an accurate solution method for the static and vibration analysis...

Research ArticleAn Accurate Solution Method for the Static and VibrationAnalysis of Functionally Graded Reissner-Mindlin RectangularPlate with General Boundary Conditions

Haichao Li Ning Liu Fuzhen Pang Yuan Du and Shuo Li

College of Shipbuilding Engineering Harbin Engineering University Harbin 150001 China

Correspondence should be addressed to Ning Liu liuninghrbeueducn

Received 30 January 2018 Accepted 2 May 2018 Published 27 June 2018

Academic Editor Tai Thai

Copyright copy 2018 Haichao Li 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

This paper presents an accurate solution method for the static and vibration analysis of functionally graded Reissner-Mindlinplate with general boundary conditions on the basis of the improved Fourier series method In the theoretical formulations thegoverning equations and the general elastic boundary equations are obtained by using Hamiltonrsquos principle The components ofadmissible displacement functions are expanded as an improved Fourier series form which contains a 2D Fourier cosine series andauxiliary function in the form of 1D series The major role of the auxiliary function is to remove the potential discontinuities ofthe displacement function and its derivatives at the edges and ensure and accelerate the convergence of the series representationThe characteristic equations are easily obtained via substituting admissible displacement functions into governing equations andthe general elastic boundary equations Several examples are made to show the excellent accuracy and convergence of the currentsolutions The results of this paper may serve as benchmark data for future research in related field

1 Introduction

The concept of functionally graded materials (FGMs) wasfirstly presented in 1987 by a group of material scientists inSendai region of Japan during the first five-year project tostudy relaxation of thermal stress of materials in high speedaerospace vehicle [1 2] Since then FGMs have received themajor attention as heat-shielding advanced structural materi-als in a variety of engineering applications andmanufacturingindustries like aerospace nuclear reactor automobile air-craft space vehicles and biomedical and steel industries Theremarkable mechanical properties of the FGMs are achievedby gradually varying the volume fraction of the constituentmaterials whose properties vary from one interface to theother continuously

As a most common infrastructure the functionallygraded rectangular plate has been widely used in practi-cal engineering applications ie aerospace naval vesselsnuclear reactor automobile robot and civil industries Onone hand the research of static and vibration analysis ofFG plate could provide the theoretical basis for practical

engineering applications On the other hand the results ofthis paper could serve as the reference data in practicalengineering applications in related field in future The staticand vibration analysis of the FG plate has been investigatedby a large number of researches in the past decade Amongthose available Abrate [3] used the CPT to study the freevibration of FG thin rectangular plates with simply supportedand clamped boundary conditions Later the free vibrationbuckling and static deflections of FG square and circularand skew plates with different combinations of boundaryconditions were analyzed by Abrate [4] using the samemethod on the basis of the CPT FSDT and TSDT Freeand forced vibration analyses of both homogeneous andFG thick plates with classical boundary conditions werecarried out by Qian et al [5 6] using the meshless localPetrovndashGalerkin method based on the higher-order shearand normal deformable plate theory Pradyumna and Bandy-opadhyay [7] used the higher-order finite element method toinvestigate the free vibration of FG square plates with simplysupported boundary condition Ferreira et al [8] employedthe global collocation method and approximated the trial

HindawiShock and VibrationVolume 2018 Article ID 4535871 21 pageshttpsdoiorg10115520184535871

2 Shock and Vibration

solution with multiquadric radial basis functions to studythe free vibration of FG square plates with different classicalboundary conditions on the basis of the FSDT and third-order shear deformation plate theory (TSDT) Praveen andReddy [9] used the finite element method to obtain thestatic and dynamic thermoelastic behavior of the FG plateswith some selected classical boundary conditions Basedon the third-order shear deformation theory and Reddyvon Karman-type geometric nonlinearity theory Reddy [10]carried out the static linear analysis and nonlinear staticand dynamic analysis by using the finite element methodCroce and Venini [11] presented a new method of finite ele-ments for analyzingReissner-Mindlin FGplateswith classicalboundary condition Cheng and Batra [12 13] applied thefirst-order shear deformation theory and a third-order sheardeformation theory to study the static buckling and steadystate vibrations of FG polygonal plate with simply supportedboundary conditions Zhao et al [14] extended the elementfree kp-Ritzmethod to study the free vibration analysis for FGsquare and skew plates with different boundary conditionson the basis of the FSDT The nonlinear free vibrationbehavior of FG square thin plates was analyzed by Woo etal [15] by presenting an analytical solution on the basis ofthe von Karman theory Based on the CPT Yang and Shen[16] presented an analytical method for free vibration andtransient response of initially stressed FG rectangular thinplates resting on Pasternak elastic foundation having classicalboundary condition subjected to impulsive lateral loadsLater Yang and Shen [17] used a semianalytical approach toinvestigate the dynamic behavior of FG plates with impulsivelateral loads and complicated environment Zhong and Yu[18] employed a state-space approach to analyze free andforced vibrations of an FG piezoelectric rectangular thickplate with simple support at its edges Roque et al [19]employed the multiquadric radial basis function method andthe HSDT to study free vibration of FG plates with differentclassical boundary conditions The natural frequencies andbuckling stresses of FG plates having simply supported edgeswere studied by Matsunaga [20] who utilized the method ofpower series expansion and two-dimensional (2D) higher-order theory Hosseini-Hashemi et al [21 22] presented anew exact closed-form procedure for free vibration analysisof FG rectangular thick plates and Reissner-Mindlin FGrectangular plates where the plate has two opposite edgesof simply supported boundary conditions Vel and Batra[23] presented an exact 3-D elasticity solution to study thethermoelastic deformation free and forced vibrations of FGsimply supported plates Based on the higher-order shear andnormal deformation theory Shiyekar et al [24] presentedthe bidirectional flexure analysis of smart FG plate subjectedto electromechanical loading by using Navierrsquos techniqueCarrera et al [25] evaluated the effect of thickness stretchingin plateshell structures made by materials which are func-tionally graded (FGM) in the thickness direction accordingto Carrerarsquos Unified Formulation Based on Carrerarsquos UnifiedFormulation Neves et al [26] presented explicit governingequations to study static and free vibration analysis of FGMplates by using meshless technique based on collocationwith radial basis functions Zhang et al [27] used the local

Kriging meshless method to study the mechanical and ther-mal buckling behaviors of ceramicndashmetal functionally gradeplates subject the classical boundary conditions Belabed etal [28] presented an efficient and simple higher-order shearand normal deformation theory for the bending and freevibration analysis of FG simply supported platesThai and Vo[29ndash34] extended a simple shear deformation theory to studythe bending buckling and vibration of functionally gradedplates with simply supported cases Valizadeh et al [35]applied isogeometric finite elementmethod to study the staticand dynamic characteristics of functionally graded material(FGM) plates with classical boundary condition Based onisogeometric approach (IGA) and higher-order deformationplate theory (HSDT) Tran et al [36] used the 1198621 continuouselement to investigate the static dynamic and bucklinganalysis of FG rectangular and circular plates with differentclassical boundary conditions Mantari et al [37] showed ananalytical solution to static analysis of functionally gradedplates (FGPs) based on a new trigonometric higher-ordertheory in which the stretching effect was considered Xiangand Kang [38] conducted bending analysis for functionallygraded plates according to a nth-order shear deformationtheory and the meshless global collocation method basedon the thin plate spline radial basis function and Reddyrsquosthird-order theory Based on various plate theory Tornabeneet al [39] applied the Generalized Differential Quadrature(GDQ) to study the bending vibration and dynamic analysisof FG plates and panels with different classical boundaryconditions Xuan et al [40] presented a simple and effectiveapproach which incorporated the isogeometric finite elementanalysis (IGA) with a refined plate theory (RPT) for staticfree vibration and buckling analysis of functionally gradedmaterial (FGM) plates with simply supported and clampedboundary conditions Akavci [41] presented a closed-formsolution to study the free vibration analysis of functionallygraded plate resting on Pasternak (two-parameter) modelsubject to classical boundary condition

From the above review we can know that most of existingliteratures on the static and vibration analysis of functionallygraded Reissner-Mindlin plate are restricted to the classicalboundary condition However the elastic edge restraintsmay be more common in the practical engineering andthe classical boundary condition is considered as a specialcase Thus to establish a unified efficient and accurateformulation for static and vibration analysis of functionallygraded Reissner-Mindlin plate is necessary and significant

To the best of authorrsquos knowledge the subject of static andvibration analysis of functionally graded Reissner-Mindlinplate with general boundary conditions has not been per-formed in a specified work yet Therefore the major focusof this paper is to present an accurate solution methodfor the static and vibration analysis of functionally gradedReissner-Mindlin plate with general boundary conditionsThe effective material properties of functionally gradedmaterials are assumed to vary continuously in the thicknessdirection according to the power-law distribution in termsof the volume fraction of the constituents and are estimatedby the Voigt model and MorindashTanaka scheme All the platedisplacements are expanded as a modified Fourier series

Shock and Vibration 3

which is made up of a standard cosine Fourier series andsome certain supplementary terms whatever the boundaryconditions are These supplementary terms contribute toremoving the potential discontinuities at the edges andthen ensure and accelerate the series convergence Thecharacteristic equations can be derived directly by solvingthe equation of motion and by combining the associatedboundary equations and the modified Fourier series Thecharacteristic equations are easily obtained via substitutingrelated modified Fourier series into governing equationsand the general elastic boundary equations A large numberof numerical examples are made to show the convergencereliability and accuracy of the present method

2 Theoretical and Numerical Formulations

21 Geometrical Configuration Consider a flat and moder-ately thick FG plate with a with uniform thickness ℎ length119886 and width 119887 of plate in the 119911- 119909- and 119910-direction asshown in Figure 1 In addition the coordinate system (119909 119910 119911)is also shown in Figure 1 which will be used in the analysisThe arbitrary boundary technique [43ndash47] is introduced toimplement the general boundary condition in which onegroup of liner spring 119896120574119908 (120574 = 1199090 119909119886 1199100 and 119910119887 denotethe location of the spring ie 1199090 represents the location ofthe edge 119909 = 0) and two groups of rotation springs 119870120574119909and 119870120574119909119910 are introduced to simulate the related boundaryforces in each boundary of a plate as shown in Figure 1 Thegeneral boundary condition is easily obtained by assigning

the stiffness of the boundary springs with various valuesTake the clamped boundary condition as an example whenthe spring stiffness is set larger enough than the bendingrigidity of plate it can be obtained essentially In addition acommon uniform pressure 119902 = 1199020 is acted on the rectangulararea [(1199091 1199092) (1199101 1199102)] and when the acting area is narroweddown to a point a special case named the point load will beobtained as illustrated in Figure 1

