moderately thick circular cylindrical shell with different circumferential wave numbers 119899119871119903 ℎ119903 = 012 ℎ119903 = 018119899 = 1 119899 = 2 119899 = 3 119899 = 4 119899 = 1 119899 = 2 119899 = 3 119899 = 42

Present 003724 002349 002449 003664 005632 003893 004939 007726Reference [31] 003730 002359 002462 003686 005652 003929 004996 007821Reference [32] 003730 002365 002470 003687 005653 003944 005009 007833

1Present 005861 004969 004765 005508 009420 008501 008995 011056

Reference [31] 005853 004978 004789 005545 009402 008545 009093 011205Reference [32] 005856 004986 004799 005550 009409 008562 009109 011202

05Present 010087 010238 010658 011463 018915 019406 020451 022180

Reference [31] 010057 010234 010688 011528 018894 019467 020616 022450Reference [32] 010047 010224 010674 011508 018832 019403 020544 022361

025Present 027310 027650 028219 029020 049751 050299 051216 052506

Reference [31] 027491 027849 028447 029287 050338 050937 051934 053325Reference [32] 027286 027641 028233 029064 049818 050418 051416 052808

(1) For low order frequencies good agreement is reachedbetween thin and moderately thick dynamic stiffnessformulations with ℎ119903 varying from 0025 to 01 Thesmaller the ℎ119903 the smaller the difference

(2) The difference between frequencies from thin andmoderately thick DSM formulations increases withthe increase of 119899 as log10(120578) always has a smallermagnitude under 119899 = 0 than those under 119899 = 1 2and 3

(3) It can be argued that the difference is insensitiveto boundary conditions The differences are roughlywithin 10minus5sim10minus2 in this investigation and the trends

of curves are similar for different boundary condi-tions

612 High Order Frequencies Investigation on the differ-ences between high order frequencies computed from thethin and moderately thick DSM formulations is examined inthis subsection Since it is known that the DSM formulationis not sensitive to boundary conditions [26] a C-S circu-lar cylindrical shell is studied The geometry and materialproperties are the same to the example in Section 611 if notspecifically stated

For high order frequencies ℎ119903 is investigated from verysmall value 0005 to a relatively large value 01 and the first

8 Shock and Vibration

005 0075 010025ℎr

minus1

minus2

minus3

minus4

minus5log 1

0(120578

)

n = 0

n = 1

n = 2

n = 3

(a) C-C

005 0075 010025ℎr

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)n = 0

n = 1

n = 2

n = 3

(b) C-F

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

n = 0

n = 1

n = 2

n = 3

005 0075 010025ℎr

(c) C-S

005 0075 010025ℎr

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

n = 0

n = 1

n = 2

n = 3

(d) S-S

Figure 6 The differences of low order frequencies between thin and moderately thick DSM formulations

ten frequencies under 119899 ranging from 0 to 10 from boththin and moderately thick DSM formulations are computedand compared The differences are shown in Figure 7 andconcluded as follows

(1) When ℎ119903 le 0025 log10(120578) is smaller than 10minus2most of the time which suggests that frequenciescomputed frommoderately thick shell theory are well

compatible with those from thin shell theory Noshear locking is observed

(2) When ℎ119903 ge 005 only frequencies with a small 119899(eg 119899 le 2) produce an acceptable difference (sim10minus2)With the increase of n the difference becomes quitenoticeable The higher the 119899 the more obvious thedifference

Shock and Vibration 9

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

(a) ℎ119903 = 0005

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

(b) ℎ119903 = 001

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

2 4 6 8 100m

(c) ℎ119903 = 0025

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

(d) ℎ119903 = 005

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

(e) ℎ119903 = 0075

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

(f) ℎ119903 = 01

Figure 7 The differences for frequencies between thin and moderately thick C-S circular cylindrical shell theories

(3) The order 119898 under a specific circumferential wavenumber 119899 does not have a significant impact on thedifferences as most curves in Figure 7 are largely flatand stable with the increase of119898

62 Ratio of Length to Radius (119871119903) Parameter 119871119903 controlsthe relative length of a circular cylindrical shell A larger119871119903 indicates a long shell and vice versa In this subsectionthe frequency trends and the compatibility between thin andmoderately thick DSM formulations with respect to 119871119903 arestudied In this subsection a C-C circular cylindrical shell

with ] = 03 is examined Figure 8 plots the trends of the firstfrequency parameterΩ under each 119899 with 119871119903 from 02 to 10

According to Figure 8 for a typical long circular cylin-drical shell (eg 119871119903 = 10) the lowest natural frequency isobtained with 119899 = 2 with the decrease of 119871119903 the circumfer-ential wave number 119899 corresponding to the lowest natural fre-quency slightly increases to 119899 = 3 or 4 when 119871119903 is very small(eg119871119903 = 02 and the shell is actually degraded to a ring) thefirst natural frequency under each 119899 increases monotonicallyand the smallest is reached at 119899 = 0The trends shown by Fig-ure 8 are also in good agreementwith other researchersrsquo work

10 Shock and Vibration

Ω

0

2

4

6

8

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(a) ℎ119903 = 0025 moderately thick shell

Ω

0

2

4

6

8

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(b) ℎ119903 = 0025 thin shell

0

3

6

9

12

Ω

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(c) ℎ119903 = 005 moderately thick shell

0

3

6

9

12

Ω

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(d) ℎ119903 = 005 thin shell

Figure 8 The trends of the first natural frequency parameters Ω under different 119899for example Markus [35] These obtained trends are applica-ble to both thin and moderately thick DSM formulations

Figure 9 shows the differences of the first frequenciesunder each 119899 computed from the thin and moderately thickDSM formulations of the circular cylindrical shell with ℎ119903 =0025 and 005 It can be understood fromFigure 9 that (1) thesmaller the ℎ119903 the smaller the difference (2) the larger the119871119903 the smaller the difference (3) conditions of ℎ119903 le 0025and 119899 le 10 need to be satisfied and a minor difference(log10(120578) le 10minus2) is required for a normal length shell thatis 119871119903 ge 2

7 Conclusions

This paper conducts comparison study on the frequenciesfrom thin and moderately thick circular cylindrical shellscomputed using the DSM formulations and comprehensiveparametric studies on ℎ119903 and 119871119903 are performed Based onthese investigations conclusions are outlined as follows

(1) Natural frequencies computed from the moder-ately thick DSM formulations can be consideredas satisfactory alternatives to those from thin DSMformulations when ℎ119903 le 0025 and 119871119903 ge 2

Shock and Vibration 11

0

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(a) ℎ119903 = 0025

0

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(b) ℎ119903 = 005

Figure 9 Differences between the first frequencies under each 119899 with different 119871119903

(differences are around or smaller than 10minus2) Sincethe DSM is exact no shear locking is observed

(2) Differences between frequencies from thin and mod-erately thick DSM formulations change with a varietyof factors The smaller ℎ119903 the smaller the differencethe smaller119871119903 the larger the difference the larger the119899 the larger the difference

It is proved that moderately thick DSM formulations canbe used to analyse the free vibration of normal length thin

circular cylindrical shells without shear locking demon-strating that the applicable range of moderately thick DSMformulations is expanded These merits should be attributedto that the dynamic stiffness method is an exact method

Though this comparison study is performed on circularcylindrical shells similar conclusions can be extended to thefree vibration of other shaped shells of revolution

Appendix

uthin =

1199061 = 1199061198991199062 = V119899

1199063 = 1199081198991199064 = 1199081015840119899

kthin =

V1 = 119903120587119864ℎ1 minus ]2[]119899119903 V119899 + ]

1119903119908119899 + 1199061015840119899]V2 = 119903120587119864ℎ2 (1 + ]) [minus119899119903 119906119899 + (1 + ℎ2121199032) V1015840119899 + 119899ℎ261199032 1199081015840119899]V3 = 119903120587119864ℎ2 (1 minus ]2) [119899ℎ

2

61199032 V1015840119899 + (2 minus ]) 1198992ℎ261199032 1199081015840119899 minus ℎ26 119908101584010158401015840119899 ]V4 = 119903120587119864ℎ12 (1 minus ]2) [minus]119899ℎ

2

1199032 V119899 minus ]1198992ℎ21199032 119908119899 + ℎ211990810158401015840119899 ]

12 Shock and Vibration

uthick =

1199061 = 1199061198991199062 = V119899

1199063 = 1199081198991199064 = 1205951199091198991199065 = 120595120579119899

kthick =

V1 = 119903120587119864ℎ1 minus ]2[1199061015840119899 + ]

119899119903 V119899 + ]119903119908119899]V2 = 12 119903120587119864ℎ1 + ]

[minus119899119903 119906119899 + (1 + ℎ2121199032) V1015840119899 minus 119899ℎ2121199032120595119909119899 + ℎ21211990321205951015840120579119899]V3 = 119903120587120581119866ℎ [1199081015840119899 + 120595119909119899]V4 = ℎ212 119903120587119864ℎ1 minus ]2

[1205951015840119909119899 + ]119899119903120595120579119899]

V5 = ℎ224 119903120587119864ℎ1 + ][V1015840119899119903 minus 119899119903120595119909119899 + 1205951015840120579119899]

[S]thin =

[[[[[[[[[[[[[[[[[[

11990411 0 0 11990414 0 11990416 0 011990422 11990423 0 11990425 0 0 1199042811990433 0 11990435 0 0 1199043811990444 0 11990446 11990447 0

11990455 0 0 0sym 11990466 0 0

0 011990488

]]]]]]]]]]]]]]]]]]

(A1)

where nonzero elements in the upper triangular domain are

11990411 = 120588120587119903ℎ1205962 minus 120587119864ℎ11989922119903 (1 minus ]) + 61198992119903120587119864ℎ(121199032 + ℎ2) (1 + ]) 11990414 = minus 1198992ℎ3120587119864(121199032 + ℎ2) (1 + ]) 11990416 = 12119899119903121199032 + ℎ2 11990422 = 120588120587119903ℎ1205962 minus 1198992120587119864ℎ (121199032 + ℎ2)

121199033 11990423 = minus120587119864ℎ121198991199032 + 1198993ℎ2121199033 11990425 = minus]119899119903 11990428 = ]

1198991199032 11990433 = 120588ℎ1199031205871205962 minus 120587119864ℎ119903 minus 1198994ℎ3120587119864121199033

11990435 = minus]119903 11990438 = ]11989921199032 11990444 = minus 21198992ℎ3119903120587119864(121199032 + ℎ2) (1 + ])

11990446 = minus 2119899ℎ2121199032 + ℎ2 11990447 = 111990455 = 1 minus ]2119903120587119864ℎ 11990466 = 24119903 (1 + ])120587119864ℎ (121199032 + ℎ2)

11990488 = 12 (1 minus ]2)119903120587119864ℎ3

[S]thick

Shock and Vibration 13

=

[[[[[[[[[[[[[[[[[[[[[[[

11990411 0 0 0 0 0 11990417 0 0 1199041lowast1011990422 11990423 0 11990425 11990426 0 0 0 011990433 0 11990435 11990436 0 0 0 0

11990444 0 0 0 11990448 0 1199044lowast1011990455 0 0 0 11990459 011990466 0 0 0 0

11990477 0 0 1199047lowast10sym 11990488 0 0

11990499 011990410lowast10

]]]]]]]]]]]]]]]]]]]]]]]

(A2)

where nonzero elements in the upper triangular domain are

11990411 = 119903120587120588ℎ120596211990417 = 119899119903

1199041lowast10 = minus 1198991199032 11990422 = 120587(119903120588ℎ1205962 minus 120581119866ℎ119903 minus 1198992119864ℎ119903 ) 11990423 = minus119899120587119903 (119864ℎ + 120581119866ℎ) 11990425 = 120587120581119866ℎ11990426 = minus]119899119903 11990433 = 120587(119903120588ℎ1205962 minus 119864ℎ119903 minus 1198992120581119866ℎ119903 ) 11990435 = 119899120587120581119866ℎ11990436 = minus]119903 11990444 = 1199031205871205881205962 ℎ312 11990448 = minus1

1199044lowast10 = 119899119903 11990455 = 120587(1199031205881205962 ℎ312 minus 1198992119864ℎ312119903 minus 119903120581119866ℎ) 11990459 = minus]119899119903 11990466 = 1 minus ]2119903120587119864ℎ 11990477 = 2 (1 + ])119903120587119864ℎ

1199047lowast10 = minus2 (1 + ])1199032120587119864ℎ 11990488 = 1119903120587120581119866ℎ 11990499 = 12 (1 minus ]2)

119903120587119864ℎ3 11990410lowast10 = 2 (1 + ])119903120587119864ℎ (12ℎ2 + 11199032 )

120581 = 120587212 (A3)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully appreciate the financial support fromNational Natural Science Foundation of China (51078198)Tsinghua University (2011THZ03) Jiangsu University ofScience and Technology (1732921402) and Shuang-ChuangProgram of Jiangsu Province for this research

References

[1] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover England UK 1927

[2] R N Arnold and G B Warburton ldquoThe flexural vibrationsof thin cylindersrdquo Proceedings of the Institution of MechanicalEngineers vol 167 no 1 pp 62ndash80 1953

[3] A W Leissa ldquoVibration of shellsrdquo Report 288 National Aero-nautics and Space Administration (NASA) Washington DCUSA 1973

[4] N S Bardell J M Dunsdon and R S Langley ldquoOn thefree vibration of completely free open cylindrically curvedisotropic shell panelsrdquo Journal of Sound and Vibration vol 207no 5 pp 647ndash669 1997

[5] D-Y Tan ldquoFree vibration analysis of shells of revolutionrdquoJournal of Sound and Vibration vol 213 no 1 pp 15ndash33 1998

[6] P M Naghdi and R M Cooper ldquoPropagation of elastic wavesin cylindrical shells including the effects of transverse shearand rotatory inertiardquo The Journal of the Acoustical Society ofAmerica vol 28 no 1 pp 56ndash63 1956

[7] I Mirsky and G Herrmann ldquoNonaxially symmetric motionsof cylindrical shellsrdquo The Journal of the Acoustical Society ofAmerica vol 29 pp 1116ndash1123 1957

[8] F Tornabene and E Viola ldquo2-D solution for free vibrationsof parabolic shells using generalized differential quadraturemethodrdquo European Journal of Mechanics ASolids vol 27 no6 pp 1001ndash1025 2008

[9] J L Mantari A S Oktem and C Guedes Soares ldquoBending andfree vibration analysis of isotropic and multilayered plates andshells by using a new accurate higher-order shear deformationtheoryrdquo Composites Part B Engineering vol 43 no 8 pp 3348ndash3360 2012

14 Shock and Vibration

[10] J G Kim J K Lee and H J Yoon ldquoFree vibration analysisfor shells of revolution based on p-version mixed finite elementformulationrdquo Finite Elements in Analysis and Design vol 95 pp12ndash19 2015

[11] F Tornabene ldquoFree vibrations of laminated composite doubly-curved shells and panels of revolution via the GDQ methodrdquoComputer Methods in Applied Mechanics and Engineering vol200 no 9ndash12 pp 931ndash952 2011

[12] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe local GDQmethod applied to general higher-order theories of doubly-curved laminated composite shells and panels the free vibrationanalysisrdquo Composite Structures vol 116 pp 637ndash660 2014

[13] A Pagani E Carrera and A Ferreira ldquoHigher-order theoriesand radial basis functions applied to free vibration analysisof thin-walled beamsrdquo Mechanics of Advanced Materials andStructures vol 23 no 9 pp 1080ndash1091 2016

[14] T Ye G Jin Y Chen and S Shi ldquoA unified formulationfor vibration analysis of open shells with arbitrary boundaryconditionsrdquo International Journal ofMechanical Sciences vol 81pp 42ndash59 2014

[15] T Ye G Jin S Shi and X Ma ldquoThree-dimensional freevibration analysis of thick cylindrical shells with general endconditions and resting on elastic foundationsrdquo InternationalJournal of Mechanical Sciences vol 84 pp 120ndash137 2014

[16] F W Williams and W H Wittrick ldquoAn automatic computa-tional procedure for calculating natural frequencies of skeletalstructuresrdquo International Journal of Mechanical Sciences vol 12no 9 pp 781ndash791 1970

[17] W H Wittrick and F W Williams ldquoA general algorithmfor computing natural frequencies of elastic structuresrdquo TheQuarterly Journal of Mechanics and Applied Mathematics vol24 pp 263ndash284 1971

[18] S Yuan K Ye F W Williams and D Kennedy ldquoRecursivesecond order convergence method for natural frequencies andmodes when using dynamic stiffness matricesrdquo InternationalJournal for Numerical Methods in Engineering vol 56 no 12pp 1795ndash1814 2003

[19] S Yuan K Ye and FWWilliams ldquoSecond order mode-findingmethod in dynamic stiffness matrix methodsrdquo Journal of Soundand Vibration vol 269 no 3ndash5 pp 689ndash708 2004

[20] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007

[21] J R Banerjee C W Cheung R Morishima M Pereraand J Njuguna ldquoFree vibration of a three-layered sandwichbeam using the dynamic stiffness method and experimentrdquoInternational Journal of Solids and Structures vol 44 no 22-23pp 7543ndash7563 2007

[22] H Su J R Banerjee and C W Cheung ldquoDynamic stiffnessformulation and free vibration analysis of functionally gradedbeamsrdquo Composite Structures vol 106 pp 854ndash862 2013

[23] A Pagani M Boscolo J R Banerjee and E Carrera ldquoExactdynamic stiffness elements based on one-dimensional higher-order theories for free vibration analysis of solid and thin-walled structuresrdquo Journal of Sound and Vibration vol 332 no23 pp 6104ndash6127 2013

[24] N El-Kaabazi and D Kennedy ldquoCalculation of natural fre-quencies and vibration modes of variable thickness cylindricalshells using the Wittrick-Williams algorithmrdquo Computers andStructures vol 104-105 pp 4ndash12 2012

[25] X Chen Research on dynamic stiffness method for free vibrationof shells of revolution [MS thesis] Tsinghua University 2009(Chinese)

[26] XChen andKYe ldquoFreeVibrationAnalysis for Shells of Revolu-tion Using an Exact Dynamic Stiffness Methodrdquo MathematicalProblems in Engineering vol 2016 Article ID 4513520 12 pages2016

[27] X Chen and K Ye ldquoAnalysis of free vibration of moderatelythick circular cylindrical shells using the dynamic stiffnessmethodrdquo Engineering Mechanics vol 33 no 9 pp 40ndash48 2016(Chinese)

[28] X M Zhang G R Liu and K Y Lam ldquoVibration analysisof thin cylindrical shells using wave propagation approachrdquoJournal of Sound and Vibration vol 239 no 3 pp 397ndash4032001

[29] L Xuebin ldquoStudy on free vibration analysis of circular cylin-drical shells using wave propagationrdquo Journal of Sound andVibration vol 311 no 3-5 pp 667ndash682 2008

[30] C B Sharma ldquoCalculation of natural frequencies of fixed-freecircular cylindrical shellsrdquo Journal of Sound and Vibration vol35 no 1 pp 55ndash76 1974

[31] A E ArmenakasDCGazis andGHerrmannFreeVibrationsof Circular Cylindrical Shells PergamonPressOxfordUK 1969

[32] A Bhimaraddi ldquoA higher order theory for free vibrationanalysis of circular cylindrical shellsrdquo International Journal ofSolids and Structures vol 20 no 7 pp 623ndash630 1984

[33] U Ascher J Christiansen and R D Russell ldquoAlgorithm 569COLSYS collocation software for boundary value ODEs [D2]rdquoACM Transactions on Mathematical Software vol 7 no 2 pp223ndash229 1981

[34] U Ascher J Christiansen and R D Russell ldquoCollocationsoftware for boundary-value ODEsrdquo ACM Transactions onMathematical Software vol 7 no 2 pp 209ndash222 1981

[35] S Markus The Mechanics of Vibrations of Cylindrical ShellsElsevier Science Limited 1988

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4 Shock and Vibration

119909119887 respectivelyThe segment-end displacement vector d119890 isdefined as

d119890thin = 1199061 (119909119886) 1199062 (119909119886) 1199063 (119909119886) 1199064 (119909119886) 1199061 (119909119887) 1199062 (119909119887) 1199063 (119909119887) 1199064 (119909119887)T

d119890thick = 1199061 (119909119886) 1199062 (119909119886) 1199063 (119909119886) 1199064 (119909119886) 1199065 (119909119886) 1199061 (119909119887) 1199062 (119909119887) 1199063 (119909119887) 1199064 (119909119887) 1199065 (119909119887)T

(9)

and the force vector F119890 isF119890thin = minusV1 (119909119886) minusV2 (119909119886) minusV3 (119909119886)

minus V4 (119909119886) V1 (119909119887) V2 (119909119887) V3 (119909119887) V4 (119909119887)T

F119890thick = minusV1 (119909119886) minusV2 (119909119886) minusV3 (119909119886) minusV4 (119909119886) minus V5 (119909119886) V1 (119909119887) V2 (119909119887) V3 (119909119887) V4 (119909119887) V5 (119909119887)T

(10)

Mathematically apply the boundary conditions in (11) or (12)to (8) in turn

d119890thin = 119890119895 119895 = 1 8 (11)

d119890thick = 119890119895 119895 = 1 10 (12)

where 119890119895 is a unit vector with the 119895th element equal to oneand the corresponding element dynamic stiffness matrix [k]119890is formulated by

[k]119890thin = [F119890119895=1 F119890119895=2 F119890119895=3 F119890119895=4 F119890119895=5 F119890119895=6 F119890119895=7 F119890119895=8]thin [k]119890thick = [F119890119895=1 F119890119895=2 F119890119895=3 F119890119895=4 F119890119895=5 F119890119895=6 F119890119895=7 F119890119895=8 F119890119895=9 F119890119895=10]thick

(13)

Due to the complexity (8) is solved computationally usingadaptive ordinary differential equations (ODEs) solver COL-SYS [33 34] The global dynamic stiffness matrix K of acircular cylindrical shell is assembled in a regular FEMway

4 Wittrick-Williams Algorithm

The dynamic stiffness method is used in conjunction withtheWittrick-Williams (W-W) algorithmTheW-Walgorithmmerely gives the number of frequencies below 120596lowast with theexpression of

119869 = 1198690 + 119869119870 (14)

where 119869 is the total number of frequencies exceeded bythe trial frequency 120596lowast 119869119870 = 119904K(120596lowast) is the sign countof the global dynamic stiffness matrix K and equal to thenumber of negative elements on the diagonal of an uppertriangular matrix obtained from K by applying standardGaussian eliminationwithout row exchange 1198690 is the numberof clamped-end frequencies exceeded by 120596lowast and can beaccumulated from 1198691198900 of each shell segment element over thewhole meridian as is expressed in

1198690 = sum1198691198900 (15)

To solve 1198691198900 a substructuring method is employed by takingadvantage of the self-adaptability of COLSYSWhen comput-ing the dynamic stiffness matrix of a shell segment elementfor example element (119890) a submesh will be generated byCOLSYS adaptively as is shown in Figure 4

Applying the Wittrick-Williams algorithm again on thesubmesh we have

1198691198900 = sum119869()0 + 119904 K (120596lowast) (16)

where notation ldquordquo stands for submesh Since COLSYS iscapable of controlling the error tolerance adaptively thenumber of clamped-end natural frequencies exceeded by 120596lowaston the submesh has to be zero suggesting 119869()0 = 0 Otherwisethe clamped-end modes can be enlarged infinitely resultingin the unsatisfaction of the given error tolerance

The following will introduce the formulation of K(120596lowast) in(16) Taking a subelement () with the coordinates (119909119894 119909119894+1)on element (119890) for instance its corresponding dynamicstiffness matrix [k]() can be obtained by linearly combiningthe eight (or ten) solutions of (8) and (11) or (12) Note thatthe enddisplacement and force vectors for this subelement ()are d() and F() respectively and the subelement stiffnessmatrix [k]() satisfies

[k]() d() = F() (17)

By applying the boundary conditions in (11) or (12) in turnagain for thin or moderately thick circular cylindrical shells(18) is obtained

[k]() [B]() = [C]() (18)

where

[B]()= [d()119895=1 d()119895=2 d()119895=3 d()119895=4 d()119895=5 d()119895=6 d()119895=7 d()119895=8]thin [C]()= [F()119895=1 F()119895=2 F()119895=3 F()119895=4 F()119895=5 F()119895=6 F()119895=7 F()119895=8]thin

(19)

Shock and Vibration 5

xa xi xi+1 xb

(e)

Figure 4 Submesh on element (119890)

for thin shells or

[B]() = [d()119895=1 d()119895=2 d()119895=3 d()119895=4 d()119895=5 d()119895=6 d()119895=7 d()119895=8 d()119895=9 d()119895=10]thick [C]() = [F()119895=1 F()119895=2 F()119895=3 F()119895=4 F()119895=5 F()119895=6 F()119895=7 F()119895=8 F()119895=9 F()119895=10]thick

(20)

for moderately thick shellsSince 119869()0 = 0 [B]() is nonsingular (otherwise120596lowast is one of

clamped-end frequencies which is conflicted with 119869()0 = 0)and (21) is obtained

[k]() = [C]() ([B]())minus1 (21)

Thus K(120596lowast) can be assembled regularly from [k]() Based onthe above theories a FORTRAN code was written and usedto solve the free vibration of both thin and moderately thickcircular cylindrical shells in this paper

5 Validation

In this section four examples on free vibration of circularcylindrical shells are presented to show the reliability andaccuracy of the proposed method in the above sectionsExamples in Sections 51 and 52 are calculated with thin shelltheory in Section 53 both thin and moderately thick shelltheories are employed to study the free vibration of the sameshell and results are given opposite for comparison examplesin Section 54 are performed with the moderately thick shelltheory

51 Free Vibration of a Thin Clamped-Clamped Shell Zhanget al [28] and Xuebin [29] analysed the natural frequenciesof a clamped-clamped (C-C) circular cylindrical shell basedon the Flugge thin shell theory by employing the FEM codeMSCNASTRAN and wave propagation approach respec-tively The clamped boundary condition is considered as

119906 = V = 119908 = 120597119908120597119909 = 0 (22)

The length of the shell is 119871 = 20m radius 119903 = 1m thicknessℎ = 001m Density 120588 = 7850 kgm3 Youngrsquos modulus119864 = 210GPa Poissonrsquos ratio ] = 03 In this example thinshell theory is used in formulating the dynamic stiffness andthe first nine frequencies are tabulated in Table 1 where 119898denotes the number of half-waves along the axial direction

It can be observed from Table 1 that the frequenciesobtained using the present method agree very well withthat from the FEM (MSCNASTRAN) and wave propagation

approach Further investigation into Table 1 shows that thelowest natural frequency of a circular cylindrical usuallycomes with a relatively larger 119899 that is 119899 = 2 in this case

52 Free Vibration of a Thin Clamped-Free Shell In Sharma[30] the free vibration of a clamped-free (C-F) circularcylindrical shell is investigated based on the Rayleigh-Ritzmethod and Flugge shell theory The geometry parametersof the shell are length 119871 = 0502m radius 119903 = 00635mand thickness ℎ = 000163m and the material propertiesare Youngrsquos modulus 119864 = 210GPa Poissonrsquos ratio ] = 028and density 120588 = 7800 kgm3 A thin shell formulation is usedin the present dynamic stiffness approach Table 2 presentsthe lowest four frequencies of the shell with a relatively largercircumferential wave number 119899 = 5 and 6

It can be concluded that the results agree very well with[30] and the errors are no larger than 04 By comparingthe results from the present method and the Rayleigh-Ritzmethod used in Sharma [30] the reliability of the presentmethod is verified

