research article transverse vibration of axially moving...

Research ArticleTransverse Vibration of Axially Moving Functionally GradedMaterials Based on Timoshenko Beam Theory

Suihan Sui Ling Chen Cheng Li and Xinpei Liu

School of Urban Rail Transportation Soochow University Suzhou 215006 China

Correspondence should be addressed to Cheng Li lichengsudaeducn

Received 16 August 2014 Accepted 18 October 2014

Academic Editor Kim M Liew

Copyright copy 2015 Suihan Sui et al 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

The transverse free vibration of an axially moving beam made of functionally graded materials (FGM) is investigated using aTimoshenko beam theory Natural frequencies vibrationmodes and critical speeds of such axially moving systems are determinedand discussed in detail Thematerial properties are assumed to vary continuously through the thickness of the beam according to apower law distribution Hamiltonrsquos principle is employed to derive the governing equation and a complexmode approach is utilizedto obtain the transverse dynamical behaviors including the vibration modes and natural frequencies Effects of the axially movingspeed and the power-law exponent on the dynamic responses are examined Some numerical examples are presented to reveal thedifferences of natural frequencies for Timoshenko beam model and Euler beam model Moreover the critical speed is determinednumerically to indicate its variation with respect to the power-law exponent axial initial stress and length to thickness ratio

1 Introduction

The axially moving systems are extensively applied inmachinery electronics and some other related fields Manyengineering devices such as band-saw blade power transmis-sion belts and crane hoist cables can be modeled as axiallymoving systems Transverse vibration of axially moving sys-tems may induce some disadvantageous effects For examplethe transverse vibration of power transmission belts causesnoise and accelerated wear of the belt the vibration of theblade in band saws leads to poor cutting quality Thereforeresearch of the transverse vibration of axially moving beamsis indispensable which has important engineering signifi-cance in controlling and optimizing the transverse dynamicsAt present the axially moving beam structures have beenwidely studied of which most literatures are based on Euler-Bernoulli beam model Oz [1] computed natural frequenciesof an axially moving beam in contact with a small stationarymass under pinned-pinned or clamped-clamped boundaryconditions Chen et al [2] studied the dynamic stability ofan axially accelerating viscoelastic beam and analyzed theeffects of the dynamic viscosity the mean axial speed andthe tension on the stability conditions Chen and Yang [3]

developed two nonlinear models for transverse vibration ofan axially accelerating viscoelastic beam and applied themethod of multiple scales to compare the correspondingsteady-state responses and their stability Chen and Yang [4]presented the first two mode frequencies of axially movingelastic and viscoelastic beams under simple supports withtorsion springs Lee and Jang [5] studied the effects ofthe continuously incoming and outgoing semi-infinite beamparts on the dynamic characteristics and stability of an axiallymoving beam by using the spectral element method Linand Qiao [6] determined some numerical results for naturalfrequency of an axially moving beam in fluid based on adifferential quadrature method Ghayesh et al [7] developedan approximate analytical solution for nonlinear dynamicresponses of a simply supported Kelvin-Voigt viscoelasticbeam with an attached heavy intraspan mass Lv et al [8]investigated natural frequency bifurcation and stability oftransverse vibration of axially accelerating moving viscoelas-tic sandwich beams with time-dependent axial tension

All the cases aforementioned are in the framework ofEuler-Bernoulli beam model However it does not containthe information about shear stress and moment of inertia inEuler-Bernoulli beammodel Consequently the Timoshenko

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 391452 9 pageshttpdxdoiorg1011552015391452

2 Mathematical Problems in Engineering

beam model has received more attention in modeling theaxially moving structures in recent years Lee et al [9]formulated the spectral element model for the transversevibration of an axially moving Timoshenko beam subjectedto a constant axial tension and verify its high accuracy bycomparing with the solutions by other methods Tang et al[10] analyzed the parametric resonance of axially movingTimoshenko beams with time-dependent speed Ghayeshand Balar [11] presented nonlinear vibration and stabilityanalysis of an axially moving Timoshenko beam for twodynamic models and employed the multiple scales methodto obtain the mode shape equations natural frequenciesand steady-state responses of the system Li et al [12]studied the nonlinear free transverse vibrations of the axiallymoving Timoshenko beam with constant speed Ghayeshand Amabili [13] investigated the nonlinear forced vibrationsand stability of an axially moving Timoshenko beam withan intraspan spring-support All the literatures using axiallymoving Timoshenko beam model receive more accurateresults and it is applicable to the stubby axially movingstructures

In engineering practices the axially moving beammay bemade by FGMThepresent study is concernedwith the axiallymoving FGM Timoshenko beam In the related literaturesmost are focused on the mechanical properties of FGMwithout axial motion but the studies of axially moving FGMTimoshenko beam are very few Ding et al [14] derived aseries of analytical solutions for anisotropic FGM beams withvarious end conditions using an Airy stress function in thegeneralized polynomial form Kang and Li [15] investigatedthe mechanical behaviors of a nonlinear FGM cantileverbeam subjected to an end force by using large and smalldeformation theories Zhou et al [16] studied the mechanicalresponses of a functionally graded cantilever beam by useof two kinds of particle with different properties based ondiscrete element method Simsek [17] investigated the forcedvibration of a functionally graded beam under a movingmass by using Euler-Bernoulli Timoshenko and the third-order shear deformation beam theories Alshorbagy et al[18] studied the free vibration of a functionally graded beamby finite element method with material graduation axiallyor transversally through the thickness Ke et al [19] investi-gated nonlinear free vibration of size-dependent functionallygraded microbeams based on the modified couple stresstheory and von Karman geometric nonlinearity Simsek etal [20] analytically examined static bending of functionallygraded microbeams based on the modified couple stresstheory and Euler-Bernoulli and Timoshenko beam theoriesrespectively

Considering the wide application of axially movingstructures and the excellent characteristics of FGM theinvestigation of transverse vibration of axially moving beamsmade of FGM is of great significance both in theoretical studyand engineering application In this paper the transversevibration of an axiallymoving initially tensioned beammadeof FGM is investigated The Timoshenko beam model isutilized and the complex mode approach is performed toobtain the natural frequencies and themodal functionsWiththe numerical example for boundary condition of simply

z

L

hb

x1205900

1205900o

Figure 1 Schematics of an axially moving FGM beam

supported on both ends the effects of power-law exponentaxial speed and initial stress to the natural frequencies areanalyzed and discussed in detail Finally the critical speedis determined to show its variation with different power-lawexponent axial initial stress and length to thickness ratio

2 Physical Model

For an axially moving FGM beam as shown in Figure 1 thebeam travels at a speed V between two boundaries separatedby distance 119871 under an applied axial initial stress 120590

0 The

cross section is a rectangle of thickness ℎ and width 119887 Thecoordinate system (119874119909119911) is defined on the middle plane ofthe beamNote that the 119909-axis is taken along themiddle planeand the 119911-axis in the thickness direction

The elastic modulus 119864 (Nm2) and density 120588 (kgm3) ofthe beamare assumed to vary through the thickness followinga simple power-law distribution They can be described by

119864 (119911) = (119864119888minus 119864119898) (

119911

ℎ+1

2)

119896

+ 119864119898

120588 (119911) = (120588119888minus 120588119898) (

119911

ℎ+1

2)

119896

+ 120588119898

minusℎ

2le 119911 le

ℎ

2

(1)

where 119896 stands for the power-law exponent As the FGMbeam is assumed to be made of pure Alumina ceramics andpure steel metal subscripts 119888 and 119898 refer to the ceramic(119864119888= 390GPa 120588

119888= 3960 kgm3) and metal (119864

119898= 210GPa

120588119898= 7800 kgm3) respectively

3 Governing Equation andTheoretical Formulation

Based on the Timoshenko beam theory the total transversedeflection and the angle of rotation due to bending aredenoted by 119908(119909 119905) and 120595(119909 119905) respectively The kineticenergy 119879 and potential energies 119881 are given by

119879 =1

2int

119871

0

1205881198790[V2 + (

120597119908

120597119905+ V

120597119908

120597119909)

2

]

+1205881198792(120597120595

120597119905+ V

120597120595

120597119909)

2

119889119909

119881 =1

2int

119871

0

(119864119868)eq (120597120595

120597119909)

2

+ 120581119866119879(120597119908

120597119909minus 120595)

2

+1205900119860(

120597119908

120597119909)

2

119889119909

(2)

Mathematical Problems in Engineering 3

where

1205881198790

= int119860

120588 (119911) 119889119860 1205881198792

= int119860

120588 (119911) 1199112119889119860

119866119879= int119860

119866 (119911) 119889119860 (119864119868)eq = int119860

119864 (119911) 1199112119889119860

119866 (119911) =119864 (119911)

2 (1 + 120583)

(3)

in which 120583 is the Poissonrsquos ratio and 120581 is the shear correctionfactor

Substitute (2) into the extended Hamiltonrsquos principle

int

1199052

1199051

(120575119879 minus 120575119881) 119889119905 = 0 (4)

The detail is shown in Appendix A from which the equationsofmotion for the axiallymoving FGMTimoshenko beam canbe derived as

120581119866119879(1205972119908

1205971199092minus120597120595

120597119909) minus 1205881198790(1205972119908

1205971199052+ 2V

1205972119908

120597119905120597119909+ V2

1205972119908

1205971199092)

+ 12059001198601205972119908

1205971199092= 0

(5)

(119864119868)eq1205972120595

1205971199092minus 1205881198792(1205972120595

1205971199052+ 2V

1205972120595

120597119905120597119909+ V2

1205972120595

1205971199092)

+ 120581119866119879(120597119908

120597119909minus 120595) = 0

(6)

Decoupling (5) and (6) yields the governing equationfor the transverse vibration of such axially moving FGMTimoshenko beams

(1 minus 1198862V2) (1 + 119886

4minus 1198861V2)

1205974119908

1205971199094

minus 2V (1198861+ 1198862+ 11988621198864minus 211988611198862V2)

1205974119908

1205971199051205971199093

+ 1198863(1198861V2 minus 119886

4)1205972119908

1205971199092minus (1198861+ 1198862+ 11988621198864minus 611988611198862V2)

times1205974119908

12059711990521205971199092+ 411988611198862V1205974119908

1205971199053120597119909+ 211988611198863V1205972119908

120597119905120597119909

+ 11988611198863

1205972119908

1205971199052+ 11988611198862

1205972119908

1205971199054= 0

(7)

where

1198861=

1205881198790

120581119866119879

1198862=

1205881198792

(119864119868)eq

1198863=

120581119866119879

(119864119868)eq 119886

4=1205900119860

120581119866119879

(8)

4 Case Studies

41 Natural Frequencies and Mode Function The boundaryconditions for the simple supports at both ends are given by

119908 (0 119905) = 0 119908 (119871 119905) = 0

1205972119908 (0 119905)