22 Material Properties It is supposed that the FGM of plateis made of two material constituents In this study the topsurface (119911 = ℎ2) of the plate is ceramic-rich (119872119888) whereasthe bottom surface (119911 = minusℎ2) is metal-rich (119872119898) Boththe Voigt model and MorindashTanaka scheme are adopted toevaluate the effective material properties [48] In the Voigtmodel it is assumed that the material properties includingYoungrsquos modulus 119864(119911) density 120588(119911) and Poissonrsquos ratio 120583(119911)are proportional to the volume fraction according to119864 (119911) = (119864119888 minus 119864119898) 119881119888 (119911) + 119864119898120588 (119911) = (120588119888 minus 120588119898)119881119888 (119911) + 120588119898120583 (119911) = (120583119888 minus 120583119898) 119881119888 (119911) + 120583119898 (1)

in which the subscripts 119898 and 119888 represent the metallic andceramic constituents respectively

Based on the MorindashTanaka scheme the effective localbulk modulus119870119888 and the shear modulus119866119888 of the FGM platecan be expressed as

119870 minus 119870119898119870119888 minus 119870119898 = 119881119888 (119911)1 + (1 minus 119881119888 (119911)) (119870119888 minus 119870119898) (119870119898 + 41198661198983)119866 minus 119866119898119866119888 minus 119866119898 = 119881119888 (119911)1 + (1 minus 119881119888 (119911)) (119866119888 minus 119866119898) (119866119898 + 119866119898 (9119870119898 + 8119866119898) [6 (119870119898 + 2119866119898)]) (2)

where119870119894 = 119864119894[3(1minus2120583119894)] and119866119894 = 119864119894[2(1+2120583119894)] (119894 = 119898 119888)The effective mass density which is defined by (1) is alsoutilized in the MorindashTanaka scheme The effective Youngrsquosmodulus 119864(119911) and Poissonrsquos ratio 120583(119911) are119864 (119911) = 91198701198663119870 + 119866

120583 (119911) = 3119870 minus 21198666119870 + 2119866 (3)

In the above FG scheme the common volume fraction119881119888may be given by

119881119888 = [1 minus 119890 (12 + 119911ℎ) + 119891(12 + 119911ℎ)119892]119901[1 minus 119890 (12 minus 119911ℎ) + 119891(12 minus 119911ℎ)119892]119901 (4)

where 119911 is the thickness coordinate and 119901 denotes the power-law exponent and only takes the positive values Thematerial

parameters 119890 119891 and 119892 determine the material variationprofile through the functionally graded plate thickness Itshould be noted that in order to make the results becomemore universal in the future the authors just choose the Voigtmodel and material parameters of the volume fraction119881119888 aregiven as 119890 = 1 and 119891 = 0 The variations of volume fraction119881119888 for different values of the parameters 119890 119891 119892 and 119901 aredepicted in Figure 2

23 Governing Equations and Boundary Conditions Basedon the Mindlin plate theory the displacement componentsof the plate are assumed to be119906 (119909 y 119911) = 119911120593119909 (119909 119910) V (119909 119910 119911)= 119911120593119910 (119909 119910) 119908 (119909 119910 119911) = 119908 (119909 119910) (5)

where 119908 is the displacements of the middle surface in119911 directions and 120595119909 and 120595119910 are the rotation functionsFollowing assumptions of the small deformation and the


z

x

y

b

a

Metal

z

h

Ceramic

y

q(xy)

1

2

3

4

y1

y2

y2

y1

x1

x2

Figure 1 Schematic diagram of a FGMindlin plate with elastically restrained edges

p=10

p=5

p=2

p=1

p=05

p=02p=01

minus05

minus04

minus03

minus02

minus01

00

01

02

03

04

05

zh

02 04 06 08 1000Vc

Figure 2 Variation of volume fraction119881119888 through the thickness fordifferent values of power-law exponent

linear strainndashdisplacement relation the strain components ofFG plates can be written as

120576119909 = 119911120597120593119909120597119909 = 119911120581119909120576119910 = 119911120597120593119910120597119910 = 119911120581119910120574119909119910 = 119911(120597120593119910120597119909 + 120597120593119909120597119910 ) = 119911120581119909119910

(6a)

120574119909119911 = 120597119908120597119909 + 120593119909120574119910119911 = 120597119908120597119910 + 120593119910 (6b)

According to Hookrsquos law the stress components of FGplates can be expressed as

120590119909120590119910120591119909119910120591119909119911120591119910119911

= [[[[[[[[[

11987611 (119911) 11987612 (119911) 0 0 011987612 (119911) 11987611 (119911) 0 0 00 0 11987666 (119911) 0 00 0 0 11987666 (119911) 00 0 0 0 11987666 (119911)]]]]]]]]]120576119909120576119910120574119909119910120574119909119911120574119910119911

(7)

where the elastic constants 119876119894119895(119911) are the functions of thick-ness coordinate 119911 11987611 (119911) = 119864 (119911)1 minus 1205832 (119911) 11987612 (119911) = 120583 (119911) 119864 (119911)1 minus 1205832 (119911) 11987666 (119911) = 119864 (119911)2 [1 + 120583 (119911)]

(8)

By carrying the integration of the stresses over the cross-section the force and moment resultants are written asfollows

119872119909 = 11986311 120597120593119909120597119909 + 11986312 120597120593119910120597119910 119872119910 = 11986312 120597120593119909120597119909 + 11986311 120597120593119910120597119910 119872119909119910 = 11986366 (120597120593119910120597119909 + 120597120593119909120597119910 )


119876119909 = 11986066 (120597119908120597119909 + 120593119909) 119876119910 = 11986066 (120597119908120597119910 + 120593119910)

(9)

where 119860 119894119895 and 119863119894119895 are the extensional bending stiffness andthey are respectively expressed as

(11986066 11986311 11986322 11986312) = intℎ2minusℎ2

(12058111987666 (119911) 11987611 (119911)sdot 1199112 11987611 (119911) 1199112 11987612 (119911) 1199112) 119889119911 (10)

The strain energy and kinetic energy of the FG plate canbe described as119880 = 12 int1198860 int1198870 (120581119909119872119909 + 120581119910119872119910 + 120581119909119910119872119909119910 + 120574119909119911119876119909

+ 120574119910119911119876119910) 119889119909119889119910 = 12 int1198860 int1198870 (119872119909 120597120593119909120597119909+119872119910 120597120593119910120597119910 +119872119909119910 (120597120593119909120597119910 + 120597120593119910120597119909 )+ (120597119908120597119909 + 120593119909)119876119909 + (120597119908120597119910 + 120593119910)119876119910)119889119909119889119910

(11)

119879 = 12 int1198860 int1198870 [1198680 (120597119908120597119905 )2+ 1198682((120597120593119909120597119905 )2 + (120597120593119910120597119905 )2)]119889119909119889119910

[1198680 1198681] = intℎ2minusℎ2

120588 (1 1199112) 119889119911(12)

In this paper to develop a unified solution of the FGplate subjected to general elastic restrains the static andvibration analysis is focused So the strain energy stored inthe boundary springs during vibration can be defined as

119881 = 12 int1198860 (1198961199100119910 1199082 + 11987011991001199101199091205932119909 + 1198701199100119910 1205932119910)10038161003816100381610038161003816119910=0 119889119909+ 12 int1198860 (119896119910119887119910 1199082 + 1198701199101198871199101199091205932119909 + 119870119910119887119910 1205932119910)10038161003816100381610038161003816119910=119887 119889119909+ 12 int1198870 (1198961199090119909 1199082 + 1198701199090119909 1205932119909 + 11987011990901199091199101205932119910)10038161003816100381610038161003816119909=0 119889119910+ 12 int1198870 (119896119909119886119909 1199082 + 119870119909119886119909 1205932119909 + 1198701199091198861199091199101205932119910)10038161003816100381610038161003816119909=119886 119889119910

(13)

Also the external force on the plate during vibration canbe defined as 119882 = ∬

119860119902 (119909 119910)119908 (119909 119910) 119889119860 (14)

To determine the static deflection the load functionshould be expressed as a Fourier cosine series and the detailedinformation can be seen in [49]

The Lagrangian energy functional (119871) of the plate iswritten as 119871 = 119879 +119882 minus 119880 minus 119881 (15)

Within arbitrary length of time 0 to 1199050 Hamiltonrsquosprinciple can be stated as follows

int11990500(120575119879 + 120575119882 minus 120575119880 minus 120575119881) 119889119905 = 0 (16)

By substituting (11)ndash(14) into (16) the variation equationcan be rewritten as

int11990500int1198860int1198870(minus120597119872119909120597119909 120575120593119909 minus 120597119872119903120579120597119910 120575120593119909 + 119876119909120575120593119909 minus 120597119872119910120597119910 120575120593119910 minus 120597119872119909119910120597119909 120575120593119910 + 119876119910120575120593119910 minus 120597119876119909120597119909 120575119908 + minus120597119876119910120597119910 120575119908)119889119909119889119910119889119905

int11990500int1198860(11987211990911991012057512059311990910038161003816100381610038161003816119910=119887119910=0 + 11987611991012057511990810038161003816100381610038161003816119910=119887119910=0 + 11987211991012057512059311991010038161003816100381610038161003816119910=119887119910=0) 119889119909119889119905 + int1199050

0int1198870(1198721199091205751205931199091003816100381610038161003816119909=119886119909=0 + 1198761199091205751199081003816100381610038161003816119909=119886119909=0 + 11987211990911991012057512059311991010038161003816100381610038161003816119909=119886119909=0) 119889119910119889119905

int11990500int1198860(1198961199100119910 119908120575119908 + 1198701199100119910119909120593119909120575120593119909 + 1198701199100119910 120593119910120575120593119910)10038161003816100381610038161003816119910=0 119889119909119889119905 + int1199050

0int1198860(119896119910119887119910 119908120575119908 + 119870119910119887119910119909120593119909120575120593119909 + 119870119910119887119910 120593119910120575120593119910)10038161003816100381610038161003816119910=119887 119889119909119889119905

+ int11990500int1198870(1198961199090119909 119908120575119908 + 1198701199090119909 120593119909120575120593119909 + 1198701199090119909119910120593119910120575120593119910)10038161003816100381610038161003816119909=0 119889119910119889119905 + int1199050

0int1198870(119896119909119886119909 119908120575119908 + 119870119909119886119909 120593119909120575120593119909 + 119870119909119886119909119910120593119910120575120593119910)10038161003816100381610038161003816119909=119886 119889119910119889119905

+ int11990500int1198860int1198870[1198680 (120597119908120597119905 )2 120575119908 + 1198682((120597120593119909120597119905 )2 120575120593119909 + (120597120593119910120597119905 )2 120575120593119910)]119889119909119889119910119889119905 minus int1199050

0int1198860int1198870119902 (119909 119910) 120575119908119889119909119889119910119889119905 = 0

(17)


Integrating by parts to relieve the virtual displacements120575120593119909 120575120593119910 and 120575119908 we have

int11990500int1198860int1198870

((((

minus120597119872119909120597119909 minus 120597119872119909119910120597119910 + 119876119909 + 1198682 (120597120593119909120597119905 )2120575120593119909minus120597119872119910120597119910 minus 120597119872119909119910120597119909 + 119876119910 + 1198682(120597120593119910120597119905 )2120575120593119910minus120597119876119909120597119909 minus 120597119876119910120597119910 minus 119902 + 1198680 (120597119908120597119905 )2120575119908))))

