research article analysis of nonlinear vibration of hard

Research ArticleAnalysis of Nonlinear Vibration of Hard Coating ThinPlate by Finite Element Iteration Method

Hui Li Liu Ying and Wei Sun

School of Mechanical Engineering and Automation Northeastern University No 3-11 Wenhua RoadHeping District Shenyang 110819 China

Correspondence should be addressed to Wei Sun weisunmailneueducn

Received 6 July 2014 Revised 16 September 2014 Accepted 20 September 2014 Published 8 December 2014

Academic Editor Ahmet S Yigit

Copyright copy 2014 Hui Li et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper studies nonlinear vibration mechanism of hard coating thin plate based on macroscopic vibration theory and proposesfinite element iteration method (FEIM) to theoretically calculate its nature frequency and vibration response First of all straindependent mechanical property of hard coating is briefly introduced and polynomial method is adopted to characterize the storageand loss modulus of coating material Then the principle formulas of inherent and dynamic response characteristics of the hardcoating composite plate are derived And consequently specific analysis procedure is proposed by combining ANSYS APDL andself-designedMATLAB program Finally a composite plate coated withMgO +Al

2O3is taken as a study object and both nonlinear

vibration test and analysis are conducted on the plate specimen with considering strain dependent mechanical parameters ofhard coating Through comparing the resulting frequency and response results the practicability and reliability of FEIM havebeen verified and the corresponding analysis results can provide an important reference for further study on nonlinear vibrationmechanism of hard coating composite structure

1 Introduction

The hard coating is a kind of coating materials prepared bythe metal ceramic or their mixtures which is mainly used inthermal barrier coatings antifriction antierosion and otherengineering application fields Recently increased attentionhas been paid to the structural vibration-suppression withhard coating because it can reduce the vibration stress underhigh temperature or highly corrosive environment [1ndash3]Several scholars and researchers [4ndash6] have experimentallystudied the vibration characteristics of thin-walled structureswith hard coating such as beams plates and shells theyfound that the coating can not only contribute to increasingstructural damping as well as reducing the amplitude ofresonant vibrations but also lead to nonlinear characteristicsto the coated structure For example Ivancic and Palazotto[4] and Blackwell et al [5] measure inherent characteristic oftitanium plate coated with magnesium aluminate spinel theyfound that its frequency response curve was not symmetricabout resonance frequency at certain mode and the valueof resonance frequency decreased with the increase of the

amplitude of the applied force indicating some characteris-tics of nonlinear stiffness and they called this phenomenonldquostrain softeningrdquo

Several papers tried to explain the above phenomenonof nonlinear vibration of hard coating composite structurefrom the scope of mesoscopic elastic materials For exampleTassini et al [7] implemented a phenomenological modelthat characterizes nonlinear mesoscopic elastic materialswhich is capable of reproducing the basic features of theobserved damping behavior for zirconia coatings preparedby air plasma spraying and electron-beam physical-vapor-deposition Green and Patsias [8] proposed a friction modelto calculate the response of a coated beamand obtain the vari-ation of damping effectiveness for several vibration modesTorvik [9] developed a slip damping model incorporatingboth microslip and macroslip in providing qualitative pre-dictions of the amplitude dependence of the dissipation in alap joint and the damping (loss modulus) was also extractedand the amplitude dependence was found to be similar tothat observed with actual coating materials Al-Rub andPalazotto [10] proposed a micromechanical theoretical and

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 941709 12 pageshttpdxdoiorg1011552014941709

2 Shock and Vibration

computational model to assess the main micromechanicalmechanisms responsible for the experimentally observednonlinear vibration in plasma sprayed hard ceramic coatingsThese studies have contributed a lot to the understanding ofenergy dissipation mechanism of hard coating as well as therelated nonlinear vibration of composite structure

However it is not realistic to completely understandnonlinear vibration characteristics of hard coating compos-ite structure from microscopic scope in order to effec-tively implement damping coating technique for vibrationsuppression it is necessary to develop mechanical modelin accordance with macrovibration theory Unfortunatelyresearch efforts spent in this area are not adequate enoughbecause it is still a challenge to present a reliable modelto describe the mechanical properties of hard coating com-posite structure when the observed response is that of anonlinear system which have generally been found to displaya strong dependence on the amplitude of strain Besidesanother challenge is to identify mechanical parameters ofthe coating such as strain dependent Youngrsquos modulus andthe loss modulus which are mainly responsible for non-linearity of the coated structure Fortunately several papershave made some progress in identifying such parametersPatsias et al [11] tested Nimonic C-263 beam specimencoatedwith aluminiumoxide under different excitation levelsand proposed an experimental procedure for extracting thecoatingrsquos damping (internal friction) and the modulus ofelasticity with the variation of strain from response decaysTorvik [12] presented amethodology suitable for determiningthe amplitude-dependent mechanical properties of coatingapplied on substrate beams and its storage (Youngrsquos) mod-ulus the loss modulus and the loss factor can be extractedby careful measurement of specimen dimensions weightsnature frequencies and loss factors before and after coatingReed et al [13] set up a novel vibration experiment system andproposed an experimental technique for accurate measure-ment of strain dependent stiffness and damping propertiesof hard coatings at high strain levels which provided animportant support for modeling coating composite structureand its nonlinear vibration analysis

Finite element iteration technique is an efficient and prac-tical method which can be used to solve vibration problemswith considering variable mass stiffness and damping in thekinetic equation For example Hernandez and Bernal [14]adopted an iterative time domain formulation for finite ele-ment model to update the stiffness and damping parametersof a twenty-degree-of-freedom shear beam Hu et al [15]considered the contact between drillstring and borehole andproposed an iterative finite element technology to analyzedrillstring dynamic Soares [16] presented an optimised FEM-BEM iterative coupling algorithm and demonstrated theeffectiveness on an elastodynamic rectangular rod and acircular cavity Filippi and Torvik [17] proposed iterativefinite element analysis method to predict the response of anactual blade with a nonlinear coating which can be usedto quickly determine the approximate resonant frequencydamping and response amplitude in the vicinity of a par-ticular resonance However linear analytic expressions areused for the nonlinear coating material and the structural

damping is expressed by loss factor whichmay help in gettingdampingmatrix in uniform form but fails to separate viscousdamping from loss factor of the material Therefore if we caneffectively express strain dependent mechanical parametersof hard coating and consequently introduce them into thefinite element equation to solve the nodal displacementsin the equation by numerical iteration method it wouldbe a good solution to theoretical analysis of the concernednonlinear dynamics mechanism of hard coating compositestructure

This paper studies nonlinear vibration mechanism ofhard coating thin plate based on macroscopic vibrationtheory and proposes finite element iteration method (FEIM)to theoretically calculate its nature frequency and vibrationresponse In Section 2 strain dependent mechanical prop-erty of hard coating is briefly introduced and polynomialmethod is adopted to characterize the storage and lossmodulus of coating material Then go on to theoreticallyderive the principle formulas of inherent and dynamicresponse characteristics of the hard coating composite platein Section 3 and consequently propose specific analysisprocedure by combining ANSYS APDL and self-designedMATLAB program in Section 4 Finally in Section 5 acomposite plate coated with MgO + Al

2O3is taken as a

study object and both nonlinear vibration test and analysisare conducted on the plate specimen with considering straindependent mechanical parameters of hard coating Throughcomparing the resulting frequency and response results thepracticability and reliability of FEIM have been verified andthe corresponding analysis results can provide an importantreference for further study onnonlinear vibrationmechanismof hard coating composite structure

2 Polynomial Characterization ofStrain Dependent Mechanical Parameters ofHard Coating

Suppose that 119864lowast119888can be used to represent elastic modulus of

hard coating which is defined as complex modulus119864lowast

119888= 1198641015840

119888(120576119890) + 119894119864

10158401015840

119888(120576119890) (1)

where ldquolowastrdquo refers to complex1198641015840119888(120576119890)11986410158401015840119888(120576119890) are storagemodu-

lus (or Youngrsquos modulus) and loss modulus respectively andthey are all functions of equivalent strain 120576

119890

For simplicity we can adopt polynomial method tocharacterize 1198641015840

119888(120576119890) and 11986410158401015840

119888(120576119890) which can be expressed as

1198641015840

119888(120576119890) = 1198641198770+ 1205761198901198641198771+ 1205762

1198901198641198772+ 1205763

1198901198641198773+ sdot sdot sdot

11986410158401015840

119888(120576119890) = 1198641198680+ 1205761198901198641198681+ 1205762

1198901198641198682+ 1205763

1198901198641198683+ sdot sdot sdot

(2)

where 1198641198770 1198641198680

are the storage and loss modulus withoutconsidering strain dependence and 119864

119877119899 119864119868119899(119899 = 1 2 3 )

are the specific coefficients of strain dependent storagemodulus and loss modulus

Substituting (2) into (1) and elastic modulus of hardcoating 119864lowast

119888can be expressed as

119864lowast

119888= 119864lowast

1198880+ 119864lowast

1198881120576119890+ 119864lowast

11988821205762

119890+ 119864lowast

11988831205763

119890+ sdot sdot sdot (3)

where 119864lowast1198880= 1198641198770+ 1198941198641198680and 119864lowast

119888119899= 119864119877119899+ 119894119864119868119899 119899 = 1 2 3

Shock and Vibration 3

0 200 400 600 800

Polynomial fittedMeasured

Stor

age m

odul

us(G

pa)

35

40

45

50

55

120583120576 (120583m120583m)

(a) Storage modulus

25

15

05

1

2

3

0

Loss

mod

ulus

(Gpa

)

200 400 600 800120583120576 (120583m120583m)


(b) Loss modulus

Figure 1 Mechanical parameters of MgO + Al2O3with strain dependent characteristics

Usually mechanical parameters of hard coating can beidentified by some experimental method which are mainlydependent on the amplitude of strain that is strain ampli-tudes and modulus values are corresponding to each otherIn an effort to effectively express their relationship it isnecessary to adopt polynomial fitting technique As reference[13] has already experimentally determined discrete storagemodulus and lossmodulus of coatingmaterial such asMgO+Al2O3 as seen in Figures 1(a) and 1(b) we can calculate the

explicit expressions by curve-fitting of cube polynomial Theresulting mathematical expressions are given in (4) and (5)and the fitted curves are also plotted in Figures 1(a) and 1(b)respectively