53 Free Vibration of a Shell with Different Boundary Condi-tions Pagani et al [23] applied the one-dimensional higher-order beam theory and exact dynamic stiffness elements tostudy the free vibration of solid and thin-walled structuresin which a thin-walled cylinder with a circular cross-section(Figure 5) is analysed In a recent paper [13] they employedboth the radial basis functions (RBFs) method and CarreraUnified Formulation (CUF) to examine the free vibrationbehaviour of both open and closed thin-walled structures

The length of the examined cylinder is 119871 = 20mouter diameter is 119889 = 2m and thickness is ℎ = 002mThe material properties are Youngrsquos modulus 119864 = 75GPaPoissonrsquos ratio ] = 033 and density 120588 = 2700 kgm3 Table 3shows the natural frequencies of this thin-walled cylindricalbeam with different boundaries namely both ends simplysupported (S-S) both ends clamped (C-C) clamped-free(C-F) and free-free (F-F) The first and second flexural(bending) shell-like and torsional natural frequencies arepresented Solutions from the present thin and moderatelythick DSM 2D MSCNASTRAN and [13 23] are given forcomparison

6 Shock and Vibration

Table 1 The lowest nine frequencies of the C-C circular cylindricalshell (Hz)

Order (119898 119899) Present Reference [28] Reference [29]1 (1 2) 1200 1225 12132 (1 3) 1956 1964 19613 (2 3) 2310 2318 23284 (2 2) 2716 2769 28065 (1 1) 2829 mdash 30096 (3 3) 3148 316 31977 (1 4) 3642 367 36488 (2 4) 3728 3755 37389 (3 4) 3960 3987 3977mdash not given

Table 2The lowest four frequencies of the C-F shell with 119899 = 5 and6 (Hz)

119899 = 5 119899 = 6Present Sharma [30] Present Sharma [30]

1 23619 23666 34628 346972 24021 24064 34990 350503 25050 25091 35768 358074 27062 27160 37120 37166

It can be observed that the natural frequencies computedfrom the present dynamic stiffness method agree very wellwith that from 2D MSCNASTRAN code for both thin andmoderately thick shell theories Comparing with the datafrom the higher-order beam theories [13 23] the presentmethod provides an excellent agreement for flexural andtorsional modes The differences of frequencies for the firstand second shell-like modes are larger and this can belargely attributed that the formulations established in [1323] are based on higher-order beam theories Obviouslybetter agreement can be achieved if exact shell theories areemployed by [13 23] Further investigation by the presentDSM approach shows that 119899 = 1 corresponds to flexural(bending) mode while 119899 = 0 and 2 correspond to tor-sional and shell-like vibration modes respectively whichdemonstrate that the current DSM approach is capable ofshowing the correct vibration modes as presented in existingliterature

54 Free Vibration of a Moderately Thick Shell Bhimaraddi[32] analysed the free vibration of a simply supported (S-S) moderately thick circular cylindrical shell by using thehigher-order displacement model with the consideration ofin-plane inertia rotary inertia and shear effect In [31]natural frequencies of the circular cylindrical shell subjectsto different ℎ119903 and 119871119903 are computed and compared withthe solutions from 3D elasticity theory Results from themoderately thick DSM approach are also presented and listedin Table 4 In this example Poissonrsquos ratio is 03 and 1205960 =(120587ℎ)radic119866120588 where 119866 is the shear modulus

According to Table 4 results from the present methodagree very well with those from [31 32] demonstrating the

h d

Figure 5 The cross-section of a thin-walled cylinder

high accuracy and adaptability of the present method forcircular cylindrical shells with different ℎ119903 and 1198711199036 Parametric Study

61 Ratio of Thickness to Radius (ℎ119903) An important param-eter to determine whether a shell is thin or moderatelythick is the thickness to radius ratio (ℎ119903) According tothe classic shell theory a critical value for ℎ119903 between thinand moderately thick shell theories is 005 In this sectioninfluences of ℎ119903 on natural frequencies are investigatedextensively from 0005 to 01 for both thin and moderatelythick DSM formulations

611 Low Order Frequencies Consider a circular cylindricalshell with 119871119903 = 12 and Poissonrsquos ratio ] = 03 Toinvestigate the differences of natural frequencies between thinand moderately thick DSM formulations a notation 120578 in (23)is introduced as

120578 = 10038161003816100381610038161003816100381610038161003816ΩthinΩthick

minus 110038161003816100381610038161003816100381610038161003816 (23)

where Ωthin and Ωthick are the nondimensional frequencyparameters defined as

Ω = 120596119903radic120588 (1 minus ]2)119864 (24)

Consequently log10(120578) is the index of differenceFigure 6 shows the log10(120578) for the first frequencies (119898 =1) under 119899 = 0 1 2 and 3 between thin and moderately thick

DSM formulations with a variety of boundary conditionsfor example clamped-clamped (C-C) clamped-free (C-F)clamped-simply supported (C-S) and both ends simplysupported (S-S) In these cases ℎ119903 is investigated from 0025to 01

If log10(120578) le minus2 (which suggests the difference is nolarger than 1) it can be considered that results from the thinand moderately thick dynamic stiffness formulations are ingood agreement with each other Furthermore frequenciescomputed from themoderately thick shell theory can be usedas good alternatives to those computed from the thin shelltheory Referring to Figure 6 the following conclusions canbe reached

Shock and Vibration 7

Table 3 Natural frequencies of the thin-walled cylinder with different boundary conditions (Hz)

BCs Model I flexural II flexural I shell-like II shell-like I torsional II torsional

S-S

Present thin 14025 51506 14851 22931 80790 161579Present thick 14021 51502 14828 22877 80792 161585NAS2D [23] 13978 51366 14913 22917 80415 160810Reference [23] 14022 51503 18405 25460 80786 161573Reference [13] 14294 51567 18608 25574 80639 162551

C-C

Present thin 28577 69110 17350 30239 80790 161579Present thick 28574 69105 17327 30193 80792 161585NAS2D [23] 28498 68960 17396 30225 80415 160810Reference [23] 28576 69110 20484 32222 80786 161573Reference [13] 28354 69096 20463 31974 80838 161579

C-F

Present thin 5081 29092 14153 17427 40395 121184Present thick 5076 29086 14140 17370 40396 121188NAS2D [23] 5059 29001 14235 17435 40209 120620Reference [23] 5077 29090 23069 25239 40394 121181Reference [13] 5047 29002 23003 24979 40431 121203

F-F

Present thin 30939 77045 14036 14105 80788 161576Present thick 30931 77038 14032 14076 80792 161585NAS2D [23] 30829 76806 14129 14171 80415 160810Reference [23] 30932 77043 22987 23053 80789 161577Reference [13] 30945 77052 22864 23048 80787 161592

Table 4 Lowest natural frequencies (times1205960) of a moderately thick circular cylindrical shell with different circumferential wave numbers 119899119871119903 ℎ119903 = 012 ℎ119903 = 018119899 = 1 119899 = 2 119899 = 3 119899 = 4 119899 = 1 119899 = 2 119899 = 3 119899 = 42

Present 003724 002349 002449 003664 005632 003893 004939 007726Reference [31] 003730 002359 002462 003686 005652 003929 004996 007821Reference [32] 003730 002365 002470 003687 005653 003944 005009 007833

1Present 005861 004969 004765 005508 009420 008501 008995 011056

Reference [31] 005853 004978 004789 005545 009402 008545 009093 011205Reference [32] 005856 004986 004799 005550 009409 008562 009109 011202

05Present 010087 010238 010658 011463 018915 019406 020451 022180

Reference [31] 010057 010234 010688 011528 018894 019467 020616 022450Reference [32] 010047 010224 010674 011508 018832 019403 020544 022361

025Present 027310 027650 028219 029020 049751 050299 051216 052506

Reference [31] 027491 027849 028447 029287 050338 050937 051934 053325Reference [32] 027286 027641 028233 029064 049818 050418 051416 052808

(1) For low order frequencies good agreement is reachedbetween thin and moderately thick dynamic stiffnessformulations with ℎ119903 varying from 0025 to 01 Thesmaller the ℎ119903 the smaller the difference

(2) The difference between frequencies from thin andmoderately thick DSM formulations increases withthe increase of 119899 as log10(120578) always has a smallermagnitude under 119899 = 0 than those under 119899 = 1 2and 3

(3) It can be argued that the difference is insensitiveto boundary conditions The differences are roughlywithin 10minus5sim10minus2 in this investigation and the trends

of curves are similar for different boundary condi-tions

612 High Order Frequencies Investigation on the differ-ences between high order frequencies computed from thethin and moderately thick DSM formulations is examined inthis subsection Since it is known that the DSM formulationis not sensitive to boundary conditions [26] a C-S circu-lar cylindrical shell is studied The geometry and materialproperties are the same to the example in Section 611 if notspecifically stated

For high order frequencies ℎ119903 is investigated from verysmall value 0005 to a relatively large value 01 and the first

8 Shock and Vibration

005 0075 010025ℎr

minus1

minus2

minus3

minus4

minus5log 1

0(120578

)

n = 0

n = 1

n = 2

n = 3

(a) C-C

005 0075 010025ℎr

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)n = 0

n = 1

n = 2

n = 3

(b) C-F

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

n = 0

n = 1

n = 2

n = 3

005 0075 010025ℎr

(c) C-S

005 0075 010025ℎr

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

n = 0

n = 1

n = 2

n = 3

(d) S-S

Figure 6 The differences of low order frequencies between thin and moderately thick DSM formulations

ten frequencies under 119899 ranging from 0 to 10 from boththin and moderately thick DSM formulations are computedand compared The differences are shown in Figure 7 andconcluded as follows

(1) When ℎ119903 le 0025 log10(120578) is smaller than 10minus2most of the time which suggests that frequenciescomputed frommoderately thick shell theory are well

compatible with those from thin shell theory Noshear locking is observed

(2) When ℎ119903 ge 005 only frequencies with a small 119899(eg 119899 le 2) produce an acceptable difference (sim10minus2)With the increase of n the difference becomes quitenoticeable The higher the 119899 the more obvious thedifference

Shock and Vibration 9

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

(a) ℎ119903 = 0005

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

(b) ℎ119903 = 001

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

2 4 6 8 100m

(c) ℎ119903 = 0025

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

(d) ℎ119903 = 005

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

(e) ℎ119903 = 0075

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

(f) ℎ119903 = 01

Figure 7 The differences for frequencies between thin and moderately thick C-S circular cylindrical shell theories

(3) The order 119898 under a specific circumferential wavenumber 119899 does not have a significant impact on thedifferences as most curves in Figure 7 are largely flatand stable with the increase of119898

62 Ratio of Length to Radius (119871119903) Parameter 119871119903 controlsthe relative length of a circular cylindrical shell A larger119871119903 indicates a long shell and vice versa In this subsectionthe frequency trends and the compatibility between thin andmoderately thick DSM formulations with respect to 119871119903 arestudied In this subsection a C-C circular cylindrical shell

with ] = 03 is examined Figure 8 plots the trends of the firstfrequency parameterΩ under each 119899 with 119871119903 from 02 to 10

According to Figure 8 for a typical long circular cylin-drical shell (eg 119871119903 = 10) the lowest natural frequency isobtained with 119899 = 2 with the decrease of 119871119903 the circumfer-ential wave number 119899 corresponding to the lowest natural fre-quency slightly increases to 119899 = 3 or 4 when 119871119903 is very small(eg119871119903 = 02 and the shell is actually degraded to a ring) thefirst natural frequency under each 119899 increases monotonicallyand the smallest is reached at 119899 = 0The trends shown by Fig-ure 8 are also in good agreementwith other researchersrsquo work

10 Shock and Vibration

Ω

0

2

4

6

8

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(a) ℎ119903 = 0025 moderately thick shell

Ω

0

2

4

6

8

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(b) ℎ119903 = 0025 thin shell

0

3

6

9

12

Ω

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(c) ℎ119903 = 005 moderately thick shell

0

3

6

9

12

Ω

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(d) ℎ119903 = 005 thin shell

Figure 8 The trends of the first natural frequency parameters Ω under different 119899for example Markus [35] These obtained trends are applica-ble to both thin and moderately thick DSM formulations

Figure 9 shows the differences of the first frequenciesunder each 119899 computed from the thin and moderately thickDSM formulations of the circular cylindrical shell with ℎ119903 =0025 and 005 It can be understood fromFigure 9 that (1) thesmaller the ℎ119903 the smaller the difference (2) the larger the119871119903 the smaller the difference (3) conditions of ℎ119903 le 0025and 119899 le 10 need to be satisfied and a minor difference(log10(120578) le 10minus2) is required for a normal length shell thatis 119871119903 ge 2

7 Conclusions

This paper conducts comparison study on the frequenciesfrom thin and moderately thick circular cylindrical shellscomputed using the DSM formulations and comprehensiveparametric studies on ℎ119903 and 119871119903 are performed Based onthese investigations conclusions are outlined as follows

(1) Natural frequencies computed from the moder-ately thick DSM formulations can be consideredas satisfactory alternatives to those from thin DSMformulations when ℎ119903 le 0025 and 119871119903 ge 2

Shock and Vibration 11

0

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(a) ℎ119903 = 0025

0

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(b) ℎ119903 = 005

Figure 9 Differences between the first frequencies under each 119899 with different 119871119903

(differences are around or smaller than 10minus2) Sincethe DSM is exact no shear locking is observed

(2) Differences between frequencies from thin and mod-erately thick DSM formulations change with a varietyof factors The smaller ℎ119903 the smaller the differencethe smaller119871119903 the larger the difference the larger the119899 the larger the difference

It is proved that moderately thick DSM formulations canbe used to analyse the free vibration of normal length thin

circular cylindrical shells without shear locking demon-strating that the applicable range of moderately thick DSMformulations is expanded These merits should be attributedto that the dynamic stiffness method is an exact method

Though this comparison study is performed on circularcylindrical shells similar conclusions can be extended to thefree vibration of other shaped shells of revolution

Appendix

uthin =

1199061 = 1199061198991199062 = V119899

1199063 = 1199081198991199064 = 1199081015840119899

kthin =

V1 = 119903120587119864ℎ1 minus ]2[]119899119903 V119899 + ]

1119903119908119899 + 1199061015840119899]V2 = 119903120587119864ℎ2 (1 + ]) [minus119899119903 119906119899 + (1 + ℎ2121199032) V1015840119899 + 119899ℎ261199032 1199081015840119899]V3 = 119903120587119864ℎ2 (1 minus ]2) [119899ℎ

2

61199032 V1015840119899 + (2 minus ]) 1198992ℎ261199032 1199081015840119899 minus ℎ26 119908101584010158401015840119899 ]V4 = 119903120587119864ℎ12 (1 minus ]2) [minus]119899ℎ

2

1199032 V119899 minus ]1198992ℎ21199032 119908119899 + ℎ211990810158401015840119899 ]

12 Shock and Vibration

uthick =

1199061 = 1199061198991199062 = V119899

1199063 = 1199081198991199064 = 1205951199091198991199065 = 120595120579119899

kthick =

V1 = 119903120587119864ℎ1 minus ]2[1199061015840119899 + ]

119899119903 V119899 + ]119903119908119899]V2 = 12 119903120587119864ℎ1 + ]

[minus119899119903 119906119899 + (1 + ℎ2121199032) V1015840119899 minus 119899ℎ2121199032120595119909119899 + ℎ21211990321205951015840120579119899]V3 = 119903120587120581119866ℎ [1199081015840119899 + 120595119909119899]V4 = ℎ212 119903120587119864ℎ1 minus ]2

[1205951015840119909119899 + ]119899119903120595120579119899]

V5 = ℎ224 119903120587119864ℎ1 + ][V1015840119899119903 minus 119899119903120595119909119899 + 1205951015840120579119899]

[S]thin =

[[[[[[[[[[[[[[[[[[

11990411 0 0 11990414 0 11990416 0 011990422 11990423 0 11990425 0 0 1199042811990433 0 11990435 0 0 1199043811990444 0 11990446 11990447 0

11990455 0 0 0sym 11990466 0 0

0 011990488

]]]]]]]]]]]]]]]]]]

(A1)

where nonzero elements in the upper triangular domain are

11990411 = 120588120587119903ℎ1205962 minus 120587119864ℎ11989922119903 (1 minus ]) + 61198992119903120587119864ℎ(121199032 + ℎ2) (1 + ]) 11990414 = minus 1198992ℎ3120587119864(121199032 + ℎ2) (1 + ]) 11990416 = 12119899119903121199032 + ℎ2 11990422 = 120588120587119903ℎ1205962 minus 1198992120587119864ℎ (121199032 + ℎ2)

121199033 11990423 = minus120587119864ℎ121198991199032 + 1198993ℎ2121199033 11990425 = minus]119899119903 11990428 = ]

1198991199032 11990433 = 120588ℎ1199031205871205962 minus 120587119864ℎ119903 minus 1198994ℎ3120587119864121199033

11990435 = minus]119903 11990438 = ]11989921199032 11990444 = minus 21198992ℎ3119903120587119864(121199032 + ℎ2) (1 + ])

11990446 = minus 2119899ℎ2121199032 + ℎ2 11990447 = 111990455 = 1 minus ]2119903120587119864ℎ 11990466 = 24119903 (1 + ])120587119864ℎ (121199032 + ℎ2)

11990488 = 12 (1 minus ]2)119903120587119864ℎ3

[S]thick

Shock and Vibration 13

=

[[[[[[[[[[[[[[[[[[[[[[[

11990411 0 0 0 0 0 11990417 0 0 1199041lowast1011990422 11990423 0 11990425 11990426 0 0 0 011990433 0 11990435 11990436 0 0 0 0

11990444 0 0 0 11990448 0 1199044lowast1011990455 0 0 0 11990459 011990466 0 0 0 0

11990477 0 0 1199047lowast10sym 11990488 0 0

11990499 011990410lowast10

]]]]]]]]]]]]]]]]]]]]]]]

(A2)

where nonzero elements in the upper triangular domain are

11990411 = 119903120587120588ℎ120596211990417 = 119899119903

1199041lowast10 = minus 1198991199032 11990422 = 120587(119903120588ℎ1205962 minus 120581119866ℎ119903 minus 1198992119864ℎ119903 ) 11990423 = minus119899120587119903 (119864ℎ + 120581119866ℎ) 11990425 = 120587120581119866ℎ11990426 = minus]119899119903 11990433 = 120587(119903120588ℎ1205962 minus 119864ℎ119903 minus 1198992120581119866ℎ119903 ) 11990435 = 119899120587120581119866ℎ11990436 = minus]119903 11990444 = 1199031205871205881205962 ℎ312 11990448 = minus1

1199044lowast10 = 119899119903 11990455 = 120587(1199031205881205962 ℎ312 minus 1198992119864ℎ312119903 minus 119903120581119866ℎ) 11990459 = minus]119899119903 11990466 = 1 minus ]2119903120587119864ℎ 11990477 = 2 (1 + ])119903120587119864ℎ

1199047lowast10 = minus2 (1 + ])1199032120587119864ℎ 11990488 = 1119903120587120581119866ℎ 11990499 = 12 (1 minus ]2)

119903120587119864ℎ3 11990410lowast10 = 2 (1 + ])119903120587119864ℎ (12ℎ2 + 11199032 )

120581 = 120587212 (A3)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully appreciate the financial support fromNational Natural Science Foundation of China (51078198)Tsinghua University (2011THZ03) Jiangsu University ofScience and Technology (1732921402) and Shuang-ChuangProgram of Jiangsu Province for this research

References

[1] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover England UK 1927

[2] R N Arnold and G B Warburton ldquoThe flexural vibrationsof thin cylindersrdquo Proceedings of the Institution of MechanicalEngineers vol 167 no 1 pp 62ndash80 1953

[3] A W Leissa ldquoVibration of shellsrdquo Report 288 National Aero-nautics and Space Administration (NASA) Washington DCUSA 1973

[4] N S Bardell J M Dunsdon and R S Langley ldquoOn thefree vibration of completely free open cylindrically curvedisotropic shell panelsrdquo Journal of Sound and Vibration vol 207no 5 pp 647ndash669 1997

[5] D-Y Tan ldquoFree vibration analysis of shells of revolutionrdquoJournal of Sound and Vibration vol 213 no 1 pp 15ndash33 1998

[6] P M Naghdi and R M Cooper ldquoPropagation of elastic wavesin cylindrical shells including the effects of transverse shearand rotatory inertiardquo The Journal of the Acoustical Society ofAmerica vol 28 no 1 pp 56ndash63 1956

[7] I Mirsky and G Herrmann ldquoNonaxially symmetric motionsof cylindrical shellsrdquo The Journal of the Acoustical Society ofAmerica vol 29 pp 1116ndash1123 1957

[8] F Tornabene and E Viola ldquo2-D solution for free vibrationsof parabolic shells using generalized differential quadraturemethodrdquo European Journal of Mechanics ASolids vol 27 no6 pp 1001ndash1025 2008

[9] J L Mantari A S Oktem and C Guedes Soares ldquoBending andfree vibration analysis of isotropic and multilayered plates andshells by using a new accurate higher-order shear deformationtheoryrdquo Composites Part B Engineering vol 43 no 8 pp 3348ndash3360 2012

14 Shock and Vibration

[10] J G Kim J K Lee and H J Yoon ldquoFree vibration analysisfor shells of revolution based on p-version mixed finite elementformulationrdquo Finite Elements in Analysis and Design vol 95 pp12ndash19 2015

[11] F Tornabene ldquoFree vibrations of laminated composite doubly-curved shells and panels of revolution via the GDQ methodrdquoComputer Methods in Applied Mechanics and Engineering vol200 no 9ndash12 pp 931ndash952 2011

[12] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe local GDQmethod applied to general higher-order theories of doubly-curved laminated composite shells and panels the free vibrationanalysisrdquo Composite Structures vol 116 pp 637ndash660 2014

[13] A Pagani E Carrera and A Ferreira ldquoHigher-order theoriesand radial basis functions applied to free vibration analysisof thin-walled beamsrdquo Mechanics of Advanced Materials andStructures vol 23 no 9 pp 1080ndash1091 2016

[14] T Ye G Jin Y Chen and S Shi ldquoA unified formulationfor vibration analysis of open shells with arbitrary boundaryconditionsrdquo International Journal ofMechanical Sciences vol 81pp 42ndash59 2014

[15] T Ye G Jin S Shi and X Ma ldquoThree-dimensional freevibration analysis of thick cylindrical shells with general endconditions and resting on elastic foundationsrdquo InternationalJournal of Mechanical Sciences vol 84 pp 120ndash137 2014

[16] F W Williams and W H Wittrick ldquoAn automatic computa-tional procedure for calculating natural frequencies of skeletalstructuresrdquo International Journal of Mechanical Sciences vol 12no 9 pp 781ndash791 1970

[17] W H Wittrick and F W Williams ldquoA general algorithmfor computing natural frequencies of elastic structuresrdquo TheQuarterly Journal of Mechanics and Applied Mathematics vol24 pp 263ndash284 1971

[18] S Yuan K Ye F W Williams and D Kennedy ldquoRecursivesecond order convergence method for natural frequencies andmodes when using dynamic stiffness matricesrdquo InternationalJournal for Numerical Methods in Engineering vol 56 no 12pp 1795ndash1814 2003

[19] S Yuan K Ye and FWWilliams ldquoSecond order mode-findingmethod in dynamic stiffness matrix methodsrdquo Journal of Soundand Vibration vol 269 no 3ndash5 pp 689ndash708 2004

[20] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007

[21] J R Banerjee C W Cheung R Morishima M Pereraand J Njuguna ldquoFree vibration of a three-layered sandwichbeam using the dynamic stiffness method and experimentrdquoInternational Journal of Solids and Structures vol 44 no 22-23pp 7543ndash7563 2007

[22] H Su J R Banerjee and C W Cheung ldquoDynamic stiffnessformulation and free vibration analysis of functionally gradedbeamsrdquo Composite Structures vol 106 pp 854ndash862 2013

[23] A Pagani M Boscolo J R Banerjee and E Carrera ldquoExactdynamic stiffness elements based on one-dimensional higher-order theories for free vibration analysis of solid and thin-walled structuresrdquo Journal of Sound and Vibration vol 332 no23 pp 6104ndash6127 2013

[24] N El-Kaabazi and D Kennedy ldquoCalculation of natural fre-quencies and vibration modes of variable thickness cylindricalshells using the Wittrick-Williams algorithmrdquo Computers andStructures vol 104-105 pp 4ndash12 2012

[25] X Chen Research on dynamic stiffness method for free vibrationof shells of revolution [MS thesis] Tsinghua University 2009(Chinese)

[26] XChen andKYe ldquoFreeVibrationAnalysis for Shells of Revolu-tion Using an Exact Dynamic Stiffness Methodrdquo MathematicalProblems in Engineering vol 2016 Article ID 4513520 12 pages2016

[27] X Chen and K Ye ldquoAnalysis of free vibration of moderatelythick circular cylindrical shells using the dynamic stiffnessmethodrdquo Engineering Mechanics vol 33 no 9 pp 40ndash48 2016(Chinese)

[28] X M Zhang G R Liu and K Y Lam ldquoVibration analysisof thin cylindrical shells using wave propagation approachrdquoJournal of Sound and Vibration vol 239 no 3 pp 397ndash4032001

[29] L Xuebin ldquoStudy on free vibration analysis of circular cylin-drical shells using wave propagationrdquo Journal of Sound andVibration vol 311 no 3-5 pp 667ndash682 2008

[30] C B Sharma ldquoCalculation of natural frequencies of fixed-freecircular cylindrical shellsrdquo Journal of Sound and Vibration vol35 no 1 pp 55ndash76 1974

[31] A E ArmenakasDCGazis andGHerrmannFreeVibrationsof Circular Cylindrical Shells PergamonPressOxfordUK 1969

[32] A Bhimaraddi ldquoA higher order theory for free vibrationanalysis of circular cylindrical shellsrdquo International Journal ofSolids and Structures vol 20 no 7 pp 623ndash630 1984

[33] U Ascher J Christiansen and R D Russell ldquoAlgorithm 569COLSYS collocation software for boundary value ODEs [D2]rdquoACM Transactions on Mathematical Software vol 7 no 2 pp223ndash229 1981

[34] U Ascher J Christiansen and R D Russell ldquoCollocationsoftware for boundary-value ODEsrdquo ACM Transactions onMathematical Software vol 7 no 2 pp 209ndash222 1981

[35] S Markus The Mechanics of Vibrations of Cylindrical ShellsElsevier Science Limited 1988

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Shock and Vibration 5

xa xi xi+1 xb

(e)

Figure 4 Submesh on element (119890)

for thin shells or

[B]() = [d()119895=1 d()119895=2 d()119895=3 d()119895=4 d()119895=5 d()119895=6 d()119895=7 d()119895=8 d()119895=9 d()119895=10]thick [C]() = [F()119895=1 F()119895=2 F()119895=3 F()119895=4 F()119895=5 F()119895=6 F()119895=7 F()119895=8 F()119895=9 F()119895=10]thick

(20)

for moderately thick shellsSince 119869()0 = 0 [B]() is nonsingular (otherwise120596lowast is one of

clamped-end frequencies which is conflicted with 119869()0 = 0)and (21) is obtained

[k]() = [C]() ([B]())minus1 (21)

Thus K(120596lowast) can be assembled regularly from [k]() Based onthe above theories a FORTRAN code was written and usedto solve the free vibration of both thin and moderately thickcircular cylindrical shells in this paper