119889119909119889119910119889119905+ int11990500int1198860((1198961199100119910 119908 minus 119876119910) 120575119908 + (1198701199100119910119909120593119909 minus119872119909119910) 120575120593119909 + (1198701199100119910 120593119910 minus119872119910) 120575120593119910)10038161003816100381610038161003816119910=0 119889119909119889119905

+ int11990500int1198860((119896119910119887119910 119908 + 119876119910) 120575119908 + (119870119910119887119910119909120593119909 +119872119909119910) 120575120593119909 + (119870119910119887119910 120593119910 +119872119910) 120575120593119910)10038161003816100381610038161003816119910=119887 119889119909119889119905

+ int11990500int1198870((1198961199090119909 119908 minus 119876119909) 120575119908 + (1198701199090119909 120593119909 minus119872119909) 120575120593119909 + (1198701199090119909119910120593119910 minus119872119909119910) 120575120593119910)10038161003816100381610038161003816119909=0 119889119910119889119905

+ int11990500int1198870((119896119909119886119909 119908 + 119876119909) 120575119908 + (119870119909119886119909 120593119909 +119872119909) 120575120593119909 + (119870119909119886119909119910120593119910 +119872119909119910) 120575120593119910)10038161003816100381610038161003816119909=119886 119889119910119889119905 = 0

(18)

Since the virtual displacements 120575120593119909 120575120593119910 and 120575119908 arearbitrary (18) can be satisfied only if the coefficients of thevirtual displacements are zero Thus the governing equationof motion and general boundary conditions for the static andvibration analysis of a FG plate can be derived as120597119872119909120597119909 + 120597119872119909119910120597119910 minus 119876119909 minus 1198682 (120597120593119909120597119905 )2 = 0

120597119872119910120597119910 + 120597119872119909119910120597119909 minus 119876119910 minus 1198682 (120597120593119910120597119905 )2 = 0120597119876119909120597119909 + 120597119876119910120597119910 + 119902 (119909 119910) minus 1198680 (120597119908120597119905 )2 = 0

(19)

119909 = 0 1198701199090119909 120593119909 = 1198721199091198701199090119909119910120593119910 = 1198721199091199101198961199090119909 119908 = 119876119909119909 = 119886 119870119909119886119909 120593119909 = minus119872119909119870119909119886119909119910120593119910 = minus119872119909119910119896119909119886119909 119908 = minus119876119909

119910 = 0 1198701199100119910 120593119910 = 1198721199101198701199100119910119909120593119909 = 1198721199091199101198961199100119910 119908 = 119876119910119910 = 119887 119870119910119887119910 120593119910 = minus119872119910119870119910119887119910119909120593119909 = minus119872119909119910119896119910119887119910 119908 = minus119876119910(20)

Further substituting (6a) (6b) and (9) into (19) and (20)the governing equation of motion can be expressed in matrixform

([[[[11987111 11987112 1198711311987121 11987122 1198712311987131 11987132 11987133

]]]] minus 1205962 [[[[11987211 0 00 11987222 00 0 11987233

]]]])[[[[120593119909120593119910119908]]]]

= [[[[00119902]]]]

(21)


where

11987111 = 11986311 12059721205971199092 minus 11986066 + 11986366 12059721205971199102 11987112 = 11986312 1205972120597119909120597119910 + 11986366 1205972120597119909120597119910 11987113 = minus11986066 12059712059711990911987121 = 11986312 1205972120597119909120597119910 + 11986366 1205972120597119909120597119910 11987122 = 11986366 12059721205971199092 minus 11986066 + 11986311 12059721205971199102 11987123 = minus11986066 12059712059711991011987131 = 11986066 120597120597119909 11987132 = 11986066 120597120597119910 11987133 = 11986066 12059721205971199092 + 11986066 12059721205971199102 11987211 = 11987222 = 119868211987233 = 1198680

(22)

Similarly the general boundary conditions can be writtenas

[[[[[11987111989511 11987111989512 1198711198951311987111989521 11987111989522 1198711198952311987111989531 11987111989532 11987111989533

]]]]][[[[120593119909120593119910119908]]]] = [[[[

000]]]] (119895 = 1199090 119909119886 1199100 119910b)

(23)

where 119871119909011 = 11986311 120597120597119909 minus 1198701199090119909 119871119909012 = 11986312 120597120597119910 119871119909013 = 0119871119909021 = 11986366 120597120597119910 119871119909022 = 11986366 120597120597119909 minus 1198701199090119909119910 119871119909023 = 0

119871119909031 = 11986066119871119909032 = 0119871119909033 = 11986066 120597120597119909 minus 1198961199090119909 119871119910011 = 11986366 120597120597119910 minus 1198701199100119910119909119871119910012 = 11986366 120597120597119909 119871119910013 = 0119871119910021 = 11986312 120597120597119909 119871119910022 = 11986311 120597120597119910 minus 1198701199100119910 119871119910023 = 0119871119910031 = 0119871119910032 = 11986066119871119910033 = 11986066 120597120597119910 minus 11989611991001199101198711199090119894119895 = 1198711199091119894119895 (119894 = 119895) 1198711199100119894119895 = 1198711199101119894119895 (119894 = 119895) 119871119909111 = 11986311 120597120597119909 + 119870119909119886119909 119871119909122 = 11986366 120597120597119909 + 119870119909119886119909119910119871119909133 = 11986066 120597120597119909 + 119896119909119886119909 119871119910111 = 11986366 120597120597119910 + 119870119910119887119910119909119871119910122 = 11986311 120597120597119910 + 119870119910119887119910 119871119910133 = 11986066 120597120597119910 + 119896119910119887119910

(24)

24 Admissible Displacement Functions In the structurevibration problem the scope of boundary condition andthe accuracy of the solution strongly depend on the choiceof the admissible function of structures Generally for thecommonly used polynomial expression their convergenceis uncertain In other words the lower-order polynomialscannot form a complete set and on the contrary it may leadto be numerically unstable owing to the computer round-off errors when the higher-order polynomials are applied


To avoid the above weakness the admissible functions canbe expanded as the form of Fourier series due to theexcellent numerical stability of the Fourier series Howeverthe conventional Fourier series just adapt to a few of simpleboundary conditions due to the convergence problem alongthe boundary conditions Recently a modified Fourier seriestechnique proposed by Li [50] has been widely applied in thevibration problems of plates and shells subject to differentboundary conditions by the Ritz method eg [42 51ndash60] In this technique each displacement of the structureunder study is written in the form of a conventional cosineFourier series and several supplementary terms The detailedprinciple and merit of the improved Fourier series can beseen in the related book [61] (entitled ldquoStructural VibrationA Uniform Accurate Solution for Laminated Beams Platesand Shells with General Boundary Conditionsrdquo) On the basisof the modified Fourier series technique Jin et al [48 62]present an exact series solution to study the free vibrationof functionally graded sandwich beams composite laminateddeep curved beams and so on Compared with most of theexisting methods the exact series solution not only ownsthe excellent convergence and accuracy but also can beapplied to general boundary conditions Therefore in thisformulation themodified Fourier series technique is adoptedand extended to conduct the static and vibration analysisof functionally graded Reissner-Mindlin plate with generalboundary conditions

Combining (6a) (6b) (9) and (19) it can be knownthat each displacementrotation component of a FG plateis required to have up to the second derivative Thereforeno matter what the boundary conditions are each displace-mentrotation component of the plate is assumed to be a two-dimensional modified Fourier series as120593119909 (119909 119910) = 119872sum

119898=0

119873sum119899=0

119860119898119899 cos 120582119886119898119909 cos 120582119887119899119910+ 2sum119897=1

119872sum119898=0

119887119897119898120577119897119887 (119910) cos120582119886119898119909+ 2sum119897=1

119873sum119899=0

119886119897119899120577119897119886 (119909) cos 120582119887119899119910120593119910 (119909 119910) = 119872sum

119898=0

119873sum119899=0

119861119898119899 cos 120582119886119898119909 cos 120582119887119899119910+ 2sum119897=1

119872sum119898=0

119889119897119898120577119897119887 (119910) cos 120582119886119898119909+ 2sum119897=1

119873sum119899=0

119888119897119899120577119897119886 (119909) cos 120582119887119899119910119908 (119909 119910) = 119872sum

119898=0

119873sum119899=0

119862119898119899 cos 120582119886119898119909 cos 120582119887119899119910+ 2sum119897=1

119872sum119898=0

119891119897119898120577119897119887 (119910) cos 120582119886119898119909+ 2sum119897=1

119873sum119899=0

119890119897119899120577119897119886 (119909) cos120582119887119899119910

(25)

where 120582119886119898 = 119898120587119886 120582119887119887 = 119899120587119887 and 119860119898119899 119861119898119899and 119862119898119899are the Fourier coefficients of two-dimensional Fourier seriesexpansions for the displacements functions respectively 119872and 119873 are the truncation numbers 120577119897119886 and 120577119897119887 119897 = 1 2represent the auxiliary functions defined over 119863([0 119886] times[0 119887]) whose major role is to eliminate all the discontinuitiespotentially associated with the first-order derivatives at theboundary and then ensure and accelerate the convergence ofthe series expansion of the plate displacement Here it shouldbe noted that the auxiliary functions just satisfy 1205771119886(0) =1205771119886(119886) = 12057711015840119886 (119886) = 0 12057711015840119886 (0) = 1 and 1205772119886(0) = 1205772119886(119886) = 12057721015840119886 (0) = 0and 12057721015840119886 (119886) = 1 however the concrete form is not the focusof attention alm b

ln c

lm d

ln e

lm and f ln are the corresponding

supplemented coefficients of the auxiliary functions where119897 = 1 2 Those auxiliary functions are defined as follows1205771119886 (119909) = 1198862120587 sin (1205871199092119886 ) + 1198862120587 sin (31205871199092119886 ) 1205772119886 (119909) = minus 1198862120587 cos (1205871199092119886 ) + 1198862120587 cos(31205871199092119886 )1205771119887 (119910) = 1198872120587 sin (1205871199102119887 ) + 1198872120587 sin (31205871199102119887 ) 1205772119887 (119910) = minus 1198872120587 cos (1205871199102119887 ) + 1198872120587 cos (31205871199102119887 )

(26)

Further the modified Fourier series expressions pre-sented in (25) can be rewritten in the matrix form as120593119909 (119909 119910) = H119909119910A + H119909a + H119910b120593119910 (119909 119910) = H119909119910B + H119909c + H119910d119908 (119909 119910) = H119909119910C + H119909e + H119910f