1198641015840

119888(120576) = 50439 minus 005386120576

119890

+ 92396 times 10minus51205762

119890minus 54190 times 10

minus81205763

119890

(4)

11986410158401015840

119888(120576) = 09122 + 001321120576

119890

minus 32997 times 10minus51205762

119890+ 22086 times 10

minus81205763

119890

(5)

3 Nonlinear Vibration Analysis Principle ofHard Coating Composite Plate by FEIM

The finite element equation of hard coating composite struc-ture such as beams plates and shells with different kinds ofmaterials covered on the surface can be expressed as

Mx + Cx + Klowastx = F (6)

where M C and Klowast are mass matrix damping matrixand complex stiffness matrix of the physical system and Fis external excitation force vector x x and x are nodaldisplacement velocity and acceleration vector in physicalcoordinate respectively Additionally Klowast = K + 119894D whereK represents the real part of the complex stiffness matrix andD is material damping matrix

Due to strain dependence characteristic complex stiff-ness matrixKlowast in (6) requires special consideration Besidesbecause each finite element is related to different strain states

it is necessary to calculate each element stiffness matrix andcombine them to get the global stiffness matrix which can beexpressed as

Klowast = Klowast0+

119873

sum

119899

K1198990(119864lowast

1198881120576119890119899+ 119864lowast

11988821205762

119890119899+ 119864lowast

11988831205763

119890119899)

1198641198770

(119899 = 1 2 3 119873)

(7)

where Klowast0are the global stiffness matrix without considering

strain dependence 120576119890119899

is equivalent strain of the 119899th elementof composite plate 119864lowast

1198881 119864lowast1198882 and 119864lowast

1198883can be obtained from (3)

by polynomial fitting technique and K1198990

is stiffness matrixwith the 120576

119890119899= 0

According to definition of strain energy density equiva-lent strain of the 119899th element of composite plate 120576

119890119899 can be

expressed as

120576119890119899= radicx119879

K1198990

1198641198770119881119899

x (8)

where 119879 represents the transpose operation and 119881119899is the

volume of the 119899th elementSuppose that modal analysis theory still can be used to

solve (6) considering strain dependence of hard coating andnormal mode shape matrix always indicates constant valuefor the different strain states of composite plate thus the rela-tion between physical coordinates and normal coordinatescan be expressed as

x = Φ119873q (9)

where q is the response in normal coordinate and Φ119873

isnormal mode shape matrix which can be obtained from thefollowing equation

Φ119879

119873MΦ119873= I (10)

where I refers to identity matrix and its dimension is equal tothat ofM


Substitute (9) into (8) 120576119890119899

can be further simplified as

120576119890119899= radic

q119879h119899q

119881119899

(11)

h119899=Φ119879

119873K1198990Φ119873

1198641198770

(12)

By transforming (6) using normal mode shape matrixΦ119873 the resulting motion equation can be expressed as

(Klowast119873+ 119894120596C

119873minus 1205962I) q = F

119873 (13)

Klowast119873= Φ119879

119873KlowastΦ119873

= Klowast0119873+

119873

sum

119899

Φ119879

119873K1198990Φ119873

1198641198770

times (119864lowast

1198881120576119890119899+ 119864lowast

11988821205762

119890119899+ 119864lowast

11988831205763

119890119899)

(14)

Klowast0119873

= Φ119879

119873Klowast0Φ119873 C

119873= Φ119879

119873CΦ119873 F

119873= Φ119879

119873F(15)

where 120596 refers to excitation angular frequency Klowast119873refers to

complex stiffness matrix in normal coordinate andC119873refers

to damping matrix in normal coordinate It should be notedthat usually the damping matrix C

119873cannot be diagonalized

by the same transformation that diagonalizes themassmatrixM or stiffness matrix Klowast

0119873 But damping matrix C

119873can often

be regarded as viscous damping matrix as long as the 119896thdamping ratio of hard coating composite structure is lessthan 02 In this case the physical system can be considereda ldquoweak damped systemrdquo and damping matrix C

119873can be

diagonalized by the 119896th nature frequency and 119896th dampingratio (we will discuss how to bring the tested damping resultsinto this system in Section 42 in detail)

In order to solve nature frequencies of hard coatingcomposite plate the following equation can be used as theprinciple formula for iterative solution

10038161003816100381610038161003816119877 (Klowast119873) minus 1205962

119899I10038161003816100381610038161003816 = 0 (16)

where operator 119877() refers to getting the real part of anexpression in the bracket notation and 120596

119899is the 119899th nature

frequency (circular frequency)When the 2-norm of the difference between nature

frequency 120596(119895+1)119899

at the 119895 + 1th time and 120596(119895)119899

at the 119895th timemeets the following equation we can finally determine 120596

119899

by setting appropriate iteration termination condition in (17)whose calculation procedure will be described in detail inSection 41 Consider

10038171003817100381710038171003817120596(119895+1)

119899minus 120596(119895)

119899

100381710038171003817100381710038172le 1198780 (17)

where 1198780is the accuracy factor for iteration calculation

By solving (16) with iteration technique we can alreadycalculate nature frequency of coated structure Next wecan further calculate its vibration response such as nodedisplacement with considering strain dependence of hard

coating which can be regarded as searching for solutions of(13) Here Newton-Raphson iteration method is employedand through transposition treatment we get the followingexpression

r = (Klowast119873+ 119894120596C

119873minus 1205962I) q minus F

119873

= (Klowast0119873+ 119894120596C


119873

+ 119864lowast

1198881

119873

sum

119899

radicq119879h119899q

119881119899

h119899q + 119864lowast1198882

119873

sum

119899

q119879h119899q

119881119899

h119899q

+ 119864lowast

1198883

119873

sum

119899

(q119879h119899q

119881119899

)

32

h119899q

(18)

where r is the residual vectorBecause r is a complex vector it is relatively difficult to

solve this complex vector by some traditional techniques Forthe reliable accuracy it is necessary to separate the real andimaginary part of r and a similar separation is done on q andthe Jacobian matrix J whose solving equation with separatedreal and imaginary part can be expressed as

J =[[[

[

119877(120597r120597q119877

) 119877(120597r120597q119868

)

119868(120597r120597q119877

) 119868(120597r120597q119868

)

]]]

]

(19)

120597r120597q119877

= Klowast0119873+ 119894120596C

119873minus 1205962I

+

119873

sum

119899

((119864lowast

1198881

119881119899120576119899

+2119864lowast

1198882

119881119899

+3119864lowast

1198883120576119899

119881119899

) h119899q119877(h119899q)119879)

(20)

120597r120597q119868

= 119894 (Klowast0119873+ 119894120596C

119873minus 1205962I)

+

119873

sum

119899

((119864lowast

1198881

119881119899120576119899

+2119864lowast

1198882

119881119899

+3119864lowast

1198883120576119899

119881119899

) h119899q119868(h119899q)119879)

(21)

where q119877and q

119868are the real and imaginary part of the

response in normal coordinate respectively and operator 119868()refers to getting the imaginary part of an expression in thebracket notation

Then by using Newton-Raphson iteration method theiteration formula of (18) can be expressed as following

r(119895) + J(119895) sdot Δq(119895) = 0 (22)

q(119895+1) = q(119895) + Δq(119895) (23)

whereΔq(119895) is response increment at the 119895th time (superscript119895 or 119895+1 represents different iteration calculation times) and


r and q are the separation vectors of r and q which can beexpressed as following

r = 119877 (r)119868 (r) (24)

q = 119877 (q)119868 (q) (25)

In (22) and (23) we can determine its iteration termina-tion by calculating 2-norm of residual vector rwith separatedreal and imaginary part which can be expressed as following

r2 = radic(100381610038161003816100381611990311003816100381610038161003816

2

+100381610038161003816100381611990321003816100381610038161003816

2

+100381610038161003816100381611990331003816100381610038161003816

2

+ sdot sdot sdot ) le 1198780 (26)

When the above termination condition is achieved (egset accuracy factor for iteration calculation 119878

0= 0001) we

can obtain q Then adopt the following equation to restoreit into complex vector and bring q to (9) and the concernednode displacement x can be completely obtained Consider

q = 119877 (q) + 119894119868 (q) (27)

4 Nonlinear Vibration Analysis Procedure ofHard Coating Composite Plate by FEIM

In this section based on the deep understanding of principleformulas associated with finite element iteration method inSection 3 by taking an example of thin plate specimen coatedwith MgO + Al

2O3 analysis procedure of nature frequency

and vibration response of hard coating composite plate areproposed which combine ANSYS APDL program with self-designed MATLAB program and can be summarized asfollows

(1) firstly write APDL program to calculate and extractmass matrix damping matrix stiffness matrix andexternal excitation force vector of coated structure

(2) secondly write finite element iteration program basedon MATLAB to obtain equivalent strain of eachelement 120576

119890119899 and consequently get complex modulus

119864lowast

119888by the known mechanical parameters of MgO +

Al2O3obtained by curve-fitting of cube polynomial

(3) finally obtain complex stiffness matrix in normalcoordinate Klowast

119873 with considering strain dependent

mechanical parameters of hard coating and thusnature frequency and vibration response can beiteratively calculated by setting appropriate iterationtermination condition

41 Analysis Procedure of Nature Frequency In order tocalculate nature frequencies of hard coating composite platethe following analysis procedure is proposed as seen inFigure 2(a) some of the key steps need to be illustrated asfollow

(1) Extract Mass Matrix Damping Matrix Stiffness Matrixand External Excitation Force Vector of Composite PlateCalculating and extracting mass matrix damping matrix

stiffness matrix and external excitation force vector of com-posite plate is the basis for analyzing its inherent and dynamicresponse characteristics and this step is accomplished usingAPDL program Firstly input dimension parameters ofcomposite plate such as length width and thickness ofthe substrate plate and the coating Then input the relatedmechanical parameters without considering strain depen-dence such as the density Poissonrsquos ratio storage (Youngrsquos)modulus and the loss factor (noting that the coatingrsquos lossfactor is the ratio of loss modulus in (5) to storage modulusin (4) with the strain 120576