5 Validation

In this section four examples on free vibration of circularcylindrical shells are presented to show the reliability andaccuracy of the proposed method in the above sectionsExamples in Sections 51 and 52 are calculated with thin shelltheory in Section 53 both thin and moderately thick shelltheories are employed to study the free vibration of the sameshell and results are given opposite for comparison examplesin Section 54 are performed with the moderately thick shelltheory

51 Free Vibration of a Thin Clamped-Clamped Shell Zhanget al [28] and Xuebin [29] analysed the natural frequenciesof a clamped-clamped (C-C) circular cylindrical shell basedon the Flugge thin shell theory by employing the FEM codeMSCNASTRAN and wave propagation approach respec-tively The clamped boundary condition is considered as

119906 = V = 119908 = 120597119908120597119909 = 0 (22)

The length of the shell is 119871 = 20m radius 119903 = 1m thicknessℎ = 001m Density 120588 = 7850 kgm3 Youngrsquos modulus119864 = 210GPa Poissonrsquos ratio ] = 03 In this example thinshell theory is used in formulating the dynamic stiffness andthe first nine frequencies are tabulated in Table 1 where 119898denotes the number of half-waves along the axial direction

It can be observed from Table 1 that the frequenciesobtained using the present method agree very well withthat from the FEM (MSCNASTRAN) and wave propagation

approach Further investigation into Table 1 shows that thelowest natural frequency of a circular cylindrical usuallycomes with a relatively larger 119899 that is 119899 = 2 in this case

52 Free Vibration of a Thin Clamped-Free Shell In Sharma[30] the free vibration of a clamped-free (C-F) circularcylindrical shell is investigated based on the Rayleigh-Ritzmethod and Flugge shell theory The geometry parametersof the shell are length 119871 = 0502m radius 119903 = 00635mand thickness ℎ = 000163m and the material propertiesare Youngrsquos modulus 119864 = 210GPa Poissonrsquos ratio ] = 028and density 120588 = 7800 kgm3 A thin shell formulation is usedin the present dynamic stiffness approach Table 2 presentsthe lowest four frequencies of the shell with a relatively largercircumferential wave number 119899 = 5 and 6

It can be concluded that the results agree very well with[30] and the errors are no larger than 04 By comparingthe results from the present method and the Rayleigh-Ritzmethod used in Sharma [30] the reliability of the presentmethod is verified

53 Free Vibration of a Shell with Different Boundary Condi-tions Pagani et al [23] applied the one-dimensional higher-order beam theory and exact dynamic stiffness elements tostudy the free vibration of solid and thin-walled structuresin which a thin-walled cylinder with a circular cross-section(Figure 5) is analysed In a recent paper [13] they employedboth the radial basis functions (RBFs) method and CarreraUnified Formulation (CUF) to examine the free vibrationbehaviour of both open and closed thin-walled structures

The length of the examined cylinder is 119871 = 20mouter diameter is 119889 = 2m and thickness is ℎ = 002mThe material properties are Youngrsquos modulus 119864 = 75GPaPoissonrsquos ratio ] = 033 and density 120588 = 2700 kgm3 Table 3shows the natural frequencies of this thin-walled cylindricalbeam with different boundaries namely both ends simplysupported (S-S) both ends clamped (C-C) clamped-free(C-F) and free-free (F-F) The first and second flexural(bending) shell-like and torsional natural frequencies arepresented Solutions from the present thin and moderatelythick DSM 2D MSCNASTRAN and [13 23] are given forcomparison

6 Shock and Vibration

Table 1 The lowest nine frequencies of the C-C circular cylindricalshell (Hz)

Order (119898 119899) Present Reference [28] Reference [29]1 (1 2) 1200 1225 12132 (1 3) 1956 1964 19613 (2 3) 2310 2318 23284 (2 2) 2716 2769 28065 (1 1) 2829 mdash 30096 (3 3) 3148 316 31977 (1 4) 3642 367 36488 (2 4) 3728 3755 37389 (3 4) 3960 3987 3977mdash not given

Table 2The lowest four frequencies of the C-F shell with 119899 = 5 and6 (Hz)

119899 = 5 119899 = 6Present Sharma [30] Present Sharma [30]

1 23619 23666 34628 346972 24021 24064 34990 350503 25050 25091 35768 358074 27062 27160 37120 37166

It can be observed that the natural frequencies computedfrom the present dynamic stiffness method agree very wellwith that from 2D MSCNASTRAN code for both thin andmoderately thick shell theories Comparing with the datafrom the higher-order beam theories [13 23] the presentmethod provides an excellent agreement for flexural andtorsional modes The differences of frequencies for the firstand second shell-like modes are larger and this can belargely attributed that the formulations established in [1323] are based on higher-order beam theories Obviouslybetter agreement can be achieved if exact shell theories areemployed by [13 23] Further investigation by the presentDSM approach shows that 119899 = 1 corresponds to flexural(bending) mode while 119899 = 0 and 2 correspond to tor-sional and shell-like vibration modes respectively whichdemonstrate that the current DSM approach is capable ofshowing the correct vibration modes as presented in existingliterature

54 Free Vibration of a Moderately Thick Shell Bhimaraddi[32] analysed the free vibration of a simply supported (S-S) moderately thick circular cylindrical shell by using thehigher-order displacement model with the consideration ofin-plane inertia rotary inertia and shear effect In [31]natural frequencies of the circular cylindrical shell subjectsto different ℎ119903 and 119871119903 are computed and compared withthe solutions from 3D elasticity theory Results from themoderately thick DSM approach are also presented and listedin Table 4 In this example Poissonrsquos ratio is 03 and 1205960 =(120587ℎ)radic119866120588 where 119866 is the shear modulus

According to Table 4 results from the present methodagree very well with those from [31 32] demonstrating the

h d

Figure 5 The cross-section of a thin-walled cylinder

high accuracy and adaptability of the present method forcircular cylindrical shells with different ℎ119903 and 1198711199036 Parametric Study

61 Ratio of Thickness to Radius (ℎ119903) An important param-eter to determine whether a shell is thin or moderatelythick is the thickness to radius ratio (ℎ119903) According tothe classic shell theory a critical value for ℎ119903 between thinand moderately thick shell theories is 005 In this sectioninfluences of ℎ119903 on natural frequencies are investigatedextensively from 0005 to 01 for both thin and moderatelythick DSM formulations

611 Low Order Frequencies Consider a circular cylindricalshell with 119871119903 = 12 and Poissonrsquos ratio ] = 03 Toinvestigate the differences of natural frequencies between thinand moderately thick DSM formulations a notation 120578 in (23)is introduced as

120578 = 10038161003816100381610038161003816100381610038161003816ΩthinΩthick

minus 110038161003816100381610038161003816100381610038161003816 (23)

where Ωthin and Ωthick are the nondimensional frequencyparameters defined as

Ω = 120596119903radic120588 (1 minus ]2)119864 (24)

Consequently log10(120578) is the index of differenceFigure 6 shows the log10(120578) for the first frequencies (119898 =1) under 119899 = 0 1 2 and 3 between thin and moderately thick

DSM formulations with a variety of boundary conditionsfor example clamped-clamped (C-C) clamped-free (C-F)clamped-simply supported (C-S) and both ends simplysupported (S-S) In these cases ℎ119903 is investigated from 0025to 01

If log10(120578) le minus2 (which suggests the difference is nolarger than 1) it can be considered that results from the thinand moderately thick dynamic stiffness formulations are ingood agreement with each other Furthermore frequenciescomputed from themoderately thick shell theory can be usedas good alternatives to those computed from the thin shelltheory Referring to Figure 6 the following conclusions canbe reached

Shock and Vibration 7

Table 3 Natural frequencies of the thin-walled cylinder with different boundary conditions (Hz)

BCs Model I flexural II flexural I shell-like II shell-like I torsional II torsional

S-S

Present thin 14025 51506 14851 22931 80790 161579Present thick 14021 51502 14828 22877 80792 161585NAS2D [23] 13978 51366 14913 22917 80415 160810Reference [23] 14022 51503 18405 25460 80786 161573Reference [13] 14294 51567 18608 25574 80639 162551

C-C

Present thin 28577 69110 17350 30239 80790 161579Present thick 28574 69105 17327 30193 80792 161585NAS2D [23] 28498 68960 17396 30225 80415 160810Reference [23] 28576 69110 20484 32222 80786 161573Reference [13] 28354 69096 20463 31974 80838 161579

C-F

Present thin 5081 29092 14153 17427 40395 121184Present thick 5076 29086 14140 17370 40396 121188NAS2D [23] 5059 29001 14235 17435 40209 120620Reference [23] 5077 29090 23069 25239 40394 121181Reference [13] 5047 29002 23003 24979 40431 121203

F-F

Present thin 30939 77045 14036 14105 80788 161576Present thick 30931 77038 14032 14076 80792 161585NAS2D [23] 30829 76806 14129 14171 80415 160810Reference [23] 30932 77043 22987 23053 80789 161577Reference [13] 30945 77052 22864 23048 80787 161592

Table 4 Lowest natural frequencies (times1205960) of a moderately thick circular cylindrical shell with different circumferential wave numbers 119899119871119903 ℎ119903 = 012 ℎ119903 = 018119899 = 1 119899 = 2 119899 = 3 119899 = 4 119899 = 1 119899 = 2 119899 = 3 119899 = 42

Present 003724 002349 002449 003664 005632 003893 004939 007726Reference [31] 003730 002359 002462 003686 005652 003929 004996 007821Reference [32] 003730 002365 002470 003687 005653 003944 005009 007833

1Present 005861 004969 004765 005508 009420 008501 008995 011056

Reference [31] 005853 004978 004789 005545 009402 008545 009093 011205Reference [32] 005856 004986 004799 005550 009409 008562 009109 011202

05Present 010087 010238 010658 011463 018915 019406 020451 022180

Reference [31] 010057 010234 010688 011528 018894 019467 020616 022450Reference [32] 010047 010224 010674 011508 018832 019403 020544 022361

025Present 027310 027650 028219 029020 049751 050299 051216 052506

Reference [31] 027491 027849 028447 029287 050338 050937 051934 053325Reference [32] 027286 027641 028233 029064 049818 050418 051416 052808

(1) For low order frequencies good agreement is reachedbetween thin and moderately thick dynamic stiffnessformulations with ℎ119903 varying from 0025 to 01 Thesmaller the ℎ119903 the smaller the difference

(2) The difference between frequencies from thin andmoderately thick DSM formulations increases withthe increase of 119899 as log10(120578) always has a smallermagnitude under 119899 = 0 than those under 119899 = 1 2and 3

(3) It can be argued that the difference is insensitiveto boundary conditions The differences are roughlywithin 10minus5sim10minus2 in this investigation and the trends

of curves are similar for different boundary condi-tions

612 High Order Frequencies Investigation on the differ-ences between high order frequencies computed from thethin and moderately thick DSM formulations is examined inthis subsection Since it is known that the DSM formulationis not sensitive to boundary conditions [26] a C-S circu-lar cylindrical shell is studied The geometry and materialproperties are the same to the example in Section 611 if notspecifically stated

For high order frequencies ℎ119903 is investigated from verysmall value 0005 to a relatively large value 01 and the first

8 Shock and Vibration

005 0075 010025ℎr

minus1

minus2

minus3

minus4

minus5log 1

0(120578

)

n = 0

n = 1

n = 2

n = 3

(a) C-C

005 0075 010025ℎr

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)n = 0

n = 1

n = 2

n = 3

(b) C-F

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

n = 0

n = 1

n = 2

n = 3

005 0075 010025ℎr

(c) C-S

005 0075 010025ℎr

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

n = 0

n = 1

n = 2

n = 3

(d) S-S

Figure 6 The differences of low order frequencies between thin and moderately thick DSM formulations

ten frequencies under 119899 ranging from 0 to 10 from boththin and moderately thick DSM formulations are computedand compared The differences are shown in Figure 7 andconcluded as follows

(1) When ℎ119903 le 0025 log10(120578) is smaller than 10minus2most of the time which suggests that frequenciescomputed frommoderately thick shell theory are well

compatible with those from thin shell theory Noshear locking is observed

(2) When ℎ119903 ge 005 only frequencies with a small 119899(eg 119899 le 2) produce an acceptable difference (sim10minus2)With the increase of n the difference becomes quitenoticeable The higher the 119899 the more obvious thedifference

Shock and Vibration 9

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

(a) ℎ119903 = 0005

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

(b) ℎ119903 = 001

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

2 4 6 8 100m

(c) ℎ119903 = 0025

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

(d) ℎ119903 = 005

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

(e) ℎ119903 = 0075

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

(f) ℎ119903 = 01

Figure 7 The differences for frequencies between thin and moderately thick C-S circular cylindrical shell theories

(3) The order 119898 under a specific circumferential wavenumber 119899 does not have a significant impact on thedifferences as most curves in Figure 7 are largely flatand stable with the increase of119898

62 Ratio of Length to Radius (119871119903) Parameter 119871119903 controlsthe relative length of a circular cylindrical shell A larger119871119903 indicates a long shell and vice versa In this subsectionthe frequency trends and the compatibility between thin andmoderately thick DSM formulations with respect to 119871119903 arestudied In this subsection a C-C circular cylindrical shell

with ] = 03 is examined Figure 8 plots the trends of the firstfrequency parameterΩ under each 119899 with 119871119903 from 02 to 10

According to Figure 8 for a typical long circular cylin-drical shell (eg 119871119903 = 10) the lowest natural frequency isobtained with 119899 = 2 with the decrease of 119871119903 the circumfer-ential wave number 119899 corresponding to the lowest natural fre-quency slightly increases to 119899 = 3 or 4 when 119871119903 is very small(eg119871119903 = 02 and the shell is actually degraded to a ring) thefirst natural frequency under each 119899 increases monotonicallyand the smallest is reached at 119899 = 0The trends shown by Fig-ure 8 are also in good agreementwith other researchersrsquo work

10 Shock and Vibration

Ω

0

2

4

6

8

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(a) ℎ119903 = 0025 moderately thick shell

Ω

0

2

4

6

8

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(b) ℎ119903 = 0025 thin shell

0

3

6

9

12

Ω

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(c) ℎ119903 = 005 moderately thick shell

0

3

6

9

12

Ω

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(d) ℎ119903 = 005 thin shell

Figure 8 The trends of the first natural frequency parameters Ω under different 119899for example Markus [35] These obtained trends are applica-ble to both thin and moderately thick DSM formulations

Figure 9 shows the differences of the first frequenciesunder each 119899 computed from the thin and moderately thickDSM formulations of the circular cylindrical shell with ℎ119903 =0025 and 005 It can be understood fromFigure 9 that (1) thesmaller the ℎ119903 the smaller the difference (2) the larger the119871119903 the smaller the difference (3) conditions of ℎ119903 le 0025and 119899 le 10 need to be satisfied and a minor difference(log10(120578) le 10minus2) is required for a normal length shell thatis 119871119903 ge 2

7 Conclusions

This paper conducts comparison study on the frequenciesfrom thin and moderately thick circular cylindrical shellscomputed using the DSM formulations and comprehensiveparametric studies on ℎ119903 and 119871119903 are performed Based onthese investigations conclusions are outlined as follows

(1) Natural frequencies computed from the moder-ately thick DSM formulations can be consideredas satisfactory alternatives to those from thin DSMformulations when ℎ119903 le 0025 and 119871119903 ge 2

Shock and Vibration 11

0

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(a) ℎ119903 = 0025

0

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(b) ℎ119903 = 005

Figure 9 Differences between the first frequencies under each 119899 with different 119871119903

(differences are around or smaller than 10minus2) Sincethe DSM is exact no shear locking is observed

(2) Differences between frequencies from thin and mod-erately thick DSM formulations change with a varietyof factors The smaller ℎ119903 the smaller the differencethe smaller119871119903 the larger the difference the larger the119899 the larger the difference

It is proved that moderately thick DSM formulations canbe used to analyse the free vibration of normal length thin

circular cylindrical shells without shear locking demon-strating that the applicable range of moderately thick DSMformulations is expanded These merits should be attributedto that the dynamic stiffness method is an exact method

Though this comparison study is performed on circularcylindrical shells similar conclusions can be extended to thefree vibration of other shaped shells of revolution

Appendix

uthin =

1199061 = 1199061198991199062 = V119899

1199063 = 1199081198991199064 = 1199081015840119899

kthin =

V1 = 119903120587119864ℎ1 minus ]2[]119899119903 V119899 + ]

1119903119908119899 + 1199061015840119899]V2 = 119903120587119864ℎ2 (1 + ]) [minus119899119903 119906119899 + (1 + ℎ2121199032) V1015840119899 + 119899ℎ261199032 1199081015840119899]V3 = 119903120587119864ℎ2 (1 minus ]2) [119899ℎ

2

61199032 V1015840119899 + (2 minus ]) 1198992ℎ261199032 1199081015840119899 minus ℎ26 119908101584010158401015840119899 ]V4 = 119903120587119864ℎ12 (1 minus ]2) [minus]119899ℎ

2

1199032 V119899 minus ]1198992ℎ21199032 119908119899 + ℎ211990810158401015840119899 ]

12 Shock and Vibration

uthick =

1199061 = 1199061198991199062 = V119899

1199063 = 1199081198991199064 = 1205951199091198991199065 = 120595120579119899

kthick =

V1 = 119903120587119864ℎ1 minus ]2[1199061015840119899 + ]

119899119903 V119899 + ]119903119908119899]V2 = 12 119903120587119864ℎ1 + ]

[minus119899119903 119906119899 + (1 + ℎ2121199032) V1015840119899 minus 119899ℎ2121199032120595119909119899 + ℎ21211990321205951015840120579119899]V3 = 119903120587120581119866ℎ [1199081015840119899 + 120595119909119899]V4 = ℎ212 119903120587119864ℎ1 minus ]2

[1205951015840119909119899 + ]119899119903120595120579119899]

V5 = ℎ224 119903120587119864ℎ1 + ][V1015840119899119903 minus 119899119903120595119909119899 + 1205951015840120579119899]

[S]thin =

[[[[[[[[[[[[[[[[[[

11990411 0 0 11990414 0 11990416 0 011990422 11990423 0 11990425 0 0 1199042811990433 0 11990435 0 0 1199043811990444 0 11990446 11990447 0

11990455 0 0 0sym 11990466 0 0

0 011990488

]]]]]]]]]]]]]]]]]]

(A1)

where nonzero elements in the upper triangular domain are

11990411 = 120588120587119903ℎ1205962 minus 120587119864ℎ11989922119903 (1 minus ]) + 61198992119903120587119864ℎ(121199032 + ℎ2) (1 + ]) 11990414 = minus 1198992ℎ3120587119864(121199032 + ℎ2) (1 + ]) 11990416 = 12119899119903121199032 + ℎ2 11990422 = 120588120587119903ℎ1205962 minus 1198992120587119864ℎ (121199032 + ℎ2)

121199033 11990423 = minus120587119864ℎ121198991199032 + 1198993ℎ2121199033 11990425 = minus]119899119903 11990428 = ]

1198991199032 11990433 = 120588ℎ1199031205871205962 minus 120587119864ℎ119903 minus 1198994ℎ3120587119864121199033

11990435 = minus]119903 11990438 = ]11989921199032 11990444 = minus 21198992ℎ3119903120587119864(121199032 + ℎ2) (1 + ])

11990446 = minus 2119899ℎ2121199032 + ℎ2 11990447 = 111990455 = 1 minus ]2119903120587119864ℎ 11990466 = 24119903 (1 + ])120587119864ℎ (121199032 + ℎ2)

11990488 = 12 (1 minus ]2)119903120587119864ℎ3

[S]thick

Shock and Vibration 13

=

[[[[[[[[[[[[[[[[[[[[[[[

11990411 0 0 0 0 0 11990417 0 0 1199041lowast1011990422 11990423 0 11990425 11990426 0 0 0 011990433 0 11990435 11990436 0 0 0 0

11990444 0 0 0 11990448 0 1199044lowast1011990455 0 0 0 11990459 011990466 0 0 0 0

11990477 0 0 1199047lowast10sym 11990488 0 0

11990499 011990410lowast10

]]]]]]]]]]]]]]]]]]]]]]]

(A2)

where nonzero elements in the upper triangular domain are

11990411 = 119903120587120588ℎ120596211990417 = 119899119903

1199041lowast10 = minus 1198991199032 11990422 = 120587(119903120588ℎ1205962 minus 120581119866ℎ119903 minus 1198992119864ℎ119903 ) 11990423 = minus119899120587119903 (119864ℎ + 120581119866ℎ) 11990425 = 120587120581119866ℎ11990426 = minus]119899119903 11990433 = 120587(119903120588ℎ1205962 minus 119864ℎ119903 minus 1198992120581119866ℎ119903 ) 11990435 = 119899120587120581119866ℎ11990436 = minus]119903 11990444 = 1199031205871205881205962 ℎ312 11990448 = minus1

1199044lowast10 = 119899119903 11990455 = 120587(1199031205881205962 ℎ312 minus 1198992119864ℎ312119903 minus 119903120581119866ℎ) 11990459 = minus]119899119903 11990466 = 1 minus ]2119903120587119864ℎ 11990477 = 2 (1 + ])119903120587119864ℎ

1199047lowast10 = minus2 (1 + ])1199032120587119864ℎ 11990488 = 1119903120587120581119866ℎ 11990499 = 12 (1 minus ]2)

119903120587119864ℎ3 11990410lowast10 = 2 (1 + ])119903120587119864ℎ (12ℎ2 + 11199032 )

120581 = 120587212 (A3)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully appreciate the financial support fromNational Natural Science Foundation of China (51078198)Tsinghua University (2011THZ03) Jiangsu University ofScience and Technology (1732921402) and Shuang-ChuangProgram of Jiangsu Province for this research

References

[1] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover England UK 1927

[2] R N Arnold and G B Warburton ldquoThe flexural vibrationsof thin cylindersrdquo Proceedings of the Institution of MechanicalEngineers vol 167 no 1 pp 62ndash80 1953

[3] A W Leissa ldquoVibration of shellsrdquo Report 288 National Aero-nautics and Space Administration (NASA) Washington DCUSA 1973

[4] N S Bardell J M Dunsdon and R S Langley ldquoOn thefree vibration of completely free open cylindrically curvedisotropic shell panelsrdquo Journal of Sound and Vibration vol 207no 5 pp 647ndash669 1997

[5] D-Y Tan ldquoFree vibration analysis of shells of revolutionrdquoJournal of Sound and Vibration vol 213 no 1 pp 15ndash33 1998

[6] P M Naghdi and R M Cooper ldquoPropagation of elastic wavesin cylindrical shells including the effects of transverse shearand rotatory inertiardquo The Journal of the Acoustical Society ofAmerica vol 28 no 1 pp 56ndash63 1956

[7] I Mirsky and G Herrmann ldquoNonaxially symmetric motionsof cylindrical shellsrdquo The Journal of the Acoustical Society ofAmerica vol 29 pp 1116ndash1123 1957

[8] F Tornabene and E Viola ldquo2-D solution for free vibrationsof parabolic shells using generalized differential quadraturemethodrdquo European Journal of Mechanics ASolids vol 27 no6 pp 1001ndash1025 2008

[9] J L Mantari A S Oktem and C Guedes Soares ldquoBending andfree vibration analysis of isotropic and multilayered plates andshells by using a new accurate higher-order shear deformationtheoryrdquo Composites Part B Engineering vol 43 no 8 pp 3348ndash3360 2012

14 Shock and Vibration

[10] J G Kim J K Lee and H J Yoon ldquoFree vibration analysisfor shells of revolution based on p-version mixed finite elementformulationrdquo Finite Elements in Analysis and Design vol 95 pp12ndash19 2015

[11] F Tornabene ldquoFree vibrations of laminated composite doubly-curved shells and panels of revolution via the GDQ methodrdquoComputer Methods in Applied Mechanics and Engineering vol200 no 9ndash12 pp 931ndash952 2011

[12] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe local GDQmethod applied to general higher-order theories of doubly-curved laminated composite shells and panels the free vibrationanalysisrdquo Composite Structures vol 116 pp 637ndash660 2014

[13] A Pagani E Carrera and A Ferreira ldquoHigher-order theoriesand radial basis functions applied to free vibration analysisof thin-walled beamsrdquo Mechanics of Advanced Materials andStructures vol 23 no 9 pp 1080ndash1091 2016

[14] T Ye G Jin Y Chen and S Shi ldquoA unified formulationfor vibration analysis of open shells with arbitrary boundaryconditionsrdquo International Journal ofMechanical Sciences vol 81pp 42ndash59 2014

[15] T Ye G Jin S Shi and X Ma ldquoThree-dimensional freevibration analysis of thick cylindrical shells with general endconditions and resting on elastic foundationsrdquo InternationalJournal of Mechanical Sciences vol 84 pp 120ndash137 2014

[16] F W Williams and W H Wittrick ldquoAn automatic computa-tional procedure for calculating natural frequencies of skeletalstructuresrdquo International Journal of Mechanical Sciences vol 12no 9 pp 781ndash791 1970

[17] W H Wittrick and F W Williams ldquoA general algorithmfor computing natural frequencies of elastic structuresrdquo TheQuarterly Journal of Mechanics and Applied Mathematics vol24 pp 263ndash284 1971

[18] S Yuan K Ye F W Williams and D Kennedy ldquoRecursivesecond order convergence method for natural frequencies andmodes when using dynamic stiffness matricesrdquo InternationalJournal for Numerical Methods in Engineering vol 56 no 12pp 1795ndash1814 2003

[19] S Yuan K Ye and FWWilliams ldquoSecond order mode-findingmethod in dynamic stiffness matrix methodsrdquo Journal of Soundand Vibration vol 269 no 3ndash5 pp 689ndash708 2004

[20] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007

[21] J R Banerjee C W Cheung R Morishima M Pereraand J Njuguna ldquoFree vibration of a three-layered sandwichbeam using the dynamic stiffness method and experimentrdquoInternational Journal of Solids and Structures vol 44 no 22-23pp 7543ndash7563 2007

[22] H Su J R Banerjee and C W Cheung ldquoDynamic stiffnessformulation and free vibration analysis of functionally gradedbeamsrdquo Composite Structures vol 106 pp 854ndash862 2013

[23] A Pagani M Boscolo J R Banerjee and E Carrera ldquoExactdynamic stiffness elements based on one-dimensional higher-order theories for free vibration analysis of solid and thin-walled structuresrdquo Journal of Sound and Vibration vol 332 no23 pp 6104ndash6127 2013

[24] N El-Kaabazi and D Kennedy ldquoCalculation of natural fre-quencies and vibration modes of variable thickness cylindricalshells using the Wittrick-Williams algorithmrdquo Computers andStructures vol 104-105 pp 4ndash12 2012

[25] X Chen Research on dynamic stiffness method for free vibrationof shells of revolution [MS thesis] Tsinghua University 2009(Chinese)

[26] XChen andKYe ldquoFreeVibrationAnalysis for Shells of Revolu-tion Using an Exact Dynamic Stiffness Methodrdquo MathematicalProblems in Engineering vol 2016 Article ID 4513520 12 pages2016

[27] X Chen and K Ye ldquoAnalysis of free vibration of moderatelythick circular cylindrical shells using the dynamic stiffnessmethodrdquo Engineering Mechanics vol 33 no 9 pp 40ndash48 2016(Chinese)

[28] X M Zhang G R Liu and K Y Lam ldquoVibration analysisof thin cylindrical shells using wave propagation approachrdquoJournal of Sound and Vibration vol 239 no 3 pp 397ndash4032001