1205971199092= 0

1205972119908 (119871 119905)

1205971199092= 0

(9)

The solution to (7) can be assumed as

119908 (119909 119905) = 120593119899(119909) 119890119894120596119899119905+ 120593119899(119909) 119890minus119894120596119899119905 (10)

where 120593119899and 120596

119899denote the 119899th mode function and natural

frequency respectively 120593119899represents the complex conjugate

of 120593119899 Substitution of (10) into (7) and (9) leads to

(1 minus 1198862V2) (1 + 119886

4minus 1198861V2) 120593(4)119899

minus 2119894120596119899V (1198861+ 1198862+ 11988621198864minus 211988611198862V2) 120593(3)119899

+ [1198863(1198861V2 minus 119886

4) + 1205962

119899(1198861+ 1198862+ 11988621198864minus 611988611198862V2)] 12059310158401015840

119899

minus 21198941198861V120596119899(211988621205962

119899minus 1198863) 1205931015840

119899minus 11988611205962

119899(1198863minus 11988621205962

119899) 120593119899= 0

(11)

120593119899(0) = 0 120593

119899(119871) = 0

12059310158401015840

119899(0) = 0 120593

10158401015840

119899(119871) = 0

(12)

where 120593(4)

119899 120593(3)119899 12059310158401015840119899 and 120593

1015840

119899denote the derivative with

respect to coordinate 119909 The solution to ordinary differentialequation (11) can be expressed by

120593119899(119909) = 119862

11198991198901205731119899119909+ 11986221198991198901205732119899119909+ 11986231198991198901205733119899119909+ 11986241198991198901205734119899119909 (13)

where 1198621119899sim1198624119899are four unknown constants and 120573

1119899sim1205734119899are

four complex eigenvalues of (11) Substituting (13) into (12)yields

1198621119899+ 1198622119899+ 1198623119899+ 1198624119899= 0 (14a)

11986211198991198901205731119899 + 119862

21198991198901205732119899 + 119862

31198991198901205733119899 + 119862

41198991198901205734119899 = 0 (14b)

11986211198991205732

1119899+ 11986221198991205732

2119899+ 11986231198991205732

3119899+ 11986241198991205732

4119899= 0 (14c)

11986211198991205732

11198991198901205731119899 + 119862

21198991205732

21198991198901205732119899 + 119862

31198991205732

31198991198901205733119899 + 119862

41198991205732

41198991198901205734119899 = 0

(14d)

Rewrite (14a) (14b) (14c) and (14d) in the form of amatrix as

(

1 1 1 1

1198901205731119899 119890

1205732119899 119890

1205733119899 119890

1205734119899

1205732

11198991205732

21198991205732

31198991205732

4119899

1205732

11198991198901205731119899 1205732

21198991198901205732119899 1205732

31198991198901205733119899 1205732

41198991198901205734119899

)(

1198621119899

1198622119899

1198623119899

1198624119899

) = 0 (15)

For the nontrivial solution of (15) the determinant of thecoefficient matrix must be zero


0 50 100 150 200

500

600

700

800

900

1000

1100

The first natural frequency

k = 1

k = 025

k = 01

k = 001

(ms)

1205961

(Hz)

(a)

0 100 200 300 400

2500

3000

3500

4000

4500The second natural frequency

k = 1

k = 025

k = 01

k = 001

(ms)

1205962

(Hz)

(b)

0 100 200 300 400

6500

7500

8500

9500

10500The third natural frequency

k = 1

k = 025

k = 01

k = 001

(ms)

1205963

(Hz)

(c)

0 100 200 300 400

12

13

14

15

16

17

18The fourth natural frequency

k = 1

k = 025

k = 01

k = 001

(ms)

1205964

(Hz)

times104

(d)

Figure 2 The natural frequencies versus axially moving speeds for different power-law exponents (1205900= 1MPa)

Consider

[119890(1205731119899+1205732119899)+ 119890(1205733119899+1205734119899)] (1205732

1119899minus 1205732

2119899) (1205732

3119899minus 1205732

4119899)

minus [119890(1205731119899+1205733119899)+ 119890(1205732119899+1205734119899)] (1205732

1119899minus 1205732

3119899) (1205732

2119899minus 1205732

4119899)

+ [119890(1205732119899+1205733119899)+ 119890(1205731119899+1205734119899)] (1205732

2119899minus 1205732

3119899) (1205732

1119899minus 1205732

4119899) = 0

(16)

Using (15) and (16) one can obtain the coefficients of (13)the expressions of 119862

2119899sim1198624119899are listed in Appendix B

The 119899th modal function of the axially moving FGM beamis determined as120593119899 (119909)

= 1198901205731119899119909minus

(1205732

4119899minus 1205732

1119899) (1198901205733119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

2119899) (1198901205733119899 minus 1198901205732119899)

1198901205732119899119909

minus

(1205732

4119899minus 1205732

1119899) (1198901205732119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

3119899) (1198901205732119899 minus 1198901205733119899)

1198901205733119899119909

+ (minus1 +

(1205732

4119899minus 1205732

1119899) (1198901205733119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

2119899) (1198901205733119899 minus 1198901205732119899)

+

(1205732

4119899minus 1205732

1119899) (1198901205732119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

3119899) (1198901205732119899 minus 1198901205733119899)

) 11989012057341198991199091198621119899

(17)


1205961

(Hz)


k

0 2 4 6 8 10 12400

500

600

700

800

900

1000

1100

1200

(a)

1205962

(Hz)

1205962

(Hz)

The second natural frequency

k

k

0 2 4 6 8 10 12

2500

3000

3500

4000

4500

4 6 8 10 12

2500

2600

2700

2800

2400

(b)

1205963

(Hz)

1205963

(Hz)

The third natural frequency

= 10ms = 100ms

= 150ms

k

k

0 2 4 6 8 10 12

6000

7000

8000

9000

10000

4 6 8 10 125600

5800

6000

6200

6300

(c)

1205964

(Hz)

times104times1041205964

(Hz)

The fourth natural frequency

k

k

0 2 4 6 8 10 12

1

11

12

13

14

15

16

17

18

4 6 8 10 121

102

104

106

108

11

= 10ms = 100ms

= 150ms

(d)

Figure 3 The natural frequencies versus power-law exponent for different axially moving speeds (1205900= 1MPa)

42 Critical Speeds The time-independent equilibrium formof the linear equation (7) can be written as

(1 minus 1198862V2) (1 + 119886

4minus 1198861V2)

1205974119908

1205971199094+ 1198863(1198861V2 minus 119886

4)1205972119908

1205971199092= 0

(18)

The characteristic equation of (18) is

(1 minus 1198862V2) (1 + 119886

4minus 1198861V2) 1199034 + 119886

3(1198861V2 minus 119886

4) 1199032= 0 (19)

When V is higher than 100ms numerical results demon-strate that (1minus119886

2V2)(1+119886

4minus1198861V2) and 119886

3(1198861V2minus1198864) are positive

numbers Consequently the solution to (19) is determined as

1199031= 1199032= 0 119903

3= 1199034= plusmn119894radic

1198863(1198861V2 minus 119886

4)

(1 minus 1198862V2) (1 + 119886

4minus 1198861V2)

(20)On the other hand the solution to (18) can be expressed

by119908 (119909) = 119862

1+ 1198622119909 + 119862

3cos (120573119909) + 119862

4sin (120573119909) (21)


x (m)

minus2

minus1

1205931

0 02 04 06 08 1

0

1

2

3

4

5

The first mode

(a)

x (m)

minus10

minus5

1205932

0 02 04 06 08 1

0

5

10

The second mode

(b)

x (m)

minus15

minus10

minus5

1205933

k = 1

k = 025

0 02 04 06 08 1

0

5

10

15

The third mode

(c)

x (m)

k = 1

k = 025

minus20

minus10

1205934

0 02 04 06 08 1

0

10

20

The fourth mode

(d)

Figure 4 The first four modal functions (V = 100ms 1205900= 1MPa)

where

120573 = radic1198863(1198861V2 minus 119886

4)

(1 minus 1198862V2) (1 + 119886

4minus 1198861V2)

(22)

Substituting (21) into the boundary condition of (9) yields

(

1 0 1 0

1 119871 cos (120573119871) sin (120573119871)0 0 minus120573

20

0 0 minus1205732 cos (120573119871) minus120573

2 sin (120573119871)

)(

1198621

1198622

1198623

1198624

) = 0 (23)

For the nontrivial solution of (23) the determinant of thecoefficient matrix must be zero which results in

1198711205734 sin (119871120573) = 0 (24)

Using (22) and (24) as well as the conditions120573119871 = 119899120587 and119899 = 1 one can get the critical speeds of axially moving FGMbeam as follows

V119888= radic

119872 minus119873

2120587211988611198862

(25)

or

V119888= radic

119872 +119873

2120587211988611198862

(26)

where119872 = 120587

2(1198861+ 1198862+ 11988621198864) + 119871211988611198863

119873 = ((1198861(1205872+ 11987121198863) + 12058721198862(1 + 119886

4))2

minus 4120587211988611198862(1205872+ 12058721198864+ 119871211988631198864) )

12

(27)


0 2 4 6 8 10 12200

250

300

350

400

c(m

s)

k

1205900 = 100MPa1205900 = 50MPa

1205900 = 1MPa

Figure 5 The critical speeds versus power-law exponent for differ-ent initial stress

10 13 16 19 22 25200

350

500

650

800

900

c(m

s)

Lh

k = 01

k = 1

k = 12

Figure 6The critical speeds versus119871ℎ ratio for different power-lawexponents (120590

0= 1MPa)

Combined with numerical results in Section 43 it can beverified that the critical speed can be determined by equation(25) only

43 Numerical Results and Discussions In the followingnumerical computation the beam is made of steel metal andAlumina ceramics several parameters are chosen to be

119871 = 1m ℎ = 004m 119887 = 004m 120583 = 03

120581 =5

6

(28)

Table 1 Effect of initial stress on first four frequencies for FGMbeam (V = 100ms)

119896 1205900(MPa) 120596

1(Hz) 120596

2(Hz) 120596

3(Hz) 120596

4(Hz)

0001

1 1075 4436 9900 1732250 1129 4490 9955 17377100 1182 4545 10010 17433

Increase rate 995 246 111 064

1

1 738 3170 7109 1245550 790 3221 7160 12506100 841 3272 7211 12559


100

1 496 2304 5209 914750 553 2357 5262 9200100 605 2409 5315 9254


Natural frequencies are found by numerically solving(11) and (16) simultaneously with different speeds differentpower-law exponents and different initial stresses Figure 2illustrates the effect of axial speed on the first four naturalfrequencies for different power-law exponents As observedwith the increase of axiallymoving speed natural frequenciesdecrease Increasing the power-law exponent on the otherhand causes a decrease in natural frequencies Figure 3illustrates the effect of power-law exponents on the first fournatural frequencies under different axially moving speeds Asobserved with the increase of power-law exponent naturalfrequencies decrease and the decrease becomes more andmore gentle