119890= 0) Next establish the finite

element model of coated plate and apply external excitationload and boundary condition to the model to carry out finiteelement modal analysis by QR damping method It should benoted that for the sake of convenient extraction of excitationforce vector (called nodal load vector) from ANSYS it isnecessary to apply external excitation load to the modelthis load could be single-point or multipoint excitation forceor base excitation In this example clamped-free boundarycondition andbase excitation are chosen because theywill notbring additional quality to the coated plate and it is relativelyeasy to verify such finite element model by experimentaltest method At last by outputting in the form of textmass matrix damping matrix stiffness matrix and externalexcitation force vector of composite plate can be obtained

(2) Calculate Iteration Initial Value of Nature FrequencyNormal Mode Shape Matrix and Normal Force In this stepiteration initial value of nature frequency

119899 is obtained by

(28) on the premise that strain dependence characteristic ofthe coating has not been considered Additionally normalmode shape matrix Φ

119873 can be obtained by performing

operation on mass matrix M in (10) and normal force F119873

can be calculated by (15) It should be noted that this andthe following steps are all accomplished by the self-designedMATLAB program

10038161003816100381610038161003816K minus

2

119899M10038161003816100381610038161003816 = 0 (28)

(3) Calculate Iteration Initial Value of Resonant Response inNormal Coordinate In this step iteration initial value of reso-nant response in normal coordinate q is obtained by (29) onthe same premise that strain dependence characteristic of thecoating has not been considered and

119899can be obtained from

step (1) andKlowast0119873

can be obtained by performing operation onthe global stiffness matrix Klowast

0 in (15) Thus select a certain

nature frequency which needs to be iteratively calculated inthe following steps and by using (29) we can obtain thecorresponding iteration initial value of resonant response q

(Klowast0119873minus 2

119899I) q = F

119873 (29)

(4) Calculate Equivalent Strain of Each Element and ComplexStiffness Matrix of Composite Plate In this step becauseiteration initial value of resonant response q has alreadybeen obtained we can further substitute q = q into (11)and combine this equation with (12) thus equivalent strainof each element of coated plate 120576

119890119899 can be calculated


Yes

No

Without consideringstrain dependence

of hard coating

With consideringstrain dependence

of hard coating

Calculate 120576en and lowastN

Calculate 120596nat certain mode

120596(j+1)n minus 120596

(j)n

2le S0

Output 120596n

Extract M C Klowast and F

Calculate nΦN and NF

Calculate orq q

K

(a) Analysis procedure of nature frequency

Yes

No


of hard coating


of hard coating

Calculate r rand

Output x


Calculate initial q

120597 I JCalculate 120597 120597 R 120597r

r qq

2 le S0r

Calculate Δ andq q q x

(b) Analysis procedure of vibration response

Figure 2 Nonlinear vibration analysis procedure of hard coating composite plate by FEIM

Next get complex modulus 119864lowast119888by the known mechanical

parameters of MgO + Al2O3obtained by curve-fitting of

cube polynomial Then bring 120576119890119899

and 119864lowast

119888into (14) and

complex stiffness matrix in normal coordinate and Klowast119873can

be calculated with considering strain dependent mechanicalparameters of hard coating which is a critical step in makingsure that iteration calculation of nature frequency can besuccessfully conducted

(5) Iteratively Calculate Nature Frequency of Composite PlateIn this step in the effort to iteratively calculate 119899th naturefrequency 120596

119899associated with the acquired complex stiffness

matrix Klowast119873in (16) it is necessary to set appropriate accuracy

factor for iteration calculation 1198780 If the resulting 120596

119899do

not meet the requirement of 1198780 it is needed to recalculate

resonant response q by using 120596119899as the excitation frequency

By carefully comparing calculation results when 1198780is equal to

0001 it is accurate enough to determine120596119899and its calculation

efficiency is relatively satisfying Finally by repeating step (3)to step (5) the first five or more nature frequencies can besequentially obtained

42 Analysis Procedure of Vibration Response In order tocalculate vibration response of hard coating composite platethe following analysis procedure is proposed as seen inFigure 2(b) some of the key steps need to be illustrated asfollows

(1) Extract Mass Matrix Damping Matrix Stiffness Matrixand External Excitation Force Vector of Composite Plate Thestep has been discussed in Section 41 in detail

(2) Calculate Iteration Initial Value of Vibration Responsein Normal Coordinate In this step iteration initial valueof vibration response in normal coordinate q is obtainedon the premise that strain dependence characteristic of thecoating has not been considered Firstly use the extractedmass matrix damping matrix stiffness matrix and externalexcitation force vector in step (1) to calculate the relatedmatrix in normal coordinate Then choose excitation fre-quency 120596 to get the corresponding iteration initial value ofvibration response q by the following equation

(Klowast0119873+ 119894120596C

119873minus 1205962I) q = F

119873 (30)

whereKlowast0119873C119873 and F

119873can be obtained by performing oper-

ation on Klowast0 C and F in (15) Because damping matrix C

119873

is regarded as viscous damping matrix without consideringstrain dependence of hard coating which is approximatelyequal to the corresponding dampingmatrix of uncoated platestructure it is actually a diagonal matrix and its expression inthe 119896th row and 119896th column can be approximately written as

C119873(119896 119896) asymp 2120596

1015840

1198961205771015840

119896 (31)

where 1205961015840

119896and 120577

1015840

119896are the 119896th nature frequency and 119896th

damping ratio of uncoated plate respectively which can beaccurately measured by experimental test

(3) Calculate Residual Vector and Separate Its Real andImaginary Part Calculating and converting residual vector toget the separated real and imaginary part is a critical step inmaking sure that iteration calculation of vibration responsecan be successfully conducted because it is necessary to


Table 1 The geometry parameters and material parameters of the composite plate

Name Material Type Free length(mm)

Clampinglength (mm)

Width(mm)

Thickness(mm)

Density(kgm3)

Youngrsquosmodulus (GPa)

Poissonrsquosratio

Substrate plate Ti-6Al-4V 134 20 110 10 4420 1103 03Hard coating MgO + Al2O3 134 20 110 002 2565 504 03

separate the real and imaginary part for residual vector rin the effort to accurately calculate vibration response innormal coordinate q Firstly bring iteration initial value ofvibration response in normal coordinate q obtained fromstep (2) into (18) residual vector r with considering straindependent mechanical parameters of hard coating can beobtained Then by using (24) we can separate its real andimaginary part to get r

(4) Calculate Jacobian Matrix In this step because iterationinitial value of vibration response q has already beenobtained we can further substitute q = q into (20) and (21)and combine these equations with (12) to get 120597r120597q

119877and

120597r120597q119868 Next by bringing the results into (19) we can obtain

Jacobian matrix J

(5) Iteratively CalculateVibrationResponse of Composite PlateFirstly substitute r obtained from step (3) and Jacobianmatrix J obtained from step (4) into (22) to get responseincrement Δq at different iteration calculation times Thenby using (23) to calculate separation vector of vibrationresponse in normal coordinate q adopt (27) to restore itinto complex vector q In these steps setting appropriateaccuracy factor for iteration calculation 119878

0in (26) is very

important if the resulting residual vector r does not meet therequirement of 119878

0 it is needed to recalculate q Similar to step

(5) in the analysis procedure of nature frequency from Sec-tion 41 by carefully comparing calculation results we choose1198780= 0001 which has been proved to be accurate enough

to determine q with relatively high calculation efficiencyFinally bring q to (9) andwe can completely obtain vibrationresponse x

5 A Study Case

In this section a composite plate coated with MgO + Al2O3

is taken as a study object as seen in Figure 3 By consideringstrain dependence of MgO + Al

2O3 nature frequencies and

vibration responses of coated plate are calculated by FEIMand its results are compared with linear results for betterclarifying analysis effect of nonlinear vibration Meanwhilein order to ensure that the finite element model and cor-responding cantilever boundary condition are trustworthyexperimental test is conducted on the plate specimen coatedwith or without hard coating to provide input parameters fortheoretical calculation At last nonlinear vibration responsescontaining certain nature frequency under different excita-tion levels are measured Through comparing the resultingfrequency and response results the practicability and relia-bility of FEIM have been verified

51 Nonlinear Vibration Test and Its Results The geometryparameters and material parameters of the composite plateare listed in Table 1 the substrate plate is Ti-6Al-4V andthe hard coating is MgO + Al

2O3 its density is already

given in [13] Besides Youngrsquos modulus can be obtained fromFigure 1(a) with strain 120576

119890= 0 and Poisson is not measured but

is assumed to be 03 It should be noted that each geometryparameter is measured by vernier caliper three times and theaverage is used as the final result and similar test is done byelectron microscope on the thickness of the coating

In order to accurately test nature frequencies and vibra-tion response of composite plate the following vibration testsystem is set up and its schematic and real test picture canbe seen in Figures 3 and 4 The instruments used in the testare as follows (I) king-design EM-1000F vibration shakersystems (II) Polytec PDV-100 laser Doppler vibrometer (III)BK 4514-001 accelerometer (VI) LMS SCADAS 16-channelmobile front end and Dell notebook computer In the testset-up vibration shaker is used to provide base excitationwithout bringing additional mass and stiffness to the testedplate laser Doppler vibrometer mounted to a rigid supportis used to measure the plate velocity at a single point alongthe excitation direction of vibration shaker while the baseresponse signal is simultaneously acquired by the accelerom-eter which is fixed at the clamping fixture Besides LMSSCADAS 16-channel data acquisition front end is responsiblefor recording these signals andDell notebook computer (withIntel Core i7 293GHz processor and 4G RAM) is used tooperate LMS Test Lab 10B software and store measured data