[29] L Xuebin ldquoStudy on free vibration analysis of circular cylin-drical shells using wave propagationrdquo Journal of Sound andVibration vol 311 no 3-5 pp 667ndash682 2008

[30] C B Sharma ldquoCalculation of natural frequencies of fixed-freecircular cylindrical shellsrdquo Journal of Sound and Vibration vol35 no 1 pp 55ndash76 1974

[31] A E ArmenakasDCGazis andGHerrmannFreeVibrationsof Circular Cylindrical Shells PergamonPressOxfordUK 1969

[32] A Bhimaraddi ldquoA higher order theory for free vibrationanalysis of circular cylindrical shellsrdquo International Journal ofSolids and Structures vol 20 no 7 pp 623ndash630 1984

[33] U Ascher J Christiansen and R D Russell ldquoAlgorithm 569COLSYS collocation software for boundary value ODEs [D2]rdquoACM Transactions on Mathematical Software vol 7 no 2 pp223ndash229 1981

[34] U Ascher J Christiansen and R D Russell ldquoCollocationsoftware for boundary-value ODEsrdquo ACM Transactions onMathematical Software vol 7 no 2 pp 209ndash222 1981

[35] S Markus The Mechanics of Vibrations of Cylindrical ShellsElsevier Science Limited 1988

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6 Shock and Vibration

Table 1 The lowest nine frequencies of the C-C circular cylindricalshell (Hz)

Order (119898 119899) Present Reference [28] Reference [29]1 (1 2) 1200 1225 12132 (1 3) 1956 1964 19613 (2 3) 2310 2318 23284 (2 2) 2716 2769 28065 (1 1) 2829 mdash 30096 (3 3) 3148 316 31977 (1 4) 3642 367 36488 (2 4) 3728 3755 37389 (3 4) 3960 3987 3977mdash not given

Table 2The lowest four frequencies of the C-F shell with 119899 = 5 and6 (Hz)

119899 = 5 119899 = 6Present Sharma [30] Present Sharma [30]

1 23619 23666 34628 346972 24021 24064 34990 350503 25050 25091 35768 358074 27062 27160 37120 37166

It can be observed that the natural frequencies computedfrom the present dynamic stiffness method agree very wellwith that from 2D MSCNASTRAN code for both thin andmoderately thick shell theories Comparing with the datafrom the higher-order beam theories [13 23] the presentmethod provides an excellent agreement for flexural andtorsional modes The differences of frequencies for the firstand second shell-like modes are larger and this can belargely attributed that the formulations established in [1323] are based on higher-order beam theories Obviouslybetter agreement can be achieved if exact shell theories areemployed by [13 23] Further investigation by the presentDSM approach shows that 119899 = 1 corresponds to flexural(bending) mode while 119899 = 0 and 2 correspond to tor-sional and shell-like vibration modes respectively whichdemonstrate that the current DSM approach is capable ofshowing the correct vibration modes as presented in existingliterature

54 Free Vibration of a Moderately Thick Shell Bhimaraddi[32] analysed the free vibration of a simply supported (S-S) moderately thick circular cylindrical shell by using thehigher-order displacement model with the consideration ofin-plane inertia rotary inertia and shear effect In [31]natural frequencies of the circular cylindrical shell subjectsto different ℎ119903 and 119871119903 are computed and compared withthe solutions from 3D elasticity theory Results from themoderately thick DSM approach are also presented and listedin Table 4 In this example Poissonrsquos ratio is 03 and 1205960 =(120587ℎ)radic119866120588 where 119866 is the shear modulus

According to Table 4 results from the present methodagree very well with those from [31 32] demonstrating the

h d

Figure 5 The cross-section of a thin-walled cylinder

high accuracy and adaptability of the present method forcircular cylindrical shells with different ℎ119903 and 1198711199036 Parametric Study

61 Ratio of Thickness to Radius (ℎ119903) An important param-eter to determine whether a shell is thin or moderatelythick is the thickness to radius ratio (ℎ119903) According tothe classic shell theory a critical value for ℎ119903 between thinand moderately thick shell theories is 005 In this sectioninfluences of ℎ119903 on natural frequencies are investigatedextensively from 0005 to 01 for both thin and moderatelythick DSM formulations

611 Low Order Frequencies Consider a circular cylindricalshell with 119871119903 = 12 and Poissonrsquos ratio ] = 03 Toinvestigate the differences of natural frequencies between thinand moderately thick DSM formulations a notation 120578 in (23)is introduced as

120578 = 10038161003816100381610038161003816100381610038161003816ΩthinΩthick

minus 110038161003816100381610038161003816100381610038161003816 (23)

where Ωthin and Ωthick are the nondimensional frequencyparameters defined as

Ω = 120596119903radic120588 (1 minus ]2)119864 (24)

Consequently log10(120578) is the index of differenceFigure 6 shows the log10(120578) for the first frequencies (119898 =1) under 119899 = 0 1 2 and 3 between thin and moderately thick

DSM formulations with a variety of boundary conditionsfor example clamped-clamped (C-C) clamped-free (C-F)clamped-simply supported (C-S) and both ends simplysupported (S-S) In these cases ℎ119903 is investigated from 0025to 01

If log10(120578) le minus2 (which suggests the difference is nolarger than 1) it can be considered that results from the thinand moderately thick dynamic stiffness formulations are ingood agreement with each other Furthermore frequenciescomputed from themoderately thick shell theory can be usedas good alternatives to those computed from the thin shelltheory Referring to Figure 6 the following conclusions canbe reached

Shock and Vibration 7

Table 3 Natural frequencies of the thin-walled cylinder with different boundary conditions (Hz)

BCs Model I flexural II flexural I shell-like II shell-like I torsional II torsional

S-S

Present thin 14025 51506 14851 22931 80790 161579Present thick 14021 51502 14828 22877 80792 161585NAS2D [23] 13978 51366 14913 22917 80415 160810Reference [23] 14022 51503 18405 25460 80786 161573Reference [13] 14294 51567 18608 25574 80639 162551

C-C

Present thin 28577 69110 17350 30239 80790 161579Present thick 28574 69105 17327 30193 80792 161585NAS2D [23] 28498 68960 17396 30225 80415 160810Reference [23] 28576 69110 20484 32222 80786 161573Reference [13] 28354 69096 20463 31974 80838 161579

C-F

Present thin 5081 29092 14153 17427 40395 121184Present thick 5076 29086 14140 17370 40396 121188NAS2D [23] 5059 29001 14235 17435 40209 120620Reference [23] 5077 29090 23069 25239 40394 121181Reference [13] 5047 29002 23003 24979 40431 121203

F-F

Present thin 30939 77045 14036 14105 80788 161576Present thick 30931 77038 14032 14076 80792 161585NAS2D [23] 30829 76806 14129 14171 80415 160810Reference [23] 30932 77043 22987 23053 80789 161577Reference [13] 30945 77052 22864 23048 80787 161592

Table 4 Lowest natural frequencies (times1205960) of a moderately thick circular cylindrical shell with different circumferential wave numbers 119899119871119903 ℎ119903 = 012 ℎ119903 = 018119899 = 1 119899 = 2 119899 = 3 119899 = 4 119899 = 1 119899 = 2 119899 = 3 119899 = 42

Present 003724 002349 002449 003664 005632 003893 004939 007726Reference [31] 003730 002359 002462 003686 005652 003929 004996 007821Reference [32] 003730 002365 002470 003687 005653 003944 005009 007833

1Present 005861 004969 004765 005508 009420 008501 008995 011056

Reference [31] 005853 004978 004789 005545 009402 008545 009093 011205Reference [32] 005856 004986 004799 005550 009409 008562 009109 011202

05Present 010087 010238 010658 011463 018915 019406 020451 022180

Reference [31] 010057 010234 010688 011528 018894 019467 020616 022450Reference [32] 010047 010224 010674 011508 018832 019403 020544 022361

025Present 027310 027650 028219 029020 049751 050299 051216 052506

Reference [31] 027491 027849 028447 029287 050338 050937 051934 053325Reference [32] 027286 027641 028233 029064 049818 050418 051416 052808

(1) For low order frequencies good agreement is reachedbetween thin and moderately thick dynamic stiffnessformulations with ℎ119903 varying from 0025 to 01 Thesmaller the ℎ119903 the smaller the difference

(2) The difference between frequencies from thin andmoderately thick DSM formulations increases withthe increase of 119899 as log10(120578) always has a smallermagnitude under 119899 = 0 than those under 119899 = 1 2and 3

(3) It can be argued that the difference is insensitiveto boundary conditions The differences are roughlywithin 10minus5sim10minus2 in this investigation and the trends

of curves are similar for different boundary condi-tions

612 High Order Frequencies Investigation on the differ-ences between high order frequencies computed from thethin and moderately thick DSM formulations is examined inthis subsection Since it is known that the DSM formulationis not sensitive to boundary conditions [26] a C-S circu-lar cylindrical shell is studied The geometry and materialproperties are the same to the example in Section 611 if notspecifically stated

For high order frequencies ℎ119903 is investigated from verysmall value 0005 to a relatively large value 01 and the first

8 Shock and Vibration

005 0075 010025ℎr

minus1

minus2

minus3

minus4

minus5log 1

0(120578

)

n = 0

n = 1

n = 2

n = 3

(a) C-C

005 0075 010025ℎr

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)n = 0

n = 1

n = 2

n = 3

(b) C-F

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

n = 0

n = 1

n = 2

n = 3

005 0075 010025ℎr

(c) C-S

005 0075 010025ℎr

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

n = 0

n = 1

n = 2

n = 3

(d) S-S

Figure 6 The differences of low order frequencies between thin and moderately thick DSM formulations

ten frequencies under 119899 ranging from 0 to 10 from boththin and moderately thick DSM formulations are computedand compared The differences are shown in Figure 7 andconcluded as follows

(1) When ℎ119903 le 0025 log10(120578) is smaller than 10minus2most of the time which suggests that frequenciescomputed frommoderately thick shell theory are well

compatible with those from thin shell theory Noshear locking is observed

(2) When ℎ119903 ge 005 only frequencies with a small 119899(eg 119899 le 2) produce an acceptable difference (sim10minus2)With the increase of n the difference becomes quitenoticeable The higher the 119899 the more obvious thedifference

Shock and Vibration 9

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

(a) ℎ119903 = 0005

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

(b) ℎ119903 = 001

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

2 4 6 8 100m

(c) ℎ119903 = 0025

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

(d) ℎ119903 = 005

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

(e) ℎ119903 = 0075

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

(f) ℎ119903 = 01

Figure 7 The differences for frequencies between thin and moderately thick C-S circular cylindrical shell theories

(3) The order 119898 under a specific circumferential wavenumber 119899 does not have a significant impact on thedifferences as most curves in Figure 7 are largely flatand stable with the increase of119898

62 Ratio of Length to Radius (119871119903) Parameter 119871119903 controlsthe relative length of a circular cylindrical shell A larger119871119903 indicates a long shell and vice versa In this subsectionthe frequency trends and the compatibility between thin andmoderately thick DSM formulations with respect to 119871119903 arestudied In this subsection a C-C circular cylindrical shell

with ] = 03 is examined Figure 8 plots the trends of the firstfrequency parameterΩ under each 119899 with 119871119903 from 02 to 10

According to Figure 8 for a typical long circular cylin-drical shell (eg 119871119903 = 10) the lowest natural frequency isobtained with 119899 = 2 with the decrease of 119871119903 the circumfer-ential wave number 119899 corresponding to the lowest natural fre-quency slightly increases to 119899 = 3 or 4 when 119871119903 is very small(eg119871119903 = 02 and the shell is actually degraded to a ring) thefirst natural frequency under each 119899 increases monotonicallyand the smallest is reached at 119899 = 0The trends shown by Fig-ure 8 are also in good agreementwith other researchersrsquo work

10 Shock and Vibration

Ω

0

2

4

6

8

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(a) ℎ119903 = 0025 moderately thick shell

Ω

0

2

4

6

8

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(b) ℎ119903 = 0025 thin shell

0

3

6

9

12

Ω

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(c) ℎ119903 = 005 moderately thick shell

0

3

6

9

12

Ω

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(d) ℎ119903 = 005 thin shell

Figure 8 The trends of the first natural frequency parameters Ω under different 119899for example Markus [35] These obtained trends are applica-ble to both thin and moderately thick DSM formulations

Figure 9 shows the differences of the first frequenciesunder each 119899 computed from the thin and moderately thickDSM formulations of the circular cylindrical shell with ℎ119903 =0025 and 005 It can be understood fromFigure 9 that (1) thesmaller the ℎ119903 the smaller the difference (2) the larger the119871119903 the smaller the difference (3) conditions of ℎ119903 le 0025and 119899 le 10 need to be satisfied and a minor difference(log10(120578) le 10minus2) is required for a normal length shell thatis 119871119903 ge 2

7 Conclusions

This paper conducts comparison study on the frequenciesfrom thin and moderately thick circular cylindrical shellscomputed using the DSM formulations and comprehensiveparametric studies on ℎ119903 and 119871119903 are performed Based onthese investigations conclusions are outlined as follows

(1) Natural frequencies computed from the moder-ately thick DSM formulations can be consideredas satisfactory alternatives to those from thin DSMformulations when ℎ119903 le 0025 and 119871119903 ge 2

Shock and Vibration 11

0

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(a) ℎ119903 = 0025

0

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(b) ℎ119903 = 005

Figure 9 Differences between the first frequencies under each 119899 with different 119871119903

(differences are around or smaller than 10minus2) Sincethe DSM is exact no shear locking is observed

(2) Differences between frequencies from thin and mod-erately thick DSM formulations change with a varietyof factors The smaller ℎ119903 the smaller the differencethe smaller119871119903 the larger the difference the larger the119899 the larger the difference

It is proved that moderately thick DSM formulations canbe used to analyse the free vibration of normal length thin

circular cylindrical shells without shear locking demon-strating that the applicable range of moderately thick DSMformulations is expanded These merits should be attributedto that the dynamic stiffness method is an exact method

Though this comparison study is performed on circularcylindrical shells similar conclusions can be extended to thefree vibration of other shaped shells of revolution

Appendix

uthin =

1199061 = 1199061198991199062 = V119899

1199063 = 1199081198991199064 = 1199081015840119899

kthin =

V1 = 119903120587119864ℎ1 minus ]2[]119899119903 V119899 + ]

1119903119908119899 + 1199061015840119899]V2 = 119903120587119864ℎ2 (1 + ]) [minus119899119903 119906119899 + (1 + ℎ2121199032) V1015840119899 + 119899ℎ261199032 1199081015840119899]V3 = 119903120587119864ℎ2 (1 minus ]2) [119899ℎ

2

61199032 V1015840119899 + (2 minus ]) 1198992ℎ261199032 1199081015840119899 minus ℎ26 119908101584010158401015840119899 ]V4 = 119903120587119864ℎ12 (1 minus ]2) [minus]119899ℎ

2

1199032 V119899 minus ]1198992ℎ21199032 119908119899 + ℎ211990810158401015840119899 ]

12 Shock and Vibration

uthick =

1199061 = 1199061198991199062 = V119899

1199063 = 1199081198991199064 = 1205951199091198991199065 = 120595120579119899

kthick =

V1 = 119903120587119864ℎ1 minus ]2[1199061015840119899 + ]

119899119903 V119899 + ]119903119908119899]V2 = 12 119903120587119864ℎ1 + ]

[minus119899119903 119906119899 + (1 + ℎ2121199032) V1015840119899 minus 119899ℎ2121199032120595119909119899 + ℎ21211990321205951015840120579119899]V3 = 119903120587120581119866ℎ [1199081015840119899 + 120595119909119899]V4 = ℎ212 119903120587119864ℎ1 minus ]2

[1205951015840119909119899 + ]119899119903120595120579119899]

V5 = ℎ224 119903120587119864ℎ1 + ][V1015840119899119903 minus 119899119903120595119909119899 + 1205951015840120579119899]

[S]thin =

[[[[[[[[[[[[[[[[[[

11990411 0 0 11990414 0 11990416 0 011990422 11990423 0 11990425 0 0 1199042811990433 0 11990435 0 0 1199043811990444 0 11990446 11990447 0

11990455 0 0 0sym 11990466 0 0

0 011990488

]]]]]]]]]]]]]]]]]]

(A1)

where nonzero elements in the upper triangular domain are

11990411 = 120588120587119903ℎ1205962 minus 120587119864ℎ11989922119903 (1 minus ]) + 61198992119903120587119864ℎ(121199032 + ℎ2) (1 + ]) 11990414 = minus 1198992ℎ3120587119864(121199032 + ℎ2) (1 + ]) 11990416 = 12119899119903121199032 + ℎ2 11990422 = 120588120587119903ℎ1205962 minus 1198992120587119864ℎ (121199032 + ℎ2)

121199033 11990423 = minus120587119864ℎ121198991199032 + 1198993ℎ2121199033 11990425 = minus]119899119903 11990428 = ]

1198991199032 11990433 = 120588ℎ1199031205871205962 minus 120587119864ℎ119903 minus 1198994ℎ3120587119864121199033

11990435 = minus]119903 11990438 = ]11989921199032 11990444 = minus 21198992ℎ3119903120587119864(121199032 + ℎ2) (1 + ])

11990446 = minus 2119899ℎ2121199032 + ℎ2 11990447 = 111990455 = 1 minus ]2119903120587119864ℎ 11990466 = 24119903 (1 + ])120587119864ℎ (121199032 + ℎ2)

11990488 = 12 (1 minus ]2)119903120587119864ℎ3

[S]thick

Shock and Vibration 13

=

[[[[[[[[[[[[[[[[[[[[[[[

11990411 0 0 0 0 0 11990417 0 0 1199041lowast1011990422 11990423 0 11990425 11990426 0 0 0 011990433 0 11990435 11990436 0 0 0 0

11990444 0 0 0 11990448 0 1199044lowast1011990455 0 0 0 11990459 011990466 0 0 0 0

11990477 0 0 1199047lowast10sym 11990488 0 0

11990499 011990410lowast10

]]]]]]]]]]]]]]]]]]]]]]]

(A2)

where nonzero elements in the upper triangular domain are

11990411 = 119903120587120588ℎ120596211990417 = 119899119903

1199041lowast10 = minus 1198991199032 11990422 = 120587(119903120588ℎ1205962 minus 120581119866ℎ119903 minus 1198992119864ℎ119903 ) 11990423 = minus119899120587119903 (119864ℎ + 120581119866ℎ) 11990425 = 120587120581119866ℎ11990426 = minus]119899119903 11990433 = 120587(119903120588ℎ1205962 minus 119864ℎ119903 minus 1198992120581119866ℎ119903 ) 11990435 = 119899120587120581119866ℎ11990436 = minus]119903 11990444 = 1199031205871205881205962 ℎ312 11990448 = minus1

1199044lowast10 = 119899119903 11990455 = 120587(1199031205881205962 ℎ312 minus 1198992119864ℎ312119903 minus 119903120581119866ℎ) 11990459 = minus]119899119903 11990466 = 1 minus ]2119903120587119864ℎ 11990477 = 2 (1 + ])119903120587119864ℎ

1199047lowast10 = minus2 (1 + ])1199032120587119864ℎ 11990488 = 1119903120587120581119866ℎ 11990499 = 12 (1 minus ]2)

119903120587119864ℎ3 11990410lowast10 = 2 (1 + ])119903120587119864ℎ (12ℎ2 + 11199032 )

120581 = 120587212 (A3)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully appreciate the financial support fromNational Natural Science Foundation of China (51078198)Tsinghua University (2011THZ03) Jiangsu University ofScience and Technology (1732921402) and Shuang-ChuangProgram of Jiangsu Province for this research

References

[1] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover England UK 1927

[2] R N Arnold and G B Warburton ldquoThe flexural vibrationsof thin cylindersrdquo Proceedings of the Institution of MechanicalEngineers vol 167 no 1 pp 62ndash80 1953

[3] A W Leissa ldquoVibration of shellsrdquo Report 288 National Aero-nautics and Space Administration (NASA) Washington DCUSA 1973

[4] N S Bardell J M Dunsdon and R S Langley ldquoOn thefree vibration of completely free open cylindrically curvedisotropic shell panelsrdquo Journal of Sound and Vibration vol 207no 5 pp 647ndash669 1997

[5] D-Y Tan ldquoFree vibration analysis of shells of revolutionrdquoJournal of Sound and Vibration vol 213 no 1 pp 15ndash33 1998

[6] P M Naghdi and R M Cooper ldquoPropagation of elastic wavesin cylindrical shells including the effects of transverse shearand rotatory inertiardquo The Journal of the Acoustical Society ofAmerica vol 28 no 1 pp 56ndash63 1956

[7] I Mirsky and G Herrmann ldquoNonaxially symmetric motionsof cylindrical shellsrdquo The Journal of the Acoustical Society ofAmerica vol 29 pp 1116ndash1123 1957

[8] F Tornabene and E Viola ldquo2-D solution for free vibrationsof parabolic shells using generalized differential quadraturemethodrdquo European Journal of Mechanics ASolids vol 27 no6 pp 1001ndash1025 2008

[9] J L Mantari A S Oktem and C Guedes Soares ldquoBending andfree vibration analysis of isotropic and multilayered plates andshells by using a new accurate higher-order shear deformationtheoryrdquo Composites Part B Engineering vol 43 no 8 pp 3348ndash3360 2012

14 Shock and Vibration

[10] J G Kim J K Lee and H J Yoon ldquoFree vibration analysisfor shells of revolution based on p-version mixed finite elementformulationrdquo Finite Elements in Analysis and Design vol 95 pp12ndash19 2015

[11] F Tornabene ldquoFree vibrations of laminated composite doubly-curved shells and panels of revolution via the GDQ methodrdquoComputer Methods in Applied Mechanics and Engineering vol200 no 9ndash12 pp 931ndash952 2011

[12] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe local GDQmethod applied to general higher-order theories of doubly-curved laminated composite shells and panels the free vibrationanalysisrdquo Composite Structures vol 116 pp 637ndash660 2014

[13] A Pagani E Carrera and A Ferreira ldquoHigher-order theoriesand radial basis functions applied to free vibration analysisof thin-walled beamsrdquo Mechanics of Advanced Materials andStructures vol 23 no 9 pp 1080ndash1091 2016

[14] T Ye G Jin Y Chen and S Shi ldquoA unified formulationfor vibration analysis of open shells with arbitrary boundaryconditionsrdquo International Journal ofMechanical Sciences vol 81pp 42ndash59 2014

[15] T Ye G Jin S Shi and X Ma ldquoThree-dimensional freevibration analysis of thick cylindrical shells with general endconditions and resting on elastic foundationsrdquo InternationalJournal of Mechanical Sciences vol 84 pp 120ndash137 2014

[16] F W Williams and W H Wittrick ldquoAn automatic computa-tional procedure for calculating natural frequencies of skeletalstructuresrdquo International Journal of Mechanical Sciences vol 12no 9 pp 781ndash791 1970

[17] W H Wittrick and F W Williams ldquoA general algorithmfor computing natural frequencies of elastic structuresrdquo TheQuarterly Journal of Mechanics and Applied Mathematics vol24 pp 263ndash284 1971

[18] S Yuan K Ye F W Williams and D Kennedy ldquoRecursivesecond order convergence method for natural frequencies andmodes when using dynamic stiffness matricesrdquo InternationalJournal for Numerical Methods in Engineering vol 56 no 12pp 1795ndash1814 2003

[19] S Yuan K Ye and FWWilliams ldquoSecond order mode-findingmethod in dynamic stiffness matrix methodsrdquo Journal of Soundand Vibration vol 269 no 3ndash5 pp 689ndash708 2004

[20] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007

[21] J R Banerjee C W Cheung R Morishima M Pereraand J Njuguna ldquoFree vibration of a three-layered sandwichbeam using the dynamic stiffness method and experimentrdquoInternational Journal of Solids and Structures vol 44 no 22-23pp 7543ndash7563 2007

[22] H Su J R Banerjee and C W Cheung ldquoDynamic stiffnessformulation and free vibration analysis of functionally gradedbeamsrdquo Composite Structures vol 106 pp 854ndash862 2013

[23] A Pagani M Boscolo J R Banerjee and E Carrera ldquoExactdynamic stiffness elements based on one-dimensional higher-order theories for free vibration analysis of solid and thin-walled structuresrdquo Journal of Sound and Vibration vol 332 no23 pp 6104ndash6127 2013

[24] N El-Kaabazi and D Kennedy ldquoCalculation of natural fre-quencies and vibration modes of variable thickness cylindricalshells using the Wittrick-Williams algorithmrdquo Computers andStructures vol 104-105 pp 4ndash12 2012

[25] X Chen Research on dynamic stiffness method for free vibrationof shells of revolution [MS thesis] Tsinghua University 2009(Chinese)

[26] XChen andKYe ldquoFreeVibrationAnalysis for Shells of Revolu-tion Using an Exact Dynamic Stiffness Methodrdquo MathematicalProblems in Engineering vol 2016 Article ID 4513520 12 pages2016

[27] X Chen and K Ye ldquoAnalysis of free vibration of moderatelythick circular cylindrical shells using the dynamic stiffnessmethodrdquo Engineering Mechanics vol 33 no 9 pp 40ndash48 2016(Chinese)

[28] X M Zhang G R Liu and K Y Lam ldquoVibration analysisof thin cylindrical shells using wave propagation approachrdquoJournal of Sound and Vibration vol 239 no 3 pp 397ndash4032001

[29] L Xuebin ldquoStudy on free vibration analysis of circular cylin-drical shells using wave propagationrdquo Journal of Sound andVibration vol 311 no 3-5 pp 667ndash682 2008

[30] C B Sharma ldquoCalculation of natural frequencies of fixed-freecircular cylindrical shellsrdquo Journal of Sound and Vibration vol35 no 1 pp 55ndash76 1974

[31] A E ArmenakasDCGazis andGHerrmannFreeVibrationsof Circular Cylindrical Shells PergamonPressOxfordUK 1969

[32] A Bhimaraddi ldquoA higher order theory for free vibrationanalysis of circular cylindrical shellsrdquo International Journal ofSolids and Structures vol 20 no 7 pp 623ndash630 1984

[33] U Ascher J Christiansen and R D Russell ldquoAlgorithm 569COLSYS collocation software for boundary value ODEs [D2]rdquoACM Transactions on Mathematical Software vol 7 no 2 pp223ndash229 1981

[34] U Ascher J Christiansen and R D Russell ldquoCollocationsoftware for boundary-value ODEsrdquo ACM Transactions onMathematical Software vol 7 no 2 pp 209ndash222 1981

[35] S Markus The Mechanics of Vibrations of Cylindrical ShellsElsevier Science Limited 1988

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International Journal of

Shock and Vibration 7

Table 3 Natural frequencies of the thin-walled cylinder with different boundary conditions (Hz)

BCs Model I flexural II flexural I shell-like II shell-like I torsional II torsional

S-S

Present thin 14025 51506 14851 22931 80790 161579Present thick 14021 51502 14828 22877 80792 161585NAS2D [23] 13978 51366 14913 22917 80415 160810Reference [23] 14022 51503 18405 25460 80786 161573Reference [13] 14294 51567 18608 25574 80639 162551

C-C

Present thin 28577 69110 17350 30239 80790 161579Present thick 28574 69105 17327 30193 80792 161585NAS2D [23] 28498 68960 17396 30225 80415 160810Reference [23] 28576 69110 20484 32222 80786 161573Reference [13] 28354 69096 20463 31974 80838 161579

C-F

Present thin 5081 29092 14153 17427 40395 121184Present thick 5076 29086 14140 17370 40396 121188NAS2D [23] 5059 29001 14235 17435 40209 120620Reference [23] 5077 29090 23069 25239 40394 121181Reference [13] 5047 29002 23003 24979 40431 121203