In Figures 2 and 3 the solid lines and dashed linesdenote the Timoshenko model and the Euler beam modelrespectively The frequencies are lower with Timoshenkomodel comparing with the Euler model as other parametersare fixed Besides the numerical results demonstrate that thedifference of the natural frequencies with the two modelsbecomes more significant as the order increases

The frequencies of a FGM beam always remain betweenthe frequencies of its constituent material of the beam thatis when 119896 is small the content of ceramic in the beam ishigh then the natural frequencies of FGM beams are close tothe natural frequencies of the pure ceramic beam When 119896 islarge the content ofmetal in the beam is high and the naturalfrequencies of FGM beams approach the natural frequenciesof the pure metal beam

Table 1 shows the first four natural frequencies with dif-ferent 120590

0and different 119896 It is seen that natural frequencies are

increasing with increasing 1205900and decreasing with an increase

in 119896Moreover with the increase of 119896 the increase rate of eachorder raises to some extent and the first frequencies raise themost

After obtaining the natural frequencies of the system thesolution 120573

1119899sim1205734119899of (11) can be obtained Subsequently using

(17) one can get the modal function of the axially movingFGM beam Figure 4 presents the first four modal functionsin which the solid lines and dashed lines denote the real and


imaginary parts of the modal functions respectively whereV = 100ms and 120590

0= 1MPa are adopted

With an increase of axially moving speed each order ofthe frequency tends to vanish The exact values at which thefirst natural frequency vanishes are called the critical speedsIf axially moving speed is higher than the critical speed thesystem is unstable Figure 5 illustrates the effect of 119896 on thecritical speeds for different initial stressThenumerical resultsindicate that the critical speeds of the beam decrease with theincreasing 119896 and the decreasing 120590

0 For a specific initial stress

for example 1205900= 1MPa with an increase of 119896 the critical

speeds begin to decrease rapidly and then gently The effectsof the 119871ℎ ratio on the critical speeds for different 119896 are shownin Figure 6 The larger 119871ℎ ratio leads to lower critical speedsfor given 119896 and the higher the critical speeds the smaller the119896 for given 119871ℎ ratio

5 Conclusions

This work is devoted to the transverse dynamic responses ofaxially moving initially tensioned Timoshenko beams madeof FGMThe complex mode approach is performed to obtainthe natural frequencies and modal functions respectivelyThe effects of some parameters including axially movingspeed and the power-law exponent on the natural frequenciesare investigated Some numerical examples are presentedto demonstrate the comparisons of natural frequencies forTimoshenko beammodel and Euler beammodelThe criticalspeeds are determined and numerically investigated Theresults show that an increase of the power-law exponent orthe axial speed results in a lower natural frequency while theaxial initial stress tends to increase the natural frequenciesWith the increase of the power-law exponent or the 119871ℎ ratiothe critical speeds decrease while the axial initial stress tendsto increase the critical speeds The results reported in thiswork could be useful for designing and optimizing the axiallymoving FGM Timoshenko beam-like structures

Appendices

A Result of Substituting (2) into (4)Leads to

Consider

int

1199052

1199051

int

119871

0

[ 120581119866119879(1205972119908

1205971199092minus120597120595

120597119909)

minus 1205881198790(1205972119908

1205971199052+ 2V

1205972119908

120597119905120597119909+ V2

1205972119908

1205971199092)

+12059001198601205972119908

1205971199092] 120575119908119889119909119889119905

+ int

1199052

1199051

int

119871

0

[(119864119868)eq1205972120595

1205971199092minus 1205881198792(1205972120595

1205971199052+ 2V

1205972120595

120597119905120597119909+ V2

1205972120595

1205971199092)

+120581119866119879(120597119908

120597119909minus 120595)]120575120595119889119909119889119905

+ 1205881198790int

1199052

1199051

int (V120597119908

120597119905120575119908 + V2

120597119908

120597119909120575119908)

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905

+ 1205881198790int

119871

0

(120597119908

120597119905120575119908 + V

120597119908

120597119909120575119908)

10038161003816100381610038161003816100381610038161003816

1199052

1199051

119889119909

+ 1205881198792int

1199052

1199051

(V120597120595

120597119905120575120595 + V2

120597120595

120597119909120575120595)

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905

+ 1205881198792int

119871

0

(120597120595

120597119905120575120595 + V

120597120595

120597119909120575120595)

10038161003816100381610038161003816100381610038161003816

1199052

1199051

119889119909

minus 120581119866119879int

1199052

1199051

(120597119908

120597119909120575119908 + 120595120575119908)

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905 minus 1205900119860int

1199052

1199051

120597119908

120597119909120575119908

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905

minus (119864119868)eq int1199052

1199051

120597120595

120597119909120575120595

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905 = 0

(A1)

B The Coefficient of (13)

Consider

1198622119899= minus

(1205732

4119899minus 1205732

1119899) (1198901205733119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

2119899) (1198901205733119899 minus 1198901205732119899)

1198621119899

1198623119899= minus

(1205732

4119899minus 1205732

1119899) (1198901205732119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

3119899) (1198901205732119899 minus 1198901205733119899)

1198621119899

1198624119899= [minus1 +

(1205732

4119899minus 1205732

1119899) (1198901205733119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

2119899) (1198901205733119899 minus 1198901205732119899)

+

(1205732

4119899minus 1205732

1119899) (1198901205732119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

3119899) (1198901205732119899 minus 1198901205733119899)

]1198621119899

(B1)

Conflict of Interests

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

Acknowledgments

This work was supported by National Natural Science Foun-dation of China (nos 11202145 51405321) the Jiangsu Provin-cial Natural Science Foundation of China (no BK2012175)and the Natural Science Foundation of the Jiangsu HigherEducation Institutions of China (no 14KJB460026)

References

[1] H R Oz ldquoNatural frequencies of axially travelling tensionedbeams in contact with a stationary massrdquo Journal of Sound andVibration vol 259 no 2 pp 445ndash456 2003

[2] L-Q Chen X-D Yang and C-J Cheng ldquoDynamic stability ofan axially accelerating viscoelastic beamrdquo European Journal ofMechanicsmdashASolids vol 23 no 4 pp 659ndash666 2004


[3] L Q Chen and X D Yang ldquoSteady-state response of axiallymoving viscoelastic beams with pulsating speed comparisonof two nonlinear modelsrdquo International Journal of Solids andStructures vol 42 no 1 pp 37ndash50 2005

[4] L-Q Chen andX-D Yang ldquoVibration and stability of an axiallymoving viscoelastic beam with hybrid supportsrdquo EuropeanJournal ofMechanics A Solids vol 25 no 6 pp 996ndash1008 2006

[5] U Lee and I Jang ldquoOn the boundary conditions for axiallymoving beamsrdquo Journal of Sound and Vibration vol 306 no3-5 pp 675ndash690 2007

[6] W Lin andNQiao ldquoVibration and stability of an axiallymovingbeam immersed in fluidrdquo International Journal of Solids andStructures vol 45 no 5 pp 1445ndash1457 2008

[7] M H Ghayesh F Alijani and M A Darabi ldquoAn analyticalsolution for nonlinear dynamics of a viscoelastic beam-heavymass systemrdquo Journal ofMechanical Science and Technology vol25 no 8 pp 1915ndash1923 2011

[8] H Lv Y Li L Li and Q Liu ldquoTransverse vibration ofviscoelastic sandwich beam with time-dependent axial tensionand axially varying moving velocityrdquo Applied MathematicalModelling vol 38 no 9-10 pp 2558ndash2585 2014

[9] U Lee J Kim and H Oh ldquoSpectral analysis for the transversevibration of an axially moving Timoshenko beamrdquo Journal ofSound and Vibration vol 271 no 3-4 pp 685ndash703 2004

[10] Y-Q Tang L-Q Chen and X-D Yang ldquoParametric resonanceof axially moving Timoshenko beams with time-dependentspeedrdquo Nonlinear Dynamics vol 58 no 4 pp 715ndash724 2009

[11] M H Ghayesh and S Balar ldquoNon-linear parametric vibrationand stability analysis for two dynamic models of axially movingTimoshenko beamsrdquo Applied Mathematical Modelling vol 34no 10 pp 2850ndash2859 2010

[12] B Li Y Tang and L Chen ldquoNonlinear free transverse vibra-tions of axially moving Timoshenko beams with two free endsrdquoScience China Technological Sciences vol 54 no 8 pp 1966ndash1976 2011

[13] M H Ghayesh and M Amabili ldquoNonlinear vibrations andstability of an axially moving Timoshenko beam with anintermediate spring supportrdquoMechanism and Machine Theoryvol 67 pp 1ndash16 2013

[14] H J Ding D J Huang and W Q Chen ldquoElasticity solutionsfor plane anisotropic functionally graded beamsrdquo InternationalJournal of Solids and Structures vol 44 no 1 pp 176ndash196 2007

[15] Y-A Kang and X-F Li ldquoBending of functionally gradedcantilever beam with power-law non-linearity subjected to anend forcerdquo International Journal of Non-Linear Mechanics vol44 no 6 pp 696ndash703 2009

[16] F-X Zhou S-R Li Y-M Lai and Y Yang ldquoAnalyzing offunctionally graded materials by discrete element methodrdquoApplied Mechanics and Materials vol 29ndash32 pp 1948ndash19532010

[17] M Simsek ldquoVibration analysis of a functionally graded beamunder a moving mass by using different beam theoriesrdquo Com-posite Structures vol 92 no 4 pp 904ndash917 2010

[18] A E Alshorbagy M A Eltaher and F F Mahmoud ldquoFreevibration characteristics of a functionally graded beam by finiteelement methodrdquo Applied Mathematical Modelling vol 35 no1 pp 412ndash425 2011

[19] L L Ke Y SWang J Yang and S Kitipornchai ldquoNonlinear freevibration of size-dependent functionally graded microbeamsrdquoInternational Journal of Engineering Science vol 50 no 1 pp256ndash267 2012

[20] M Simsek T Kocaturk and S D Akbas ldquoStatic bending of afunctionally gradedmicroscale Timoshenko beam based on themodified couple stress theoryrdquoComposite Structures vol 95 pp740ndash747 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


beam model has received more attention in modeling theaxially moving structures in recent years Lee et al [9]formulated the spectral element model for the transversevibration of an axially moving Timoshenko beam subjectedto a constant axial tension and verify its high accuracy bycomparing with the solutions by other methods Tang et al[10] analyzed the parametric resonance of axially movingTimoshenko beams with time-dependent speed Ghayeshand Balar [11] presented nonlinear vibration and stabilityanalysis of an axially moving Timoshenko beam for twodynamic models and employed the multiple scales methodto obtain the mode shape equations natural frequenciesand steady-state responses of the system Li et al [12]studied the nonlinear free transverse vibrations of the axiallymoving Timoshenko beam with constant speed Ghayeshand Amabili [13] investigated the nonlinear forced vibrationsand stability of an axially moving Timoshenko beam withan intraspan spring-support All the literatures using axiallymoving Timoshenko beam model receive more accurateresults and it is applicable to the stubby axially movingstructures