(27)

where

H119909119910 = [cos1205821198860119909 cos 1205821198870119910 cos 120582119886119898119909sdot cos120582119887119899119910 cos120582119886119872119909 cos 120582119887119873119910]H119909 = [1205771119886 (119909) cos1205821198870119910 120577119897119886 (119909) cos120582119887119899119910 1205772119886 (119909)sdot cos120582119887119873119910]H119910 = [1205771119887 (119910) cos 1205821198860119909 120577119897119887 (119910) cos 120582119886119898119909 1205772119887 (119910)sdot cos120582119886119872119909]A = [11986000 sdot sdot sdot 119860119898119899 sdot sdot sdot 119860119872119873]119879 a = [11988610 sdot sdot sdot 119886119897119899 sdot sdot sdot 1198862119873]119879 b = [11988710 sdot sdot sdot 119887119897119898 sdot sdot sdot 1198872119872]119879B = [11986100 sdot sdot sdot 119861119898119899 sdot sdot sdot 119861119872119873]119879 c = [11988810 sdot sdot sdot 119888119897119899 sdot sdot sdot 1198882119873]119879


d = [11988910 sdot sdot sdot 119889119897119898 sdot sdot sdot 1198892119872]119879C = [11986200 sdot sdot sdot 119862119898119899 sdot sdot sdot 119862119872119873]119879 e = [11989010 sdot sdot sdot 119890119897119899 sdot sdot sdot 1198902119873]119879 f = [11989110 sdot sdot sdot 119891119897119898 sdot sdot sdot 1198912119872]119879

(28)

25 Governing Eigenvalue Equations Substituting (27) into(21) results in

L119909119910Θ119909119910 + L119909Θ119909 + L119910Θ119910minus 1205962 (M119909119910Θ119909119910 + M119909Θ119909 + M119910Θ119910) = F(29)

where

L119894 = [[[11987111H119894 11987112H119894 11987113H11989411987121H119894 11987122H119894 11987123H11989411987131H119894 11987132H119894 11987133H119894]]] (119894 = 119909119910 119909 119910)

M119894 = [[[11987211H119894 0 00 11987222H119894 00 0 11987233H119894]]] (119894 = 119909119910 119909 119910)

Θ119909119910 = [A B C]119879 Θ119909 = [a c e]119879 Θ119910 = [b d f]119879

F = [0 0 q]119879

(30)

In the same way substituting (27) into (23) the generalboundary conditions of the plate can be rewritten as

L119895119909119910Θ119909119910 + L119895119909Θ119909 + L119895119910Θ119910 = 0 (119895 = 1199090 1199100 119909119886 119910119887) (31)

where

L119895119894 = [[[[11987111989511H119894 11987111989512H119894 11987111989513H11989411987111989521H119894 11987111989522H119894 11987111989523H11989411987111989531H119894 11987111989532H119894 11987111989533H119894

]]]] (119894 = 119909119910 119909 119910 119895 = 1199090 1199100 119909119886 119910119887)

(32)

To derive the constraint equations of the unknownFourier coefficients all the sine terms the auxiliary polyno-mial functions and their derivatives in (29) and (31) will beexpanded into Fourier cosine series letting

C119909119910 = [cos 1205821198860119909 cos 1205821198870119910 sdot sdot sdot cos 120582119886119898119909sdot cos 120582119887119899119910 sdot sdot sdot cos 120582119886119872119909 cos120582119887119873119910]C119909 = [cos 1205821198860119909 sdot sdot sdot cos 120582119886119898119909 sdot sdot sdot cos 120582119886119872119909]C119910 = [cos1205821198870119910 sdot sdot sdot cos 120582119887119899119910 sdot sdot sdot cos 120582119887119873119910]

(33)

Multiplying (29) with 119862119909119910 in the left side and integratingit from 0 to 119886 and 0 to 119887 separately with respect to 119909 and 119910obtains

L119909119910Θ119909119910 + [L119909 L119910] [Θ119909Θ119910

]minus 1205962 (M119909119910Θ119909119910 + [M119909 M119910] [Θ119909

Θ119910

]) = F

(34)

where

L119909119910 = int1198860int1198870

C119909119910L119909119910119889119909119889119910M119909119910 = int119886

0int1198870

C119909119910M119909119910119889119909119889119910L119909 = int119886

0int1198870

C119909119910L119909119889119909119889119910M119909 = int119886

0int1198870

C119909119910M119909119889119909119889119910L119910 = int119886

0int1198870

C119909119910L119910119889119909119889119910M119910 = int119886

0int1198870

C119909119910M119910119889119909119889119910

(35)

Similarlymultiplying (31) with119862119910 in the left side and thenintegrating it from 0 to 119887 with respect to 119910 along the edges119909 = 1199090 and 119909119886 and multiplying (31) with 119862119909 in the left sideand then integrating it from 0 to 119886 with respect to 119909 at theedges 119910 = 1199100 and 119910119887 we have

L119895119909119910Θ119909119910 + L119895119909Θ119909 + L119895119910Θ119910 = 0 (119895 = 1199090 1199100 1199091 1199101) (36)

where

L1199090119909119910 = int1198870

C119910L1199090119909119910119889119910

L119909119886119909119910 = int1198870

C119910L119909119886119909119910119889119910

L1199090119909 = int1198870

C119910L1199090119909 119889119910

L119909119886119909 = int1198870

C119910L119909119886119909 119889119910

L1199090119910 = int1198870

C119910L1199090119910 119889119910

L119909119886119910 = int1198870

C119910L119909119886119910 119889119910

L1199100119909119910 = int1198860

C119909L1199100119909119910119889119909

L119910119887119909119910 = int1198860

C119909L119910119887119909119910119889119909


Table 1Material properties used in the FGplate Table 1 is reproduced fromWang [42] et al (2017) (under theCreative CommonsAttributionLicensepublic domain)

Properties Metal CeramicSteel Aluminum (Al) Alumina (Al2O3) Zirconia (ZrO2)119864 (Gpa) 207 70 380 200120583 03 03 03 03120588 (kgm3) 7800 2702 3800 5700

L1199100119909 = int1198860

C119909L1199100119909 119889119909

L119910119887119909 = int1198860

C119909L119910119887119909 119889119909

L1199100119910 = int1198860

C119909L1199100119910 119889119909

L119910119887119910 = int1198860

C119909L119910119887119910 119889119909

(37)Thus (36) can be rewritten as

[Θ119909Θ119910

] = [[[[[[[[L1199090119909 L1199090119910L119909119886119909 L119909119886119910L1199100119909 L1199100119910L119910119886119909 L119910119887119910

]]]]]]]]

minus1 [[[[[[[[L1199090119909119910L119909119886119909119910L1199100119909119910L119910119887119909119910

]]]]]]]]Θ119909119910 (38)

Finally combining (34) and (37) results in(K minus 1205962M)Θ119909119910 = F (39)

where

K = L119909119910 minus [L119909 L119910] [[[[[[[[L1199090119909 L1199090119910L119909119886119909 L119909119886119910L1199100119909 L1199100119910L119910119886119909 L119910119887119910

]]]]]]]]

minus1 [[[[[[[[L1199090119909119910L119909119886119909119910L1199100119909119910L119910119887119909119910

]]]]]]]]

M = M119909119910 minus [M119909 M119910] [[[[[[[[L1199090119909 L1199090119910L119909119886119909 L119909119886119910L1199100119909 L1199100119910L119910119886119909 L119910119887119910

]]]]]]]]

minus1 [[[[[[[[L1199090119909119910L119909119886119909119910L1199100119909119910L119910119887119909119910

]]]]]]]]

(40)

In (40) K is the stiffness matrix for the plate and the Mis the mass matrix F is the load vector In the static analysisthe Fourier coefficients Θ119909119910 will be firstly solved from (39)by setting 120596 = 0 and then the remaining Fourier coefficientswill be calculated by using (38) The actual displacementfunction can then be easily determined from (37) While theload vector is equal to zero (F = 0) the vibration behaviorwhich consists of the natural frequencies (or eigenvalues) andassociated mode shapes (or eigenvectors) of FG plates can bereadily obtained

3 Numerical Results and Discussion

In order to demonstrate the present method the static andfree vibration analysis of FG plates with different boundaryconditions will be considered in the following examples Fourtypes of material properties will be used in the followingexamples as seen in Table 1 For the purpose of describingthe boundary condition sequence of the FG plate a simpleletter string is employed to simplify this study as shownin Figure 1 The corresponding stiffness for the restrainingsprings is specified in Table 2 In addition the new functionof the shear correction factor will be introduced in the FGplate which can adapt to the actual model and the detailedexpression is defined as120581 (119901 120599) = 56 + 1198621 (119890minus1198622119901 minus 119890minus1198623119901) (10120599 minus 2)minus 1198624 (119890minus1198625119901 minus 119890minus1198626119901) (10120599 minus 1) (41)

where 120599 is the thickness-to-length ratio and ℎ119886 and 119862119894 arethe corresponding constant coefficients values of which arelisted in Table 3 If you need more detailed information aboutthe principle and reason of the new shear correction factormodel you can read [63]

31 Static Deflections The first example considers thedeformations of the isotropic plates under the action ofa uniform pressure 1199010 = 1000 Pa with various boundaryconditions ie CCCC CFCF CCCF CSCS SSSS andCSFF and different action regions as [(0 1) (0 1)] [(0109) (01 09)] [(02 08) (02 08)] [(03 07) (03 07)]and [(04 06) (04 06)] and the results are shown inTable 4 Also the deformations of the isotropic plates withcentral point loading 1199010 = 01 N are also considered in theTable 4 The locations of the maximum deflections obviouslydepend on the boundary condition while the maximumdeflection obviously occurs at the center of the clampedplate but its locations will not be so clear for other boundaryconditions For comparison the maximum deflectionspredicted by the ABAQUS based on the finite elementmethod (FEM) are also given in Table 4 due to the lack ofthe reference data Besides the deflection fields of isotropicplate for the CCCC CCCF CFCF and CSCF by means of thepresent method and finite element method are presented inFigure 3 The element type and mesh sizes of the ABAQUSmodel are the S4R and 001 m times 001 m respectively Thegeometrical parameters of the plate are defined as 119886119887 = 1ℎ119886 = 005 and 119886 = 1 m and the material parameter ischosen as the steel From Table 4 and Figure 3 it is obvious


Table 2 The corresponding spring stiffness values for various boundary conditions

Edges Boundarycondition Essential conditions Corresponding spring stiffness valuesΓ119908 Γ119909or 119910 Γ119909119910119909 = constantFree (F) 119876119909 = 119872119909 = 119872119909119910 = 0 0 0 0

Clamped (C) 119908 = 120593119909 = 120593119910 = 0 1015 1015 1015

Simply supported (S) 119908 = 119872119909 = 120593119910 = 0 1015 0 1015

119910 = constantFree (F) 119876119909 = 119872119910 = 119872119909119910 = 0 0 0 0

Clamped (C) 119908 = 120593119910 = 120593119909 = 0 1015 1015 1015

Simply supported (S) 119908 = 119872119910 = 120593119909 1015 0 1015

Table 3 The values of the constant coefficients used in 120581 function for two FGMs