There are two holes with diameter of 9mm in the platewhich are machined to be clamped by two M8 bolts in theclamping fixture so that clamped-free boundary conditioncan be simulated The length width and thickness of theclamping fixture are about 200mm 20mm and 20mmrespectively so the clamping length of the plate is 20mmThelaser point is about 22mm above the constraint end and thehorizontal distance between this point and the left free edgeof the plate is about 24mm In practice in order to ensurethe tested plate is effectively clamped the first three naturefrequencies are measured under different tightening torqueThrough comparisonwith each other the best repeatability offrequency values can be reached when the tightening torqueis 34Nm so this torque value is used in the formal testregardless of whether the plate specimen is coated with orwithout hard coating

On the one hand in order to accurately analyze nonlinearvibration response of hard coating composite plate it isnecessary to measure each nature frequency and dampingratio of uncoated plate to provide input parameters foriteration calculation Firstly sine sweep test is done onthe uncoated plate by selecting sweep frequency range of 0sim


Table 2 The first 8 nature frequencies and damping ratios of uncoated plate obtained by experimental test

Modal order 1 2 3 4 5 6 7 8Nature frequency (Hz) 468 1326 2894 4734 5354 8249 9192 9953Damping ratio () 080 019 004 003 004 007 005 004

Composite plateAccelerometerLMS mobile front end

Laser Doppler vibrometer

Baseexcitation

Data processing

Responsesignal

ComputerTime processing Frequency analysis

Baseresponse

Vibration shaker

Laser point

Accelerometerpoint

Figure 3 Schematic of vibration test system of hard coating composite plate

Hard coating

Vibration shaker

Laser Doppler

LMS mobile front end

composite plate

vibrometer

Figure 4 Real test picture of vibration test system of hard coatingcomposite plate

1000Hz base excitation amplitude of 01 g and quick sweepspeed of 5Hzs and consequently the raw sweep signal canbe recorded Next employ FFT processing technique to getfrequency spectrum of the signal and the first 8 naturefrequencies can be roughly determined by identifying theresponse peak in the related frequency spectrum so thatsweep frequency range which contains each imprecise valueof nature frequency can be determined

Because too rapid sweep speed will lead to higher damp-ing values for the coated or uncoated plate according to the

literature [18] the reasonable sweep speed 119878 (Hzs) bettermeets the following equation in (32) so that the effect oftransient vibration in sweep test can be reduced to the lowestlevel If the 6th mode of the coated plate is our concern therequired sweep speed 119878 should less than 119878max asymp (8249 times

00007)2= 033Hz But considering sweep efficiency and

testing experience in our test series sweep speed of 1Hzs isfinally chosen to conduct sweep test and the first 8 naturefrequencies and damping ratios of the uncoated plate can bedetermined by half-power bandwidth technique as listed inTable 2 Consider

119878 lt 119878max = (120578119891119896

2)

2

asymp 1205772

1198961198912

119896 (32)

where 119891119896and 120577


damping ratio of the coated or uncoated plate respectivelyand 120578 is the loss factor (approximately two times largerthan 120577

119896) These parameters can be measured by half-power

bandwidth technique during sweep experimentOn the other hand in order to verify the practicability

and reliability of FEIM nature frequencies of coated plateunder lower excitation level of 03 g are also obtained andthe results are listed in Table 4 (for the convenience ofcomparing the analysis errors of iteration calculation) Andby taking the 6th nature frequency and vibration response asan example nonlinear vibration test on coated plate is done


Table 3 Nature frequencies of the uncoated plate by FEM calculation

Modal order 1 2 3 4 5 6 7 8Nature frequency (Hz) 467 1329 2890 4639 5328 8213 9148 9978

Table 4 The first 8 nature frequencies of hard coating composite plate under 03 g excitation level obtained by finite element iterationcalculation and experimental test

Modal order 1 2 3 4 5 6 7 8Experimental test (Hz) 470 1321 2875 4704 5361 8234 9183 9972Finite element iteration calculation (Hz) 466 1326 2884 4634 5305 8198 9112 9953Analysis errors () 085 minus038 minus031 149 104 043 099 minus011

Table 5The 6th nature frequencies and resonant responses of composite plate under different excitation levels obtained by experimental testand theoretical calculation

Excitationlevel (g)

Experimental test Linear calculation Finite element iteration calculationNature

frequency(Hz)

Resonantresponse 119860(mms)

Naturefrequency

(Hz)


Errors(119861 minus 119860)119860

()

Naturefrequency

(Hz)


Errors(119862 minus 119860)119860

()03 8234 467 8198 502 75 8198 482 3204 8232 624 8198 655 50 8196 632 1305 8230 726 8198 829 142 8195 779 7306 8229 856 8198 1025 197 8193 932 8907 8228 951 8198 1178 239 8191 1044 98

012

009

006

003

0821 823 825 827

07 g06 g05 g

04 g03 g

Frequency (Hz)

Vibr

atio

n re

spon

se (m

s)

Figure 5 Frequency response spectrums containing the 6th naturefrequency under different excitation levels obtained by experimentaltest

under different base excitation levels Firstly set excitationamplitude as total of 5 excitation levels in the softwarenamely 03 g 04 g 05 g 06 g and 07 g and then choose

much slower sweep speed such as 1Hzs to accurately obtainthe corresponding frequency response spectrum containingthe 6th nature frequency under each excitation level as seenin Figure 5 Consequently the resulting nature frequenciesand resonant responses under different excitation levels canbe identified as listed in Table 5

52 Nonlinear Vibration Analysis and Its Results SHELL281is selected to establish finite element model of the plate withor without hard coating based on ANSYS APDL program asseen in Figures 6(a) and 6(b) and all the nodes along oneside of the plate are constrained in all six degrees of freedomto represent clamped-free boundary condition in the ANSYSsoftware Besides the substrate plate and the coating are bothmeshed with 10 elements along the length or width directionand the corresponding nodes between them are all coupledtogether Firstly employ uncoated platemodel to calculate thefirst 8 nature frequencies by QR damping or Block Lanczosmethod as listed in Table 3 By comparing the resultingfrequencieswith test results in Table 2 it can be found out thatthey are very close to each otherTherefore we can ensure thatthis model and corresponding cantilever boundary conditionare trustworthyThen we can use coated plate model to carryoutmodal analysis byQRdampingmethod and consequentlyextract mass matrix damping matrix stiffness matrix andexternal excitation force vector of composite plate Becausethe 6th mode of the coated plate is our concern the relatedmodal shape is also calculated and extracted as seen inFigure 6(c)

According to the analysis procedure of nature frequencyproposed in Section 41 with considering strain dependence


Substrate plate

(a) Uncoated plate model

Composite plate

Calculated response point

Hard coating

(b) Coated plate model

(c) The 6th modal shape of the coated plate

Figure 6 Finite element model and modal shape of hard coating composite plate

Vibr

atio

n re

spon

se (m

s)

012

009

006

003

0816 818 820 822 824

Frequency (Hz)

07 g06 g05 g

04 g03 g

Figure 7 The 6th frequency response spectrums of hard coatingcomposite plate obtained by linear calculation method

of MgO + Al2O3 the first 8 nature frequencies of coated

plate are calculated by FEIM based on theMATLAB program(under 03 g excitation level similar to the one in experimentaltest) as listed in Table 4 Besides the corresponding analysis

errors of iteration calculation from 1st to 8th mode areobtained and listed in the same table

After finishing calculation work of nature frequencies ofcoated plate we can further iteratively calculate its responsesbased on the analysis procedure of vibration response pro-posed in Section 42 Similar to experimental test includingthe same excitation levels (03 g 04 g 05 g 06 g and 07 g)and the same response point nonlinear vibration responsescontaining the 6th nature frequency under different excita-tion levels are obtained and the resulting frequencies andresonant responses are listed in Table 5 Then for betterclarifying analysis effect of nonlinear vibration the relatedlinear calculation results under the above excitation levels arealso given in the same table (in this circumstances storagemodulus 1198641015840

119888(120576119890) and loss modulus 11986410158401015840

119888(120576119890) are linear in (4)

and (5) with equivalent strain 120576119890= 0) and consequently

analysis error results between these two theoretical calcu-lation method and experimental method are obtained andlisted in Table 5 At last by applying finite element iterationcalculation as well as linear calculation method frequencyresponse spectrums in the vicinity of the 6th resonantresponse are plotted with frequency interval of 001Hz in therange of 816sim824Hz and the resulting spectrums can be seenin Figures 7 and 8

53 Result Analysis and Discussion We analyze and discussthe results as follows

(1) As can be seen from Table 4 the first 8 naturefrequencies of coated plate under 03 g excitation levelobtained by finite element iteration calculation andexperimental test are in good agreement and the

Shock and Vibration 11Vi

brat

ion

resp

onse

(ms

)

012

009

006

003

0817 819 821 823

Frequency (Hz)

07 g06 g05 g

04 g03 g

Figure 8 The 6th frequency response spectrums of hard coatingcomposite plate obtained by finite element iteration method

maximum analysis errors are less than 2Thereforethis nonlinear calculation method can be used toanalyze nature frequency of hard coating compositeplate with high preciseness

(2) As can be seen from Table 5 the resulting 6th naturefrequencies obtained under different excitation levelsby finite element iteration calculation can complywith the same rule discovered in experimental testthat is nature frequency of coated plate will decreasewith the increase of base excitation level whichshows soft stiffness nonlinear or ldquostrain softeningrdquoAdditionally the results also show that the reso-nant responses obtained by iteration calculation arerelatively lower than the ones obtained by linearcalculation and linear calculation errors are largerthan the corresponding errors of iteration calculationFor example themaximum error of linear calculationis about 24 while the one of iteration calculationis less than 10 which are within an acceptablerange Therefore the reliability of FEIM has beenverified which can be used to analyze nonlinearvibration response of composite plate with consider-ing strain dependent mechanical parameters of hardcoating