In engineering practices the axially moving beammay bemade by FGMThepresent study is concernedwith the axiallymoving FGM Timoshenko beam In the related literaturesmost are focused on the mechanical properties of FGMwithout axial motion but the studies of axially moving FGMTimoshenko beam are very few Ding et al [14] derived aseries of analytical solutions for anisotropic FGM beams withvarious end conditions using an Airy stress function in thegeneralized polynomial form Kang and Li [15] investigatedthe mechanical behaviors of a nonlinear FGM cantileverbeam subjected to an end force by using large and smalldeformation theories Zhou et al [16] studied the mechanicalresponses of a functionally graded cantilever beam by useof two kinds of particle with different properties based ondiscrete element method Simsek [17] investigated the forcedvibration of a functionally graded beam under a movingmass by using Euler-Bernoulli Timoshenko and the third-order shear deformation beam theories Alshorbagy et al[18] studied the free vibration of a functionally graded beamby finite element method with material graduation axiallyor transversally through the thickness Ke et al [19] investi-gated nonlinear free vibration of size-dependent functionallygraded microbeams based on the modified couple stresstheory and von Karman geometric nonlinearity Simsek etal [20] analytically examined static bending of functionallygraded microbeams based on the modified couple stresstheory and Euler-Bernoulli and Timoshenko beam theoriesrespectively

Considering the wide application of axially movingstructures and the excellent characteristics of FGM theinvestigation of transverse vibration of axially moving beamsmade of FGM is of great significance both in theoretical studyand engineering application In this paper the transversevibration of an axiallymoving initially tensioned beammadeof FGM is investigated The Timoshenko beam model isutilized and the complex mode approach is performed toobtain the natural frequencies and themodal functionsWiththe numerical example for boundary condition of simply

z

L

hb

x1205900

1205900o

Figure 1 Schematics of an axially moving FGM beam

supported on both ends the effects of power-law exponentaxial speed and initial stress to the natural frequencies areanalyzed and discussed in detail Finally the critical speedis determined to show its variation with different power-lawexponent axial initial stress and length to thickness ratio

2 Physical Model

For an axially moving FGM beam as shown in Figure 1 thebeam travels at a speed V between two boundaries separatedby distance 119871 under an applied axial initial stress 120590

0 The

cross section is a rectangle of thickness ℎ and width 119887 Thecoordinate system (119874119909119911) is defined on the middle plane ofthe beamNote that the 119909-axis is taken along themiddle planeand the 119911-axis in the thickness direction

The elastic modulus 119864 (Nm2) and density 120588 (kgm3) ofthe beamare assumed to vary through the thickness followinga simple power-law distribution They can be described by

119864 (119911) = (119864119888minus 119864119898) (

119911

ℎ+1

2)

119896

+ 119864119898

120588 (119911) = (120588119888minus 120588119898) (

119911

ℎ+1

2)

119896

+ 120588119898

minusℎ

2le 119911 le

ℎ

2

(1)

where 119896 stands for the power-law exponent As the FGMbeam is assumed to be made of pure Alumina ceramics andpure steel metal subscripts 119888 and 119898 refer to the ceramic(119864119888= 390GPa 120588

119888= 3960 kgm3) and metal (119864

119898= 210GPa

120588119898= 7800 kgm3) respectively

3 Governing Equation andTheoretical Formulation

Based on the Timoshenko beam theory the total transversedeflection and the angle of rotation due to bending aredenoted by 119908(119909 119905) and 120595(119909 119905) respectively The kineticenergy 119879 and potential energies 119881 are given by

119879 =1

2int

119871

0

1205881198790[V2 + (

120597119908

120597119905+ V

120597119908

120597119909)

2

]

+1205881198792(120597120595

120597119905+ V

120597120595

120597119909)

2

119889119909

119881 =1

2int

119871

0

(119864119868)eq (120597120595

120597119909)

2

+ 120581119866119879(120597119908

120597119909minus 120595)

2

+1205900119860(

120597119908

120597119909)

2

119889119909

(2)


where

1205881198790

= int119860

120588 (119911) 119889119860 1205881198792

= int119860

120588 (119911) 1199112119889119860

119866119879= int119860

119866 (119911) 119889119860 (119864119868)eq = int119860

119864 (119911) 1199112119889119860

119866 (119911) =119864 (119911)

2 (1 + 120583)

(3)



int

1199052

1199051

(120575119879 minus 120575119881) 119889119905 = 0 (4)


120581119866119879(1205972119908

1205971199092minus120597120595

120597119909) minus 1205881198790(1205972119908

1205971199052+ 2V

1205972119908

120597119905120597119909+ V2

1205972119908

1205971199092)

+ 12059001198601205972119908

1205971199092= 0

(5)

(119864119868)eq1205972120595

1205971199092minus 1205881198792(1205972120595

1205971199052+ 2V

1205972120595

120597119905120597119909+ V2

1205972120595

1205971199092)

+ 120581119866119879(120597119908

120597119909minus 120595) = 0

(6)


(1 minus 1198862V2) (1 + 119886

4minus 1198861V2)

1205974119908

1205971199094

minus 2V (1198861+ 1198862+ 11988621198864minus 211988611198862V2)

1205974119908

1205971199051205971199093

+ 1198863(1198861V2 minus 119886

4)1205972119908

1205971199092minus (1198861+ 1198862+ 11988621198864minus 611988611198862V2)

times1205974119908

12059711990521205971199092+ 411988611198862V1205974119908

1205971199053120597119909+ 211988611198863V1205972119908

120597119905120597119909

+ 11988611198863

1205972119908

1205971199052+ 11988611198862

1205972119908

1205971199054= 0

(7)

where

1198861=

1205881198790

120581119866119879

1198862=

1205881198792

(119864119868)eq

1198863=

120581119866119879

(119864119868)eq 119886

4=1205900119860

120581119866119879

(8)

4 Case Studies


119908 (0 119905) = 0 119908 (119871 119905) = 0

1205972119908 (0 119905)

1205971199092= 0

1205972119908 (119871 119905)

1205971199092= 0

(9)


119908 (119909 119905) = 120593119899(119909) 119890119894120596119899119905+ 120593119899(119909) 119890minus119894120596119899119905 (10)

where 120593119899and 120596




(1 minus 1198862V2) (1 + 119886

4minus 1198861V2) 120593(4)119899

minus 2119894120596119899V (1198861+ 1198862+ 11988621198864minus 211988611198862V2) 120593(3)119899

+ [1198863(1198861V2 minus 119886

4) + 1205962

119899(1198861+ 1198862+ 11988621198864minus 611988611198862V2)] 12059310158401015840

119899

minus 21198941198861V120596119899(211988621205962

119899minus 1198863) 1205931015840

119899minus 11988611205962

119899(1198863minus 11988621205962

119899) 120593119899= 0

(11)

120593119899(0) = 0 120593

119899(119871) = 0

12059310158401015840

119899(0) = 0 120593

10158401015840

119899(119871) = 0

(12)

where 120593(4)

119899 120593(3)119899 12059310158401015840119899 and 120593

1015840



120593119899(119909) = 119862

11198991198901205731119899119909+ 11986221198991198901205732119899119909+ 11986231198991198901205733119899119909+ 11986241198991198901205734119899119909 (13)


1119899sim1205734119899are


1198621119899+ 1198622119899+ 1198623119899+ 1198624119899= 0 (14a)

11986211198991198901205731119899 + 119862

21198991198901205732119899 + 119862

31198991198901205733119899 + 119862

41198991198901205734119899 = 0 (14b)

11986211198991205732

1119899+ 11986221198991205732

2119899+ 11986231198991205732

3119899+ 11986241198991205732

4119899= 0 (14c)

11986211198991205732

11198991198901205731119899 + 119862

21198991205732

21198991198901205732119899 + 119862

31198991205732

31198991198901205733119899 + 119862

41198991205732

41198991198901205734119899 = 0

(14d)


(

1 1 1 1

1198901205731119899 119890

1205732119899 119890

1205733119899 119890

1205734119899

1205732

11198991205732

21198991205732

31198991205732

4119899

1205732

11198991198901205731119899 1205732

21198991198901205732119899 1205732

31198991198901205733119899 1205732

41198991198901205734119899

)(

1198621119899

1198622119899

1198623119899

1198624119899

) = 0 (15)



0 50 100 150 200

500

600

700

800

900

1000

1100


k = 1

k = 025

k = 01

k = 001

(ms)

1205961

(Hz)

(a)

0 100 200 300 400

2500

3000

3500

4000


k = 1

k = 025

k = 01

k = 001

(ms)

1205962

(Hz)

(b)

0 100 200 300 400

6500

7500

8500

9500


k = 1

k = 025

k = 01

k = 001

(ms)

1205963

(Hz)

(c)

0 100 200 300 400

12

13

14

15

16

17


k = 1

k = 025

k = 01

k = 001

(ms)

1205964

(Hz)

times104

(d)


Consider

[119890(1205731119899+1205732119899)+ 119890(1205733119899+1205734119899)] (1205732

1119899minus 1205732

2119899) (1205732

3119899minus 1205732

4119899)

minus [119890(1205731119899+1205733119899)+ 119890(1205732119899+1205734119899)] (1205732

1119899minus 1205732

3119899) (1205732

2119899minus 1205732

4119899)

+ [119890(1205732119899+1205733119899)+ 119890(1205731119899+1205734119899)] (1205732

2119899minus 1205732

3119899) (1205732

1119899minus 1205732

4119899) = 0

(16)




= 1198901205731119899119909minus

(1205732

4119899minus 1205732

1119899) (1198901205733119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

2119899) (1198901205733119899 minus 1198901205732119899)

1198901205732119899119909

minus

(1205732

4119899minus 1205732

1119899) (1198901205732119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

3119899) (1198901205732119899 minus 1198901205733119899)

1198901205733119899119909

+ (minus1 +

(1205732

4119899minus 1205732

1119899) (1198901205733119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

2119899) (1198901205733119899 minus 1198901205732119899)

+

(1205732

4119899minus 1205732

1119899) (1198901205732119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

3119899) (1198901205732119899 minus 1198901205733119899)

) 11989012057341198991199091198621119899

(17)


1205961

(Hz)


k

0 2 4 6 8 10 12400

500

600

700

800

900

1000

1100

1200

(a)

1205962

(Hz)

1205962

(Hz)


k

k

0 2 4 6 8 10 12

2500

3000

3500

4000

4500

4 6 8 10 12

2500

2600

2700

2800

2400

(b)