FGMs Constant coefficients1198621 1198622 1198623 1198624 1198625 1198626AlAl2O3 0750 0025 2000 0640 0060 1000AlZrO2 0560 0001 5450 0420 0095 1175

that the present solution has excellent prediction accuracyfor the static deflection

Based on the verification some new results of the staticdeflection of the FG plate with different boundary conditionsand FGs type will been shown in the Tables 5ndash7 Table 5shows the maximum deflection for AlAl2O3 and AlZrO2plate having CCCC CSCS CFCF and SSSS boundary casesunder a central concentrated force 1199010 = 1 N Also themaximum deflections of AlAl2O3 and AlZrO2 plate havingdifferent boundary conditions with the uniform pressureacting on entire rectangular area and part rectangular area[(02 08) (05 15)] are respectively performed in Tables 6and 7 The geometrical parameters of the above Tables areused as follows 119887119886 = 2 ℎ119886 = 01 and 119886 = 1 m Fromthe above tables we can see that the maximum deflectionincreases with the increase of the power-law exponent 119901regardless of the boundary conditions and load functionsIn order to complete this study and to further enhance theunderstanding of this phenomenon the relations between themaximum deflection and power-law exponent 119901 are shownin Figures 4 and 5 The geometrical parameters of those arethe same as those of Tables 5ndash7 The variation of the staticdeflection versus the power-law exponent 119901 for FG platewith central concentrated force subject to different bound-ary condition is presented in Figure 4 Two FG materialsAlAl2O3 and AlZrO2 are considered in Figure 4 For theAlAl2O3 plate no matter what the boundary condition isthe maximum deflection keeps increasing when the power-law exponent 119901 increases However for the AlZrO2 platethe maximum deflection trace climbs up then declines andreaches its crest around in the critical value Figure 5 showsthe variations of the static deflection versus the power-lawexponent 119901 for AlAl2O3 plate having the uniform pressurewith different acting regions which are the same as those ofTables 6 and 7 Four kinds boundary condition ie CCCCCSCS CFCF and SSSS are consideredWe can see clearly thatthemaximum deflection always climbs up versus the increaseof the power-law exponent 119901 regardless of the boundarycondition and load type Finally those results in Tables 5ndash7

and Figures 4-5 may serve as the benchmark data of FG platefor the future works in this filed

32 Free Vibration Analysis In dynamic analysis like thesteady state response and transient response the free vibra-tion plays an important role Thus in this subsection ourattention will be focused on the modal results To startwith a verification study is given to display the accuracyand reliability of the current method In Table 8 the presentmethod is verified by comparing the evaluation of fundamen-tal frequency parameters Ω = 120596ℎ(120588119898119864119898)12 for a simplysupported AlZrO2 square plate with those of the finiteelement HSDT method [7] finite element FSDT method[7] two-dimensional higher-order theory [20] an analyticalFSDT solution [63] an exact closed-form procedure on basisFSDT [22] and three-dimensional theory by employing thepower series method [23] From Table 8 it is not hard to seethat the results obtained by the present method are in closeagreement with those obtained by other methods For thesake of completeness the comparison for the fundamentalfrequency Ω = 120596ℎ(120588119888119864119888)12 of AlAl2O3 and AlZrO2 platesunder six combinations of boundary conditions ie SSSCSCSC SSSF SCSF and SFSF is given in Tables 9 and 10respectivelyThe results reported byHashemi et al [63] on thebasis of the FSDT are included in the comparison Althoughdifferent solution approaches are used in the literature it isstill clearly seen that the present results and referential dataagree well with each other

After the verification some new vibration results of FGplate with a variety of boundary conditions will be obtainedin Tables 11 and 12 by using the present solution In cases ofTable 11 the FG plates are fabricated from aluminum (metal)and alumina (ceramic) In cases of Table 12 the FG plates arefabricated from aluminum (metal) and zirconia (ceramic)The geometrical parameters and power-law exponents of theFG plate are taken to be 119887119886 = 2 ℎ119886 = 005 01 and02 and 119901 = 0 05 1 5 and 10 Some selected modeshapes of the AlAl2O3 plate are as shown in Figure 6 FromTables 11-12 the power-law exponent significantly affects the


Table4Com

paris

onof

them

axim

umdeflectionforisotro

pics

quarep

latesh

avingvario

usbo

undary

cond

ition

swith

different

load

functio

ns

Boun

dary

cond

ition

sMetho

dUniform

pressure

PointL

oad(0505

)[(01)

(01)][(010

9)(0109)]

[(0208)(02

08)][(030

7)(0307)]

[(0406)(04

06)]CCCC

Present

5602E

-07

5438E

-07

4530E

-07

2790E

-07

8901E-08

2522E

-10

FEM

5602E

-07

5438E

-07

4530E

-07

2789E

-07

8896E

-08

2563E

-10

CFCF

Present

1278E-06

9716E-07

7275E-07

4116

E-07

1237E-07

340

1E-10

FEM

1278E-06

9716E-07

7273E-07

4115

E-07

1236E-07

3431E-10

CCCF

Present

1301E-09

8452E

-07

6152E

-07

3502E

-07

1066

E-07

2950E

-10

FEM

1302E-09

8452E

-07

6150E

-07

3501E-07

1065E-07

2981E-10

CSCS

Present

840

6E-07

7930E-07

6272E

-07

3677E

-07

1129E-07

3157E

-10

FEM

840

6E-07

7930E-07

6272E

-07

3677E

-07

1129E-07

3174E

-10

SSSS

Present

1737E-06

1584E-06

1175E-06

644

8E-07

1874E-07

506

4E-10

FEM

1737E-06

1585E-06

1740

E-06

644

7E-07

1873E-07

5076E

-10

CSFF

Present

3084E

-05

1858E-05

9955E-06

4267E

-06

1043E-06

260

4E-09

FEM

3084E

-05

1858E-05

9955E-06

4267E

-06

1043E-06

2618E

-09


CCCC[(0208) (0208)]

CFCF [(0406) (0406)]

CCCF [(01) (01)]

CSCF (0505) (Point Load)

(a) (b)

Figure 3 Static deflection of an isotropic plate under different loads (a) finite element (ABAQUS) (b) current solution


Table 5 Maximum deflectionW for FG plates having various boundary conditions with the central point load119901 AlAl2O3 AlZrO2CCCC CSCS CFCF SSSS CCCC CSCS CFCF SSSS

0 2542E-10 4867E-10 1349E-09 5183E-10 4830E-10 9248E-10 2564E-09 9847E-1005 4429E-10 7747E-10 2051E-09 8179E-10 8237E-10 1402E-09 3657E-09 1477E-092 1040E-09 1486E-09 3442E-09 1536E-09 1123E-09 1840E-09 4683E-09 1928E-095 1201E-09 1746E-09 4100E-09 1806E-09 1322E-09 2138E-09 5400E-09 2234E-0910 1205E-09 1873E-09 4621E-09 1952E-09 1472E-09 2390E-09 6048E-09 2498E-0950 1223E-09 2236E-09 6065E-09 2371E-09 1701E-09 2841E-09 7304E-09 2982E-09100 1264E-09 2381E-09 6557E-09 2532E-09 1709E-09 2896E-09 7539E-09 3049E-09

Table 6 Maximum deflectionW for FG plates having various boundary conditions with the uniform pressure in the entire rectangular area119901 AlAl2O3 AlZrO2CCCC CSCS CFCF SSSS CCCC CSCS CFCF SSSS


Table 7 Maximum deflection for FG plates having various boundary conditions with the uniform pressure in the part rectangular area[(02 08) (05 15)]119901 AlAl2O3 AlZrO2

CCCC CSCS CFCF SSSS CCCC CSCS CFCF SSSS0 6583E-08 1713E-07 6209E-07 1897E-07 1251E-07 3254E-07 1180E-06 3604E-0705 1072E-07 2583E-07 9251E-07 2834E-07 1954E-07 4590E-07 1637E-06 5021E-072 2160E-07 4213E-07 1452E-06 4507E-07 2585E-07 5861E-07 2078E-06 6384E-075 2530E-07 5043E-07 1745E-06 5408E-07 3012E-07 6742E-07 2384E-06 7333E-0710 2669E-07 5723E-07 2009E-06 6192E-07 3369E-07 7555E-07 2673E-06 8219E-0750 3060E-07 7668E-07 2764E-06 8454E-07 3972E-07 9160E-07 3257E-06 9999E-07100 3233E-07 8314E-07 3009E-06 9193E-07 4035E-07 9461E-07 3373E-06 1035E-06

fundamental frequency of the FG plate To have a moreintuitive understanding the variation of the fundamentalfrequencies of FG plate with power-law exponents anddifferent boundary conditions is depicted in Figures 7-8From Figure 7 we can see that the fundamental frequenciesdecreasemonotonouslywhile increasing the power-law index119901 Moreover the fundamental frequencies rapidly decreasethen increase and finally decrease with the power-law index119901 increasing when the FG plate is fabricated from aluminum(metal) and zirconia (ceramic)

4 Conclusions

This paper presents a unified functionally graded Reissner-Mindlin rectangular plate model with general boundaryconditions for static and vibration analysisTheunifiedmodelis based on the Reissner-Mindlin plate theory and the generalboundary condition is implemented by using the arbitraryboundary technique which consists of one group of liner

spring and two groups of rotation springs along all edges Animproved Fourier series method which contains a 2D Fouriercosine series and auxiliary function in the form of 1D seriesis introduced to formulate the components of admissibledisplacement functions The target of the auxiliary functionis to remove the potential discontinuities of the displacementfunction and their derivative at the edges and ensure andaccelerate the convergence of the series representation Thissolution method is exact in the sense that the governingdifferential equation and the boundary conditions are simul-taneously satisfied on a point-wise basis to any specifiedaccuracy The excellent accuracy and reliability of the presentmethod are displayed by comparing its static and vibrationpredictions with those of other techniques The researchof static and vibration analysis of FG plate could providethe theoretical basis for practical engineering applicationsAnd the results of this paper could serve as the referencedata in practical engineering applications in related field infuture


00

Max

imum

defl

ectio

n W

Power-law exponent p

CCCCSSSS

CFCFCSCS

20 times 10minus9

40 times 10minus9

60 times 10minus9

80 times 10minus9

10minus3 10minus2 10minus1 100 101 102 103

(a)

00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus9

40 times 10minus9

60 times 10minus9

80 times 10minus9


(b)

Figure 4 Variation of the static deflection versus the power-law exponent 119901 for FG plate with central point load (a) AlAl2O3 plate (b)AlZrO2 plate

00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus6

40 times 10minus6

60 times 10minus6

80 times 10minus6


(a)

00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

10 times 10minus6

20 times 10minus6

30 times 10minus6


(b)

Figure 5 Variation of the static deflection versus the power-law exponent 119901 for AlAl2O3 plate with different uniform press (a) entirerectangular plate (b) part rectangular area [(02 08) (05 15)]