(3) However the maximum error of iteratively calculatedresonant response reaches 10 and the amplitudedifferences in frequency response spectrums betweenthe experiment and theoretical calculation from Fig-ures 5 and 8 are still existed which might be due to

the following reasons (I) the laser point in experi-mental test and response point chosen in theoreticalcalculation are not closely coincident (II) the nonlin-earities of hard coating composite plate are caused bynot only strain dependence of the coating but alsononlinearity of clamping fixture shear deformation ofthe uncoated plate and other factors which are notconsidered in the above nonlinear vibration analysisprocedure (III) comparing with the real values thereare some errors in geometry parameters and materialparameters used in theoretical calculation for exam-ple Youngrsquos modulus loss modulus and the thicknessof the coating and so forth

6 Conclusions

This research proposes finite element iteration method toanalyze nonlinear vibration of hard coating thin plate andexperimental test is used to verify the practicability andreliability of this nonlinear calculation method Based on theanalysis and experimental results the following conclusionscan be drawn

(1) Polynomial method can be used to characterize thestorage and loss modulus of coating material whichare mainly responsible for nonlinearity of hard coat-ing composite plate

(2) FEIM combines the advantages of ANSYS APDLprogram and self-designed MATLAB program withthe following analysis techniques (I) firstly writeAPDL program to calculate and extract mass matrixdampingmatrix stiffness matrix and external excita-tion force vector of finite element model of compositeplate (II) secondly write finite element iterationprogram based on MATLAB to calculate equivalentstrain of each element and consequently get complexmodulus by the known mechanical parameters ofhard coating obtained by polynomial method (III)finally obtain complex stiffness matrix in normalcoordinate and thus nature frequency and vibra-tion response can be iteratively calculated by settingappropriate iteration termination condition

(3) FEIM is an effective theoretical analysis method toanalyze nonlinear dynamics mechanism of hard coat-ing composite plate because the calculation resultsare in good agreement with the experimental resultsFor example the maximum analysis errors of thefirst 8 nature frequencies is less than 2 and theerrors of the 6th resonant response under differentexcitation levels are less than 10 Besides the iter-atively calculated results can comply with the samerule discovered in experimental test that is naturefrequency of composite plate will decrease with theincrease of external excitation level which shows softstiffness nonlinear or ldquostrain softeningrdquo


Conflict of Interests

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

Acknowledgment

This study was supported by the National Nature ScienceFoundation of China 51375079

References

[1] H-Y Yen andM-H Herman Shen ldquoPassive vibration suppres-sion of beams and blades using magnetomechanical coatingrdquoJournal of Sound andVibration vol 245 no 4 pp 701ndash714 2001

[2] G Gregori L Lı J A Nychka and D R Clarke ldquoVibrationdamping of superalloys and thermal barrier coatings at high-temperaturesrdquoMaterials Science andEngineeringA vol 466 no1-2 pp 256ndash264 2007

[3] G Y Du D C Ba Z Tan W Sun K Liu and Q K HanldquoVibration damping performance of ZrTiN coating depositedby arc ion plating on TC4 Titanium alloyrdquo Surface and CoatingsTechnology vol 229 pp 172ndash175 2013

[4] F Ivancic and A Palazotto ldquoExperimental considerations fordetermining the damping coefficients of hard coatingsrdquo Journalof Aerospace Engineering vol 18 no 1 pp 8ndash17 2005

[5] C Blackwell A Palazotto T J George and C J Cross ldquoTheevaluation of the damping characteristics of a hard coating ontitaniumrdquo Shock and Vibration vol 14 no 1 pp 37ndash51 2007

[6] J E HanselThe influence of thickness on the complex modulus ofair plasma sprayed ceramic blend coatings [MS thesis] WrightState University Dayton Ohio USA 2008

[7] N Tassini S Patsias and K Lambrinou ldquoCeramic coatings aphenomenological modeling for damping behavior related tomicrostructural featuresrdquo Materials Science and Engineering Avol 442 no 1-2 pp 509ndash513 2006

[8] J Green and S Patsias ldquoA preliminary approach for themodeling of a hard damping coating using friction elementsrdquoin Proceedings of 7th National Turbine Engine High Cycle FatigueConference pp 1ndash9 Palm Beach Gardens Fla USA 2002

[9] P J Torvik ldquoA slip damping model for plasma sprayed ceram-icsrdquo Journal of Applied Mechanics vol 76 no 6 Article ID061018 8 pages 2009

[10] R K A Al-Rub and A N Palazotto ldquoMicromechanicaltheoretical and computational modeling of energy dissipationdue to nonlinear vibration of hard ceramic coatings withmicrostructural recursive faultsrdquo International Journal of Solidsand Structures vol 47 no 16 pp 2131ndash2142 2010

[11] S Patsias C Saxton and M Shipton ldquoHard damping coat-ings an experimental procedure for extraction of dampingcharacteristics and modulus of elasticityrdquoMaterials Science andEngineering A vol 370 no 1-2 pp 412ndash416 2004

[12] P J Torvik ldquoDetermination of mechanical properties of non-linear coatings from measurements with coated beamsrdquo Inter-national Journal of Solids and Structures vol 46 no 5 pp 1066ndash1077 2009

[13] S A Reed A N Palazotto and W P Baker ldquoAn experimentaltechnique for the evaluation of strain dependent materialproperties of hard coatingsrdquo Shock and Vibration vol 15 no6 pp 697ndash712 2008

[14] E M Hernandez and D Bernal ldquoIterative finite element modelupdating in the time domainrdquo Mechanical Systems and SignalProcessing vol 34 no 1-2 pp 39ndash46 2013

[15] Y Hu Q Di W Zhu Z Chen and W Wang ldquoDynamiccharacteristics analysis of drillstring in the ultra-deep well withspatial curved beamfinite elementrdquo Journal of PetroleumScienceand Engineering vol 82-83 pp 166ndash173 2012

[16] D Soares Jr ldquoAn optimised FEM-BEM time-domain iterativecoupling algorithm for dynamic analysesrdquo Computers amp Struc-tures vol 86 no 19-20 pp 1839ndash1844 2008

[17] S Filippi and P J Torvik ldquoA methodology for predictingthe response of blades with nonlinear coatingsrdquo Journal ofEngineering for Gas Turbines and Power vol 133 no 4 ArticleID 042503 2011

[18] P J Torvik ldquoOn estimating system damping from frequencyresponse bandwidthsrdquo Journal of Sound and Vibration vol 330no 25 pp 6088ndash6097 2011

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computational model to assess the main micromechanicalmechanisms responsible for the experimentally observednonlinear vibration in plasma sprayed hard ceramic coatingsThese studies have contributed a lot to the understanding ofenergy dissipation mechanism of hard coating as well as therelated nonlinear vibration of composite structure

However it is not realistic to completely understandnonlinear vibration characteristics of hard coating compos-ite structure from microscopic scope in order to effec-tively implement damping coating technique for vibrationsuppression it is necessary to develop mechanical modelin accordance with macrovibration theory Unfortunatelyresearch efforts spent in this area are not adequate enoughbecause it is still a challenge to present a reliable modelto describe the mechanical properties of hard coating com-posite structure when the observed response is that of anonlinear system which have generally been found to displaya strong dependence on the amplitude of strain Besidesanother challenge is to identify mechanical parameters ofthe coating such as strain dependent Youngrsquos modulus andthe loss modulus which are mainly responsible for non-linearity of the coated structure Fortunately several papershave made some progress in identifying such parametersPatsias et al [11] tested Nimonic C-263 beam specimencoatedwith aluminiumoxide under different excitation levelsand proposed an experimental procedure for extracting thecoatingrsquos damping (internal friction) and the modulus ofelasticity with the variation of strain from response decaysTorvik [12] presented amethodology suitable for determiningthe amplitude-dependent mechanical properties of coatingapplied on substrate beams and its storage (Youngrsquos) mod-ulus the loss modulus and the loss factor can be extractedby careful measurement of specimen dimensions weightsnature frequencies and loss factors before and after coatingReed et al [13] set up a novel vibration experiment system andproposed an experimental technique for accurate measure-ment of strain dependent stiffness and damping propertiesof hard coatings at high strain levels which provided animportant support for modeling coating composite structureand its nonlinear vibration analysis

Finite element iteration technique is an efficient and prac-tical method which can be used to solve vibration problemswith considering variable mass stiffness and damping in thekinetic equation For example Hernandez and Bernal [14]adopted an iterative time domain formulation for finite ele-ment model to update the stiffness and damping parametersof a twenty-degree-of-freedom shear beam Hu et al [15]considered the contact between drillstring and borehole andproposed an iterative finite element technology to analyzedrillstring dynamic Soares [16] presented an optimised FEM-BEM iterative coupling algorithm and demonstrated theeffectiveness on an elastodynamic rectangular rod and acircular cavity Filippi and Torvik [17] proposed iterativefinite element analysis method to predict the response of anactual blade with a nonlinear coating which can be usedto quickly determine the approximate resonant frequencydamping and response amplitude in the vicinity of a par-ticular resonance However linear analytic expressions areused for the nonlinear coating material and the structural

damping is expressed by loss factor whichmay help in gettingdampingmatrix in uniform form but fails to separate viscousdamping from loss factor of the material Therefore if we caneffectively express strain dependent mechanical parametersof hard coating and consequently introduce them into thefinite element equation to solve the nodal displacementsin the equation by numerical iteration method it wouldbe a good solution to theoretical analysis of the concernednonlinear dynamics mechanism of hard coating compositestructure

This paper studies nonlinear vibration mechanism ofhard coating thin plate based on macroscopic vibrationtheory and proposes finite element iteration method (FEIM)to theoretically calculate its nature frequency and vibrationresponse In Section 2 strain dependent mechanical prop-erty of hard coating is briefly introduced and polynomialmethod is adopted to characterize the storage and lossmodulus of coating material Then go on to theoreticallyderive the principle formulas of inherent and dynamicresponse characteristics of the hard coating composite platein Section 3 and consequently propose specific analysisprocedure by combining ANSYS APDL and self-designedMATLAB program in Section 4 Finally in Section 5 acomposite plate coated with MgO + Al

2O3is taken as a

study object and both nonlinear vibration test and analysisare conducted on the plate specimen with considering straindependent mechanical parameters of hard coating Throughcomparing the resulting frequency and response results thepracticability and reliability of FEIM have been verified andthe corresponding analysis results can provide an importantreference for further study onnonlinear vibrationmechanismof hard coating composite structure