1205963

(Hz)

1205963

(Hz)


= 10ms = 100ms

= 150ms

k

k

0 2 4 6 8 10 12

6000

7000

8000

9000

10000

4 6 8 10 125600

5800

6000

6200

6300

(c)

1205964

(Hz)


(Hz)


k

k

0 2 4 6 8 10 12

1

11

12

13

14

15

16

17

18

4 6 8 10 121

102

104

106

108

11

= 10ms = 100ms

= 150ms

(d)



(1 minus 1198862V2) (1 + 119886

4minus 1198861V2)

1205974119908

1205971199094+ 1198863(1198861V2 minus 119886

4)1205972119908

1205971199092= 0

(18)


(1 minus 1198862V2) (1 + 119886

4minus 1198861V2) 1199034 + 119886

3(1198861V2 minus 119886

4) 1199032= 0 (19)


2V2)(1+119886

4minus1198861V2) and 119886



1199031= 1199032= 0 119903

3= 1199034= plusmn119894radic

1198863(1198861V2 minus 119886

4)

(1 minus 1198862V2) (1 + 119886

4minus 1198861V2)


by119908 (119909) = 119862

1+ 1198622119909 + 119862

3cos (120573119909) + 119862

4sin (120573119909) (21)


x (m)

minus2

minus1

1205931

0 02 04 06 08 1

0

1

2

3

4

5

The first mode

(a)

x (m)

minus10

minus5

1205932

0 02 04 06 08 1

0

5

10

The second mode

(b)

x (m)

minus15

minus10

minus5

1205933

k = 1

k = 025

0 02 04 06 08 1

0

5

10

15

The third mode

(c)

x (m)

k = 1

k = 025

minus20

minus10

1205934

0 02 04 06 08 1

0

10

20

The fourth mode

(d)


where

120573 = radic1198863(1198861V2 minus 119886

4)

(1 minus 1198862V2) (1 + 119886

4minus 1198861V2)

(22)


(

1 0 1 0

1 119871 cos (120573119871) sin (120573119871)0 0 minus120573

20

0 0 minus1205732 cos (120573119871) minus120573

2 sin (120573119871)

)(

1198621

1198622

1198623

1198624

) = 0 (23)


1198711205734 sin (119871120573) = 0 (24)


V119888= radic

119872 minus119873

2120587211988611198862

(25)

or

V119888= radic

119872 +119873

2120587211988611198862

(26)

where119872 = 120587

2(1198861+ 1198862+ 11988621198864) + 119871211988611198863

119873 = ((1198861(1205872+ 11987121198863) + 12058721198862(1 + 119886

4))2

minus 4120587211988611198862(1205872+ 12058721198864+ 119871211988631198864) )

12

(27)


0 2 4 6 8 10 12200

250

300

350

400

c(m

s)

k

1205900 = 100MPa1205900 = 50MPa

1205900 = 1MPa


10 13 16 19 22 25200

350

500

650

800

900

c(m

s)

Lh

k = 01

k = 1

k = 12


0= 1MPa)



119871 = 1m ℎ = 004m 119887 = 004m 120583 = 03

120581 =5

6

(28)


119896 1205900(MPa) 120596

1(Hz) 120596

2(Hz) 120596

3(Hz) 120596

4(Hz)

0001

1 1075 4436 9900 1732250 1129 4490 9955 17377100 1182 4545 10010 17433


1

1 738 3170 7109 1245550 790 3221 7160 12506100 841 3272 7211 12559


100

1 496 2304 5209 914750 553 2357 5262 9200100 605 2409 5315 9254














0= 1MPa are adopted





5 Conclusions


Appendices


Consider

int

1199052

1199051

int

119871

0

[ 120581119866119879(1205972119908

1205971199092minus120597120595

120597119909)

minus 1205881198790(1205972119908

1205971199052+ 2V

1205972119908

120597119905120597119909+ V2

1205972119908

1205971199092)

+12059001198601205972119908

1205971199092] 120575119908119889119909119889119905

+ int

1199052

1199051

int

119871

0

[(119864119868)eq1205972120595

1205971199092minus 1205881198792(1205972120595

1205971199052+ 2V

1205972120595

120597119905120597119909+ V2

1205972120595

1205971199092)

+120581119866119879(120597119908

120597119909minus 120595)]120575120595119889119909119889119905

+ 1205881198790int

1199052

1199051

int (V120597119908

120597119905120575119908 + V2

120597119908

120597119909120575119908)

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905

+ 1205881198790int

119871

0

(120597119908

120597119905120575119908 + V

120597119908

120597119909120575119908)

10038161003816100381610038161003816100381610038161003816

1199052

1199051

119889119909

+ 1205881198792int

1199052

1199051

(V120597120595

120597119905120575120595 + V2

120597120595

120597119909120575120595)

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905

+ 1205881198792int

119871

0

(120597120595

120597119905120575120595 + V

120597120595

120597119909120575120595)

10038161003816100381610038161003816100381610038161003816

1199052

1199051

119889119909

minus 120581119866119879int

1199052

1199051

(120597119908

120597119909120575119908 + 120595120575119908)

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905 minus 1205900119860int

1199052

1199051

120597119908

120597119909120575119908

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905

minus (119864119868)eq int1199052

1199051

120597120595

120597119909120575120595

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905 = 0

(A1)


Consider

1198622119899= minus

(1205732

4119899minus 1205732

1119899) (1198901205733119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

2119899) (1198901205733119899 minus 1198901205732119899)

1198621119899

1198623119899= minus

(1205732

4119899minus 1205732

1119899) (1198901205732119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

3119899) (1198901205732119899 minus 1198901205733119899)

1198621119899

1198624119899= [minus1 +

(1205732

4119899minus 1205732

1119899) (1198901205733119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

2119899) (1198901205733119899 minus 1198901205732119899)

+

(1205732

4119899minus 1205732

1119899) (1198901205732119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

3119899) (1198901205732119899 minus 1198901205733119899)

]1198621119899

(B1)



Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

1205881198790

= int119860

120588 (119911) 119889119860 1205881198792

= int119860

120588 (119911) 1199112119889119860

119866119879= int119860

119866 (119911) 119889119860 (119864119868)eq = int119860

119864 (119911) 1199112119889119860

119866 (119911) =119864 (119911)

2 (1 + 120583)

(3)



int

1199052

1199051

(120575119879 minus 120575119881) 119889119905 = 0 (4)


120581119866119879(1205972119908

1205971199092minus120597120595

120597119909) minus 1205881198790(1205972119908

1205971199052+ 2V

1205972119908

120597119905120597119909+ V2

1205972119908

1205971199092)

+ 12059001198601205972119908

1205971199092= 0

(5)

(119864119868)eq1205972120595

1205971199092minus 1205881198792(1205972120595

1205971199052+ 2V

1205972120595

120597119905120597119909+ V2

1205972120595

1205971199092)

+ 120581119866119879(120597119908

120597119909minus 120595) = 0

(6)


(1 minus 1198862V2) (1 + 119886

4minus 1198861V2)

1205974119908

1205971199094

minus 2V (1198861+ 1198862+ 11988621198864minus 211988611198862V2)

1205974119908

1205971199051205971199093

+ 1198863(1198861V2 minus 119886

4)1205972119908

1205971199092minus (1198861+ 1198862+ 11988621198864minus 611988611198862V2)

times1205974119908

12059711990521205971199092+ 411988611198862V1205974119908

1205971199053120597119909+ 211988611198863V1205972119908

120597119905120597119909

+ 11988611198863

1205972119908

1205971199052+ 11988611198862

1205972119908

1205971199054= 0

(7)

where

1198861=

1205881198790

120581119866119879

1198862=

1205881198792

(119864119868)eq

1198863=

120581119866119879

(119864119868)eq 119886

4=1205900119860

120581119866119879

(8)

4 Case Studies


119908 (0 119905) = 0 119908 (119871 119905) = 0

1205972119908 (0 119905)

1205971199092= 0

1205972119908 (119871 119905)

1205971199092= 0

(9)


119908 (119909 119905) = 120593119899(119909) 119890119894120596119899119905+ 120593119899(119909) 119890minus119894120596119899119905 (10)

where 120593119899and 120596




(1 minus 1198862V2) (1 + 119886

4minus 1198861V2) 120593(4)119899

minus 2119894120596119899V (1198861+ 1198862+ 11988621198864minus 211988611198862V2) 120593(3)119899

+ [1198863(1198861V2 minus 119886

4) + 1205962

119899(1198861+ 1198862+ 11988621198864minus 611988611198862V2)] 12059310158401015840

119899

minus 21198941198861V120596119899(211988621205962

119899minus 1198863) 1205931015840

119899minus 11988611205962

119899(1198863minus 11988621205962

119899) 120593119899= 0

(11)

120593119899(0) = 0 120593

119899(119871) = 0

12059310158401015840

119899(0) = 0 120593

10158401015840

119899(119871) = 0

(12)

where 120593(4)

119899 120593(3)119899 12059310158401015840119899 and 120593

1015840



120593119899(119909) = 119862

11198991198901205731119899119909+ 11986221198991198901205732119899119909+ 11986231198991198901205733119899119909+ 11986241198991198901205734119899119909 (13)


1119899sim1205734119899are


1198621119899+ 1198622119899+ 1198623119899+ 1198624119899= 0 (14a)

11986211198991198901205731119899 + 119862

21198991198901205732119899 + 119862

31198991198901205733119899 + 119862

41198991198901205734119899 = 0 (14b)

11986211198991205732

1119899+ 11986221198991205732

2119899+ 11986231198991205732

3119899+ 11986241198991205732

4119899= 0 (14c)

11986211198991205732

11198991198901205731119899 + 119862

21198991205732

21198991198901205732119899 + 119862

31198991205732

31198991198901205733119899 + 119862

41198991205732

41198991198901205734119899 = 0

(14d)


(

1 1 1 1

1198901205731119899 119890

1205732119899 119890

1205733119899 119890

1205734119899

1205732

11198991205732

21198991205732

31198991205732

4119899

1205732

11198991198901205731119899 1205732

21198991198901205732119899 1205732

31198991198901205733119899 1205732

41198991198901205734119899

)(

1198621119899

1198622119899

1198623119899

1198624119899

) = 0 (15)



0 50 100 150 200

500

600

700

800

900

1000

1100


k = 1

k = 025

k = 01

k = 001

(ms)

1205961

(Hz)

(a)

0 100 200 300 400

2500

3000

3500

4000


k = 1

k = 025

k = 01

k = 001

(ms)

1205962

(Hz)

(b)

0 100 200 300 400

6500

7500

8500

9500


k = 1

k = 025

k = 01

k = 001

(ms)

1205963

(Hz)

(c)