CCCC CCCS CFCF CFFF

CSCF CSFF FFFF SSSFFigure 6 Fundamental mode shape of AlAl2O3 plates with different boundary conditions


Table 8 Comparison of fundamental frequency parameter Ω = 120596ℎ(120588119898119864119898)12 for SSSS AlZrO2 square plates Table 8 is reproduced fromWang [42] et al (2017) (under the Creative Commons Attribution Licensepublic domain)

Method 119901 = infin 119901 = 1 ℎ119886 = 02ℎ119886 = 1radic10 ℎ119886 = 01 ℎ119886 = 005 ℎ119886 = 01 ℎ119886 = 02 119901 = 2 119901 = 3 119901 = 5present 04618 00577 00159 00611 02270 02249 02254 02265HSDT [7] 04658 00578 00157 00613 02257 02237 02243 02253FSDT [7] 04619 00577 00162 00633 02323 02325 02334 02334HSDT [20] 04658 00577 00158 00619 02285 02264 02270 02281FSDT [23] 04618 00576 00158 00611 0227 02249 02254 02265FSDT [24] 04618 00577 00158 00619 02276 02264 02276 022913-D [27] 04658 00577 00153 00596 02192 02197 02211 02225

0 20 40 60 80 100

004

005

006

007

frequ

ency

par

amet

ers Ω

power-law exponent p

CCCC

0 20 40 60 80 100power-law exponent p

CFCF

0008

0010

0012

0014

0016

frequ

ency

par

amet

ers Ω


CFFF

00016

00020

00024

frequ

ency

par

amet

ers Ω


CSCF

0012

0016

0020

frequ

ency

par

amet

ers Ω

Figure 7 Variations of fundamental frequency parameter Ω versus the power-law exponent 119901 for AlAl2O3 rectangular plate with differentboundary conditions

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This study was funded by High Technology Ship Fundsof Ministry of Industry and Information of ChinaPhD Student Research and Innovation Fund of theFundamental Research Funds for the Central Universities(HEUGIP201801) Assembly Advanced Research Fundof China (6140210020105) Major Innovation Projects of


Table 9 Comparison of fundamental frequency parameterΩ = 120596ℎ(120588119888119864119888)12 for AlAl2O3 plates when ℎ119886 = 015119887119886 119901 SSSC SCSC SSSF SFSF SCSFExact Present Exact Present Exact Present Exact Present Exact Present

2

0 008325 008325 008729 008729 006713 006713 006364 006365 006781 006781025 007600 007600 007950 007950 006145 006145 005829 005829 006205 0062051 006541 006541 006790 006790 005346 005346 005080 005080 005391 0053915 005524 005524 005695 005695 004568 004568 004349 004349 004600 004600infin 004263 004237 004443 004443 003417 003417 003239 003240 003451 003452

1

0 014378 014377 016713 016712 007537 007537 006290 006290 008062 008062025 012974 012973 014927 014926 006890 006890 005761 005761 007351 0073501 010725 010725 011955 011955 005968 005968 005021 005021 006308 0063085 008720 008720 009479 009479 005078 005078 004301 004300 005322 005322infin 007318 007318 008507 008507 003836 003836 003202 003202 004104 004104

05

0 035045 035037 041996 041980 010065 010065 006217 006216 013484 013483025 030709 030703 036112 036102 009170 009170 005695 005695 012160 0121591 023262 023260 026091 026088 007851 007851 004970 004970 010066 0100665 017691 017690 019258 019257 006610 006610 004262 004262 008226 008226infin 017921 017837 021375 021374 005123 005123 003164 003164 006863 006863

0 20 40 60 80 100

006

007

frequ

ency

par

amet

ers Ω


CCCC


CFCF

0014

0015

0016

frequ

ency

par

amet

ers Ω


CFFF

00024

00026

frequ

ency

par

amet

ers Ω


CSCF

0017

0016

0018

0019

frequ

ency

par

amet

ers Ω

Figure 8 Variations of fundamental frequency parameter Ω versus the power-law exponent 119901 for AlZrO2 rectangular plate with differentboundary conditions


Table 10 Comparison of fundamental frequency parameterΩ = 120596ℎ(120588119888119864119888)12 for AlZrO2 plates when 119886119887 = 15ℎ119886 119901 SSSC SCSC SSSF SFSF SCSFExact Present Exact Present Exact Present Exact Present Exact Present

005

0 003129 003130 004076 004076 001024 001024 000719 000719 001249 0012492025 002899 002899 003664 003664 000981 000981 000692 000692 001185 00118511 002667 002667 003250 003250 000948 000948 000674 000674 001132 00113165 002677 002677 003239 003239 000963 000963 000685 000685 001146 001146infin 002689 002689 003502 003502 000880 000880 000618 000618 001073 0010734

01

0 011639 011638 014580 014580 004001 004001 002835 002835 004817 004817025 010561 010561 012781 012780 003810 003810 002717 002717 004532 00453151 009734 009733 011453 011453 003679 003679 002641 002641 004327 00432675 009646 009646 011234 011234 003718 003718 002677 002677 004352 0043517infin 010001 010001 012528 012528 003438 003438 002426 002436 004139 0041391

02

0 037876 037870 043939 043930 014871 014871 010795 010795 017323 0173218025 036117 036113 041624 041617 014354 014354 010436 010436 016671 016671 033549 033547 037962 037958 013851 013851 010127 010127 015937 01593665 032783 032781 036695 036692 013888 013887 010200 010200 015878 0158779infin 032545 032544 037755 037753 012779 012778 009276 009276 014885 0148845

Table 11 Fundamental frequency parameterΩ = 120596ℎ(120588119888119864119888)12 for AlAl2O3 plates with different boundary conditionsℎ119886 119901 Boundary conditionsCCCC CCCS CFCF FFFF CSCF CSFF CFFF SSSF

005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001541 001175 000360 000355 000429 000139 000057 0002641 001291 001027 000329 000333 000389 000129 000054 0002465 001111 000893 000290 000296 000342 000114 000048 00021810 001107 000868 000274 000274 000324 000107 000044 000203

01

0 006897 018772 001632 001608 001941 000630 000259 00120005 005652 015857 001406 001405 001663 000547 000227 0010431 004862 014291 001291 001320 001519 000509 000214 0009745 003953 011409 001113 001166 001303 000446 000190 00085610 003929 011157 001054 001082 001239 000417 000175 000798

02

0 023286 018772 006198 006260 007269 002439 001026 00465705 019360 015857 005338 005472 006242 002121 000899 0040551 017136 014291 004927 005147 005740 001983 000848 0037985 013242 011409 004126 004505 004776 001714 000749 00329510 013167 011157 003935 004191 004569 001606 000694 003081

Table 12 Fundamental frequency parameterΩ = 120596ℎ(120588119888119864119888)12 for AlZrO2 plates with different boundary conditions

ha p Boundary conditionsCCCC CCCS CFCF FFFF CSCF CSFF CFFF SSSF

005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001606 001246 000388 000386 000461 000150 000062 0002871 001588 001235 000386 000385 000458 000150 000062 0002865 001598 001249 000393 000392 000465 000153 000063 00029110 001481 001179 000378 000383 000447 000148 000062 000283

01

0 006897 005286 001632 001608 001941 000630 000259 00120005 006029 004770 001524 001532 001801 000595 000247 0011351 005888 004690 001511 001526 001782 000591 000246 0011295 005858 004702 001529 001553 001801 000601 000251 00114810 005748 004611 001498 001521 001765 000588 000246 001124

02

0 023288 018773 006198 006260 007269 002439 001026 00465705 021550 017558 005870 005988 006871 002324 000983 0044421 020752 017074 005782 005952 006754 002304 000979 0044075 020352 016912 005810 006038 006770 002332 000996 00446410 020672 017013 005770 005935 006737 002300 000978 004398


High Technology Ship Funds of Ministry of Industryand Information of China National key Research andDevelopment Program (2016YFC0303406) Naval Pre-Research Project Fundamental Research Funds for theCentral University (HEUCFD1515 HEUCFM170113) ChinaPostdoctoral Science Foundation (2014M552661) andNational Natural Science Foundation of China (5120905251709063)

References

[1] M Koizumi ldquoFGM activities in Japanrdquo Composites Part BEngineering vol 28 no 1-2 pp 1ndash4 1997

[2] C T Loy K Y Lam and J N Reddy ldquoVibration of functionallygraded cylindrical shellsrdquo International Journal of MechanicalSciences vol 41 no 3 pp 309ndash324 1999

[3] S Abrate ldquoFunctionally graded plates behave like homogeneousplatesrdquoComposites Part B Engineering vol 39 no 1 pp 151ndash1582008

[4] S Abrate ldquoFree vibration buckling and static deflections offunctionally graded platesrdquo Composites Science and Technologyvol 66 no 14 pp 2383ndash2394 2006

[5] L F Qian R C Batra and L M Chen ldquoFree and forcedvibrations of thick rectangular plates using higher-order shearand normal deformable plate theory and Meshless Petrov-Galerkin (MLPG) methodrdquo Computer Modeling in Engineeringand Sciences vol 4 no 5 pp 519ndash534 2003

[6] L F Qian R C Batra and L M Chen ldquoStatic and dynamicdeformations of thick functionally graded elastic plates by usinghigher order shear and normal deformable plate theory andmeshless local Petrov-Galerkin methodrdquo Composites Part BEngineering vol 35 no 6ndash8 pp 685ndash697 2004

[7] S Pradyumna and JN Bandyopadhyay ldquoFree vibration analysisof functionally graded curved panels using a higher-order finiteelement formulationrdquo Journal of Sound and Vibration vol 318no 1-2 pp 176ndash192 2008

[8] A J M Ferreira R C Batra C M C Roque L F Qian and RM N Jorge ldquoNatural frequencies of functionally graded platesby ameshless methodrdquoComposite Structures vol 75 no 1-4 pp593ndash600 2006

[9] G N Praveen and J N Reddy ldquoNonlinear transient ther-moelastic analysis of functionally graded ceramic-metal platesrdquoInternational Journal of Solids and Structures vol 35 no 33 pp4457ndash4476 1998

[10] J N Reddy ldquoAnalysis of functionally graded platesrdquo Interna-tional Journal for Numerical Methods in Engineering vol 47 no1ndash3 pp 663ndash684 2000

[11] L Della Croce and P Venini ldquoFinite elements for functionallygraded Reissner-Mindlin platesrdquo Computer Methods AppliedMechanics and Engineering vol 193 no 9-11 pp 705ndash725 2004

[12] Z-Q Cheng and R C Batra ldquoDeflection relationships betweenthe homogeneousKirchhoffplate theory anddifferent function-ally graded plate theoriesrdquo Archives of Mechanics vol 52 no 1pp 143ndash158 2000