F-F

Present thin 30939 77045 14036 14105 80788 161576Present thick 30931 77038 14032 14076 80792 161585NAS2D [23] 30829 76806 14129 14171 80415 160810Reference [23] 30932 77043 22987 23053 80789 161577Reference [13] 30945 77052 22864 23048 80787 161592

Table 4 Lowest natural frequencies (times1205960) of a moderately thick circular cylindrical shell with different circumferential wave numbers 119899119871119903 ℎ119903 = 012 ℎ119903 = 018119899 = 1 119899 = 2 119899 = 3 119899 = 4 119899 = 1 119899 = 2 119899 = 3 119899 = 42

Present 003724 002349 002449 003664 005632 003893 004939 007726Reference [31] 003730 002359 002462 003686 005652 003929 004996 007821Reference [32] 003730 002365 002470 003687 005653 003944 005009 007833

1Present 005861 004969 004765 005508 009420 008501 008995 011056

Reference [31] 005853 004978 004789 005545 009402 008545 009093 011205Reference [32] 005856 004986 004799 005550 009409 008562 009109 011202

05Present 010087 010238 010658 011463 018915 019406 020451 022180

Reference [31] 010057 010234 010688 011528 018894 019467 020616 022450Reference [32] 010047 010224 010674 011508 018832 019403 020544 022361

025Present 027310 027650 028219 029020 049751 050299 051216 052506

Reference [31] 027491 027849 028447 029287 050338 050937 051934 053325Reference [32] 027286 027641 028233 029064 049818 050418 051416 052808

(1) For low order frequencies good agreement is reachedbetween thin and moderately thick dynamic stiffnessformulations with ℎ119903 varying from 0025 to 01 Thesmaller the ℎ119903 the smaller the difference

(2) The difference between frequencies from thin andmoderately thick DSM formulations increases withthe increase of 119899 as log10(120578) always has a smallermagnitude under 119899 = 0 than those under 119899 = 1 2and 3

(3) It can be argued that the difference is insensitiveto boundary conditions The differences are roughlywithin 10minus5sim10minus2 in this investigation and the trends

of curves are similar for different boundary condi-tions

612 High Order Frequencies Investigation on the differ-ences between high order frequencies computed from thethin and moderately thick DSM formulations is examined inthis subsection Since it is known that the DSM formulationis not sensitive to boundary conditions [26] a C-S circu-lar cylindrical shell is studied The geometry and materialproperties are the same to the example in Section 611 if notspecifically stated

For high order frequencies ℎ119903 is investigated from verysmall value 0005 to a relatively large value 01 and the first

8 Shock and Vibration

005 0075 010025ℎr

minus1

minus2

minus3

minus4

minus5log 1

0(120578

)

n = 0

n = 1

n = 2

n = 3

(a) C-C

005 0075 010025ℎr

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)n = 0

n = 1

n = 2

n = 3

(b) C-F

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

n = 0

n = 1

n = 2

n = 3

005 0075 010025ℎr

(c) C-S

005 0075 010025ℎr

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

n = 0

n = 1

n = 2

n = 3

(d) S-S

Figure 6 The differences of low order frequencies between thin and moderately thick DSM formulations

ten frequencies under 119899 ranging from 0 to 10 from boththin and moderately thick DSM formulations are computedand compared The differences are shown in Figure 7 andconcluded as follows

(1) When ℎ119903 le 0025 log10(120578) is smaller than 10minus2most of the time which suggests that frequenciescomputed frommoderately thick shell theory are well

compatible with those from thin shell theory Noshear locking is observed

(2) When ℎ119903 ge 005 only frequencies with a small 119899(eg 119899 le 2) produce an acceptable difference (sim10minus2)With the increase of n the difference becomes quitenoticeable The higher the 119899 the more obvious thedifference

Shock and Vibration 9

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

(a) ℎ119903 = 0005

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

(b) ℎ119903 = 001

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

2 4 6 8 100m

(c) ℎ119903 = 0025

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

(d) ℎ119903 = 005

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

(e) ℎ119903 = 0075

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

(f) ℎ119903 = 01

Figure 7 The differences for frequencies between thin and moderately thick C-S circular cylindrical shell theories

(3) The order 119898 under a specific circumferential wavenumber 119899 does not have a significant impact on thedifferences as most curves in Figure 7 are largely flatand stable with the increase of119898

62 Ratio of Length to Radius (119871119903) Parameter 119871119903 controlsthe relative length of a circular cylindrical shell A larger119871119903 indicates a long shell and vice versa In this subsectionthe frequency trends and the compatibility between thin andmoderately thick DSM formulations with respect to 119871119903 arestudied In this subsection a C-C circular cylindrical shell

with ] = 03 is examined Figure 8 plots the trends of the firstfrequency parameterΩ under each 119899 with 119871119903 from 02 to 10

According to Figure 8 for a typical long circular cylin-drical shell (eg 119871119903 = 10) the lowest natural frequency isobtained with 119899 = 2 with the decrease of 119871119903 the circumfer-ential wave number 119899 corresponding to the lowest natural fre-quency slightly increases to 119899 = 3 or 4 when 119871119903 is very small(eg119871119903 = 02 and the shell is actually degraded to a ring) thefirst natural frequency under each 119899 increases monotonicallyand the smallest is reached at 119899 = 0The trends shown by Fig-ure 8 are also in good agreementwith other researchersrsquo work

10 Shock and Vibration

Ω

0

2

4

6

8

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(a) ℎ119903 = 0025 moderately thick shell

Ω

0

2

4

6

8

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(b) ℎ119903 = 0025 thin shell

0

3

6

9

12

Ω

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(c) ℎ119903 = 005 moderately thick shell

0

3

6

9

12

Ω

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(d) ℎ119903 = 005 thin shell

Figure 8 The trends of the first natural frequency parameters Ω under different 119899for example Markus [35] These obtained trends are applica-ble to both thin and moderately thick DSM formulations

Figure 9 shows the differences of the first frequenciesunder each 119899 computed from the thin and moderately thickDSM formulations of the circular cylindrical shell with ℎ119903 =0025 and 005 It can be understood fromFigure 9 that (1) thesmaller the ℎ119903 the smaller the difference (2) the larger the119871119903 the smaller the difference (3) conditions of ℎ119903 le 0025and 119899 le 10 need to be satisfied and a minor difference(log10(120578) le 10minus2) is required for a normal length shell thatis 119871119903 ge 2

7 Conclusions

This paper conducts comparison study on the frequenciesfrom thin and moderately thick circular cylindrical shellscomputed using the DSM formulations and comprehensiveparametric studies on ℎ119903 and 119871119903 are performed Based onthese investigations conclusions are outlined as follows

(1) Natural frequencies computed from the moder-ately thick DSM formulations can be consideredas satisfactory alternatives to those from thin DSMformulations when ℎ119903 le 0025 and 119871119903 ge 2

Shock and Vibration 11

0

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(a) ℎ119903 = 0025

0

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(b) ℎ119903 = 005

Figure 9 Differences between the first frequencies under each 119899 with different 119871119903

(differences are around or smaller than 10minus2) Sincethe DSM is exact no shear locking is observed

(2) Differences between frequencies from thin and mod-erately thick DSM formulations change with a varietyof factors The smaller ℎ119903 the smaller the differencethe smaller119871119903 the larger the difference the larger the119899 the larger the difference

It is proved that moderately thick DSM formulations canbe used to analyse the free vibration of normal length thin

circular cylindrical shells without shear locking demon-strating that the applicable range of moderately thick DSMformulations is expanded These merits should be attributedto that the dynamic stiffness method is an exact method

Though this comparison study is performed on circularcylindrical shells similar conclusions can be extended to thefree vibration of other shaped shells of revolution

Appendix

uthin =

1199061 = 1199061198991199062 = V119899

1199063 = 1199081198991199064 = 1199081015840119899

kthin =

V1 = 119903120587119864ℎ1 minus ]2[]119899119903 V119899 + ]

1119903119908119899 + 1199061015840119899]V2 = 119903120587119864ℎ2 (1 + ]) [minus119899119903 119906119899 + (1 + ℎ2121199032) V1015840119899 + 119899ℎ261199032 1199081015840119899]V3 = 119903120587119864ℎ2 (1 minus ]2) [119899ℎ

2

61199032 V1015840119899 + (2 minus ]) 1198992ℎ261199032 1199081015840119899 minus ℎ26 119908101584010158401015840119899 ]V4 = 119903120587119864ℎ12 (1 minus ]2) [minus]119899ℎ

2

1199032 V119899 minus ]1198992ℎ21199032 119908119899 + ℎ211990810158401015840119899 ]

12 Shock and Vibration

uthick =

1199061 = 1199061198991199062 = V119899

1199063 = 1199081198991199064 = 1205951199091198991199065 = 120595120579119899

kthick =

V1 = 119903120587119864ℎ1 minus ]2[1199061015840119899 + ]

119899119903 V119899 + ]119903119908119899]V2 = 12 119903120587119864ℎ1 + ]

[minus119899119903 119906119899 + (1 + ℎ2121199032) V1015840119899 minus 119899ℎ2121199032120595119909119899 + ℎ21211990321205951015840120579119899]V3 = 119903120587120581119866ℎ [1199081015840119899 + 120595119909119899]V4 = ℎ212 119903120587119864ℎ1 minus ]2

[1205951015840119909119899 + ]119899119903120595120579119899]

V5 = ℎ224 119903120587119864ℎ1 + ][V1015840119899119903 minus 119899119903120595119909119899 + 1205951015840120579119899]

[S]thin =

[[[[[[[[[[[[[[[[[[

11990411 0 0 11990414 0 11990416 0 011990422 11990423 0 11990425 0 0 1199042811990433 0 11990435 0 0 1199043811990444 0 11990446 11990447 0

11990455 0 0 0sym 11990466 0 0

0 011990488

]]]]]]]]]]]]]]]]]]

(A1)

where nonzero elements in the upper triangular domain are

11990411 = 120588120587119903ℎ1205962 minus 120587119864ℎ11989922119903 (1 minus ]) + 61198992119903120587119864ℎ(121199032 + ℎ2) (1 + ]) 11990414 = minus 1198992ℎ3120587119864(121199032 + ℎ2) (1 + ]) 11990416 = 12119899119903121199032 + ℎ2 11990422 = 120588120587119903ℎ1205962 minus 1198992120587119864ℎ (121199032 + ℎ2)

121199033 11990423 = minus120587119864ℎ121198991199032 + 1198993ℎ2121199033 11990425 = minus]119899119903 11990428 = ]

1198991199032 11990433 = 120588ℎ1199031205871205962 minus 120587119864ℎ119903 minus 1198994ℎ3120587119864121199033

11990435 = minus]119903 11990438 = ]11989921199032 11990444 = minus 21198992ℎ3119903120587119864(121199032 + ℎ2) (1 + ])

11990446 = minus 2119899ℎ2121199032 + ℎ2 11990447 = 111990455 = 1 minus ]2119903120587119864ℎ 11990466 = 24119903 (1 + ])120587119864ℎ (121199032 + ℎ2)

11990488 = 12 (1 minus ]2)119903120587119864ℎ3

[S]thick

Shock and Vibration 13

=

[[[[[[[[[[[[[[[[[[[[[[[

11990411 0 0 0 0 0 11990417 0 0 1199041lowast1011990422 11990423 0 11990425 11990426 0 0 0 011990433 0 11990435 11990436 0 0 0 0

11990444 0 0 0 11990448 0 1199044lowast1011990455 0 0 0 11990459 011990466 0 0 0 0

11990477 0 0 1199047lowast10sym 11990488 0 0

11990499 011990410lowast10

]]]]]]]]]]]]]]]]]]]]]]]

(A2)

where nonzero elements in the upper triangular domain are

11990411 = 119903120587120588ℎ120596211990417 = 119899119903

1199041lowast10 = minus 1198991199032 11990422 = 120587(119903120588ℎ1205962 minus 120581119866ℎ119903 minus 1198992119864ℎ119903 ) 11990423 = minus119899120587119903 (119864ℎ + 120581119866ℎ) 11990425 = 120587120581119866ℎ11990426 = minus]119899119903 11990433 = 120587(119903120588ℎ1205962 minus 119864ℎ119903 minus 1198992120581119866ℎ119903 ) 11990435 = 119899120587120581119866ℎ11990436 = minus]119903 11990444 = 1199031205871205881205962 ℎ312 11990448 = minus1

1199044lowast10 = 119899119903 11990455 = 120587(1199031205881205962 ℎ312 minus 1198992119864ℎ312119903 minus 119903120581119866ℎ) 11990459 = minus]119899119903 11990466 = 1 minus ]2119903120587119864ℎ 11990477 = 2 (1 + ])119903120587119864ℎ

1199047lowast10 = minus2 (1 + ])1199032120587119864ℎ 11990488 = 1119903120587120581119866ℎ 11990499 = 12 (1 minus ]2)

119903120587119864ℎ3 11990410lowast10 = 2 (1 + ])119903120587119864ℎ (12ℎ2 + 11199032 )

120581 = 120587212 (A3)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully appreciate the financial support fromNational Natural Science Foundation of China (51078198)Tsinghua University (2011THZ03) Jiangsu University ofScience and Technology (1732921402) and Shuang-ChuangProgram of Jiangsu Province for this research

References

[1] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover England UK 1927

[2] R N Arnold and G B Warburton ldquoThe flexural vibrationsof thin cylindersrdquo Proceedings of the Institution of MechanicalEngineers vol 167 no 1 pp 62ndash80 1953

[3] A W Leissa ldquoVibration of shellsrdquo Report 288 National Aero-nautics and Space Administration (NASA) Washington DCUSA 1973

[4] N S Bardell J M Dunsdon and R S Langley ldquoOn thefree vibration of completely free open cylindrically curvedisotropic shell panelsrdquo Journal of Sound and Vibration vol 207no 5 pp 647ndash669 1997

[5] D-Y Tan ldquoFree vibration analysis of shells of revolutionrdquoJournal of Sound and Vibration vol 213 no 1 pp 15ndash33 1998

[6] P M Naghdi and R M Cooper ldquoPropagation of elastic wavesin cylindrical shells including the effects of transverse shearand rotatory inertiardquo The Journal of the Acoustical Society ofAmerica vol 28 no 1 pp 56ndash63 1956

[7] I Mirsky and G Herrmann ldquoNonaxially symmetric motionsof cylindrical shellsrdquo The Journal of the Acoustical Society ofAmerica vol 29 pp 1116ndash1123 1957

[8] F Tornabene and E Viola ldquo2-D solution for free vibrationsof parabolic shells using generalized differential quadraturemethodrdquo European Journal of Mechanics ASolids vol 27 no6 pp 1001ndash1025 2008

[9] J L Mantari A S Oktem and C Guedes Soares ldquoBending andfree vibration analysis of isotropic and multilayered plates andshells by using a new accurate higher-order shear deformationtheoryrdquo Composites Part B Engineering vol 43 no 8 pp 3348ndash3360 2012

14 Shock and Vibration

[10] J G Kim J K Lee and H J Yoon ldquoFree vibration analysisfor shells of revolution based on p-version mixed finite elementformulationrdquo Finite Elements in Analysis and Design vol 95 pp12ndash19 2015

[11] F Tornabene ldquoFree vibrations of laminated composite doubly-curved shells and panels of revolution via the GDQ methodrdquoComputer Methods in Applied Mechanics and Engineering vol200 no 9ndash12 pp 931ndash952 2011

[12] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe local GDQmethod applied to general higher-order theories of doubly-curved laminated composite shells and panels the free vibrationanalysisrdquo Composite Structures vol 116 pp 637ndash660 2014

[13] A Pagani E Carrera and A Ferreira ldquoHigher-order theoriesand radial basis functions applied to free vibration analysisof thin-walled beamsrdquo Mechanics of Advanced Materials andStructures vol 23 no 9 pp 1080ndash1091 2016

[14] T Ye G Jin Y Chen and S Shi ldquoA unified formulationfor vibration analysis of open shells with arbitrary boundaryconditionsrdquo International Journal ofMechanical Sciences vol 81pp 42ndash59 2014

[15] T Ye G Jin S Shi and X Ma ldquoThree-dimensional freevibration analysis of thick cylindrical shells with general endconditions and resting on elastic foundationsrdquo InternationalJournal of Mechanical Sciences vol 84 pp 120ndash137 2014

[16] F W Williams and W H Wittrick ldquoAn automatic computa-tional procedure for calculating natural frequencies of skeletalstructuresrdquo International Journal of Mechanical Sciences vol 12no 9 pp 781ndash791 1970

[17] W H Wittrick and F W Williams ldquoA general algorithmfor computing natural frequencies of elastic structuresrdquo TheQuarterly Journal of Mechanics and Applied Mathematics vol24 pp 263ndash284 1971

[18] S Yuan K Ye F W Williams and D Kennedy ldquoRecursivesecond order convergence method for natural frequencies andmodes when using dynamic stiffness matricesrdquo InternationalJournal for Numerical Methods in Engineering vol 56 no 12pp 1795ndash1814 2003

[19] S Yuan K Ye and FWWilliams ldquoSecond order mode-findingmethod in dynamic stiffness matrix methodsrdquo Journal of Soundand Vibration vol 269 no 3ndash5 pp 689ndash708 2004

[20] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007

[21] J R Banerjee C W Cheung R Morishima M Pereraand J Njuguna ldquoFree vibration of a three-layered sandwichbeam using the dynamic stiffness method and experimentrdquoInternational Journal of Solids and Structures vol 44 no 22-23pp 7543ndash7563 2007

[22] H Su J R Banerjee and C W Cheung ldquoDynamic stiffnessformulation and free vibration analysis of functionally gradedbeamsrdquo Composite Structures vol 106 pp 854ndash862 2013

[23] A Pagani M Boscolo J R Banerjee and E Carrera ldquoExactdynamic stiffness elements based on one-dimensional higher-order theories for free vibration analysis of solid and thin-walled structuresrdquo Journal of Sound and Vibration vol 332 no23 pp 6104ndash6127 2013

[24] N El-Kaabazi and D Kennedy ldquoCalculation of natural fre-quencies and vibration modes of variable thickness cylindricalshells using the Wittrick-Williams algorithmrdquo Computers andStructures vol 104-105 pp 4ndash12 2012

[25] X Chen Research on dynamic stiffness method for free vibrationof shells of revolution [MS thesis] Tsinghua University 2009(Chinese)

[26] XChen andKYe ldquoFreeVibrationAnalysis for Shells of Revolu-tion Using an Exact Dynamic Stiffness Methodrdquo MathematicalProblems in Engineering vol 2016 Article ID 4513520 12 pages2016

[27] X Chen and K Ye ldquoAnalysis of free vibration of moderatelythick circular cylindrical shells using the dynamic stiffnessmethodrdquo Engineering Mechanics vol 33 no 9 pp 40ndash48 2016(Chinese)

[28] X M Zhang G R Liu and K Y Lam ldquoVibration analysisof thin cylindrical shells using wave propagation approachrdquoJournal of Sound and Vibration vol 239 no 3 pp 397ndash4032001

[29] L Xuebin ldquoStudy on free vibration analysis of circular cylin-drical shells using wave propagationrdquo Journal of Sound andVibration vol 311 no 3-5 pp 667ndash682 2008

[30] C B Sharma ldquoCalculation of natural frequencies of fixed-freecircular cylindrical shellsrdquo Journal of Sound and Vibration vol35 no 1 pp 55ndash76 1974

[31] A E ArmenakasDCGazis andGHerrmannFreeVibrationsof Circular Cylindrical Shells PergamonPressOxfordUK 1969

[32] A Bhimaraddi ldquoA higher order theory for free vibrationanalysis of circular cylindrical shellsrdquo International Journal ofSolids and Structures vol 20 no 7 pp 623ndash630 1984

[33] U Ascher J Christiansen and R D Russell ldquoAlgorithm 569COLSYS collocation software for boundary value ODEs [D2]rdquoACM Transactions on Mathematical Software vol 7 no 2 pp223ndash229 1981

[34] U Ascher J Christiansen and R D Russell ldquoCollocationsoftware for boundary-value ODEsrdquo ACM Transactions onMathematical Software vol 7 no 2 pp 209ndash222 1981

[35] S Markus The Mechanics of Vibrations of Cylindrical ShellsElsevier Science Limited 1988

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International Journal of

8 Shock and Vibration

005 0075 010025ℎr

minus1

minus2

minus3

minus4

minus5log 1

0(120578

)

n = 0

n = 1

n = 2

n = 3

(a) C-C

005 0075 010025ℎr

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)n = 0

n = 1

n = 2

n = 3

(b) C-F

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

n = 0

n = 1

n = 2

n = 3

005 0075 010025ℎr

(c) C-S

005 0075 010025ℎr

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

n = 0

n = 1

n = 2

n = 3

(d) S-S

Figure 6 The differences of low order frequencies between thin and moderately thick DSM formulations

ten frequencies under 119899 ranging from 0 to 10 from boththin and moderately thick DSM formulations are computedand compared The differences are shown in Figure 7 andconcluded as follows

(1) When ℎ119903 le 0025 log10(120578) is smaller than 10minus2most of the time which suggests that frequenciescomputed frommoderately thick shell theory are well

compatible with those from thin shell theory Noshear locking is observed

(2) When ℎ119903 ge 005 only frequencies with a small 119899(eg 119899 le 2) produce an acceptable difference (sim10minus2)With the increase of n the difference becomes quitenoticeable The higher the 119899 the more obvious thedifference

Shock and Vibration 9

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

(a) ℎ119903 = 0005

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

(b) ℎ119903 = 001

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

2 4 6 8 100m

(c) ℎ119903 = 0025

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

(d) ℎ119903 = 005

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

(e) ℎ119903 = 0075

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

(f) ℎ119903 = 01

Figure 7 The differences for frequencies between thin and moderately thick C-S circular cylindrical shell theories

(3) The order 119898 under a specific circumferential wavenumber 119899 does not have a significant impact on thedifferences as most curves in Figure 7 are largely flatand stable with the increase of119898

62 Ratio of Length to Radius (119871119903) Parameter 119871119903 controlsthe relative length of a circular cylindrical shell A larger119871119903 indicates a long shell and vice versa In this subsectionthe frequency trends and the compatibility between thin andmoderately thick DSM formulations with respect to 119871119903 arestudied In this subsection a C-C circular cylindrical shell

with ] = 03 is examined Figure 8 plots the trends of the firstfrequency parameterΩ under each 119899 with 119871119903 from 02 to 10

According to Figure 8 for a typical long circular cylin-drical shell (eg 119871119903 = 10) the lowest natural frequency isobtained with 119899 = 2 with the decrease of 119871119903 the circumfer-ential wave number 119899 corresponding to the lowest natural fre-quency slightly increases to 119899 = 3 or 4 when 119871119903 is very small(eg119871119903 = 02 and the shell is actually degraded to a ring) thefirst natural frequency under each 119899 increases monotonicallyand the smallest is reached at 119899 = 0The trends shown by Fig-ure 8 are also in good agreementwith other researchersrsquo work

10 Shock and Vibration

Ω

0

2

4

6

8

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(a) ℎ119903 = 0025 moderately thick shell

Ω

0

2

4

6

8

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(b) ℎ119903 = 0025 thin shell

0

3

6

9

12

Ω

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(c) ℎ119903 = 005 moderately thick shell

0

3

6

9

12

Ω

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(d) ℎ119903 = 005 thin shell

Figure 8 The trends of the first natural frequency parameters Ω under different 119899for example Markus [35] These obtained trends are applica-ble to both thin and moderately thick DSM formulations

Figure 9 shows the differences of the first frequenciesunder each 119899 computed from the thin and moderately thickDSM formulations of the circular cylindrical shell with ℎ119903 =0025 and 005 It can be understood fromFigure 9 that (1) thesmaller the ℎ119903 the smaller the difference (2) the larger the119871119903 the smaller the difference (3) conditions of ℎ119903 le 0025and 119899 le 10 need to be satisfied and a minor difference(log10(120578) le 10minus2) is required for a normal length shell thatis 119871119903 ge 2

7 Conclusions

This paper conducts comparison study on the frequenciesfrom thin and moderately thick circular cylindrical shellscomputed using the DSM formulations and comprehensiveparametric studies on ℎ119903 and 119871119903 are performed Based onthese investigations conclusions are outlined as follows

(1) Natural frequencies computed from the moder-ately thick DSM formulations can be consideredas satisfactory alternatives to those from thin DSMformulations when ℎ119903 le 0025 and 119871119903 ge 2

Shock and Vibration 11

0

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(a) ℎ119903 = 0025

0

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(b) ℎ119903 = 005

Figure 9 Differences between the first frequencies under each 119899 with different 119871119903

(differences are around or smaller than 10minus2) Sincethe DSM is exact no shear locking is observed

(2) Differences between frequencies from thin and mod-erately thick DSM formulations change with a varietyof factors The smaller ℎ119903 the smaller the differencethe smaller119871119903 the larger the difference the larger the119899 the larger the difference

It is proved that moderately thick DSM formulations canbe used to analyse the free vibration of normal length thin

circular cylindrical shells without shear locking demon-strating that the applicable range of moderately thick DSMformulations is expanded These merits should be attributedto that the dynamic stiffness method is an exact method

Though this comparison study is performed on circularcylindrical shells similar conclusions can be extended to thefree vibration of other shaped shells of revolution

Appendix

uthin =

1199061 = 1199061198991199062 = V119899

1199063 = 1199081198991199064 = 1199081015840119899

kthin =

V1 = 119903120587119864ℎ1 minus ]2[]119899119903 V119899 + ]

1119903119908119899 + 1199061015840119899]V2 = 119903120587119864ℎ2 (1 + ]) [minus119899119903 119906119899 + (1 + ℎ2121199032) V1015840119899 + 119899ℎ261199032 1199081015840119899]V3 = 119903120587119864ℎ2 (1 minus ]2) [119899ℎ

2

61199032 V1015840119899 + (2 minus ]) 1198992ℎ261199032 1199081015840119899 minus ℎ26 119908101584010158401015840119899 ]V4 = 119903120587119864ℎ12 (1 minus ]2) [minus]119899ℎ

2

1199032 V119899 minus ]1198992ℎ21199032 119908119899 + ℎ211990810158401015840119899 ]

12 Shock and Vibration

uthick =

1199061 = 1199061198991199062 = V119899

1199063 = 1199081198991199064 = 1205951199091198991199065 = 120595120579119899

kthick =

V1 = 119903120587119864ℎ1 minus ]2[1199061015840119899 + ]

119899119903 V119899 + ]119903119908119899]V2 = 12 119903120587119864ℎ1 + ]

[minus119899119903 119906119899 + (1 + ℎ2121199032) V1015840119899 minus 119899ℎ2121199032120595119909119899 + ℎ21211990321205951015840120579119899]V3 = 119903120587120581119866ℎ [1199081015840119899 + 120595119909119899]V4 = ℎ212 119903120587119864ℎ1 minus ]2

[1205951015840119909119899 + ]119899119903120595120579119899]

V5 = ℎ224 119903120587119864ℎ1 + ][V1015840119899119903 minus 119899119903120595119909119899 + 1205951015840120579119899]

[S]thin =

[[[[[[[[[[[[[[[[[[

11990411 0 0 11990414 0 11990416 0 011990422 11990423 0 11990425 0 0 1199042811990433 0 11990435 0 0 1199043811990444 0 11990446 11990447 0

11990455 0 0 0sym 11990466 0 0

0 011990488

]]]]]]]]]]]]]]]]]]