0 100 200 300 400

12

13

14

15

16

17


k = 1

k = 025

k = 01

k = 001

(ms)

1205964

(Hz)

times104

(d)


Consider

[119890(1205731119899+1205732119899)+ 119890(1205733119899+1205734119899)] (1205732

1119899minus 1205732

2119899) (1205732

3119899minus 1205732

4119899)

minus [119890(1205731119899+1205733119899)+ 119890(1205732119899+1205734119899)] (1205732

1119899minus 1205732

3119899) (1205732

2119899minus 1205732

4119899)

+ [119890(1205732119899+1205733119899)+ 119890(1205731119899+1205734119899)] (1205732

2119899minus 1205732

3119899) (1205732

1119899minus 1205732

4119899) = 0

(16)




= 1198901205731119899119909minus

(1205732

4119899minus 1205732

1119899) (1198901205733119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

2119899) (1198901205733119899 minus 1198901205732119899)

1198901205732119899119909

minus

(1205732

4119899minus 1205732

1119899) (1198901205732119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

3119899) (1198901205732119899 minus 1198901205733119899)

1198901205733119899119909

+ (minus1 +

(1205732

4119899minus 1205732

1119899) (1198901205733119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

2119899) (1198901205733119899 minus 1198901205732119899)

+

(1205732

4119899minus 1205732

1119899) (1198901205732119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

3119899) (1198901205732119899 minus 1198901205733119899)

) 11989012057341198991199091198621119899

(17)


1205961

(Hz)


k

0 2 4 6 8 10 12400

500

600

700

800

900

1000

1100

1200

(a)

1205962

(Hz)

1205962

(Hz)


k

k

0 2 4 6 8 10 12

2500

3000

3500

4000

4500

4 6 8 10 12

2500

2600

2700

2800

2400

(b)

1205963

(Hz)

1205963

(Hz)


= 10ms = 100ms

= 150ms

k

k

0 2 4 6 8 10 12

6000

7000

8000

9000

10000

4 6 8 10 125600

5800

6000

6200

6300

(c)

1205964

(Hz)


(Hz)


k

k

0 2 4 6 8 10 12

1

11

12

13

14

15

16

17

18

4 6 8 10 121

102

104

106

108

11

= 10ms = 100ms

= 150ms

(d)



(1 minus 1198862V2) (1 + 119886

4minus 1198861V2)

1205974119908

1205971199094+ 1198863(1198861V2 minus 119886

4)1205972119908

1205971199092= 0

(18)


(1 minus 1198862V2) (1 + 119886

4minus 1198861V2) 1199034 + 119886

3(1198861V2 minus 119886

4) 1199032= 0 (19)


2V2)(1+119886

4minus1198861V2) and 119886



1199031= 1199032= 0 119903

3= 1199034= plusmn119894radic

1198863(1198861V2 minus 119886

4)

(1 minus 1198862V2) (1 + 119886

4minus 1198861V2)


by119908 (119909) = 119862

1+ 1198622119909 + 119862

3cos (120573119909) + 119862

4sin (120573119909) (21)


x (m)

minus2

minus1

1205931

0 02 04 06 08 1

0

1

2

3

4

5

The first mode

(a)

x (m)

minus10

minus5

1205932

0 02 04 06 08 1

0

5

10

The second mode

(b)

x (m)

minus15

minus10

minus5

1205933

k = 1

k = 025

0 02 04 06 08 1

0

5

10

15

The third mode

(c)

x (m)

k = 1

k = 025

minus20

minus10

1205934

0 02 04 06 08 1

0

10

20

The fourth mode

(d)


where

120573 = radic1198863(1198861V2 minus 119886

4)

(1 minus 1198862V2) (1 + 119886

4minus 1198861V2)

(22)


(

1 0 1 0

1 119871 cos (120573119871) sin (120573119871)0 0 minus120573

20

0 0 minus1205732 cos (120573119871) minus120573

2 sin (120573119871)

)(

1198621

1198622

1198623

1198624

) = 0 (23)


1198711205734 sin (119871120573) = 0 (24)


V119888= radic

119872 minus119873

2120587211988611198862

(25)

or

V119888= radic

119872 +119873

2120587211988611198862

(26)

where119872 = 120587

2(1198861+ 1198862+ 11988621198864) + 119871211988611198863

119873 = ((1198861(1205872+ 11987121198863) + 12058721198862(1 + 119886

4))2

minus 4120587211988611198862(1205872+ 12058721198864+ 119871211988631198864) )

12

(27)


0 2 4 6 8 10 12200

250

300

350

400

c(m

s)

k

1205900 = 100MPa1205900 = 50MPa

1205900 = 1MPa


10 13 16 19 22 25200

350

500

650

800

900

c(m

s)

Lh

k = 01

k = 1

k = 12


0= 1MPa)



119871 = 1m ℎ = 004m 119887 = 004m 120583 = 03

120581 =5

6

(28)


119896 1205900(MPa) 120596

1(Hz) 120596

2(Hz) 120596

3(Hz) 120596

4(Hz)

0001

1 1075 4436 9900 1732250 1129 4490 9955 17377100 1182 4545 10010 17433


1

1 738 3170 7109 1245550 790 3221 7160 12506100 841 3272 7211 12559


100

1 496 2304 5209 914750 553 2357 5262 9200100 605 2409 5315 9254














0= 1MPa are adopted





5 Conclusions


Appendices


Consider

int

1199052

1199051

int

119871

0

[ 120581119866119879(1205972119908

1205971199092minus120597120595

120597119909)

minus 1205881198790(1205972119908

1205971199052+ 2V

1205972119908

120597119905120597119909+ V2

1205972119908

1205971199092)

+12059001198601205972119908

1205971199092] 120575119908119889119909119889119905

+ int

1199052

1199051

int

119871

0

[(119864119868)eq1205972120595

1205971199092minus 1205881198792(1205972120595

1205971199052+ 2V

1205972120595

120597119905120597119909+ V2

1205972120595

1205971199092)

+120581119866119879(120597119908

120597119909minus 120595)]120575120595119889119909119889119905

+ 1205881198790int

1199052

1199051

int (V120597119908

120597119905120575119908 + V2

120597119908

120597119909120575119908)

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905

+ 1205881198790int

119871

0

(120597119908

120597119905120575119908 + V

120597119908

120597119909120575119908)

10038161003816100381610038161003816100381610038161003816

1199052

1199051

119889119909

+ 1205881198792int

1199052

1199051

(V120597120595

120597119905120575120595 + V2

120597120595

120597119909120575120595)

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905

+ 1205881198792int

119871

0

(120597120595

120597119905120575120595 + V

120597120595

120597119909120575120595)

10038161003816100381610038161003816100381610038161003816

1199052

1199051

119889119909

minus 120581119866119879int

1199052

1199051

(120597119908

120597119909120575119908 + 120595120575119908)

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905 minus 1205900119860int

1199052

1199051

120597119908

120597119909120575119908

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905

minus (119864119868)eq int1199052

1199051

120597120595

120597119909120575120595

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905 = 0

(A1)


Consider

1198622119899= minus

(1205732

4119899minus 1205732

1119899) (1198901205733119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

2119899) (1198901205733119899 minus 1198901205732119899)

1198621119899

1198623119899= minus

(1205732

4119899minus 1205732

1119899) (1198901205732119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

3119899) (1198901205732119899 minus 1198901205733119899)

1198621119899

1198624119899= [minus1 +

(1205732

4119899minus 1205732

1119899) (1198901205733119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

2119899) (1198901205733119899 minus 1198901205732119899)

+

(1205732

4119899minus 1205732

1119899) (1198901205732119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

3119899) (1198901205732119899 minus 1198901205733119899)

]1198621119899

(B1)



Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50 100 150 200

500

600

700

800

900

1000

1100


k = 1

k = 025

k = 01

k = 001

(ms)

1205961

(Hz)

(a)

0 100 200 300 400

2500

3000

3500

4000


k = 1

k = 025

k = 01

k = 001

(ms)

1205962

(Hz)

(b)

0 100 200 300 400

6500

7500

8500

9500


k = 1

k = 025

k = 01

k = 001

(ms)

1205963

(Hz)

(c)

0 100 200 300 400

12

13

14

15

16

17


k = 1

k = 025

k = 01

k = 001

(ms)

1205964

(Hz)

times104

(d)


Consider

[119890(1205731119899+1205732119899)+ 119890(1205733119899+1205734119899)] (1205732

1119899minus 1205732

2119899) (1205732

3119899minus 1205732

4119899)

minus [119890(1205731119899+1205733119899)+ 119890(1205732119899+1205734119899)] (1205732

1119899minus 1205732

3119899) (1205732

2119899minus 1205732

4119899)

+ [119890(1205732119899+1205733119899)+ 119890(1205731119899+1205734119899)] (1205732

2119899minus 1205732

3119899) (1205732

1119899minus 1205732

4119899) = 0

(16)




= 1198901205731119899119909minus

(1205732

4119899minus 1205732

1119899) (1198901205733119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

2119899) (1198901205733119899 minus 1198901205732119899)

1198901205732119899119909

minus

(1205732

4119899minus 1205732

1119899) (1198901205732119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

3119899) (1198901205732119899 minus 1198901205733119899)

1198901205733119899119909

+ (minus1 +

(1205732

4119899minus 1205732

1119899) (1198901205733119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

2119899) (1198901205733119899 minus 1198901205732119899)

+

(1205732

4119899minus 1205732

1119899) (1198901205732119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

3119899) (1198901205732119899 minus 1198901205733119899)

) 11989012057341198991199091198621119899

(17)


1205961

(Hz)


k

0 2 4 6 8 10 12400

500

600

700

800

900

1000

1100

1200

(a)

1205962

(Hz)

1205962

(Hz)


k

k

0 2 4 6 8 10 12

2500

3000

3500

4000

4500

4 6 8 10 12

2500

2600

2700

2800

2400

(b)

1205963

(Hz)

1205963

(Hz)


= 10ms = 100ms

= 150ms

k

k

0 2 4 6 8 10 12

6000

7000

8000

9000

10000

4 6 8 10 125600

5800

6000

6200

6300

(c)

1205964

(Hz)


(Hz)


k

k

0 2 4 6 8 10 12

1

11

12

13

14

15

16

17

18

4 6 8 10 121

102

104

106

108

11

= 10ms = 100ms

= 150ms

(d)



(1 minus 1198862V2) (1 + 119886

4minus 1198861V2)

1205974119908

1205971199094+ 1198863(1198861V2 minus 119886

4)1205972119908

1205971199092= 0

(18)


(1 minus 1198862V2) (1 + 119886

4minus 1198861V2) 1199034 + 119886

3(1198861V2 minus 119886

4) 1199032= 0 (19)


2V2)(1+119886

4minus1198861V2) and 119886



1199031= 1199032= 0 119903

3= 1199034= plusmn119894radic

1198863(1198861V2 minus 119886

4)