[13] Z-Q Cheng and R C Batra ldquoExact correspondence betweeneigenvalues of membranes and functionally graded simplysupported polygonal platesrdquo Journal of Sound and Vibrationvol 229 no 4 pp 879ndash895 2000

[14] X Zhao Y Y Lee and K M Liew ldquoFree vibration analysisof functionally graded plates using the element-free kp-Ritz

methodrdquo Journal of Sound and Vibration vol 319 no 3ndash5 pp918ndash939 2009

[15] J Woo S A Meguid and L S Ong ldquoNonlinear free vibrationbehavior of functionally graded platesrdquo Journal of Sound andVibration vol 289 no 3 pp 595ndash611 2006

[16] J Yang and H-S Shen ldquoDynamic response of initially stressedfunctionally graded rectangular thin platesrdquo Composite Struc-tures vol 54 no 4 pp 497ndash508 2001

[17] J Yang and H-S Shen ldquoVibration characteristics and transientresponse of shear-deformable functionally graded plates inthermal environmentsrdquo Journal of Sound and Vibration vol255 no 3 pp 579ndash602 2002

[18] Z Zhong and T Yu ldquoVibration of a simply supported function-ally graded piezoelectric rectangular platerdquo SmartMaterials andStructures vol 15 no 5 article no 029 pp 1404ndash1412 2006

[19] C M C Roque A J M Ferreira and R M N Jorge ldquoAradial basis function approach for the free vibration analysisof functionally graded plates using a refined theoryrdquo Journal ofSound and Vibration vol 300 no 3-5 pp 1048ndash1070 2007

[20] H Matsunaga ldquoFree vibration and stability of functionallygraded plates according to a 2-D higher-order deformationtheoryrdquo Composite Structures vol 82 no 4 pp 499ndash512 2008

[21] P Yousefi M H Kargarnovin and S H Hosseini-HashemildquoFree vibration of generally laminated plates with variousshapesrdquo Polymer Composites vol 32 no 3 pp 445ndash454 2011

[22] S Hosseini-Hashemi M Fadaee and S R Atashipour ldquoA newexact analytical approach for free vibration of ReissnerndashMindlinfunctionally graded rectangular platesrdquo International Journal ofMechanical Sciences vol 53 no 1 pp 11ndash22 2011

[23] S S Vel and R C Batra ldquoThree-dimensional exact solution forthe vibration of functionally graded rectangular platesrdquo Journalof Sound and Vibration vol 272 no 3ndash5 pp 703ndash730 2004

[24] S M Shiyekar and T Kant ldquoAn electromechanical higher ordermodel for piezoelectric functionally graded platesrdquo Interna-tional Journal of Mechanics and Materials in Design vol 6 no2 pp 163ndash174 2010

[25] E Carrera S Brischetto M Cinefra and M Soave ldquoEffects ofthickness stretching in functionally graded plates and shellsrdquoComposites Part B Engineering vol 42 no 2 pp 123ndash133 2011

[26] A M A Neves A J M Ferreira E Carrera et al ldquoStaticfree vibration and buckling analysis of isotropic and sandwichfunctionally graded plates using a quasi-3D higher-order sheardeformation theory and ameshless techniquerdquoComposites PartB Engineering vol 44 no 1 pp 657ndash674 2013

[27] L W Zhang P Zhu and K M Liew ldquoThermal bucklingof functionally graded plates using a local Kriging meshlessmethodrdquo Composite Structures vol 108 pp 472ndash492 2014

[28] Z Belabed M S A Houari A Tounsi S R Mahmoud andO Anwar Beg ldquoAn efficient and simple higher order shear andnormal deformation theory for functionally graded material(FGM) platesrdquoComposites Part B Engineering vol 60 pp 274ndash283 2014

[29] H-T Thai and D-H Choi ldquoA refined shear deformationtheory for free vibration of functionally graded plates on elasticfoundationrdquo Composites Part B Engineering vol 43 no 5 pp2335ndash2347 2012

[30] H-T Thai and D-H Choi ldquoA simple first-order shear defor-mation theory for the bending and free vibration analysis offunctionally graded platesrdquo Composite Structures vol 101 pp332ndash340 2013


[31] H-T Thai and S-E Kim ldquoA simple higher-order shear defor-mation theory for bending and free vibration analysis offunctionally graded platesrdquo Composite Structures vol 96 pp165ndash173 2013

[32] H-T Thai and S-E Kim ldquoA simple quasi-3D sinusoidal sheardeformation theory for functionally graded platesrdquo CompositeStructures vol 99 pp 172ndash180 2013

[33] H-T Thai and S-E Kim ldquoClosed-form solution for bucklinganalysis of thick functionally graded plates on elastic founda-tionrdquo International Journal of Mechanical Sciences vol 75 pp34ndash44 2013

[34] H-T Thai and T P Vo ldquoA new sinusoidal shear deformationtheory for bending buckling and vibration of functionallygraded platesrdquoAppliedMathematical Modelling Simulation andComputation for Engineering and Environmental Systems vol37 no 5 pp 3269ndash3281 2013

[35] N Valizadeh S NatarajanO A Gonzalez-Estrada T RabczukT Q Bui and S P A Bordas ldquoNURBS-based finite elementanalysis of functionally graded plates Static bending vibrationbuckling and flutterrdquoComposite Structures vol 99 pp 309ndash3262013

[36] L V Tran A J M Ferreira and H Nguyen-Xuan ldquoIsogeo-metric analysis of functionally graded plates using higher-ordershear deformation theoryrdquo Composites Part B Engineering vol51 no 4 pp 368ndash383 2013

[37] J L Mantari and C Guedes Soares ldquoA novel higher-ordershear deformation theory with stretching effect for functionallygraded platesrdquoComposites Part B Engineering vol 45 no 1 pp268ndash281 2013

[38] S Xiang and G-w Kang ldquoA nth-order shear deformationtheory for the bending analysis on the functionally gradedplatesrdquo European Journal of Mechanics - ASolids vol 37 pp336ndash343 2013

[39] N Fantuzzi F Tornabene and E Viola ldquoFour-parameter func-tionally graded cracked plates of arbitrary shape a GDQFEMsolution for free vibrationsrdquo Mechanics of Advanced Materialsand Structures vol 23 no 1 pp 89ndash107 2016

[40] H Nguyen-Xuan L V Tran C H Thai S Kulasegaramand S P A Bordas ldquoIsogeometric analysis of functionallygraded plates using a refined plate theoryrdquo Composites Part BEngineering vol 64 pp 222ndash234 2014

[41] S S Akavci ldquoAn efficient shear deformation theory for freevibration of functionally graded thick rectangular plates onelastic foundationrdquoComposite Structures vol 108 no 1 pp 667ndash676 2014

[42] Q Wang D Shi Q Liang and F Pang ldquoA unified solutionfor vibration analysis of moderately thick functionally gradedrectangular plateswith general boundary restraints and internalline supportsrdquoMechanics of Advanced Materials and Structuresvol 24 no 11 pp 961ndash973 2017

[43] F Pang et al ldquoA semi analytical method for the free vibration ofdoubly-curved shells of revolutionrdquo Computers amp Mathematicswith Applications vol 75 no 9 pp 3249ndash3268 2018

[44] F Pang H Li X Miao and X Wang ldquoA modified Fouriersolution for vibration analysis of moderately thick laminatedannular sector plates with general boundary conditions inter-nal radial line and circumferential arc supportsrdquo Curved andLayered Structures vol 4 no 1 2017

[45] F Pang H Li K Choe D Shi and K Kim ldquoFree and ForcedVibration Analysis of Airtight Cylindrical Vessels with DoublyCurved Shells of Revolution by Using Jacobi-Ritz MethodrdquoShock and Vibration vol 2017 pp 1ndash20 2017

[46] Q Wang D Shao and B Qin ldquoA simple first-order sheardeformation shell theory for vibration analysis of compositelaminated open cylindrical shells with general boundary con-ditionsrdquo Composite Structures vol 184 pp 211ndash232 2018

[47] Q Wang K Choe D Shi and K Sin ldquoVibration analysisof the coupled doubly-curved revolution shell structures byusing Jacobi-Ritz methodrdquo International Journal of MechanicalSciences vol 135 pp 517ndash531 2018

[48] Z Su G Jin YWang and X Ye ldquoA general Fourier formulationfor vibration analysis of functionally graded sandwich beamswith arbitrary boundary condition and resting on elastic foun-dationsrdquo Acta Mechanica vol 227 no 5 pp 1493ndash1514 2016

[49] H Khov W L Li and R F Gibson ldquoAn accurate solutionmethod for the static and dynamic deflections of orthotropicplates with general boundary conditionsrdquoComposite Structuresvol 90 no 4 pp 474ndash481 2009

[50] W L Li ldquoFree vibrations of beams with general boundaryconditionsrdquo Journal of Sound and Vibration vol 237 no 4 pp709ndash725 2000

[51] G Jin T Ye X Wang and X Miao ldquoA unified solutionfor the vibration analysis of FGM doubly-curved shells ofrevolution with arbitrary boundary conditionsrdquo CompositesPart B Engineering vol 89 pp 230ndash252 2016

[52] L Li H Li F Pang X Wang Y Du and S Li ldquoThe modifiedFourier-Ritz approach for the free vibration of functionallygraded cylindrical conical spherical panels and shells ofrevolution with general boundary conditionrdquo MathematicalProblems in Engineering Art ID 9183924 32 pages 2017

[53] H Li F Pang X Wang and S Li ldquoBenchmark Solutionfor Free Vibration of Moderately Thick Functionally GradedSandwich Sector Plates on Two-Parameter Elastic Foundationwith General Boundary Conditionsrdquo Shock and Vibration vol2017 Article ID 4018629 35 pages 2017

[54] Y Zhou Q Wang D Shi Q Liang and Z Zhang ldquoExactsolutions for the free in-plane vibrations of rectangular plateswith arbitrary boundary conditionsrdquo International Journal ofMechanical Sciences vol 130 pp 1ndash10 2017

[55] Q Wang B Qin D Shi and Q Liang ldquoA semi-analyticalmethod for vibration analysis of functionally graded carbonnanotube reinforced composite doubly-curved panels andshells of revolutionrdquo Composite Structures vol 174 pp 87ndash1092017

[56] Q Wang X Cui B Qin Q Liang and J Tang ldquoA semi-analytical method for vibration analysis of functionally graded(FG) sandwich doubly-curved panels and shells of revolutionrdquoInternational Journal of Mechanical Sciences vol 134 pp 479ndash499 2017

[57] Q Wang X Cui B Qin and Q Liang ldquoVibration analysis ofthe functionally graded carbon nanotube reinforced compositeshallow shells with arbitrary boundary conditionsrdquo CompositeStructures vol 182 pp 364ndash379 2017

[58] F Pang H Li Y Du Y Shan and F Ji ldquoFree Vibration OfFunctionally Graded Carbon Nanotube Reinforced Compos-ite Annular Sector Plate With General Boundary SupportsrdquoCurved and Layered Structures vol 5 no 1 pp 49ndash67 2018