2 Polynomial Characterization ofStrain Dependent Mechanical Parameters ofHard Coating

Suppose that 119864lowast119888can be used to represent elastic modulus of

hard coating which is defined as complex modulus119864lowast

119888= 1198641015840

119888(120576119890) + 119894119864

10158401015840

119888(120576119890) (1)

where ldquolowastrdquo refers to complex1198641015840119888(120576119890)11986410158401015840119888(120576119890) are storagemodu-

lus (or Youngrsquos modulus) and loss modulus respectively andthey are all functions of equivalent strain 120576

119890

For simplicity we can adopt polynomial method tocharacterize 1198641015840

119888(120576119890) and 11986410158401015840

119888(120576119890) which can be expressed as

1198641015840

119888(120576119890) = 1198641198770+ 1205761198901198641198771+ 1205762

1198901198641198772+ 1205763

1198901198641198773+ sdot sdot sdot

11986410158401015840

119888(120576119890) = 1198641198680+ 1205761198901198641198681+ 1205762

1198901198641198682+ 1205763

1198901198641198683+ sdot sdot sdot

(2)

where 1198641198770 1198641198680

are the storage and loss modulus withoutconsidering strain dependence and 119864

119877119899 119864119868119899(119899 = 1 2 3 )

are the specific coefficients of strain dependent storagemodulus and loss modulus

Substituting (2) into (1) and elastic modulus of hardcoating 119864lowast

119888can be expressed as

119864lowast

119888= 119864lowast

1198880+ 119864lowast

1198881120576119890+ 119864lowast

11988821205762

119890+ 119864lowast

11988831205763

119890+ sdot sdot sdot (3)

where 119864lowast1198880= 1198641198770+ 1198941198641198680and 119864lowast

119888119899= 119864119877119899+ 119894119864119868119899 119899 = 1 2 3


0 200 400 600 800


Stor

age m

odul

us(G

pa)

35

40

45

50

55

120583120576 (120583m120583m)

(a) Storage modulus

25

15

05

1

2

3

0

Loss

mod

ulus

(Gpa

)

200 400 600 800120583120576 (120583m120583m)


(b) Loss modulus




1198641015840

119888(120576) = 50439 minus 005386120576

119890

+ 92396 times 10minus51205762

119890minus 54190 times 10

minus81205763

119890

(4)

11986410158401015840

119888(120576) = 09122 + 001321120576

119890


119890+ 22086 times 10

minus81205763

119890

(5)







Klowast = Klowast0+

119873

sum

119899

K1198990(119864lowast

1198881120576119890119899+ 119864lowast

11988821205762

119890119899+ 119864lowast

11988831205763

119890119899)

1198641198770

(119899 = 1 2 3 119873)

(7)








119890119899= 0


119890119899 can be

expressed as

120576119890119899= radicx119879

K1198990

1198641198770119881119899

x (8)




x = Φ119873q (9)



Φ119879

119873MΦ119873= I (10)



Substitute (9) into (8) 120576119890119899


120576119890119899= radic

q119879h119899q

119881119899

(11)

h119899=Φ119879

119873K1198990Φ119873

1198641198770

(12)


(Klowast119873+ 119894120596C

119873minus 1205962I) q = F

119873 (13)

Klowast119873= Φ119879


= Klowast0119873+

119873

sum

119899

Φ119879

119873K1198990Φ119873

1198641198770

times (119864lowast

1198881120576119890119899+ 119864lowast

11988821205762

119890119899+ 119864lowast

11988831205763

119890119899)

(14)

Klowast0119873

= Φ119879

119873Klowast0Φ119873 C

119873= Φ119879

119873CΦ119873 F

119873= Φ119879

119873F(15)







119873can often


119873can be



10038161003816100381610038161003816119877 (Klowast119873) minus 1205962

119899I10038161003816100381610038161003816 = 0 (16)




frequency 120596(119895+1)119899

at the 119895 + 1th time and 120596(119895)119899


119899


10038171003817100381710038171003817120596(119895+1)

119899minus 120596(119895)

119899

100381710038171003817100381710038172le 1198780 (17)




r = (Klowast119873+ 119894120596C


119873

= (Klowast0119873+ 119894120596C


119873

+ 119864lowast

1198881

119873

sum

119899


119881119899

h119899q + 119864lowast1198882

119873

sum

119899

q119879h119899q

119881119899

h119899q

+ 119864lowast

1198883

119873

sum

119899

(q119879h119899q

119881119899

)

32

h119899q

(18)



J =[[[

[

119877(120597r120597q119877

) 119877(120597r120597q119868

)

119868(120597r120597q119877

) 119868(120597r120597q119868

)

]]]

]

(19)

120597r120597q119877

= Klowast0119873+ 119894120596C

119873minus 1205962I

+

119873

sum

119899

((119864lowast

1198881

119881119899120576119899

+2119864lowast

1198882

119881119899

+3119864lowast

1198883120576119899

119881119899

) h119899q119877(h119899q)119879)

(20)

120597r120597q119868

= 119894 (Klowast0119873+ 119894120596C

119873minus 1205962I)

+

119873

sum

119899

((119864lowast

1198881

119881119899120576119899

+2119864lowast

1198882

119881119899

+3119864lowast

1198883120576119899

119881119899

) h119899q119868(h119899q)119879)

(21)

where q119877and q




r(119895) + J(119895) sdot Δq(119895) = 0 (22)

q(119895+1) = q(119895) + Δq(119895) (23)




r = 119877 (r)119868 (r) (24)

q = 119877 (q)119868 (q) (25)


r2 = radic(100381610038161003816100381611990311003816100381610038161003816

2

+100381610038161003816100381611990321003816100381610038161003816

2

+100381610038161003816100381611990331003816100381610038161003816

2



0= 0001) we


q = 119877 (q) + 119894119868 (q) (27)








119864lowast

















10038161003816100381610038161003816K minus

2

119899M10038161003816100381610038161003816 = 0 (28)








119899I) q = F

119873 (29)




Yes

No


of hard coating


of hard coating



120596(j+1)n minus 120596

(j)n

2le S0

Output 120596n



Calculate orq q

K


Yes

No


of hard coating


of hard coating

Calculate r rand

Output x


Calculate initial q

120597 I JCalculate 120597 120597 R 120597r

r qq

2 le S0r







and 119864lowast

119888into (14) and







119899do









(Klowast0119873+ 119894120596C

119873minus 1205962I) q = F

119873 (30)




119873


C119873(119896 119896) asymp 2120596

1015840

1198961205771015840

119896 (31)

where 1205961015840

119896and 120577

1015840







Clampinglength (mm)

Width(mm)

Thickness(mm)

Density(kgm3)


Poissonrsquosratio




119877and


Jacobian matrix J


0in (26) is very





5 A Study Case


















Baseexcitation

Data processing

Responsesignal


Baseresponse

Vibration shaker

Laser point

Accelerometerpoint


Hard coating

Vibration shaker

Laser Doppler


composite plate

vibrometer







119878 lt 119878max = (120578119891119896

2)

2

asymp 1205772

1198961198912

119896 (32)

where 119891119896and 120577












Excitationlevel (g)


frequency(Hz)


Naturefrequency

(Hz)


Errors(119861 minus 119860)119860

()

Naturefrequency

(Hz)


Errors(119862 minus 119860)119860

()03 8234 467 8198 502 75 8198 482 3204 8232 624 8198 655 50 8196 632 1305 8230 726 8198 829 142 8195 779 7306 8229 856 8198 1025 197 8193 932 8907 8228 951 8198 1178 239 8191 1044 98

012

009

006

003

0821 823 825 827

07 g06 g05 g

04 g03 g

Frequency (Hz)

Vibr

atio

n re

spon

se (m

s)







Substrate plate


Composite plate


Hard coating




Vibr

atio

n re

spon

se (m

s)

012

009

006

003

0816 818 820 822 824

Frequency (Hz)

07 g06 g05 g

04 g03 g






119888(120576119890) and loss modulus 11986410158401015840

119888(120576119890) are linear in (4)






brat

ion

resp

onse

(ms

)

012

009

006

003

0817 819 821 823

Frequency (Hz)

07 g06 g05 g

04 g03 g






6 Conclusions








Acknowledgment


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 200 400 600 800


Stor

age m

odul

us(G

pa)

35

40

45

50

55

120583120576 (120583m120583m)

(a) Storage modulus

25

15

05

1

2

3

0

Loss

mod

ulus

(Gpa

)

200 400 600 800120583120576 (120583m120583m)


(b) Loss modulus




1198641015840

119888(120576) = 50439 minus 005386120576

119890

+ 92396 times 10minus51205762

119890minus 54190 times 10

minus81205763

119890

(4)

11986410158401015840

119888(120576) = 09122 + 001321120576

119890


119890+ 22086 times 10

minus81205763

119890

(5)







Klowast = Klowast0+

119873

sum

119899

K1198990(119864lowast

1198881120576119890119899+ 119864lowast

11988821205762

119890119899+ 119864lowast

11988831205763

119890119899)

1198641198770

(119899 = 1 2 3 119873)

(7)








119890119899= 0


119890119899 can be

expressed as

120576119890119899= radicx119879

K1198990

1198641198770119881119899

x (8)




x = Φ119873q (9)



Φ119879

119873MΦ119873= I (10)



Substitute (9) into (8) 120576119890119899


120576119890119899= radic

q119879h119899q

119881119899

(11)

h119899=Φ119879

119873K1198990Φ119873

1198641198770

(12)


(Klowast119873+ 119894120596C

119873minus 1205962I) q = F

119873 (13)

Klowast119873= Φ119879


= Klowast0119873+

119873

sum

119899

Φ119879

119873K1198990Φ119873

1198641198770

times (119864lowast

1198881120576119890119899+ 119864lowast

11988821205762

119890119899+ 119864lowast

11988831205763

119890119899)

(14)