(1 minus 1198862V2) (1 + 119886

4minus 1198861V2)


by119908 (119909) = 119862

1+ 1198622119909 + 119862

3cos (120573119909) + 119862

4sin (120573119909) (21)


x (m)

minus2

minus1

1205931

0 02 04 06 08 1

0

1

2

3

4

5

The first mode

(a)

x (m)

minus10

minus5

1205932

0 02 04 06 08 1

0

5

10

The second mode

(b)

x (m)

minus15

minus10

minus5

1205933

k = 1

k = 025

0 02 04 06 08 1

0

5

10

15

The third mode

(c)

x (m)

k = 1

k = 025

minus20

minus10

1205934

0 02 04 06 08 1

0

10

20

The fourth mode

(d)


where

120573 = radic1198863(1198861V2 minus 119886

4)

(1 minus 1198862V2) (1 + 119886

4minus 1198861V2)

(22)


(

1 0 1 0

1 119871 cos (120573119871) sin (120573119871)0 0 minus120573

20

0 0 minus1205732 cos (120573119871) minus120573

2 sin (120573119871)

)(

1198621

1198622

1198623

1198624

) = 0 (23)


1198711205734 sin (119871120573) = 0 (24)


V119888= radic

119872 minus119873

2120587211988611198862

(25)

or

V119888= radic

119872 +119873

2120587211988611198862

(26)

where119872 = 120587

2(1198861+ 1198862+ 11988621198864) + 119871211988611198863

119873 = ((1198861(1205872+ 11987121198863) + 12058721198862(1 + 119886

4))2

minus 4120587211988611198862(1205872+ 12058721198864+ 119871211988631198864) )

12

(27)


0 2 4 6 8 10 12200

250

300

350

400

c(m

s)

k

1205900 = 100MPa1205900 = 50MPa

1205900 = 1MPa


10 13 16 19 22 25200

350

500

650

800

900

c(m

s)

Lh

k = 01

k = 1

k = 12


0= 1MPa)



119871 = 1m ℎ = 004m 119887 = 004m 120583 = 03

120581 =5

6

(28)


119896 1205900(MPa) 120596

1(Hz) 120596

2(Hz) 120596

3(Hz) 120596

4(Hz)

0001

1 1075 4436 9900 1732250 1129 4490 9955 17377100 1182 4545 10010 17433


1

1 738 3170 7109 1245550 790 3221 7160 12506100 841 3272 7211 12559


100

1 496 2304 5209 914750 553 2357 5262 9200100 605 2409 5315 9254














0= 1MPa are adopted





5 Conclusions


Appendices


Consider

int

1199052

1199051

int

119871

0

[ 120581119866119879(1205972119908

1205971199092minus120597120595

120597119909)

minus 1205881198790(1205972119908

1205971199052+ 2V

1205972119908

120597119905120597119909+ V2

1205972119908

1205971199092)

+12059001198601205972119908

1205971199092] 120575119908119889119909119889119905

+ int

1199052

1199051

int

119871

0

[(119864119868)eq1205972120595

1205971199092minus 1205881198792(1205972120595

1205971199052+ 2V

1205972120595

120597119905120597119909+ V2

1205972120595

1205971199092)

+120581119866119879(120597119908

120597119909minus 120595)]120575120595119889119909119889119905

+ 1205881198790int

1199052

1199051

int (V120597119908

120597119905120575119908 + V2

120597119908

120597119909120575119908)

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905

+ 1205881198790int

119871

0

(120597119908

120597119905120575119908 + V

120597119908

120597119909120575119908)

10038161003816100381610038161003816100381610038161003816

1199052

1199051

119889119909

+ 1205881198792int

1199052

1199051

(V120597120595

120597119905120575120595 + V2

120597120595

120597119909120575120595)

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905

+ 1205881198792int

119871

0

(120597120595

120597119905120575120595 + V

120597120595

120597119909120575120595)

10038161003816100381610038161003816100381610038161003816

1199052

1199051

119889119909

minus 120581119866119879int

1199052

1199051

(120597119908

120597119909120575119908 + 120595120575119908)

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905 minus 1205900119860int

1199052

1199051

120597119908

120597119909120575119908

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905

minus (119864119868)eq int1199052

1199051

120597120595

120597119909120575120595

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905 = 0

(A1)


Consider

1198622119899= minus

(1205732

4119899minus 1205732

1119899) (1198901205733119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

2119899) (1198901205733119899 minus 1198901205732119899)

1198621119899

1198623119899= minus

(1205732

4119899minus 1205732

1119899) (1198901205732119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

3119899) (1198901205732119899 minus 1198901205733119899)

1198621119899

1198624119899= [minus1 +

(1205732

4119899minus 1205732

1119899) (1198901205733119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

2119899) (1198901205733119899 minus 1198901205732119899)

+

(1205732

4119899minus 1205732

1119899) (1198901205732119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

3119899) (1198901205732119899 minus 1198901205733119899)

]1198621119899

(B1)



Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1205961

(Hz)


k

0 2 4 6 8 10 12400

500

600

700

800

900

1000

1100

1200

(a)

1205962

(Hz)

1205962

(Hz)


k

k

0 2 4 6 8 10 12

2500

3000

3500

4000

4500

4 6 8 10 12

2500

2600

2700

2800

2400

(b)

1205963

(Hz)

1205963

(Hz)


= 10ms = 100ms

= 150ms

k

k

0 2 4 6 8 10 12

6000

7000

8000

9000

10000

4 6 8 10 125600

5800

6000

6200

6300

(c)

1205964

(Hz)


(Hz)


k

k

0 2 4 6 8 10 12

1

11

12

13

14

15

16

17

18

4 6 8 10 121

102

104

106

108

11

= 10ms = 100ms

= 150ms

(d)



(1 minus 1198862V2) (1 + 119886

4minus 1198861V2)

1205974119908

1205971199094+ 1198863(1198861V2 minus 119886

4)1205972119908

1205971199092= 0

(18)


(1 minus 1198862V2) (1 + 119886

4minus 1198861V2) 1199034 + 119886

3(1198861V2 minus 119886

4) 1199032= 0 (19)


2V2)(1+119886

4minus1198861V2) and 119886



1199031= 1199032= 0 119903

3= 1199034= plusmn119894radic

1198863(1198861V2 minus 119886

4)

(1 minus 1198862V2) (1 + 119886

4minus 1198861V2)


by119908 (119909) = 119862

1+ 1198622119909 + 119862

3cos (120573119909) + 119862

4sin (120573119909) (21)


x (m)

minus2

minus1

1205931

0 02 04 06 08 1

0

1

2

3

4

5

The first mode

(a)

x (m)

minus10

minus5

1205932

0 02 04 06 08 1

0

5

10

The second mode

(b)

x (m)

minus15

minus10

minus5

1205933

k = 1

k = 025

0 02 04 06 08 1

0

5

10

15

The third mode

(c)

x (m)

k = 1

k = 025

minus20

minus10

1205934

0 02 04 06 08 1

0

10

20

The fourth mode

(d)


where

120573 = radic1198863(1198861V2 minus 119886

4)

(1 minus 1198862V2) (1 + 119886

4minus 1198861V2)

(22)


(

1 0 1 0

1 119871 cos (120573119871) sin (120573119871)0 0 minus120573

20

0 0 minus1205732 cos (120573119871) minus120573

2 sin (120573119871)

)(

1198621

1198622

1198623

1198624

) = 0 (23)


1198711205734 sin (119871120573) = 0 (24)


V119888= radic

119872 minus119873

2120587211988611198862

(25)

or

V119888= radic

119872 +119873

2120587211988611198862

(26)

where119872 = 120587

2(1198861+ 1198862+ 11988621198864) + 119871211988611198863

119873 = ((1198861(1205872+ 11987121198863) + 12058721198862(1 + 119886

4))2

minus 4120587211988611198862(1205872+ 12058721198864+ 119871211988631198864) )

12

(27)


0 2 4 6 8 10 12200

250

300

350

400

c(m

s)

k

1205900 = 100MPa1205900 = 50MPa

1205900 = 1MPa


10 13 16 19 22 25200

350

500

650

800

900

c(m

s)

Lh

k = 01

k = 1

k = 12


0= 1MPa)



119871 = 1m ℎ = 004m 119887 = 004m 120583 = 03

120581 =5

6

(28)


119896 1205900(MPa) 120596

1(Hz) 120596

2(Hz) 120596

3(Hz) 120596

4(Hz)

0001

1 1075 4436 9900 1732250 1129 4490 9955 17377100 1182 4545 10010 17433


1

1 738 3170 7109 1245550 790 3221 7160 12506100 841 3272 7211 12559


100

1 496 2304 5209 914750 553 2357 5262 9200100 605 2409 5315 9254














0= 1MPa are adopted





5 Conclusions


Appendices


Consider

int

1199052

1199051

int

119871

0

[ 120581119866119879(1205972119908

1205971199092minus120597120595

120597119909)

minus 1205881198790(1205972119908

1205971199052+ 2V

1205972119908

120597119905120597119909+ V2

1205972119908

1205971199092)

+12059001198601205972119908

1205971199092] 120575119908119889119909119889119905

+ int

1199052

1199051

int

119871

0

[(119864119868)eq1205972120595

1205971199092minus 1205881198792(1205972120595

1205971199052+ 2V

1205972120595

120597119905120597119909+ V2

1205972120595

1205971199092)

+120581119866119879(120597119908

120597119909minus 120595)]120575120595119889119909119889119905

+ 1205881198790int

1199052

1199051

int (V120597119908

120597119905120575119908 + V2

120597119908

120597119909120575119908)

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905

+ 1205881198790int

119871

0

(120597119908

120597119905120575119908 + V

120597119908

120597119909120575119908)

10038161003816100381610038161003816100381610038161003816

1199052

1199051

119889119909

+ 1205881198792int

1199052

1199051

(V120597120595

120597119905120575120595 + V2

120597120595

120597119909120575120595)

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905

+ 1205881198792int

119871

0

(120597120595

120597119905120575120595 + V

120597120595

120597119909120575120595)

10038161003816100381610038161003816100381610038161003816

1199052

1199051

119889119909

minus 120581119866119879int

1199052

1199051

(120597119908

120597119909120575119908 + 120595120575119908)

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905 minus 1205900119860int

1199052

1199051

120597119908

120597119909120575119908

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905

minus (119864119868)eq int1199052

1199051

120597120595

120597119909120575120595

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905 = 0

(A1)


Consider

1198622119899= minus

(1205732

4119899minus 1205732

1119899) (1198901205733119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

2119899) (1198901205733119899 minus 1198901205732119899)

1198621119899

1198623119899= minus

(1205732

4119899minus 1205732

1119899) (1198901205732119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

3119899) (1198901205732119899 minus 1198901205733119899)