(A1)

where nonzero elements in the upper triangular domain are

11990411 = 120588120587119903ℎ1205962 minus 120587119864ℎ11989922119903 (1 minus ]) + 61198992119903120587119864ℎ(121199032 + ℎ2) (1 + ]) 11990414 = minus 1198992ℎ3120587119864(121199032 + ℎ2) (1 + ]) 11990416 = 12119899119903121199032 + ℎ2 11990422 = 120588120587119903ℎ1205962 minus 1198992120587119864ℎ (121199032 + ℎ2)

121199033 11990423 = minus120587119864ℎ121198991199032 + 1198993ℎ2121199033 11990425 = minus]119899119903 11990428 = ]

1198991199032 11990433 = 120588ℎ1199031205871205962 minus 120587119864ℎ119903 minus 1198994ℎ3120587119864121199033

11990435 = minus]119903 11990438 = ]11989921199032 11990444 = minus 21198992ℎ3119903120587119864(121199032 + ℎ2) (1 + ])

11990446 = minus 2119899ℎ2121199032 + ℎ2 11990447 = 111990455 = 1 minus ]2119903120587119864ℎ 11990466 = 24119903 (1 + ])120587119864ℎ (121199032 + ℎ2)

11990488 = 12 (1 minus ]2)119903120587119864ℎ3

[S]thick

Shock and Vibration 13

=

[[[[[[[[[[[[[[[[[[[[[[[

11990411 0 0 0 0 0 11990417 0 0 1199041lowast1011990422 11990423 0 11990425 11990426 0 0 0 011990433 0 11990435 11990436 0 0 0 0

11990444 0 0 0 11990448 0 1199044lowast1011990455 0 0 0 11990459 011990466 0 0 0 0

11990477 0 0 1199047lowast10sym 11990488 0 0

11990499 011990410lowast10

]]]]]]]]]]]]]]]]]]]]]]]

(A2)

where nonzero elements in the upper triangular domain are

11990411 = 119903120587120588ℎ120596211990417 = 119899119903

1199041lowast10 = minus 1198991199032 11990422 = 120587(119903120588ℎ1205962 minus 120581119866ℎ119903 minus 1198992119864ℎ119903 ) 11990423 = minus119899120587119903 (119864ℎ + 120581119866ℎ) 11990425 = 120587120581119866ℎ11990426 = minus]119899119903 11990433 = 120587(119903120588ℎ1205962 minus 119864ℎ119903 minus 1198992120581119866ℎ119903 ) 11990435 = 119899120587120581119866ℎ11990436 = minus]119903 11990444 = 1199031205871205881205962 ℎ312 11990448 = minus1

1199044lowast10 = 119899119903 11990455 = 120587(1199031205881205962 ℎ312 minus 1198992119864ℎ312119903 minus 119903120581119866ℎ) 11990459 = minus]119899119903 11990466 = 1 minus ]2119903120587119864ℎ 11990477 = 2 (1 + ])119903120587119864ℎ

1199047lowast10 = minus2 (1 + ])1199032120587119864ℎ 11990488 = 1119903120587120581119866ℎ 11990499 = 12 (1 minus ]2)

119903120587119864ℎ3 11990410lowast10 = 2 (1 + ])119903120587119864ℎ (12ℎ2 + 11199032 )

120581 = 120587212 (A3)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully appreciate the financial support fromNational Natural Science Foundation of China (51078198)Tsinghua University (2011THZ03) Jiangsu University ofScience and Technology (1732921402) and Shuang-ChuangProgram of Jiangsu Province for this research

References

[1] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover England UK 1927

[2] R N Arnold and G B Warburton ldquoThe flexural vibrationsof thin cylindersrdquo Proceedings of the Institution of MechanicalEngineers vol 167 no 1 pp 62ndash80 1953

[3] A W Leissa ldquoVibration of shellsrdquo Report 288 National Aero-nautics and Space Administration (NASA) Washington DCUSA 1973

[4] N S Bardell J M Dunsdon and R S Langley ldquoOn thefree vibration of completely free open cylindrically curvedisotropic shell panelsrdquo Journal of Sound and Vibration vol 207no 5 pp 647ndash669 1997

[5] D-Y Tan ldquoFree vibration analysis of shells of revolutionrdquoJournal of Sound and Vibration vol 213 no 1 pp 15ndash33 1998

[6] P M Naghdi and R M Cooper ldquoPropagation of elastic wavesin cylindrical shells including the effects of transverse shearand rotatory inertiardquo The Journal of the Acoustical Society ofAmerica vol 28 no 1 pp 56ndash63 1956

[7] I Mirsky and G Herrmann ldquoNonaxially symmetric motionsof cylindrical shellsrdquo The Journal of the Acoustical Society ofAmerica vol 29 pp 1116ndash1123 1957

[8] F Tornabene and E Viola ldquo2-D solution for free vibrationsof parabolic shells using generalized differential quadraturemethodrdquo European Journal of Mechanics ASolids vol 27 no6 pp 1001ndash1025 2008

[9] J L Mantari A S Oktem and C Guedes Soares ldquoBending andfree vibration analysis of isotropic and multilayered plates andshells by using a new accurate higher-order shear deformationtheoryrdquo Composites Part B Engineering vol 43 no 8 pp 3348ndash3360 2012

14 Shock and Vibration

[10] J G Kim J K Lee and H J Yoon ldquoFree vibration analysisfor shells of revolution based on p-version mixed finite elementformulationrdquo Finite Elements in Analysis and Design vol 95 pp12ndash19 2015

[11] F Tornabene ldquoFree vibrations of laminated composite doubly-curved shells and panels of revolution via the GDQ methodrdquoComputer Methods in Applied Mechanics and Engineering vol200 no 9ndash12 pp 931ndash952 2011

[12] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe local GDQmethod applied to general higher-order theories of doubly-curved laminated composite shells and panels the free vibrationanalysisrdquo Composite Structures vol 116 pp 637ndash660 2014

[13] A Pagani E Carrera and A Ferreira ldquoHigher-order theoriesand radial basis functions applied to free vibration analysisof thin-walled beamsrdquo Mechanics of Advanced Materials andStructures vol 23 no 9 pp 1080ndash1091 2016

[14] T Ye G Jin Y Chen and S Shi ldquoA unified formulationfor vibration analysis of open shells with arbitrary boundaryconditionsrdquo International Journal ofMechanical Sciences vol 81pp 42ndash59 2014

[15] T Ye G Jin S Shi and X Ma ldquoThree-dimensional freevibration analysis of thick cylindrical shells with general endconditions and resting on elastic foundationsrdquo InternationalJournal of Mechanical Sciences vol 84 pp 120ndash137 2014

[16] F W Williams and W H Wittrick ldquoAn automatic computa-tional procedure for calculating natural frequencies of skeletalstructuresrdquo International Journal of Mechanical Sciences vol 12no 9 pp 781ndash791 1970

[17] W H Wittrick and F W Williams ldquoA general algorithmfor computing natural frequencies of elastic structuresrdquo TheQuarterly Journal of Mechanics and Applied Mathematics vol24 pp 263ndash284 1971

[18] S Yuan K Ye F W Williams and D Kennedy ldquoRecursivesecond order convergence method for natural frequencies andmodes when using dynamic stiffness matricesrdquo InternationalJournal for Numerical Methods in Engineering vol 56 no 12pp 1795ndash1814 2003

[19] S Yuan K Ye and FWWilliams ldquoSecond order mode-findingmethod in dynamic stiffness matrix methodsrdquo Journal of Soundand Vibration vol 269 no 3ndash5 pp 689ndash708 2004

[20] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007

[21] J R Banerjee C W Cheung R Morishima M Pereraand J Njuguna ldquoFree vibration of a three-layered sandwichbeam using the dynamic stiffness method and experimentrdquoInternational Journal of Solids and Structures vol 44 no 22-23pp 7543ndash7563 2007

[22] H Su J R Banerjee and C W Cheung ldquoDynamic stiffnessformulation and free vibration analysis of functionally gradedbeamsrdquo Composite Structures vol 106 pp 854ndash862 2013

[23] A Pagani M Boscolo J R Banerjee and E Carrera ldquoExactdynamic stiffness elements based on one-dimensional higher-order theories for free vibration analysis of solid and thin-walled structuresrdquo Journal of Sound and Vibration vol 332 no23 pp 6104ndash6127 2013

[24] N El-Kaabazi and D Kennedy ldquoCalculation of natural fre-quencies and vibration modes of variable thickness cylindricalshells using the Wittrick-Williams algorithmrdquo Computers andStructures vol 104-105 pp 4ndash12 2012

[25] X Chen Research on dynamic stiffness method for free vibrationof shells of revolution [MS thesis] Tsinghua University 2009(Chinese)

[26] XChen andKYe ldquoFreeVibrationAnalysis for Shells of Revolu-tion Using an Exact Dynamic Stiffness Methodrdquo MathematicalProblems in Engineering vol 2016 Article ID 4513520 12 pages2016

[27] X Chen and K Ye ldquoAnalysis of free vibration of moderatelythick circular cylindrical shells using the dynamic stiffnessmethodrdquo Engineering Mechanics vol 33 no 9 pp 40ndash48 2016(Chinese)

[28] X M Zhang G R Liu and K Y Lam ldquoVibration analysisof thin cylindrical shells using wave propagation approachrdquoJournal of Sound and Vibration vol 239 no 3 pp 397ndash4032001

[29] L Xuebin ldquoStudy on free vibration analysis of circular cylin-drical shells using wave propagationrdquo Journal of Sound andVibration vol 311 no 3-5 pp 667ndash682 2008

[30] C B Sharma ldquoCalculation of natural frequencies of fixed-freecircular cylindrical shellsrdquo Journal of Sound and Vibration vol35 no 1 pp 55ndash76 1974

[31] A E ArmenakasDCGazis andGHerrmannFreeVibrationsof Circular Cylindrical Shells PergamonPressOxfordUK 1969

[32] A Bhimaraddi ldquoA higher order theory for free vibrationanalysis of circular cylindrical shellsrdquo International Journal ofSolids and Structures vol 20 no 7 pp 623ndash630 1984

[33] U Ascher J Christiansen and R D Russell ldquoAlgorithm 569COLSYS collocation software for boundary value ODEs [D2]rdquoACM Transactions on Mathematical Software vol 7 no 2 pp223ndash229 1981

[34] U Ascher J Christiansen and R D Russell ldquoCollocationsoftware for boundary-value ODEsrdquo ACM Transactions onMathematical Software vol 7 no 2 pp 209ndash222 1981

[35] S Markus The Mechanics of Vibrations of Cylindrical ShellsElsevier Science Limited 1988

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Shock and Vibration

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International Journal of

Shock and Vibration 9

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

(a) ℎ119903 = 0005

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

(b) ℎ119903 = 001

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

2 4 6 8 100m

(c) ℎ119903 = 0025

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

(d) ℎ119903 = 005

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

(e) ℎ119903 = 0075

2 4 6 8 100m

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

n = 6

n = 7

n = 8

n = 9

n = 10

0

minus1

minus2

minus3

minus4

minus5

minus6

log 1

0(120578

)

(f) ℎ119903 = 01

Figure 7 The differences for frequencies between thin and moderately thick C-S circular cylindrical shell theories

(3) The order 119898 under a specific circumferential wavenumber 119899 does not have a significant impact on thedifferences as most curves in Figure 7 are largely flatand stable with the increase of119898

62 Ratio of Length to Radius (119871119903) Parameter 119871119903 controlsthe relative length of a circular cylindrical shell A larger119871119903 indicates a long shell and vice versa In this subsectionthe frequency trends and the compatibility between thin andmoderately thick DSM formulations with respect to 119871119903 arestudied In this subsection a C-C circular cylindrical shell

with ] = 03 is examined Figure 8 plots the trends of the firstfrequency parameterΩ under each 119899 with 119871119903 from 02 to 10

According to Figure 8 for a typical long circular cylin-drical shell (eg 119871119903 = 10) the lowest natural frequency isobtained with 119899 = 2 with the decrease of 119871119903 the circumfer-ential wave number 119899 corresponding to the lowest natural fre-quency slightly increases to 119899 = 3 or 4 when 119871119903 is very small(eg119871119903 = 02 and the shell is actually degraded to a ring) thefirst natural frequency under each 119899 increases monotonicallyand the smallest is reached at 119899 = 0The trends shown by Fig-ure 8 are also in good agreementwith other researchersrsquo work

10 Shock and Vibration

Ω

0

2

4

6

8

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(a) ℎ119903 = 0025 moderately thick shell

Ω

0

2

4

6

8

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(b) ℎ119903 = 0025 thin shell

0

3

6

9

12

Ω

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(c) ℎ119903 = 005 moderately thick shell

0

3

6

9

12

Ω

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(d) ℎ119903 = 005 thin shell

Figure 8 The trends of the first natural frequency parameters Ω under different 119899for example Markus [35] These obtained trends are applica-ble to both thin and moderately thick DSM formulations

Figure 9 shows the differences of the first frequenciesunder each 119899 computed from the thin and moderately thickDSM formulations of the circular cylindrical shell with ℎ119903 =0025 and 005 It can be understood fromFigure 9 that (1) thesmaller the ℎ119903 the smaller the difference (2) the larger the119871119903 the smaller the difference (3) conditions of ℎ119903 le 0025and 119899 le 10 need to be satisfied and a minor difference(log10(120578) le 10minus2) is required for a normal length shell thatis 119871119903 ge 2

7 Conclusions

This paper conducts comparison study on the frequenciesfrom thin and moderately thick circular cylindrical shellscomputed using the DSM formulations and comprehensiveparametric studies on ℎ119903 and 119871119903 are performed Based onthese investigations conclusions are outlined as follows

(1) Natural frequencies computed from the moder-ately thick DSM formulations can be consideredas satisfactory alternatives to those from thin DSMformulations when ℎ119903 le 0025 and 119871119903 ge 2

Shock and Vibration 11

0

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(a) ℎ119903 = 0025

0

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(b) ℎ119903 = 005

Figure 9 Differences between the first frequencies under each 119899 with different 119871119903

(differences are around or smaller than 10minus2) Sincethe DSM is exact no shear locking is observed

(2) Differences between frequencies from thin and mod-erately thick DSM formulations change with a varietyof factors The smaller ℎ119903 the smaller the differencethe smaller119871119903 the larger the difference the larger the119899 the larger the difference

It is proved that moderately thick DSM formulations canbe used to analyse the free vibration of normal length thin

circular cylindrical shells without shear locking demon-strating that the applicable range of moderately thick DSMformulations is expanded These merits should be attributedto that the dynamic stiffness method is an exact method

Though this comparison study is performed on circularcylindrical shells similar conclusions can be extended to thefree vibration of other shaped shells of revolution

Appendix

uthin =

1199061 = 1199061198991199062 = V119899

1199063 = 1199081198991199064 = 1199081015840119899

kthin =

V1 = 119903120587119864ℎ1 minus ]2[]119899119903 V119899 + ]

1119903119908119899 + 1199061015840119899]V2 = 119903120587119864ℎ2 (1 + ]) [minus119899119903 119906119899 + (1 + ℎ2121199032) V1015840119899 + 119899ℎ261199032 1199081015840119899]V3 = 119903120587119864ℎ2 (1 minus ]2) [119899ℎ

2

61199032 V1015840119899 + (2 minus ]) 1198992ℎ261199032 1199081015840119899 minus ℎ26 119908101584010158401015840119899 ]V4 = 119903120587119864ℎ12 (1 minus ]2) [minus]119899ℎ

2

1199032 V119899 minus ]1198992ℎ21199032 119908119899 + ℎ211990810158401015840119899 ]

12 Shock and Vibration

uthick =

1199061 = 1199061198991199062 = V119899

1199063 = 1199081198991199064 = 1205951199091198991199065 = 120595120579119899

kthick =

V1 = 119903120587119864ℎ1 minus ]2[1199061015840119899 + ]

119899119903 V119899 + ]119903119908119899]V2 = 12 119903120587119864ℎ1 + ]

[minus119899119903 119906119899 + (1 + ℎ2121199032) V1015840119899 minus 119899ℎ2121199032120595119909119899 + ℎ21211990321205951015840120579119899]V3 = 119903120587120581119866ℎ [1199081015840119899 + 120595119909119899]V4 = ℎ212 119903120587119864ℎ1 minus ]2

[1205951015840119909119899 + ]119899119903120595120579119899]

V5 = ℎ224 119903120587119864ℎ1 + ][V1015840119899119903 minus 119899119903120595119909119899 + 1205951015840120579119899]

[S]thin =

[[[[[[[[[[[[[[[[[[

11990411 0 0 11990414 0 11990416 0 011990422 11990423 0 11990425 0 0 1199042811990433 0 11990435 0 0 1199043811990444 0 11990446 11990447 0

11990455 0 0 0sym 11990466 0 0

0 011990488

]]]]]]]]]]]]]]]]]]

(A1)

where nonzero elements in the upper triangular domain are

11990411 = 120588120587119903ℎ1205962 minus 120587119864ℎ11989922119903 (1 minus ]) + 61198992119903120587119864ℎ(121199032 + ℎ2) (1 + ]) 11990414 = minus 1198992ℎ3120587119864(121199032 + ℎ2) (1 + ]) 11990416 = 12119899119903121199032 + ℎ2 11990422 = 120588120587119903ℎ1205962 minus 1198992120587119864ℎ (121199032 + ℎ2)

121199033 11990423 = minus120587119864ℎ121198991199032 + 1198993ℎ2121199033 11990425 = minus]119899119903 11990428 = ]

1198991199032 11990433 = 120588ℎ1199031205871205962 minus 120587119864ℎ119903 minus 1198994ℎ3120587119864121199033

11990435 = minus]119903 11990438 = ]11989921199032 11990444 = minus 21198992ℎ3119903120587119864(121199032 + ℎ2) (1 + ])

11990446 = minus 2119899ℎ2121199032 + ℎ2 11990447 = 111990455 = 1 minus ]2119903120587119864ℎ 11990466 = 24119903 (1 + ])120587119864ℎ (121199032 + ℎ2)

11990488 = 12 (1 minus ]2)119903120587119864ℎ3

[S]thick

Shock and Vibration 13

=

[[[[[[[[[[[[[[[[[[[[[[[

11990411 0 0 0 0 0 11990417 0 0 1199041lowast1011990422 11990423 0 11990425 11990426 0 0 0 011990433 0 11990435 11990436 0 0 0 0

11990444 0 0 0 11990448 0 1199044lowast1011990455 0 0 0 11990459 011990466 0 0 0 0

11990477 0 0 1199047lowast10sym 11990488 0 0

11990499 011990410lowast10

]]]]]]]]]]]]]]]]]]]]]]]

(A2)

where nonzero elements in the upper triangular domain are

11990411 = 119903120587120588ℎ120596211990417 = 119899119903

1199041lowast10 = minus 1198991199032 11990422 = 120587(119903120588ℎ1205962 minus 120581119866ℎ119903 minus 1198992119864ℎ119903 ) 11990423 = minus119899120587119903 (119864ℎ + 120581119866ℎ) 11990425 = 120587120581119866ℎ11990426 = minus]119899119903 11990433 = 120587(119903120588ℎ1205962 minus 119864ℎ119903 minus 1198992120581119866ℎ119903 ) 11990435 = 119899120587120581119866ℎ11990436 = minus]119903 11990444 = 1199031205871205881205962 ℎ312 11990448 = minus1

1199044lowast10 = 119899119903 11990455 = 120587(1199031205881205962 ℎ312 minus 1198992119864ℎ312119903 minus 119903120581119866ℎ) 11990459 = minus]119899119903 11990466 = 1 minus ]2119903120587119864ℎ 11990477 = 2 (1 + ])119903120587119864ℎ

1199047lowast10 = minus2 (1 + ])1199032120587119864ℎ 11990488 = 1119903120587120581119866ℎ 11990499 = 12 (1 minus ]2)

119903120587119864ℎ3 11990410lowast10 = 2 (1 + ])119903120587119864ℎ (12ℎ2 + 11199032 )

120581 = 120587212 (A3)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully appreciate the financial support fromNational Natural Science Foundation of China (51078198)Tsinghua University (2011THZ03) Jiangsu University ofScience and Technology (1732921402) and Shuang-ChuangProgram of Jiangsu Province for this research

References

[1] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover England UK 1927

[2] R N Arnold and G B Warburton ldquoThe flexural vibrationsof thin cylindersrdquo Proceedings of the Institution of MechanicalEngineers vol 167 no 1 pp 62ndash80 1953

[3] A W Leissa ldquoVibration of shellsrdquo Report 288 National Aero-nautics and Space Administration (NASA) Washington DCUSA 1973

[4] N S Bardell J M Dunsdon and R S Langley ldquoOn thefree vibration of completely free open cylindrically curvedisotropic shell panelsrdquo Journal of Sound and Vibration vol 207no 5 pp 647ndash669 1997

[5] D-Y Tan ldquoFree vibration analysis of shells of revolutionrdquoJournal of Sound and Vibration vol 213 no 1 pp 15ndash33 1998

[6] P M Naghdi and R M Cooper ldquoPropagation of elastic wavesin cylindrical shells including the effects of transverse shearand rotatory inertiardquo The Journal of the Acoustical Society ofAmerica vol 28 no 1 pp 56ndash63 1956

[7] I Mirsky and G Herrmann ldquoNonaxially symmetric motionsof cylindrical shellsrdquo The Journal of the Acoustical Society ofAmerica vol 29 pp 1116ndash1123 1957

[8] F Tornabene and E Viola ldquo2-D solution for free vibrationsof parabolic shells using generalized differential quadraturemethodrdquo European Journal of Mechanics ASolids vol 27 no6 pp 1001ndash1025 2008

[9] J L Mantari A S Oktem and C Guedes Soares ldquoBending andfree vibration analysis of isotropic and multilayered plates andshells by using a new accurate higher-order shear deformationtheoryrdquo Composites Part B Engineering vol 43 no 8 pp 3348ndash3360 2012

14 Shock and Vibration

[10] J G Kim J K Lee and H J Yoon ldquoFree vibration analysisfor shells of revolution based on p-version mixed finite elementformulationrdquo Finite Elements in Analysis and Design vol 95 pp12ndash19 2015

[11] F Tornabene ldquoFree vibrations of laminated composite doubly-curved shells and panels of revolution via the GDQ methodrdquoComputer Methods in Applied Mechanics and Engineering vol200 no 9ndash12 pp 931ndash952 2011

[12] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe local GDQmethod applied to general higher-order theories of doubly-curved laminated composite shells and panels the free vibrationanalysisrdquo Composite Structures vol 116 pp 637ndash660 2014

[13] A Pagani E Carrera and A Ferreira ldquoHigher-order theoriesand radial basis functions applied to free vibration analysisof thin-walled beamsrdquo Mechanics of Advanced Materials andStructures vol 23 no 9 pp 1080ndash1091 2016

[14] T Ye G Jin Y Chen and S Shi ldquoA unified formulationfor vibration analysis of open shells with arbitrary boundaryconditionsrdquo International Journal ofMechanical Sciences vol 81pp 42ndash59 2014

[15] T Ye G Jin S Shi and X Ma ldquoThree-dimensional freevibration analysis of thick cylindrical shells with general endconditions and resting on elastic foundationsrdquo InternationalJournal of Mechanical Sciences vol 84 pp 120ndash137 2014

[16] F W Williams and W H Wittrick ldquoAn automatic computa-tional procedure for calculating natural frequencies of skeletalstructuresrdquo International Journal of Mechanical Sciences vol 12no 9 pp 781ndash791 1970

[17] W H Wittrick and F W Williams ldquoA general algorithmfor computing natural frequencies of elastic structuresrdquo TheQuarterly Journal of Mechanics and Applied Mathematics vol24 pp 263ndash284 1971

[18] S Yuan K Ye F W Williams and D Kennedy ldquoRecursivesecond order convergence method for natural frequencies andmodes when using dynamic stiffness matricesrdquo InternationalJournal for Numerical Methods in Engineering vol 56 no 12pp 1795ndash1814 2003

[19] S Yuan K Ye and FWWilliams ldquoSecond order mode-findingmethod in dynamic stiffness matrix methodsrdquo Journal of Soundand Vibration vol 269 no 3ndash5 pp 689ndash708 2004

[20] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007

[21] J R Banerjee C W Cheung R Morishima M Pereraand J Njuguna ldquoFree vibration of a three-layered sandwichbeam using the dynamic stiffness method and experimentrdquoInternational Journal of Solids and Structures vol 44 no 22-23pp 7543ndash7563 2007

[22] H Su J R Banerjee and C W Cheung ldquoDynamic stiffnessformulation and free vibration analysis of functionally gradedbeamsrdquo Composite Structures vol 106 pp 854ndash862 2013

[23] A Pagani M Boscolo J R Banerjee and E Carrera ldquoExactdynamic stiffness elements based on one-dimensional higher-order theories for free vibration analysis of solid and thin-walled structuresrdquo Journal of Sound and Vibration vol 332 no23 pp 6104ndash6127 2013

[24] N El-Kaabazi and D Kennedy ldquoCalculation of natural fre-quencies and vibration modes of variable thickness cylindricalshells using the Wittrick-Williams algorithmrdquo Computers andStructures vol 104-105 pp 4ndash12 2012

[25] X Chen Research on dynamic stiffness method for free vibrationof shells of revolution [MS thesis] Tsinghua University 2009(Chinese)

[26] XChen andKYe ldquoFreeVibrationAnalysis for Shells of Revolu-tion Using an Exact Dynamic Stiffness Methodrdquo MathematicalProblems in Engineering vol 2016 Article ID 4513520 12 pages2016

[27] X Chen and K Ye ldquoAnalysis of free vibration of moderatelythick circular cylindrical shells using the dynamic stiffnessmethodrdquo Engineering Mechanics vol 33 no 9 pp 40ndash48 2016(Chinese)

[28] X M Zhang G R Liu and K Y Lam ldquoVibration analysisof thin cylindrical shells using wave propagation approachrdquoJournal of Sound and Vibration vol 239 no 3 pp 397ndash4032001

[29] L Xuebin ldquoStudy on free vibration analysis of circular cylin-drical shells using wave propagationrdquo Journal of Sound andVibration vol 311 no 3-5 pp 667ndash682 2008

[30] C B Sharma ldquoCalculation of natural frequencies of fixed-freecircular cylindrical shellsrdquo Journal of Sound and Vibration vol35 no 1 pp 55ndash76 1974

[31] A E ArmenakasDCGazis andGHerrmannFreeVibrationsof Circular Cylindrical Shells PergamonPressOxfordUK 1969

[32] A Bhimaraddi ldquoA higher order theory for free vibrationanalysis of circular cylindrical shellsrdquo International Journal ofSolids and Structures vol 20 no 7 pp 623ndash630 1984

[33] U Ascher J Christiansen and R D Russell ldquoAlgorithm 569COLSYS collocation software for boundary value ODEs [D2]rdquoACM Transactions on Mathematical Software vol 7 no 2 pp223ndash229 1981

[34] U Ascher J Christiansen and R D Russell ldquoCollocationsoftware for boundary-value ODEsrdquo ACM Transactions onMathematical Software vol 7 no 2 pp 209ndash222 1981

[35] S Markus The Mechanics of Vibrations of Cylindrical ShellsElsevier Science Limited 1988

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Shock and Vibration

Ω

0

2

4

6

8

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(a) ℎ119903 = 0025 moderately thick shell

Ω

0

2

4

6

8

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(b) ℎ119903 = 0025 thin shell

0

3

6

9

12

Ω

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(c) ℎ119903 = 005 moderately thick shell