Klowast0119873

= Φ119879

119873Klowast0Φ119873 C

119873= Φ119879

119873CΦ119873 F

119873= Φ119879

119873F(15)







119873can often


119873can be



10038161003816100381610038161003816119877 (Klowast119873) minus 1205962

119899I10038161003816100381610038161003816 = 0 (16)




frequency 120596(119895+1)119899

at the 119895 + 1th time and 120596(119895)119899


119899


10038171003817100381710038171003817120596(119895+1)

119899minus 120596(119895)

119899

100381710038171003817100381710038172le 1198780 (17)




r = (Klowast119873+ 119894120596C


119873

= (Klowast0119873+ 119894120596C


119873

+ 119864lowast

1198881

119873

sum

119899


119881119899

h119899q + 119864lowast1198882

119873

sum

119899

q119879h119899q

119881119899

h119899q

+ 119864lowast

1198883

119873

sum

119899

(q119879h119899q

119881119899

)

32

h119899q

(18)



J =[[[

[

119877(120597r120597q119877

) 119877(120597r120597q119868

)

119868(120597r120597q119877

) 119868(120597r120597q119868

)

]]]

]

(19)

120597r120597q119877

= Klowast0119873+ 119894120596C

119873minus 1205962I

+

119873

sum

119899

((119864lowast

1198881

119881119899120576119899

+2119864lowast

1198882

119881119899

+3119864lowast

1198883120576119899

119881119899

) h119899q119877(h119899q)119879)

(20)

120597r120597q119868

= 119894 (Klowast0119873+ 119894120596C

119873minus 1205962I)

+

119873

sum

119899

((119864lowast

1198881

119881119899120576119899

+2119864lowast

1198882

119881119899

+3119864lowast

1198883120576119899

119881119899

) h119899q119868(h119899q)119879)

(21)

where q119877and q




r(119895) + J(119895) sdot Δq(119895) = 0 (22)

q(119895+1) = q(119895) + Δq(119895) (23)




r = 119877 (r)119868 (r) (24)

q = 119877 (q)119868 (q) (25)


r2 = radic(100381610038161003816100381611990311003816100381610038161003816

2

+100381610038161003816100381611990321003816100381610038161003816

2

+100381610038161003816100381611990331003816100381610038161003816

2



0= 0001) we


q = 119877 (q) + 119894119868 (q) (27)








119864lowast

















10038161003816100381610038161003816K minus

2

119899M10038161003816100381610038161003816 = 0 (28)








119899I) q = F

119873 (29)




Yes

No


of hard coating


of hard coating



120596(j+1)n minus 120596

(j)n

2le S0

Output 120596n



Calculate orq q

K


Yes

No


of hard coating


of hard coating

Calculate r rand

Output x


Calculate initial q

120597 I JCalculate 120597 120597 R 120597r

r qq

2 le S0r







and 119864lowast

119888into (14) and







119899do









(Klowast0119873+ 119894120596C

119873minus 1205962I) q = F

119873 (30)




119873


C119873(119896 119896) asymp 2120596

1015840

1198961205771015840

119896 (31)

where 1205961015840

119896and 120577

1015840







Clampinglength (mm)

Width(mm)

Thickness(mm)

Density(kgm3)


Poissonrsquosratio




119877and


Jacobian matrix J


0in (26) is very





5 A Study Case


















Baseexcitation

Data processing

Responsesignal


Baseresponse

Vibration shaker

Laser point

Accelerometerpoint


Hard coating

Vibration shaker

Laser Doppler


composite plate

vibrometer







119878 lt 119878max = (120578119891119896

2)

2

asymp 1205772

1198961198912

119896 (32)

where 119891119896and 120577












Excitationlevel (g)


frequency(Hz)


Naturefrequency

(Hz)


Errors(119861 minus 119860)119860

()

Naturefrequency

(Hz)


Errors(119862 minus 119860)119860

()03 8234 467 8198 502 75 8198 482 3204 8232 624 8198 655 50 8196 632 1305 8230 726 8198 829 142 8195 779 7306 8229 856 8198 1025 197 8193 932 8907 8228 951 8198 1178 239 8191 1044 98

012

009

006

003

0821 823 825 827

07 g06 g05 g

04 g03 g

Frequency (Hz)

Vibr

atio

n re

spon

se (m

s)







Substrate plate


Composite plate


Hard coating




Vibr

atio

n re

spon

se (m

s)

012

009

006

003

0816 818 820 822 824

Frequency (Hz)

07 g06 g05 g

04 g03 g






119888(120576119890) and loss modulus 11986410158401015840

119888(120576119890) are linear in (4)






brat

ion

resp

onse

(ms

)

012

009

006

003

0817 819 821 823

Frequency (Hz)

07 g06 g05 g

04 g03 g






6 Conclusions








Acknowledgment


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Substitute (9) into (8) 120576119890119899


120576119890119899= radic

q119879h119899q

119881119899

(11)

h119899=Φ119879

119873K1198990Φ119873

1198641198770

(12)


(Klowast119873+ 119894120596C

119873minus 1205962I) q = F

119873 (13)

Klowast119873= Φ119879


= Klowast0119873+

119873

sum

119899

Φ119879

119873K1198990Φ119873

1198641198770

times (119864lowast

1198881120576119890119899+ 119864lowast

11988821205762

119890119899+ 119864lowast

11988831205763

119890119899)

(14)

Klowast0119873

= Φ119879

119873Klowast0Φ119873 C

119873= Φ119879

119873CΦ119873 F

119873= Φ119879

119873F(15)







119873can often


119873can be



10038161003816100381610038161003816119877 (Klowast119873) minus 1205962

119899I10038161003816100381610038161003816 = 0 (16)




frequency 120596(119895+1)119899

at the 119895 + 1th time and 120596(119895)119899


119899


10038171003817100381710038171003817120596(119895+1)

119899minus 120596(119895)

119899

100381710038171003817100381710038172le 1198780 (17)




r = (Klowast119873+ 119894120596C


119873

= (Klowast0119873+ 119894120596C


119873

+ 119864lowast

1198881

119873

sum

119899


119881119899

h119899q + 119864lowast1198882

119873

sum

119899

q119879h119899q

119881119899

h119899q

+ 119864lowast

1198883

119873

sum

119899

(q119879h119899q

119881119899

)

32

h119899q

(18)



J =[[[

[

119877(120597r120597q119877

) 119877(120597r120597q119868

)

119868(120597r120597q119877

) 119868(120597r120597q119868

)

]]]

]

(19)

120597r120597q119877

= Klowast0119873+ 119894120596C

119873minus 1205962I

+

119873

sum

119899

((119864lowast

1198881

119881119899120576119899

+2119864lowast

1198882

119881119899

+3119864lowast

1198883120576119899

119881119899

) h119899q119877(h119899q)119879)

(20)

120597r120597q119868

= 119894 (Klowast0119873+ 119894120596C

119873minus 1205962I)

+

119873

sum

119899

((119864lowast

1198881

119881119899120576119899

+2119864lowast

1198882

119881119899

+3119864lowast

1198883120576119899

119881119899

) h119899q119868(h119899q)119879)

(21)

where q119877and q




r(119895) + J(119895) sdot Δq(119895) = 0 (22)

q(119895+1) = q(119895) + Δq(119895) (23)




r = 119877 (r)119868 (r) (24)

q = 119877 (q)119868 (q) (25)


r2 = radic(100381610038161003816100381611990311003816100381610038161003816

2

+100381610038161003816100381611990321003816100381610038161003816

2

+100381610038161003816100381611990331003816100381610038161003816

2



0= 0001) we


q = 119877 (q) + 119894119868 (q) (27)








119864lowast

















10038161003816100381610038161003816K minus

2

119899M10038161003816100381610038161003816 = 0 (28)








119899I) q = F

119873 (29)




Yes

No


of hard coating


of hard coating



120596(j+1)n minus 120596

(j)n

2le S0

Output 120596n



Calculate orq q

K


Yes

No


of hard coating


of hard coating

Calculate r rand

Output x


Calculate initial q

120597 I JCalculate 120597 120597 R 120597r

r qq

2 le S0r







and 119864lowast

119888into (14) and







119899do









(Klowast0119873+ 119894120596C

119873minus 1205962I) q = F

119873 (30)




119873


C119873(119896 119896) asymp 2120596

1015840

1198961205771015840

119896 (31)

where 1205961015840

119896and 120577

1015840







Clampinglength (mm)

Width(mm)

Thickness(mm)

Density(kgm3)


Poissonrsquosratio




119877and


Jacobian matrix J


0in (26) is very





5 A Study Case


















Baseexcitation

Data processing

Responsesignal


Baseresponse

Vibration shaker

Laser point

Accelerometerpoint


Hard coating

Vibration shaker

Laser Doppler


composite plate

vibrometer







119878 lt 119878max = (120578119891119896

2)

2

asymp 1205772

1198961198912

119896 (32)

where 119891119896and 120577












Excitationlevel (g)


frequency(Hz)


Naturefrequency

(Hz)


Errors(119861 minus 119860)119860

()

Naturefrequency

(Hz)


Errors(119862 minus 119860)119860

()03 8234 467 8198 502 75 8198 482 3204 8232 624 8198 655 50 8196 632 1305 8230 726 8198 829 142 8195 779 7306 8229 856 8198 1025 197 8193 932 8907 8228 951 8198 1178 239 8191 1044 98

012

009

006

003

0821 823 825 827

07 g06 g05 g

04 g03 g

Frequency (Hz)

Vibr

atio

n re

spon

se (m

s)







Substrate plate


Composite plate


Hard coating




Vibr

atio

n re

spon

se (m

s)

012

009

006

003

0816 818 820 822 824

Frequency (Hz)

07 g06 g05 g

04 g03 g






119888(120576119890) and loss modulus 11986410158401015840

119888(120576119890) are linear in (4)






brat

ion

resp

onse

(ms

)

012

009

006

003

0817 819 821 823

Frequency (Hz)