1198621119899

1198624119899= [minus1 +

(1205732

4119899minus 1205732

1119899) (1198901205733119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

2119899) (1198901205733119899 minus 1198901205732119899)

+

(1205732

4119899minus 1205732

1119899) (1198901205732119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

3119899) (1198901205732119899 minus 1198901205733119899)

]1198621119899

(B1)



Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










x (m)

minus2

minus1

1205931

0 02 04 06 08 1

0

1

2

3

4

5

The first mode

(a)

x (m)

minus10

minus5

1205932

0 02 04 06 08 1

0

5

10

The second mode

(b)

x (m)

minus15

minus10

minus5

1205933

k = 1

k = 025

0 02 04 06 08 1

0

5

10

15

The third mode

(c)

x (m)

k = 1

k = 025

minus20

minus10

1205934

0 02 04 06 08 1

0

10

20

The fourth mode

(d)


where

120573 = radic1198863(1198861V2 minus 119886

4)

(1 minus 1198862V2) (1 + 119886

4minus 1198861V2)

(22)


(

1 0 1 0

1 119871 cos (120573119871) sin (120573119871)0 0 minus120573

20

0 0 minus1205732 cos (120573119871) minus120573

2 sin (120573119871)

)(

1198621

1198622

1198623

1198624

) = 0 (23)


1198711205734 sin (119871120573) = 0 (24)


V119888= radic

119872 minus119873

2120587211988611198862

(25)

or

V119888= radic

119872 +119873

2120587211988611198862

(26)

where119872 = 120587

2(1198861+ 1198862+ 11988621198864) + 119871211988611198863

119873 = ((1198861(1205872+ 11987121198863) + 12058721198862(1 + 119886

4))2

minus 4120587211988611198862(1205872+ 12058721198864+ 119871211988631198864) )

12

(27)


0 2 4 6 8 10 12200

250

300

350

400

c(m

s)

k

1205900 = 100MPa1205900 = 50MPa

1205900 = 1MPa


10 13 16 19 22 25200

350

500

650

800

900

c(m

s)

Lh

k = 01

k = 1

k = 12


0= 1MPa)



119871 = 1m ℎ = 004m 119887 = 004m 120583 = 03

120581 =5

6

(28)


119896 1205900(MPa) 120596

1(Hz) 120596

2(Hz) 120596

3(Hz) 120596

4(Hz)

0001

1 1075 4436 9900 1732250 1129 4490 9955 17377100 1182 4545 10010 17433


1

1 738 3170 7109 1245550 790 3221 7160 12506100 841 3272 7211 12559


100

1 496 2304 5209 914750 553 2357 5262 9200100 605 2409 5315 9254














0= 1MPa are adopted





5 Conclusions


Appendices


Consider

int

1199052

1199051

int

119871

0

[ 120581119866119879(1205972119908

1205971199092minus120597120595

120597119909)

minus 1205881198790(1205972119908

1205971199052+ 2V

1205972119908

120597119905120597119909+ V2

1205972119908

1205971199092)

+12059001198601205972119908

1205971199092] 120575119908119889119909119889119905

+ int

1199052

1199051

int

119871

0

[(119864119868)eq1205972120595

1205971199092minus 1205881198792(1205972120595

1205971199052+ 2V

1205972120595

120597119905120597119909+ V2

1205972120595

1205971199092)

+120581119866119879(120597119908

120597119909minus 120595)]120575120595119889119909119889119905

+ 1205881198790int

1199052

1199051

int (V120597119908

120597119905120575119908 + V2

120597119908

120597119909120575119908)

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905

+ 1205881198790int

119871

0

(120597119908

120597119905120575119908 + V

120597119908

120597119909120575119908)

10038161003816100381610038161003816100381610038161003816

1199052

1199051

119889119909

+ 1205881198792int

1199052

1199051

(V120597120595

120597119905120575120595 + V2

120597120595

120597119909120575120595)

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905

+ 1205881198792int

119871

0

(120597120595

120597119905120575120595 + V

120597120595

120597119909120575120595)

10038161003816100381610038161003816100381610038161003816

1199052

1199051

119889119909

minus 120581119866119879int

1199052

1199051

(120597119908

120597119909120575119908 + 120595120575119908)

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905 minus 1205900119860int

1199052

1199051

120597119908

120597119909120575119908

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905

minus (119864119868)eq int1199052

1199051

120597120595

120597119909120575120595

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905 = 0

(A1)


Consider

1198622119899= minus

(1205732

4119899minus 1205732

1119899) (1198901205733119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

2119899) (1198901205733119899 minus 1198901205732119899)

1198621119899

1198623119899= minus

(1205732

4119899minus 1205732

1119899) (1198901205732119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

3119899) (1198901205732119899 minus 1198901205733119899)

1198621119899

1198624119899= [minus1 +

(1205732

4119899minus 1205732

1119899) (1198901205733119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

2119899) (1198901205733119899 minus 1198901205732119899)

+

(1205732

4119899minus 1205732

1119899) (1198901205732119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

3119899) (1198901205732119899 minus 1198901205733119899)

]1198621119899

(B1)



Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8 10 12200

250

300

350

400

c(m

s)

k

1205900 = 100MPa1205900 = 50MPa

1205900 = 1MPa


10 13 16 19 22 25200

350

500

650

800

900

c(m

s)

Lh

k = 01

k = 1

k = 12


0= 1MPa)



119871 = 1m ℎ = 004m 119887 = 004m 120583 = 03

120581 =5

6

(28)


119896 1205900(MPa) 120596

1(Hz) 120596

2(Hz) 120596

3(Hz) 120596

4(Hz)

0001

1 1075 4436 9900 1732250 1129 4490 9955 17377100 1182 4545 10010 17433


1

1 738 3170 7109 1245550 790 3221 7160 12506100 841 3272 7211 12559


100

1 496 2304 5209 914750 553 2357 5262 9200100 605 2409 5315 9254














0= 1MPa are adopted





5 Conclusions


Appendices


Consider

int

1199052

1199051

int

119871

0

[ 120581119866119879(1205972119908

1205971199092minus120597120595

120597119909)

minus 1205881198790(1205972119908

1205971199052+ 2V

1205972119908

120597119905120597119909+ V2

1205972119908

1205971199092)

+12059001198601205972119908

1205971199092] 120575119908119889119909119889119905

+ int

1199052

1199051

int

119871

0

[(119864119868)eq1205972120595

1205971199092minus 1205881198792(1205972120595

1205971199052+ 2V

1205972120595

120597119905120597119909+ V2

1205972120595

1205971199092)

+120581119866119879(120597119908

120597119909minus 120595)]120575120595119889119909119889119905

+ 1205881198790int

1199052

1199051

int (V120597119908

120597119905120575119908 + V2

120597119908

120597119909120575119908)

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905

+ 1205881198790int

119871

0

(120597119908

120597119905120575119908 + V

120597119908

120597119909120575119908)

10038161003816100381610038161003816100381610038161003816

1199052

1199051

119889119909

+ 1205881198792int

1199052

1199051

(V120597120595

120597119905120575120595 + V2

120597120595

120597119909120575120595)

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905

+ 1205881198792int

119871

0

(120597120595

120597119905120575120595 + V

120597120595

120597119909120575120595)

10038161003816100381610038161003816100381610038161003816

1199052

1199051

119889119909

minus 120581119866119879int

1199052

1199051

(120597119908

120597119909120575119908 + 120595120575119908)

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905 minus 1205900119860int

1199052

1199051

120597119908

120597119909120575119908

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905

minus (119864119868)eq int1199052

1199051

120597120595

120597119909120575120595

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905 = 0

(A1)


Consider

1198622119899= minus

(1205732

4119899minus 1205732

1119899) (1198901205733119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

2119899) (1198901205733119899 minus 1198901205732119899)

1198621119899

1198623119899= minus

(1205732

4119899minus 1205732

1119899) (1198901205732119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

3119899) (1198901205732119899 minus 1198901205733119899)

1198621119899

1198624119899= [minus1 +

(1205732

4119899minus 1205732

1119899) (1198901205733119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

2119899) (1198901205733119899 minus 1198901205732119899)

+

(1205732

4119899minus 1205732

1119899) (1198901205732119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

3119899) (1198901205732119899 minus 1198901205733119899)

]1198621119899

(B1)



Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0= 1MPa are adopted





5 Conclusions


Appendices


Consider

int

1199052

1199051

int

119871

0

[ 120581119866119879(1205972119908

1205971199092minus120597120595

120597119909)

minus 1205881198790(1205972119908

1205971199052+ 2V

1205972119908

120597119905120597119909+ V2

1205972119908

1205971199092)

+12059001198601205972119908

1205971199092] 120575119908119889119909119889119905

+ int

1199052

1199051

int

119871

0

[(119864119868)eq1205972120595

1205971199092minus 1205881198792(1205972120595

1205971199052+ 2V

1205972120595

120597119905120597119909+ V2

1205972120595

1205971199092)

+120581119866119879(120597119908

120597119909minus 120595)]120575120595119889119909119889119905

+ 1205881198790int

1199052

1199051

int (V120597119908

120597119905120575119908 + V2

120597119908

120597119909120575119908)

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905

+ 1205881198790int

119871

0

(120597119908

120597119905120575119908 + V

120597119908

120597119909120575119908)

10038161003816100381610038161003816100381610038161003816

1199052

1199051

119889119909

+ 1205881198792int

1199052

1199051

(V120597120595

120597119905120575120595 + V2

120597120595

120597119909120575120595)

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905

+ 1205881198792int

119871

0

(120597120595

120597119905120575120595 + V

120597120595

120597119909120575120595)

10038161003816100381610038161003816100381610038161003816

1199052

1199051

119889119909

minus 120581119866119879int

1199052

1199051

(120597119908

120597119909120575119908 + 120595120575119908)

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905 minus 1205900119860int

1199052

1199051

120597119908

120597119909120575119908

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905

minus (119864119868)eq int1199052

1199051

120597120595

120597119909120575120595

10038161003816100381610038161003816100381610038161003816

119871

0

119889119905 = 0

(A1)


Consider

1198622119899= minus

(1205732

4119899minus 1205732

1119899) (1198901205733119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

2119899) (1198901205733119899 minus 1198901205732119899)

1198621119899

1198623119899= minus

(1205732

4119899minus 1205732

1119899) (1198901205732119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

3119899) (1198901205732119899 minus 1198901205733119899)

1198621119899

1198624119899= [minus1 +

(1205732

4119899minus 1205732

1119899) (1198901205733119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

2119899) (1198901205733119899 minus 1198901205732119899)

+

(1205732

4119899minus 1205732

1119899) (1198901205732119899 minus 1198901205731119899)

(1205732

4119899minus 1205732

3119899) (1198901205732119899 minus 1198901205733119899)

]1198621119899

(B1)



Acknowledgments


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article transverse vibration of axially moving...

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