0

3

6

9

12

Ω

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(d) ℎ119903 = 005 thin shell

Figure 8 The trends of the first natural frequency parameters Ω under different 119899for example Markus [35] These obtained trends are applica-ble to both thin and moderately thick DSM formulations

Figure 9 shows the differences of the first frequenciesunder each 119899 computed from the thin and moderately thickDSM formulations of the circular cylindrical shell with ℎ119903 =0025 and 005 It can be understood fromFigure 9 that (1) thesmaller the ℎ119903 the smaller the difference (2) the larger the119871119903 the smaller the difference (3) conditions of ℎ119903 le 0025and 119899 le 10 need to be satisfied and a minor difference(log10(120578) le 10minus2) is required for a normal length shell thatis 119871119903 ge 2

7 Conclusions

This paper conducts comparison study on the frequenciesfrom thin and moderately thick circular cylindrical shellscomputed using the DSM formulations and comprehensiveparametric studies on ℎ119903 and 119871119903 are performed Based onthese investigations conclusions are outlined as follows

(1) Natural frequencies computed from the moder-ately thick DSM formulations can be consideredas satisfactory alternatives to those from thin DSMformulations when ℎ119903 le 0025 and 119871119903 ge 2

Shock and Vibration 11

0

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(a) ℎ119903 = 0025

0

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(b) ℎ119903 = 005

Figure 9 Differences between the first frequencies under each 119899 with different 119871119903

(differences are around or smaller than 10minus2) Sincethe DSM is exact no shear locking is observed

(2) Differences between frequencies from thin and mod-erately thick DSM formulations change with a varietyof factors The smaller ℎ119903 the smaller the differencethe smaller119871119903 the larger the difference the larger the119899 the larger the difference

It is proved that moderately thick DSM formulations canbe used to analyse the free vibration of normal length thin

circular cylindrical shells without shear locking demon-strating that the applicable range of moderately thick DSMformulations is expanded These merits should be attributedto that the dynamic stiffness method is an exact method

Though this comparison study is performed on circularcylindrical shells similar conclusions can be extended to thefree vibration of other shaped shells of revolution

Appendix

uthin =

1199061 = 1199061198991199062 = V119899

1199063 = 1199081198991199064 = 1199081015840119899

kthin =

V1 = 119903120587119864ℎ1 minus ]2[]119899119903 V119899 + ]

1119903119908119899 + 1199061015840119899]V2 = 119903120587119864ℎ2 (1 + ]) [minus119899119903 119906119899 + (1 + ℎ2121199032) V1015840119899 + 119899ℎ261199032 1199081015840119899]V3 = 119903120587119864ℎ2 (1 minus ]2) [119899ℎ

2

61199032 V1015840119899 + (2 minus ]) 1198992ℎ261199032 1199081015840119899 minus ℎ26 119908101584010158401015840119899 ]V4 = 119903120587119864ℎ12 (1 minus ]2) [minus]119899ℎ

2

1199032 V119899 minus ]1198992ℎ21199032 119908119899 + ℎ211990810158401015840119899 ]

12 Shock and Vibration

uthick =

1199061 = 1199061198991199062 = V119899

1199063 = 1199081198991199064 = 1205951199091198991199065 = 120595120579119899

kthick =

V1 = 119903120587119864ℎ1 minus ]2[1199061015840119899 + ]

119899119903 V119899 + ]119903119908119899]V2 = 12 119903120587119864ℎ1 + ]

[minus119899119903 119906119899 + (1 + ℎ2121199032) V1015840119899 minus 119899ℎ2121199032120595119909119899 + ℎ21211990321205951015840120579119899]V3 = 119903120587120581119866ℎ [1199081015840119899 + 120595119909119899]V4 = ℎ212 119903120587119864ℎ1 minus ]2

[1205951015840119909119899 + ]119899119903120595120579119899]

V5 = ℎ224 119903120587119864ℎ1 + ][V1015840119899119903 minus 119899119903120595119909119899 + 1205951015840120579119899]

[S]thin =

[[[[[[[[[[[[[[[[[[

11990411 0 0 11990414 0 11990416 0 011990422 11990423 0 11990425 0 0 1199042811990433 0 11990435 0 0 1199043811990444 0 11990446 11990447 0

11990455 0 0 0sym 11990466 0 0

0 011990488

]]]]]]]]]]]]]]]]]]

(A1)

where nonzero elements in the upper triangular domain are

11990411 = 120588120587119903ℎ1205962 minus 120587119864ℎ11989922119903 (1 minus ]) + 61198992119903120587119864ℎ(121199032 + ℎ2) (1 + ]) 11990414 = minus 1198992ℎ3120587119864(121199032 + ℎ2) (1 + ]) 11990416 = 12119899119903121199032 + ℎ2 11990422 = 120588120587119903ℎ1205962 minus 1198992120587119864ℎ (121199032 + ℎ2)

121199033 11990423 = minus120587119864ℎ121198991199032 + 1198993ℎ2121199033 11990425 = minus]119899119903 11990428 = ]

1198991199032 11990433 = 120588ℎ1199031205871205962 minus 120587119864ℎ119903 minus 1198994ℎ3120587119864121199033

11990435 = minus]119903 11990438 = ]11989921199032 11990444 = minus 21198992ℎ3119903120587119864(121199032 + ℎ2) (1 + ])

11990446 = minus 2119899ℎ2121199032 + ℎ2 11990447 = 111990455 = 1 minus ]2119903120587119864ℎ 11990466 = 24119903 (1 + ])120587119864ℎ (121199032 + ℎ2)

11990488 = 12 (1 minus ]2)119903120587119864ℎ3

[S]thick

Shock and Vibration 13

=

[[[[[[[[[[[[[[[[[[[[[[[

11990411 0 0 0 0 0 11990417 0 0 1199041lowast1011990422 11990423 0 11990425 11990426 0 0 0 011990433 0 11990435 11990436 0 0 0 0

11990444 0 0 0 11990448 0 1199044lowast1011990455 0 0 0 11990459 011990466 0 0 0 0

11990477 0 0 1199047lowast10sym 11990488 0 0

11990499 011990410lowast10

]]]]]]]]]]]]]]]]]]]]]]]

(A2)

where nonzero elements in the upper triangular domain are

11990411 = 119903120587120588ℎ120596211990417 = 119899119903

1199041lowast10 = minus 1198991199032 11990422 = 120587(119903120588ℎ1205962 minus 120581119866ℎ119903 minus 1198992119864ℎ119903 ) 11990423 = minus119899120587119903 (119864ℎ + 120581119866ℎ) 11990425 = 120587120581119866ℎ11990426 = minus]119899119903 11990433 = 120587(119903120588ℎ1205962 minus 119864ℎ119903 minus 1198992120581119866ℎ119903 ) 11990435 = 119899120587120581119866ℎ11990436 = minus]119903 11990444 = 1199031205871205881205962 ℎ312 11990448 = minus1

1199044lowast10 = 119899119903 11990455 = 120587(1199031205881205962 ℎ312 minus 1198992119864ℎ312119903 minus 119903120581119866ℎ) 11990459 = minus]119899119903 11990466 = 1 minus ]2119903120587119864ℎ 11990477 = 2 (1 + ])119903120587119864ℎ

1199047lowast10 = minus2 (1 + ])1199032120587119864ℎ 11990488 = 1119903120587120581119866ℎ 11990499 = 12 (1 minus ]2)

119903120587119864ℎ3 11990410lowast10 = 2 (1 + ])119903120587119864ℎ (12ℎ2 + 11199032 )

120581 = 120587212 (A3)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully appreciate the financial support fromNational Natural Science Foundation of China (51078198)Tsinghua University (2011THZ03) Jiangsu University ofScience and Technology (1732921402) and Shuang-ChuangProgram of Jiangsu Province for this research

References

[1] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover England UK 1927

[2] R N Arnold and G B Warburton ldquoThe flexural vibrationsof thin cylindersrdquo Proceedings of the Institution of MechanicalEngineers vol 167 no 1 pp 62ndash80 1953

[3] A W Leissa ldquoVibration of shellsrdquo Report 288 National Aero-nautics and Space Administration (NASA) Washington DCUSA 1973

[4] N S Bardell J M Dunsdon and R S Langley ldquoOn thefree vibration of completely free open cylindrically curvedisotropic shell panelsrdquo Journal of Sound and Vibration vol 207no 5 pp 647ndash669 1997

[5] D-Y Tan ldquoFree vibration analysis of shells of revolutionrdquoJournal of Sound and Vibration vol 213 no 1 pp 15ndash33 1998

[6] P M Naghdi and R M Cooper ldquoPropagation of elastic wavesin cylindrical shells including the effects of transverse shearand rotatory inertiardquo The Journal of the Acoustical Society ofAmerica vol 28 no 1 pp 56ndash63 1956

[7] I Mirsky and G Herrmann ldquoNonaxially symmetric motionsof cylindrical shellsrdquo The Journal of the Acoustical Society ofAmerica vol 29 pp 1116ndash1123 1957

[8] F Tornabene and E Viola ldquo2-D solution for free vibrationsof parabolic shells using generalized differential quadraturemethodrdquo European Journal of Mechanics ASolids vol 27 no6 pp 1001ndash1025 2008

[9] J L Mantari A S Oktem and C Guedes Soares ldquoBending andfree vibration analysis of isotropic and multilayered plates andshells by using a new accurate higher-order shear deformationtheoryrdquo Composites Part B Engineering vol 43 no 8 pp 3348ndash3360 2012

14 Shock and Vibration

[10] J G Kim J K Lee and H J Yoon ldquoFree vibration analysisfor shells of revolution based on p-version mixed finite elementformulationrdquo Finite Elements in Analysis and Design vol 95 pp12ndash19 2015

[11] F Tornabene ldquoFree vibrations of laminated composite doubly-curved shells and panels of revolution via the GDQ methodrdquoComputer Methods in Applied Mechanics and Engineering vol200 no 9ndash12 pp 931ndash952 2011

[12] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe local GDQmethod applied to general higher-order theories of doubly-curved laminated composite shells and panels the free vibrationanalysisrdquo Composite Structures vol 116 pp 637ndash660 2014

[13] A Pagani E Carrera and A Ferreira ldquoHigher-order theoriesand radial basis functions applied to free vibration analysisof thin-walled beamsrdquo Mechanics of Advanced Materials andStructures vol 23 no 9 pp 1080ndash1091 2016

[14] T Ye G Jin Y Chen and S Shi ldquoA unified formulationfor vibration analysis of open shells with arbitrary boundaryconditionsrdquo International Journal ofMechanical Sciences vol 81pp 42ndash59 2014

[15] T Ye G Jin S Shi and X Ma ldquoThree-dimensional freevibration analysis of thick cylindrical shells with general endconditions and resting on elastic foundationsrdquo InternationalJournal of Mechanical Sciences vol 84 pp 120ndash137 2014

[16] F W Williams and W H Wittrick ldquoAn automatic computa-tional procedure for calculating natural frequencies of skeletalstructuresrdquo International Journal of Mechanical Sciences vol 12no 9 pp 781ndash791 1970

[17] W H Wittrick and F W Williams ldquoA general algorithmfor computing natural frequencies of elastic structuresrdquo TheQuarterly Journal of Mechanics and Applied Mathematics vol24 pp 263ndash284 1971

[18] S Yuan K Ye F W Williams and D Kennedy ldquoRecursivesecond order convergence method for natural frequencies andmodes when using dynamic stiffness matricesrdquo InternationalJournal for Numerical Methods in Engineering vol 56 no 12pp 1795ndash1814 2003

[19] S Yuan K Ye and FWWilliams ldquoSecond order mode-findingmethod in dynamic stiffness matrix methodsrdquo Journal of Soundand Vibration vol 269 no 3ndash5 pp 689ndash708 2004

[20] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007

[21] J R Banerjee C W Cheung R Morishima M Pereraand J Njuguna ldquoFree vibration of a three-layered sandwichbeam using the dynamic stiffness method and experimentrdquoInternational Journal of Solids and Structures vol 44 no 22-23pp 7543ndash7563 2007

[22] H Su J R Banerjee and C W Cheung ldquoDynamic stiffnessformulation and free vibration analysis of functionally gradedbeamsrdquo Composite Structures vol 106 pp 854ndash862 2013

[23] A Pagani M Boscolo J R Banerjee and E Carrera ldquoExactdynamic stiffness elements based on one-dimensional higher-order theories for free vibration analysis of solid and thin-walled structuresrdquo Journal of Sound and Vibration vol 332 no23 pp 6104ndash6127 2013

[24] N El-Kaabazi and D Kennedy ldquoCalculation of natural fre-quencies and vibration modes of variable thickness cylindricalshells using the Wittrick-Williams algorithmrdquo Computers andStructures vol 104-105 pp 4ndash12 2012

[25] X Chen Research on dynamic stiffness method for free vibrationof shells of revolution [MS thesis] Tsinghua University 2009(Chinese)

[26] XChen andKYe ldquoFreeVibrationAnalysis for Shells of Revolu-tion Using an Exact Dynamic Stiffness Methodrdquo MathematicalProblems in Engineering vol 2016 Article ID 4513520 12 pages2016

[27] X Chen and K Ye ldquoAnalysis of free vibration of moderatelythick circular cylindrical shells using the dynamic stiffnessmethodrdquo Engineering Mechanics vol 33 no 9 pp 40ndash48 2016(Chinese)

[28] X M Zhang G R Liu and K Y Lam ldquoVibration analysisof thin cylindrical shells using wave propagation approachrdquoJournal of Sound and Vibration vol 239 no 3 pp 397ndash4032001

[29] L Xuebin ldquoStudy on free vibration analysis of circular cylin-drical shells using wave propagationrdquo Journal of Sound andVibration vol 311 no 3-5 pp 667ndash682 2008

[30] C B Sharma ldquoCalculation of natural frequencies of fixed-freecircular cylindrical shellsrdquo Journal of Sound and Vibration vol35 no 1 pp 55ndash76 1974

[31] A E ArmenakasDCGazis andGHerrmannFreeVibrationsof Circular Cylindrical Shells PergamonPressOxfordUK 1969

[32] A Bhimaraddi ldquoA higher order theory for free vibrationanalysis of circular cylindrical shellsrdquo International Journal ofSolids and Structures vol 20 no 7 pp 623ndash630 1984

[33] U Ascher J Christiansen and R D Russell ldquoAlgorithm 569COLSYS collocation software for boundary value ODEs [D2]rdquoACM Transactions on Mathematical Software vol 7 no 2 pp223ndash229 1981

[34] U Ascher J Christiansen and R D Russell ldquoCollocationsoftware for boundary-value ODEsrdquo ACM Transactions onMathematical Software vol 7 no 2 pp 209ndash222 1981

[35] S Markus The Mechanics of Vibrations of Cylindrical ShellsElsevier Science Limited 1988

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 11

0

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(a) ℎ119903 = 0025

0

minus1

minus2

minus3

minus4

minus5

log 1

0(120578

)

5 10 15 200n

Lr = 2

Lr = 5

Lr = 10

Lr = 02

Lr = 05

(b) ℎ119903 = 005

Figure 9 Differences between the first frequencies under each 119899 with different 119871119903

(differences are around or smaller than 10minus2) Sincethe DSM is exact no shear locking is observed

(2) Differences between frequencies from thin and mod-erately thick DSM formulations change with a varietyof factors The smaller ℎ119903 the smaller the differencethe smaller119871119903 the larger the difference the larger the119899 the larger the difference

It is proved that moderately thick DSM formulations canbe used to analyse the free vibration of normal length thin

circular cylindrical shells without shear locking demon-strating that the applicable range of moderately thick DSMformulations is expanded These merits should be attributedto that the dynamic stiffness method is an exact method

Though this comparison study is performed on circularcylindrical shells similar conclusions can be extended to thefree vibration of other shaped shells of revolution

Appendix

uthin =

1199061 = 1199061198991199062 = V119899

1199063 = 1199081198991199064 = 1199081015840119899

kthin =

V1 = 119903120587119864ℎ1 minus ]2[]119899119903 V119899 + ]

1119903119908119899 + 1199061015840119899]V2 = 119903120587119864ℎ2 (1 + ]) [minus119899119903 119906119899 + (1 + ℎ2121199032) V1015840119899 + 119899ℎ261199032 1199081015840119899]V3 = 119903120587119864ℎ2 (1 minus ]2) [119899ℎ

2

61199032 V1015840119899 + (2 minus ]) 1198992ℎ261199032 1199081015840119899 minus ℎ26 119908101584010158401015840119899 ]V4 = 119903120587119864ℎ12 (1 minus ]2) [minus]119899ℎ

2

1199032 V119899 minus ]1198992ℎ21199032 119908119899 + ℎ211990810158401015840119899 ]

12 Shock and Vibration

uthick =

1199061 = 1199061198991199062 = V119899

1199063 = 1199081198991199064 = 1205951199091198991199065 = 120595120579119899

kthick =

V1 = 119903120587119864ℎ1 minus ]2[1199061015840119899 + ]

119899119903 V119899 + ]119903119908119899]V2 = 12 119903120587119864ℎ1 + ]

[minus119899119903 119906119899 + (1 + ℎ2121199032) V1015840119899 minus 119899ℎ2121199032120595119909119899 + ℎ21211990321205951015840120579119899]V3 = 119903120587120581119866ℎ [1199081015840119899 + 120595119909119899]V4 = ℎ212 119903120587119864ℎ1 minus ]2

[1205951015840119909119899 + ]119899119903120595120579119899]

V5 = ℎ224 119903120587119864ℎ1 + ][V1015840119899119903 minus 119899119903120595119909119899 + 1205951015840120579119899]

[S]thin =

[[[[[[[[[[[[[[[[[[

11990411 0 0 11990414 0 11990416 0 011990422 11990423 0 11990425 0 0 1199042811990433 0 11990435 0 0 1199043811990444 0 11990446 11990447 0

11990455 0 0 0sym 11990466 0 0

0 011990488

]]]]]]]]]]]]]]]]]]

(A1)

where nonzero elements in the upper triangular domain are

11990411 = 120588120587119903ℎ1205962 minus 120587119864ℎ11989922119903 (1 minus ]) + 61198992119903120587119864ℎ(121199032 + ℎ2) (1 + ]) 11990414 = minus 1198992ℎ3120587119864(121199032 + ℎ2) (1 + ]) 11990416 = 12119899119903121199032 + ℎ2 11990422 = 120588120587119903ℎ1205962 minus 1198992120587119864ℎ (121199032 + ℎ2)

121199033 11990423 = minus120587119864ℎ121198991199032 + 1198993ℎ2121199033 11990425 = minus]119899119903 11990428 = ]

1198991199032 11990433 = 120588ℎ1199031205871205962 minus 120587119864ℎ119903 minus 1198994ℎ3120587119864121199033

11990435 = minus]119903 11990438 = ]11989921199032 11990444 = minus 21198992ℎ3119903120587119864(121199032 + ℎ2) (1 + ])

11990446 = minus 2119899ℎ2121199032 + ℎ2 11990447 = 111990455 = 1 minus ]2119903120587119864ℎ 11990466 = 24119903 (1 + ])120587119864ℎ (121199032 + ℎ2)

11990488 = 12 (1 minus ]2)119903120587119864ℎ3

[S]thick

Shock and Vibration 13

=

[[[[[[[[[[[[[[[[[[[[[[[

11990411 0 0 0 0 0 11990417 0 0 1199041lowast1011990422 11990423 0 11990425 11990426 0 0 0 011990433 0 11990435 11990436 0 0 0 0

11990444 0 0 0 11990448 0 1199044lowast1011990455 0 0 0 11990459 011990466 0 0 0 0

11990477 0 0 1199047lowast10sym 11990488 0 0

11990499 011990410lowast10

]]]]]]]]]]]]]]]]]]]]]]]

(A2)

where nonzero elements in the upper triangular domain are

11990411 = 119903120587120588ℎ120596211990417 = 119899119903

1199041lowast10 = minus 1198991199032 11990422 = 120587(119903120588ℎ1205962 minus 120581119866ℎ119903 minus 1198992119864ℎ119903 ) 11990423 = minus119899120587119903 (119864ℎ + 120581119866ℎ) 11990425 = 120587120581119866ℎ11990426 = minus]119899119903 11990433 = 120587(119903120588ℎ1205962 minus 119864ℎ119903 minus 1198992120581119866ℎ119903 ) 11990435 = 119899120587120581119866ℎ11990436 = minus]119903 11990444 = 1199031205871205881205962 ℎ312 11990448 = minus1

1199044lowast10 = 119899119903 11990455 = 120587(1199031205881205962 ℎ312 minus 1198992119864ℎ312119903 minus 119903120581119866ℎ) 11990459 = minus]119899119903 11990466 = 1 minus ]2119903120587119864ℎ 11990477 = 2 (1 + ])119903120587119864ℎ

1199047lowast10 = minus2 (1 + ])1199032120587119864ℎ 11990488 = 1119903120587120581119866ℎ 11990499 = 12 (1 minus ]2)

119903120587119864ℎ3 11990410lowast10 = 2 (1 + ])119903120587119864ℎ (12ℎ2 + 11199032 )

120581 = 120587212 (A3)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully appreciate the financial support fromNational Natural Science Foundation of China (51078198)Tsinghua University (2011THZ03) Jiangsu University ofScience and Technology (1732921402) and Shuang-ChuangProgram of Jiangsu Province for this research

References

[1] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover England UK 1927

[2] R N Arnold and G B Warburton ldquoThe flexural vibrationsof thin cylindersrdquo Proceedings of the Institution of MechanicalEngineers vol 167 no 1 pp 62ndash80 1953

[3] A W Leissa ldquoVibration of shellsrdquo Report 288 National Aero-nautics and Space Administration (NASA) Washington DCUSA 1973

[4] N S Bardell J M Dunsdon and R S Langley ldquoOn thefree vibration of completely free open cylindrically curvedisotropic shell panelsrdquo Journal of Sound and Vibration vol 207no 5 pp 647ndash669 1997

[5] D-Y Tan ldquoFree vibration analysis of shells of revolutionrdquoJournal of Sound and Vibration vol 213 no 1 pp 15ndash33 1998

[6] P M Naghdi and R M Cooper ldquoPropagation of elastic wavesin cylindrical shells including the effects of transverse shearand rotatory inertiardquo The Journal of the Acoustical Society ofAmerica vol 28 no 1 pp 56ndash63 1956

[7] I Mirsky and G Herrmann ldquoNonaxially symmetric motionsof cylindrical shellsrdquo The Journal of the Acoustical Society ofAmerica vol 29 pp 1116ndash1123 1957

[8] F Tornabene and E Viola ldquo2-D solution for free vibrationsof parabolic shells using generalized differential quadraturemethodrdquo European Journal of Mechanics ASolids vol 27 no6 pp 1001ndash1025 2008

[9] J L Mantari A S Oktem and C Guedes Soares ldquoBending andfree vibration analysis of isotropic and multilayered plates andshells by using a new accurate higher-order shear deformationtheoryrdquo Composites Part B Engineering vol 43 no 8 pp 3348ndash3360 2012

14 Shock and Vibration

[10] J G Kim J K Lee and H J Yoon ldquoFree vibration analysisfor shells of revolution based on p-version mixed finite elementformulationrdquo Finite Elements in Analysis and Design vol 95 pp12ndash19 2015

[11] F Tornabene ldquoFree vibrations of laminated composite doubly-curved shells and panels of revolution via the GDQ methodrdquoComputer Methods in Applied Mechanics and Engineering vol200 no 9ndash12 pp 931ndash952 2011

[12] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe local GDQmethod applied to general higher-order theories of doubly-curved laminated composite shells and panels the free vibrationanalysisrdquo Composite Structures vol 116 pp 637ndash660 2014

[13] A Pagani E Carrera and A Ferreira ldquoHigher-order theoriesand radial basis functions applied to free vibration analysisof thin-walled beamsrdquo Mechanics of Advanced Materials andStructures vol 23 no 9 pp 1080ndash1091 2016

[14] T Ye G Jin Y Chen and S Shi ldquoA unified formulationfor vibration analysis of open shells with arbitrary boundaryconditionsrdquo International Journal ofMechanical Sciences vol 81pp 42ndash59 2014

[15] T Ye G Jin S Shi and X Ma ldquoThree-dimensional freevibration analysis of thick cylindrical shells with general endconditions and resting on elastic foundationsrdquo InternationalJournal of Mechanical Sciences vol 84 pp 120ndash137 2014

[16] F W Williams and W H Wittrick ldquoAn automatic computa-tional procedure for calculating natural frequencies of skeletalstructuresrdquo International Journal of Mechanical Sciences vol 12no 9 pp 781ndash791 1970

[17] W H Wittrick and F W Williams ldquoA general algorithmfor computing natural frequencies of elastic structuresrdquo TheQuarterly Journal of Mechanics and Applied Mathematics vol24 pp 263ndash284 1971

[18] S Yuan K Ye F W Williams and D Kennedy ldquoRecursivesecond order convergence method for natural frequencies andmodes when using dynamic stiffness matricesrdquo InternationalJournal for Numerical Methods in Engineering vol 56 no 12pp 1795ndash1814 2003

[19] S Yuan K Ye and FWWilliams ldquoSecond order mode-findingmethod in dynamic stiffness matrix methodsrdquo Journal of Soundand Vibration vol 269 no 3ndash5 pp 689ndash708 2004

[20] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007

[21] J R Banerjee C W Cheung R Morishima M Pereraand J Njuguna ldquoFree vibration of a three-layered sandwichbeam using the dynamic stiffness method and experimentrdquoInternational Journal of Solids and Structures vol 44 no 22-23pp 7543ndash7563 2007

[22] H Su J R Banerjee and C W Cheung ldquoDynamic stiffnessformulation and free vibration analysis of functionally gradedbeamsrdquo Composite Structures vol 106 pp 854ndash862 2013

[23] A Pagani M Boscolo J R Banerjee and E Carrera ldquoExactdynamic stiffness elements based on one-dimensional higher-order theories for free vibration analysis of solid and thin-walled structuresrdquo Journal of Sound and Vibration vol 332 no23 pp 6104ndash6127 2013

[24] N El-Kaabazi and D Kennedy ldquoCalculation of natural fre-quencies and vibration modes of variable thickness cylindricalshells using the Wittrick-Williams algorithmrdquo Computers andStructures vol 104-105 pp 4ndash12 2012

[25] X Chen Research on dynamic stiffness method for free vibrationof shells of revolution [MS thesis] Tsinghua University 2009(Chinese)

[26] XChen andKYe ldquoFreeVibrationAnalysis for Shells of Revolu-tion Using an Exact Dynamic Stiffness Methodrdquo MathematicalProblems in Engineering vol 2016 Article ID 4513520 12 pages2016

[27] X Chen and K Ye ldquoAnalysis of free vibration of moderatelythick circular cylindrical shells using the dynamic stiffnessmethodrdquo Engineering Mechanics vol 33 no 9 pp 40ndash48 2016(Chinese)

[28] X M Zhang G R Liu and K Y Lam ldquoVibration analysisof thin cylindrical shells using wave propagation approachrdquoJournal of Sound and Vibration vol 239 no 3 pp 397ndash4032001

[29] L Xuebin ldquoStudy on free vibration analysis of circular cylin-drical shells using wave propagationrdquo Journal of Sound andVibration vol 311 no 3-5 pp 667ndash682 2008

[30] C B Sharma ldquoCalculation of natural frequencies of fixed-freecircular cylindrical shellsrdquo Journal of Sound and Vibration vol35 no 1 pp 55ndash76 1974