07 g06 g05 g

04 g03 g






6 Conclusions








Acknowledgment


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











r = 119877 (r)119868 (r) (24)

q = 119877 (q)119868 (q) (25)


r2 = radic(100381610038161003816100381611990311003816100381610038161003816

2

+100381610038161003816100381611990321003816100381610038161003816

2

+100381610038161003816100381611990331003816100381610038161003816

2



0= 0001) we


q = 119877 (q) + 119894119868 (q) (27)








119864lowast

















10038161003816100381610038161003816K minus

2

119899M10038161003816100381610038161003816 = 0 (28)








119899I) q = F

119873 (29)




Yes

No


of hard coating


of hard coating



120596(j+1)n minus 120596

(j)n

2le S0

Output 120596n



Calculate orq q

K


Yes

No


of hard coating


of hard coating

Calculate r rand

Output x


Calculate initial q

120597 I JCalculate 120597 120597 R 120597r

r qq

2 le S0r







and 119864lowast

119888into (14) and







119899do









(Klowast0119873+ 119894120596C

119873minus 1205962I) q = F

119873 (30)




119873


C119873(119896 119896) asymp 2120596

1015840

1198961205771015840

119896 (31)

where 1205961015840

119896and 120577

1015840







Clampinglength (mm)

Width(mm)

Thickness(mm)

Density(kgm3)


Poissonrsquosratio




119877and


Jacobian matrix J


0in (26) is very





5 A Study Case


















Baseexcitation

Data processing

Responsesignal


Baseresponse

Vibration shaker

Laser point

Accelerometerpoint


Hard coating

Vibration shaker

Laser Doppler


composite plate

vibrometer







119878 lt 119878max = (120578119891119896

2)

2

asymp 1205772

1198961198912

119896 (32)

where 119891119896and 120577












Excitationlevel (g)


frequency(Hz)


Naturefrequency

(Hz)


Errors(119861 minus 119860)119860

()

Naturefrequency

(Hz)


Errors(119862 minus 119860)119860

()03 8234 467 8198 502 75 8198 482 3204 8232 624 8198 655 50 8196 632 1305 8230 726 8198 829 142 8195 779 7306 8229 856 8198 1025 197 8193 932 8907 8228 951 8198 1178 239 8191 1044 98

012

009

006

003

0821 823 825 827

07 g06 g05 g

04 g03 g

Frequency (Hz)

Vibr

atio

n re

spon

se (m

s)







Substrate plate


Composite plate


Hard coating




Vibr

atio

n re

spon

se (m

s)

012

009

006

003

0816 818 820 822 824

Frequency (Hz)

07 g06 g05 g

04 g03 g






119888(120576119890) and loss modulus 11986410158401015840

119888(120576119890) are linear in (4)






brat

ion

resp

onse

(ms

)

012

009

006

003

0817 819 821 823

Frequency (Hz)

07 g06 g05 g

04 g03 g






6 Conclusions








Acknowledgment


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Yes

No


of hard coating


of hard coating



120596(j+1)n minus 120596

(j)n

2le S0

Output 120596n



Calculate orq q

K


Yes

No


of hard coating


of hard coating

Calculate r rand

Output x


Calculate initial q

120597 I JCalculate 120597 120597 R 120597r

r qq

2 le S0r







and 119864lowast

119888into (14) and







119899do









(Klowast0119873+ 119894120596C

119873minus 1205962I) q = F

119873 (30)




119873


C119873(119896 119896) asymp 2120596

1015840

1198961205771015840

119896 (31)

where 1205961015840

119896and 120577

1015840







Clampinglength (mm)

Width(mm)

Thickness(mm)

Density(kgm3)


Poissonrsquosratio




119877and


Jacobian matrix J


0in (26) is very





5 A Study Case


















Baseexcitation

Data processing

Responsesignal


Baseresponse

Vibration shaker

Laser point

Accelerometerpoint


Hard coating

Vibration shaker

Laser Doppler


composite plate

vibrometer







119878 lt 119878max = (120578119891119896

2)

2

asymp 1205772

1198961198912

119896 (32)

where 119891119896and 120577












Excitationlevel (g)


frequency(Hz)


Naturefrequency

(Hz)


Errors(119861 minus 119860)119860

()

Naturefrequency

(Hz)


Errors(119862 minus 119860)119860

()03 8234 467 8198 502 75 8198 482 3204 8232 624 8198 655 50 8196 632 1305 8230 726 8198 829 142 8195 779 7306 8229 856 8198 1025 197 8193 932 8907 8228 951 8198 1178 239 8191 1044 98

012

009

006

003

0821 823 825 827

07 g06 g05 g

04 g03 g

Frequency (Hz)

Vibr

atio

n re

spon

se (m

s)







Substrate plate


Composite plate


Hard coating




Vibr

atio

n re

spon

se (m

s)

012

009

006

003

0816 818 820 822 824

Frequency (Hz)

07 g06 g05 g

04 g03 g






119888(120576119890) and loss modulus 11986410158401015840

119888(120576119890) are linear in (4)






brat

ion

resp

onse

(ms

)

012

009

006

003

0817 819 821 823

Frequency (Hz)

07 g06 g05 g

04 g03 g






6 Conclusions








Acknowledgment


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Clampinglength (mm)

Width(mm)

Thickness(mm)

Density(kgm3)


Poissonrsquosratio




119877and


Jacobian matrix J


0in (26) is very





5 A Study Case


















Baseexcitation

Data processing

Responsesignal


Baseresponse

Vibration shaker

Laser point

Accelerometerpoint


Hard coating

Vibration shaker

Laser Doppler


composite plate

vibrometer







119878 lt 119878max = (120578119891119896

2)

2

asymp 1205772

1198961198912

119896 (32)

where 119891119896and 120577












Excitationlevel (g)


frequency(Hz)


Naturefrequency

(Hz)


Errors(119861 minus 119860)119860

()

Naturefrequency

(Hz)


Errors(119862 minus 119860)119860

()03 8234 467 8198 502 75 8198 482 3204 8232 624 8198 655 50 8196 632 1305 8230 726 8198 829 142 8195 779 7306 8229 856 8198 1025 197 8193 932 8907 8228 951 8198 1178 239 8191 1044 98

012

009

006

003

0821 823 825 827

07 g06 g05 g

04 g03 g

Frequency (Hz)

Vibr

atio

n re

spon

se (m

s)







Substrate plate


Composite plate


Hard coating




Vibr

atio

n re

spon

se (m

s)

012

009

006

003

0816 818 820 822 824

Frequency (Hz)

07 g06 g05 g

04 g03 g






119888(120576119890) and loss modulus 11986410158401015840

119888(120576119890) are linear in (4)






brat

ion

resp

onse

(ms

)

012

009

006

003

0817 819 821 823

Frequency (Hz)

07 g06 g05 g

04 g03 g






6 Conclusions








Acknowledgment


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














Baseexcitation

Data processing

Responsesignal


Baseresponse

Vibration shaker

Laser point

Accelerometerpoint


Hard coating

Vibration shaker

Laser Doppler


composite plate

vibrometer







119878 lt 119878max = (120578119891119896

2)

2

asymp 1205772

1198961198912

119896 (32)

where 119891119896and 120577












Excitationlevel (g)


frequency(Hz)


Naturefrequency

(Hz)


Errors(119861 minus 119860)119860

()

Naturefrequency

(Hz)


Errors(119862 minus 119860)119860

()03 8234 467 8198 502 75 8198 482 3204 8232 624 8198 655 50 8196 632 1305 8230 726 8198 829 142 8195 779 7306 8229 856 8198 1025 197 8193 932 8907 8228 951 8198 1178 239 8191 1044 98

012

009

006

003

0821 823 825 827

07 g06 g05 g

04 g03 g

Frequency (Hz)

Vibr

atio

n re

spon

se (m

s)







Substrate plate


Composite plate


Hard coating




Vibr

atio

n re

spon

se (m

s)

012

009

006

003

0816 818 820 822 824

Frequency (Hz)

07 g06 g05 g

04 g03 g






119888(120576119890) and loss modulus 11986410158401015840

119888(120576119890) are linear in (4)






brat

ion

resp

onse

(ms

)

012

009

006

003

0817 819 821 823

Frequency (Hz)

07 g06 g05 g

04 g03 g






6 Conclusions








Acknowledgment


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















Excitationlevel (g)


frequency(Hz)


Naturefrequency

(Hz)


Errors(119861 minus 119860)119860

()

Naturefrequency

(Hz)


Errors(119862 minus 119860)119860

()03 8234 467 8198 502 75 8198 482 3204 8232 624 8198 655 50 8196 632 1305 8230 726 8198 829 142 8195 779 7306 8229 856 8198 1025 197 8193 932 8907 8228 951 8198 1178 239 8191 1044 98

012

009

006

003

0821 823 825 827

07 g06 g05 g

04 g03 g

Frequency (Hz)

Vibr

atio

n re

spon

se (m

s)







Substrate plate


Composite plate


Hard coating




Vibr

atio

n re

spon

se (m

s)

012

009

006

003

0816 818 820 822 824

Frequency (Hz)

07 g06 g05 g

04 g03 g






119888(120576119890) and loss modulus 11986410158401015840

119888(120576119890) are linear in (4)






brat

ion

resp

onse

(ms

)

012

009

006

003

0817 819 821 823

Frequency (Hz)

07 g06 g05 g

04 g03 g






6 Conclusions








Acknowledgment


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Substrate plate


Composite plate


Hard coating




Vibr

atio

n re

spon

se (m

s)

012

009

006

003

0816 818 820 822 824

Frequency (Hz)

07 g06 g05 g

04 g03 g






119888(120576119890) and loss modulus 11986410158401015840

119888(120576119890) are linear in (4)






brat

ion

resp

onse

(ms

)

012

009

006

003

0817 819 821 823

Frequency (Hz)

07 g06 g05 g

04 g03 g






6 Conclusions








Acknowledgment


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










brat

ion

resp

onse

(ms

)

012

009

006

003

0817 819 821 823

Frequency (Hz)

07 g06 g05 g

04 g03 g






6 Conclusions








Acknowledgment


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Acknowledgment


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article analysis of nonlinear vibration of hard

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