[31] A E ArmenakasDCGazis andGHerrmannFreeVibrationsof Circular Cylindrical Shells PergamonPressOxfordUK 1969

[32] A Bhimaraddi ldquoA higher order theory for free vibrationanalysis of circular cylindrical shellsrdquo International Journal ofSolids and Structures vol 20 no 7 pp 623ndash630 1984

[33] U Ascher J Christiansen and R D Russell ldquoAlgorithm 569COLSYS collocation software for boundary value ODEs [D2]rdquoACM Transactions on Mathematical Software vol 7 no 2 pp223ndash229 1981

[34] U Ascher J Christiansen and R D Russell ldquoCollocationsoftware for boundary-value ODEsrdquo ACM Transactions onMathematical Software vol 7 no 2 pp 209ndash222 1981

[35] S Markus The Mechanics of Vibrations of Cylindrical ShellsElsevier Science Limited 1988

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 Shock and Vibration

uthick =

1199061 = 1199061198991199062 = V119899

1199063 = 1199081198991199064 = 1205951199091198991199065 = 120595120579119899

kthick =

V1 = 119903120587119864ℎ1 minus ]2[1199061015840119899 + ]

119899119903 V119899 + ]119903119908119899]V2 = 12 119903120587119864ℎ1 + ]

[minus119899119903 119906119899 + (1 + ℎ2121199032) V1015840119899 minus 119899ℎ2121199032120595119909119899 + ℎ21211990321205951015840120579119899]V3 = 119903120587120581119866ℎ [1199081015840119899 + 120595119909119899]V4 = ℎ212 119903120587119864ℎ1 minus ]2

[1205951015840119909119899 + ]119899119903120595120579119899]

V5 = ℎ224 119903120587119864ℎ1 + ][V1015840119899119903 minus 119899119903120595119909119899 + 1205951015840120579119899]

[S]thin =

[[[[[[[[[[[[[[[[[[

11990411 0 0 11990414 0 11990416 0 011990422 11990423 0 11990425 0 0 1199042811990433 0 11990435 0 0 1199043811990444 0 11990446 11990447 0

11990455 0 0 0sym 11990466 0 0

0 011990488

]]]]]]]]]]]]]]]]]]

(A1)

where nonzero elements in the upper triangular domain are

11990411 = 120588120587119903ℎ1205962 minus 120587119864ℎ11989922119903 (1 minus ]) + 61198992119903120587119864ℎ(121199032 + ℎ2) (1 + ]) 11990414 = minus 1198992ℎ3120587119864(121199032 + ℎ2) (1 + ]) 11990416 = 12119899119903121199032 + ℎ2 11990422 = 120588120587119903ℎ1205962 minus 1198992120587119864ℎ (121199032 + ℎ2)

121199033 11990423 = minus120587119864ℎ121198991199032 + 1198993ℎ2121199033 11990425 = minus]119899119903 11990428 = ]

1198991199032 11990433 = 120588ℎ1199031205871205962 minus 120587119864ℎ119903 minus 1198994ℎ3120587119864121199033

11990435 = minus]119903 11990438 = ]11989921199032 11990444 = minus 21198992ℎ3119903120587119864(121199032 + ℎ2) (1 + ])

11990446 = minus 2119899ℎ2121199032 + ℎ2 11990447 = 111990455 = 1 minus ]2119903120587119864ℎ 11990466 = 24119903 (1 + ])120587119864ℎ (121199032 + ℎ2)

11990488 = 12 (1 minus ]2)119903120587119864ℎ3

[S]thick

Shock and Vibration 13

=

[[[[[[[[[[[[[[[[[[[[[[[

11990411 0 0 0 0 0 11990417 0 0 1199041lowast1011990422 11990423 0 11990425 11990426 0 0 0 011990433 0 11990435 11990436 0 0 0 0

11990444 0 0 0 11990448 0 1199044lowast1011990455 0 0 0 11990459 011990466 0 0 0 0

11990477 0 0 1199047lowast10sym 11990488 0 0

11990499 011990410lowast10

]]]]]]]]]]]]]]]]]]]]]]]

(A2)

where nonzero elements in the upper triangular domain are

11990411 = 119903120587120588ℎ120596211990417 = 119899119903

1199041lowast10 = minus 1198991199032 11990422 = 120587(119903120588ℎ1205962 minus 120581119866ℎ119903 minus 1198992119864ℎ119903 ) 11990423 = minus119899120587119903 (119864ℎ + 120581119866ℎ) 11990425 = 120587120581119866ℎ11990426 = minus]119899119903 11990433 = 120587(119903120588ℎ1205962 minus 119864ℎ119903 minus 1198992120581119866ℎ119903 ) 11990435 = 119899120587120581119866ℎ11990436 = minus]119903 11990444 = 1199031205871205881205962 ℎ312 11990448 = minus1

1199044lowast10 = 119899119903 11990455 = 120587(1199031205881205962 ℎ312 minus 1198992119864ℎ312119903 minus 119903120581119866ℎ) 11990459 = minus]119899119903 11990466 = 1 minus ]2119903120587119864ℎ 11990477 = 2 (1 + ])119903120587119864ℎ

1199047lowast10 = minus2 (1 + ])1199032120587119864ℎ 11990488 = 1119903120587120581119866ℎ 11990499 = 12 (1 minus ]2)

119903120587119864ℎ3 11990410lowast10 = 2 (1 + ])119903120587119864ℎ (12ℎ2 + 11199032 )

120581 = 120587212 (A3)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully appreciate the financial support fromNational Natural Science Foundation of China (51078198)Tsinghua University (2011THZ03) Jiangsu University ofScience and Technology (1732921402) and Shuang-ChuangProgram of Jiangsu Province for this research

References

[1] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover England UK 1927

[2] R N Arnold and G B Warburton ldquoThe flexural vibrationsof thin cylindersrdquo Proceedings of the Institution of MechanicalEngineers vol 167 no 1 pp 62ndash80 1953

[3] A W Leissa ldquoVibration of shellsrdquo Report 288 National Aero-nautics and Space Administration (NASA) Washington DCUSA 1973

[4] N S Bardell J M Dunsdon and R S Langley ldquoOn thefree vibration of completely free open cylindrically curvedisotropic shell panelsrdquo Journal of Sound and Vibration vol 207no 5 pp 647ndash669 1997

[5] D-Y Tan ldquoFree vibration analysis of shells of revolutionrdquoJournal of Sound and Vibration vol 213 no 1 pp 15ndash33 1998

[6] P M Naghdi and R M Cooper ldquoPropagation of elastic wavesin cylindrical shells including the effects of transverse shearand rotatory inertiardquo The Journal of the Acoustical Society ofAmerica vol 28 no 1 pp 56ndash63 1956

[7] I Mirsky and G Herrmann ldquoNonaxially symmetric motionsof cylindrical shellsrdquo The Journal of the Acoustical Society ofAmerica vol 29 pp 1116ndash1123 1957

[8] F Tornabene and E Viola ldquo2-D solution for free vibrationsof parabolic shells using generalized differential quadraturemethodrdquo European Journal of Mechanics ASolids vol 27 no6 pp 1001ndash1025 2008

[9] J L Mantari A S Oktem and C Guedes Soares ldquoBending andfree vibration analysis of isotropic and multilayered plates andshells by using a new accurate higher-order shear deformationtheoryrdquo Composites Part B Engineering vol 43 no 8 pp 3348ndash3360 2012

14 Shock and Vibration

[10] J G Kim J K Lee and H J Yoon ldquoFree vibration analysisfor shells of revolution based on p-version mixed finite elementformulationrdquo Finite Elements in Analysis and Design vol 95 pp12ndash19 2015

[11] F Tornabene ldquoFree vibrations of laminated composite doubly-curved shells and panels of revolution via the GDQ methodrdquoComputer Methods in Applied Mechanics and Engineering vol200 no 9ndash12 pp 931ndash952 2011

[12] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe local GDQmethod applied to general higher-order theories of doubly-curved laminated composite shells and panels the free vibrationanalysisrdquo Composite Structures vol 116 pp 637ndash660 2014

[13] A Pagani E Carrera and A Ferreira ldquoHigher-order theoriesand radial basis functions applied to free vibration analysisof thin-walled beamsrdquo Mechanics of Advanced Materials andStructures vol 23 no 9 pp 1080ndash1091 2016

[14] T Ye G Jin Y Chen and S Shi ldquoA unified formulationfor vibration analysis of open shells with arbitrary boundaryconditionsrdquo International Journal ofMechanical Sciences vol 81pp 42ndash59 2014

[15] T Ye G Jin S Shi and X Ma ldquoThree-dimensional freevibration analysis of thick cylindrical shells with general endconditions and resting on elastic foundationsrdquo InternationalJournal of Mechanical Sciences vol 84 pp 120ndash137 2014

[16] F W Williams and W H Wittrick ldquoAn automatic computa-tional procedure for calculating natural frequencies of skeletalstructuresrdquo International Journal of Mechanical Sciences vol 12no 9 pp 781ndash791 1970

[17] W H Wittrick and F W Williams ldquoA general algorithmfor computing natural frequencies of elastic structuresrdquo TheQuarterly Journal of Mechanics and Applied Mathematics vol24 pp 263ndash284 1971

[18] S Yuan K Ye F W Williams and D Kennedy ldquoRecursivesecond order convergence method for natural frequencies andmodes when using dynamic stiffness matricesrdquo InternationalJournal for Numerical Methods in Engineering vol 56 no 12pp 1795ndash1814 2003

[19] S Yuan K Ye and FWWilliams ldquoSecond order mode-findingmethod in dynamic stiffness matrix methodsrdquo Journal of Soundand Vibration vol 269 no 3ndash5 pp 689ndash708 2004

[20] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007

[21] J R Banerjee C W Cheung R Morishima M Pereraand J Njuguna ldquoFree vibration of a three-layered sandwichbeam using the dynamic stiffness method and experimentrdquoInternational Journal of Solids and Structures vol 44 no 22-23pp 7543ndash7563 2007

[22] H Su J R Banerjee and C W Cheung ldquoDynamic stiffnessformulation and free vibration analysis of functionally gradedbeamsrdquo Composite Structures vol 106 pp 854ndash862 2013

[23] A Pagani M Boscolo J R Banerjee and E Carrera ldquoExactdynamic stiffness elements based on one-dimensional higher-order theories for free vibration analysis of solid and thin-walled structuresrdquo Journal of Sound and Vibration vol 332 no23 pp 6104ndash6127 2013

[24] N El-Kaabazi and D Kennedy ldquoCalculation of natural fre-quencies and vibration modes of variable thickness cylindricalshells using the Wittrick-Williams algorithmrdquo Computers andStructures vol 104-105 pp 4ndash12 2012

[25] X Chen Research on dynamic stiffness method for free vibrationof shells of revolution [MS thesis] Tsinghua University 2009(Chinese)

[26] XChen andKYe ldquoFreeVibrationAnalysis for Shells of Revolu-tion Using an Exact Dynamic Stiffness Methodrdquo MathematicalProblems in Engineering vol 2016 Article ID 4513520 12 pages2016

[27] X Chen and K Ye ldquoAnalysis of free vibration of moderatelythick circular cylindrical shells using the dynamic stiffnessmethodrdquo Engineering Mechanics vol 33 no 9 pp 40ndash48 2016(Chinese)

[28] X M Zhang G R Liu and K Y Lam ldquoVibration analysisof thin cylindrical shells using wave propagation approachrdquoJournal of Sound and Vibration vol 239 no 3 pp 397ndash4032001

[29] L Xuebin ldquoStudy on free vibration analysis of circular cylin-drical shells using wave propagationrdquo Journal of Sound andVibration vol 311 no 3-5 pp 667ndash682 2008

[30] C B Sharma ldquoCalculation of natural frequencies of fixed-freecircular cylindrical shellsrdquo Journal of Sound and Vibration vol35 no 1 pp 55ndash76 1974

[31] A E ArmenakasDCGazis andGHerrmannFreeVibrationsof Circular Cylindrical Shells PergamonPressOxfordUK 1969

[32] A Bhimaraddi ldquoA higher order theory for free vibrationanalysis of circular cylindrical shellsrdquo International Journal ofSolids and Structures vol 20 no 7 pp 623ndash630 1984

[33] U Ascher J Christiansen and R D Russell ldquoAlgorithm 569COLSYS collocation software for boundary value ODEs [D2]rdquoACM Transactions on Mathematical Software vol 7 no 2 pp223ndash229 1981

[34] U Ascher J Christiansen and R D Russell ldquoCollocationsoftware for boundary-value ODEsrdquo ACM Transactions onMathematical Software vol 7 no 2 pp 209ndash222 1981

[35] S Markus The Mechanics of Vibrations of Cylindrical ShellsElsevier Science Limited 1988

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 13

=

[[[[[[[[[[[[[[[[[[[[[[[

11990411 0 0 0 0 0 11990417 0 0 1199041lowast1011990422 11990423 0 11990425 11990426 0 0 0 011990433 0 11990435 11990436 0 0 0 0

11990444 0 0 0 11990448 0 1199044lowast1011990455 0 0 0 11990459 011990466 0 0 0 0

11990477 0 0 1199047lowast10sym 11990488 0 0

11990499 011990410lowast10

]]]]]]]]]]]]]]]]]]]]]]]

(A2)

where nonzero elements in the upper triangular domain are

11990411 = 119903120587120588ℎ120596211990417 = 119899119903

1199041lowast10 = minus 1198991199032 11990422 = 120587(119903120588ℎ1205962 minus 120581119866ℎ119903 minus 1198992119864ℎ119903 ) 11990423 = minus119899120587119903 (119864ℎ + 120581119866ℎ) 11990425 = 120587120581119866ℎ11990426 = minus]119899119903 11990433 = 120587(119903120588ℎ1205962 minus 119864ℎ119903 minus 1198992120581119866ℎ119903 ) 11990435 = 119899120587120581119866ℎ11990436 = minus]119903 11990444 = 1199031205871205881205962 ℎ312 11990448 = minus1

1199044lowast10 = 119899119903 11990455 = 120587(1199031205881205962 ℎ312 minus 1198992119864ℎ312119903 minus 119903120581119866ℎ) 11990459 = minus]119899119903 11990466 = 1 minus ]2119903120587119864ℎ 11990477 = 2 (1 + ])119903120587119864ℎ

1199047lowast10 = minus2 (1 + ])1199032120587119864ℎ 11990488 = 1119903120587120581119866ℎ 11990499 = 12 (1 minus ]2)

119903120587119864ℎ3 11990410lowast10 = 2 (1 + ])119903120587119864ℎ (12ℎ2 + 11199032 )

120581 = 120587212 (A3)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully appreciate the financial support fromNational Natural Science Foundation of China (51078198)Tsinghua University (2011THZ03) Jiangsu University ofScience and Technology (1732921402) and Shuang-ChuangProgram of Jiangsu Province for this research

References

[1] A E H Love A Treatise on the Mathematical Theory ofElasticity Dover England UK 1927

[2] R N Arnold and G B Warburton ldquoThe flexural vibrationsof thin cylindersrdquo Proceedings of the Institution of MechanicalEngineers vol 167 no 1 pp 62ndash80 1953

[3] A W Leissa ldquoVibration of shellsrdquo Report 288 National Aero-nautics and Space Administration (NASA) Washington DCUSA 1973

[4] N S Bardell J M Dunsdon and R S Langley ldquoOn thefree vibration of completely free open cylindrically curvedisotropic shell panelsrdquo Journal of Sound and Vibration vol 207no 5 pp 647ndash669 1997

[5] D-Y Tan ldquoFree vibration analysis of shells of revolutionrdquoJournal of Sound and Vibration vol 213 no 1 pp 15ndash33 1998

[6] P M Naghdi and R M Cooper ldquoPropagation of elastic wavesin cylindrical shells including the effects of transverse shearand rotatory inertiardquo The Journal of the Acoustical Society ofAmerica vol 28 no 1 pp 56ndash63 1956

[7] I Mirsky and G Herrmann ldquoNonaxially symmetric motionsof cylindrical shellsrdquo The Journal of the Acoustical Society ofAmerica vol 29 pp 1116ndash1123 1957

[8] F Tornabene and E Viola ldquo2-D solution for free vibrationsof parabolic shells using generalized differential quadraturemethodrdquo European Journal of Mechanics ASolids vol 27 no6 pp 1001ndash1025 2008

[9] J L Mantari A S Oktem and C Guedes Soares ldquoBending andfree vibration analysis of isotropic and multilayered plates andshells by using a new accurate higher-order shear deformationtheoryrdquo Composites Part B Engineering vol 43 no 8 pp 3348ndash3360 2012

14 Shock and Vibration

[10] J G Kim J K Lee and H J Yoon ldquoFree vibration analysisfor shells of revolution based on p-version mixed finite elementformulationrdquo Finite Elements in Analysis and Design vol 95 pp12ndash19 2015

[11] F Tornabene ldquoFree vibrations of laminated composite doubly-curved shells and panels of revolution via the GDQ methodrdquoComputer Methods in Applied Mechanics and Engineering vol200 no 9ndash12 pp 931ndash952 2011

[12] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe local GDQmethod applied to general higher-order theories of doubly-curved laminated composite shells and panels the free vibrationanalysisrdquo Composite Structures vol 116 pp 637ndash660 2014

[13] A Pagani E Carrera and A Ferreira ldquoHigher-order theoriesand radial basis functions applied to free vibration analysisof thin-walled beamsrdquo Mechanics of Advanced Materials andStructures vol 23 no 9 pp 1080ndash1091 2016

[14] T Ye G Jin Y Chen and S Shi ldquoA unified formulationfor vibration analysis of open shells with arbitrary boundaryconditionsrdquo International Journal ofMechanical Sciences vol 81pp 42ndash59 2014

[15] T Ye G Jin S Shi and X Ma ldquoThree-dimensional freevibration analysis of thick cylindrical shells with general endconditions and resting on elastic foundationsrdquo InternationalJournal of Mechanical Sciences vol 84 pp 120ndash137 2014

[16] F W Williams and W H Wittrick ldquoAn automatic computa-tional procedure for calculating natural frequencies of skeletalstructuresrdquo International Journal of Mechanical Sciences vol 12no 9 pp 781ndash791 1970

[17] W H Wittrick and F W Williams ldquoA general algorithmfor computing natural frequencies of elastic structuresrdquo TheQuarterly Journal of Mechanics and Applied Mathematics vol24 pp 263ndash284 1971

[18] S Yuan K Ye F W Williams and D Kennedy ldquoRecursivesecond order convergence method for natural frequencies andmodes when using dynamic stiffness matricesrdquo InternationalJournal for Numerical Methods in Engineering vol 56 no 12pp 1795ndash1814 2003

[19] S Yuan K Ye and FWWilliams ldquoSecond order mode-findingmethod in dynamic stiffness matrix methodsrdquo Journal of Soundand Vibration vol 269 no 3ndash5 pp 689ndash708 2004

[20] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007

[21] J R Banerjee C W Cheung R Morishima M Pereraand J Njuguna ldquoFree vibration of a three-layered sandwichbeam using the dynamic stiffness method and experimentrdquoInternational Journal of Solids and Structures vol 44 no 22-23pp 7543ndash7563 2007

[22] H Su J R Banerjee and C W Cheung ldquoDynamic stiffnessformulation and free vibration analysis of functionally gradedbeamsrdquo Composite Structures vol 106 pp 854ndash862 2013

[23] A Pagani M Boscolo J R Banerjee and E Carrera ldquoExactdynamic stiffness elements based on one-dimensional higher-order theories for free vibration analysis of solid and thin-walled structuresrdquo Journal of Sound and Vibration vol 332 no23 pp 6104ndash6127 2013

[24] N El-Kaabazi and D Kennedy ldquoCalculation of natural fre-quencies and vibration modes of variable thickness cylindricalshells using the Wittrick-Williams algorithmrdquo Computers andStructures vol 104-105 pp 4ndash12 2012

[25] X Chen Research on dynamic stiffness method for free vibrationof shells of revolution [MS thesis] Tsinghua University 2009(Chinese)

[26] XChen andKYe ldquoFreeVibrationAnalysis for Shells of Revolu-tion Using an Exact Dynamic Stiffness Methodrdquo MathematicalProblems in Engineering vol 2016 Article ID 4513520 12 pages2016

[27] X Chen and K Ye ldquoAnalysis of free vibration of moderatelythick circular cylindrical shells using the dynamic stiffnessmethodrdquo Engineering Mechanics vol 33 no 9 pp 40ndash48 2016(Chinese)

[28] X M Zhang G R Liu and K Y Lam ldquoVibration analysisof thin cylindrical shells using wave propagation approachrdquoJournal of Sound and Vibration vol 239 no 3 pp 397ndash4032001

[29] L Xuebin ldquoStudy on free vibration analysis of circular cylin-drical shells using wave propagationrdquo Journal of Sound andVibration vol 311 no 3-5 pp 667ndash682 2008

[30] C B Sharma ldquoCalculation of natural frequencies of fixed-freecircular cylindrical shellsrdquo Journal of Sound and Vibration vol35 no 1 pp 55ndash76 1974

[31] A E ArmenakasDCGazis andGHerrmannFreeVibrationsof Circular Cylindrical Shells PergamonPressOxfordUK 1969

[32] A Bhimaraddi ldquoA higher order theory for free vibrationanalysis of circular cylindrical shellsrdquo International Journal ofSolids and Structures vol 20 no 7 pp 623ndash630 1984

[33] U Ascher J Christiansen and R D Russell ldquoAlgorithm 569COLSYS collocation software for boundary value ODEs [D2]rdquoACM Transactions on Mathematical Software vol 7 no 2 pp223ndash229 1981

[34] U Ascher J Christiansen and R D Russell ldquoCollocationsoftware for boundary-value ODEsrdquo ACM Transactions onMathematical Software vol 7 no 2 pp 209ndash222 1981

[35] S Markus The Mechanics of Vibrations of Cylindrical ShellsElsevier Science Limited 1988

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

14 Shock and Vibration

[10] J G Kim J K Lee and H J Yoon ldquoFree vibration analysisfor shells of revolution based on p-version mixed finite elementformulationrdquo Finite Elements in Analysis and Design vol 95 pp12ndash19 2015

[11] F Tornabene ldquoFree vibrations of laminated composite doubly-curved shells and panels of revolution via the GDQ methodrdquoComputer Methods in Applied Mechanics and Engineering vol200 no 9ndash12 pp 931ndash952 2011

[12] F Tornabene N Fantuzzi and M Bacciocchi ldquoThe local GDQmethod applied to general higher-order theories of doubly-curved laminated composite shells and panels the free vibrationanalysisrdquo Composite Structures vol 116 pp 637ndash660 2014

[13] A Pagani E Carrera and A Ferreira ldquoHigher-order theoriesand radial basis functions applied to free vibration analysisof thin-walled beamsrdquo Mechanics of Advanced Materials andStructures vol 23 no 9 pp 1080ndash1091 2016

[14] T Ye G Jin Y Chen and S Shi ldquoA unified formulationfor vibration analysis of open shells with arbitrary boundaryconditionsrdquo International Journal ofMechanical Sciences vol 81pp 42ndash59 2014

[15] T Ye G Jin S Shi and X Ma ldquoThree-dimensional freevibration analysis of thick cylindrical shells with general endconditions and resting on elastic foundationsrdquo InternationalJournal of Mechanical Sciences vol 84 pp 120ndash137 2014

[16] F W Williams and W H Wittrick ldquoAn automatic computa-tional procedure for calculating natural frequencies of skeletalstructuresrdquo International Journal of Mechanical Sciences vol 12no 9 pp 781ndash791 1970

[17] W H Wittrick and F W Williams ldquoA general algorithmfor computing natural frequencies of elastic structuresrdquo TheQuarterly Journal of Mechanics and Applied Mathematics vol24 pp 263ndash284 1971

[18] S Yuan K Ye F W Williams and D Kennedy ldquoRecursivesecond order convergence method for natural frequencies andmodes when using dynamic stiffness matricesrdquo InternationalJournal for Numerical Methods in Engineering vol 56 no 12pp 1795ndash1814 2003

[19] S Yuan K Ye and FWWilliams ldquoSecond order mode-findingmethod in dynamic stiffness matrix methodsrdquo Journal of Soundand Vibration vol 269 no 3ndash5 pp 689ndash708 2004

[20] S Yuan K Ye C Xiao F W Williams and D Kennedy ldquoExactdynamic stiffness method for non-uniform Timoshenko beamvibrations and Bernoulli-Euler column bucklingrdquo Journal ofSound and Vibration vol 303 no 3ndash5 pp 526ndash537 2007

[21] J R Banerjee C W Cheung R Morishima M Pereraand J Njuguna ldquoFree vibration of a three-layered sandwichbeam using the dynamic stiffness method and experimentrdquoInternational Journal of Solids and Structures vol 44 no 22-23pp 7543ndash7563 2007

[22] H Su J R Banerjee and C W Cheung ldquoDynamic stiffnessformulation and free vibration analysis of functionally gradedbeamsrdquo Composite Structures vol 106 pp 854ndash862 2013

[23] A Pagani M Boscolo J R Banerjee and E Carrera ldquoExactdynamic stiffness elements based on one-dimensional higher-order theories for free vibration analysis of solid and thin-walled structuresrdquo Journal of Sound and Vibration vol 332 no23 pp 6104ndash6127 2013

[24] N El-Kaabazi and D Kennedy ldquoCalculation of natural fre-quencies and vibration modes of variable thickness cylindricalshells using the Wittrick-Williams algorithmrdquo Computers andStructures vol 104-105 pp 4ndash12 2012

[25] X Chen Research on dynamic stiffness method for free vibrationof shells of revolution [MS thesis] Tsinghua University 2009(Chinese)

[26] XChen andKYe ldquoFreeVibrationAnalysis for Shells of Revolu-tion Using an Exact Dynamic Stiffness Methodrdquo MathematicalProblems in Engineering vol 2016 Article ID 4513520 12 pages2016

[27] X Chen and K Ye ldquoAnalysis of free vibration of moderatelythick circular cylindrical shells using the dynamic stiffnessmethodrdquo Engineering Mechanics vol 33 no 9 pp 40ndash48 2016(Chinese)

[28] X M Zhang G R Liu and K Y Lam ldquoVibration analysisof thin cylindrical shells using wave propagation approachrdquoJournal of Sound and Vibration vol 239 no 3 pp 397ndash4032001

[29] L Xuebin ldquoStudy on free vibration analysis of circular cylin-drical shells using wave propagationrdquo Journal of Sound andVibration vol 311 no 3-5 pp 667ndash682 2008

[30] C B Sharma ldquoCalculation of natural frequencies of fixed-freecircular cylindrical shellsrdquo Journal of Sound and Vibration vol35 no 1 pp 55ndash76 1974

[31] A E ArmenakasDCGazis andGHerrmannFreeVibrationsof Circular Cylindrical Shells PergamonPressOxfordUK 1969

[32] A Bhimaraddi ldquoA higher order theory for free vibrationanalysis of circular cylindrical shellsrdquo International Journal ofSolids and Structures vol 20 no 7 pp 623ndash630 1984

[33] U Ascher J Christiansen and R D Russell ldquoAlgorithm 569COLSYS collocation software for boundary value ODEs [D2]rdquoACM Transactions on Mathematical Software vol 7 no 2 pp223ndash229 1981

[34] U Ascher J Christiansen and R D Russell ldquoCollocationsoftware for boundary-value ODEsrdquo ACM Transactions onMathematical Software vol 7 no 2 pp 209ndash222 1981

[35] S Markus The Mechanics of Vibrations of Cylindrical ShellsElsevier Science Limited 1988

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of