[59] H Li F Pang X Miao Y Du and H Tian ldquoA semi-analyticalmethod for vibration analysis of stepped doubly-curved shellsof revolution with arbitrary boundary conditionsrdquoThin-WalledStructures vol 129 pp 125ndash144 2018

[60] J Guo D Shi Q Wang J Tang and C Shuai ldquoDynamicanalysis of laminated doubly-curved shells with general bound-ary conditions by means of a domain decomposition methodrdquo


International Journal of Mechanical Sciences vol 138-139 pp159ndash186 2018

[61] G Jin TYe andZ Su ldquoStructural vibrationAuniformaccuratesolution for laminated beams plates and shells with generalboundary conditionsrdquo Structural Vibration AUniform AccurateSolution for Laminated Beams Plates and Shells with GeneralBoundary Conditions pp 1ndash312 2015

[62] T Ye G Jin X Ye and X Wang ldquoA series solution for thevibrations of composite laminated deep curved beams withgeneral boundariesrdquoComposite Structures vol 127 pp 450ndash4652015

[63] S Hosseini-Hashemi H Rokni Damavandi Taher H Akhavanand M Omidi ldquoFree vibration of functionally graded rectan-gular plates using first-order shear deformation plate theoryrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 34 no 5 pp1276ndash1291 2010

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solution with multiquadric radial basis functions to studythe free vibration of FG square plates with different classicalboundary conditions on the basis of the FSDT and third-order shear deformation plate theory (TSDT) Praveen andReddy [9] used the finite element method to obtain thestatic and dynamic thermoelastic behavior of the FG plateswith some selected classical boundary conditions Basedon the third-order shear deformation theory and Reddyvon Karman-type geometric nonlinearity theory Reddy [10]carried out the static linear analysis and nonlinear staticand dynamic analysis by using the finite element methodCroce and Venini [11] presented a new method of finite ele-ments for analyzingReissner-Mindlin FGplateswith classicalboundary condition Cheng and Batra [12 13] applied thefirst-order shear deformation theory and a third-order sheardeformation theory to study the static buckling and steadystate vibrations of FG polygonal plate with simply supportedboundary conditions Zhao et al [14] extended the elementfree kp-Ritzmethod to study the free vibration analysis for FGsquare and skew plates with different boundary conditionson the basis of the FSDT The nonlinear free vibrationbehavior of FG square thin plates was analyzed by Woo etal [15] by presenting an analytical solution on the basis ofthe von Karman theory Based on the CPT Yang and Shen[16] presented an analytical method for free vibration andtransient response of initially stressed FG rectangular thinplates resting on Pasternak elastic foundation having classicalboundary condition subjected to impulsive lateral loadsLater Yang and Shen [17] used a semianalytical approach toinvestigate the dynamic behavior of FG plates with impulsivelateral loads and complicated environment Zhong and Yu[18] employed a state-space approach to analyze free andforced vibrations of an FG piezoelectric rectangular thickplate with simple support at its edges Roque et al [19]employed the multiquadric radial basis function method andthe HSDT to study free vibration of FG plates with differentclassical boundary conditions The natural frequencies andbuckling stresses of FG plates having simply supported edgeswere studied by Matsunaga [20] who utilized the method ofpower series expansion and two-dimensional (2D) higher-order theory Hosseini-Hashemi et al [21 22] presented anew exact closed-form procedure for free vibration analysisof FG rectangular thick plates and Reissner-Mindlin FGrectangular plates where the plate has two opposite edgesof simply supported boundary conditions Vel and Batra[23] presented an exact 3-D elasticity solution to study thethermoelastic deformation free and forced vibrations of FGsimply supported plates Based on the higher-order shear andnormal deformation theory Shiyekar et al [24] presentedthe bidirectional flexure analysis of smart FG plate subjectedto electromechanical loading by using Navierrsquos techniqueCarrera et al [25] evaluated the effect of thickness stretchingin plateshell structures made by materials which are func-tionally graded (FGM) in the thickness direction accordingto Carrerarsquos Unified Formulation Based on Carrerarsquos UnifiedFormulation Neves et al [26] presented explicit governingequations to study static and free vibration analysis of FGMplates by using meshless technique based on collocationwith radial basis functions Zhang et al [27] used the local

Kriging meshless method to study the mechanical and ther-mal buckling behaviors of ceramicndashmetal functionally gradeplates subject the classical boundary conditions Belabed etal [28] presented an efficient and simple higher-order shearand normal deformation theory for the bending and freevibration analysis of FG simply supported platesThai and Vo[29ndash34] extended a simple shear deformation theory to studythe bending buckling and vibration of functionally gradedplates with simply supported cases Valizadeh et al [35]applied isogeometric finite elementmethod to study the staticand dynamic characteristics of functionally graded material(FGM) plates with classical boundary condition Based onisogeometric approach (IGA) and higher-order deformationplate theory (HSDT) Tran et al [36] used the 1198621 continuouselement to investigate the static dynamic and bucklinganalysis of FG rectangular and circular plates with differentclassical boundary conditions Mantari et al [37] showed ananalytical solution to static analysis of functionally gradedplates (FGPs) based on a new trigonometric higher-ordertheory in which the stretching effect was considered Xiangand Kang [38] conducted bending analysis for functionallygraded plates according to a nth-order shear deformationtheory and the meshless global collocation method basedon the thin plate spline radial basis function and Reddyrsquosthird-order theory Based on various plate theory Tornabeneet al [39] applied the Generalized Differential Quadrature(GDQ) to study the bending vibration and dynamic analysisof FG plates and panels with different classical boundaryconditions Xuan et al [40] presented a simple and effectiveapproach which incorporated the isogeometric finite elementanalysis (IGA) with a refined plate theory (RPT) for staticfree vibration and buckling analysis of functionally gradedmaterial (FGM) plates with simply supported and clampedboundary conditions Akavci [41] presented a closed-formsolution to study the free vibration analysis of functionallygraded plate resting on Pasternak (two-parameter) modelsubject to classical boundary condition

From the above review we can know that most of existingliteratures on the static and vibration analysis of functionallygraded Reissner-Mindlin plate are restricted to the classicalboundary condition However the elastic edge restraintsmay be more common in the practical engineering andthe classical boundary condition is considered as a specialcase Thus to establish a unified efficient and accurateformulation for static and vibration analysis of functionallygraded Reissner-Mindlin plate is necessary and significant

To the best of authorrsquos knowledge the subject of static andvibration analysis of functionally graded Reissner-Mindlinplate with general boundary conditions has not been per-formed in a specified work yet Therefore the major focusof this paper is to present an accurate solution methodfor the static and vibration analysis of functionally gradedReissner-Mindlin plate with general boundary conditionsThe effective material properties of functionally gradedmaterials are assumed to vary continuously in the thicknessdirection according to the power-law distribution in termsof the volume fraction of the constituents and are estimatedby the Voigt model and MorindashTanaka scheme All the platedisplacements are expanded as a modified Fourier series











120583 (119911) = 3119870 minus 21198666119870 + 2119866 (3)








z

x

y

b

a

Metal

z

h

Ceramic

y

q(xy)

1

2

3

4

y1

y2

y2

y1

x1

x2


p=10

p=5

p=2

p=1

p=05

p=02p=01

minus05

minus04

minus03

minus02

minus01

00

01

02

03

04

05

zh

02 04 06 08 1000Vc



120576119909 = 119911120597120593119909120597119909 = 119911120581119909120576119910 = 119911120597120593119910120597119910 = 119911120581119910120574119909119910 = 119911(120597120593119910120597119909 + 120597120593119909120597119910 ) = 119911120581119909119910

(6a)

120574119909119911 = 120597119908120597119909 + 120593119909120574119910119911 = 120597119908120597119910 + 120593119910 (6b)


120590119909120590119910120591119909119910120591119909119911120591119910119911

= [[[[[[[[[

11987611 (119911) 11987612 (119911) 0 0 011987612 (119911) 11987611 (119911) 0 0 00 0 11987666 (119911) 0 00 0 0 11987666 (119911) 00 0 0 0 11987666 (119911)]]]]]]]]]120576119909120576119910120574119909119910120574119909119911120574119910119911

(7)


(8)


119872119909 = 11986311 120597120593119909120597119909 + 11986312 120597120593119910120597119910 119872119910 = 11986312 120597120593119909120597119909 + 11986311 120597120593119910120597119910 119872119909119910 = 11986366 (120597120593119910120597119909 + 120597120593119909120597119910 )


119876119909 = 11986066 (120597119908120597119909 + 120593119909) 119876119910 = 11986066 (120597119908120597119910 + 120593119910)

(9)


(11986066 11986311 11986322 11986312) = intℎ2minusℎ2

(12058111987666 (119911) 11987611 (119911)sdot 1199112 11987611 (119911) 1199112 11987612 (119911) 1199112) 119889119911 (10)


+ 120574119910119911119876119910) 119889119909119889119910 = 12 int1198860 int1198870 (119872119909 120597120593119909120597119909+119872119910 120597120593119910120597119910 +119872119909119910 (120597120593119909120597119910 + 120597120593119910120597119909 )+ (120597119908120597119909 + 120593119909)119876119909 + (120597119908120597119910 + 120593119910)119876119910)119889119909119889119910

(11)

119879 = 12 int1198860 int1198870 [1198680 (120597119908120597119905 )2+ 1198682((120597120593119909120597119905 )2 + (120597120593119910120597119905 )2)]119889119909119889119910

[1198680 1198681] = intℎ2minusℎ2

120588 (1 1199112) 119889119911(12)


119881 = 12 int1198860 (1198961199100119910 1199082 + 11987011991001199101199091205932119909 + 1198701199100119910 1205932119910)10038161003816100381610038161003816119910=0 119889119909+ 12 int1198860 (119896119910119887119910 1199082 + 1198701199101198871199101199091205932119909 + 119870119910119887119910 1205932119910)10038161003816100381610038161003816119910=119887 119889119909+ 12 int1198870 (1198961199090119909 1199082 + 1198701199090119909 1205932119909 + 11987011990901199091199101205932119910)10038161003816100381610038161003816119909=0 119889119910+ 12 int1198870 (119896119909119886119909 1199082 + 119870119909119886119909 1205932119909 + 1198701199091198861199091199101205932119910)10038161003816100381610038161003816119909=119886 119889119910

(13)


119860119902 (119909 119910)119908 (119909 119910) 119889119860 (14)




int11990500(120575119879 + 120575119882 minus 120575119880 minus 120575119881) 119889119905 = 0 (16)



int11990500int1198860(11987211990911991012057512059311990910038161003816100381610038161003816119910=119887119910=0 + 11987611991012057511990810038161003816100381610038161003816119910=119887119910=0 + 11987211991012057512059311991010038161003816100381610038161003816119910=119887119910=0) 119889119909119889119905 + int1199050

0int1198870(1198721199091205751205931199091003816100381610038161003816119909=119886119909=0 + 1198761199091205751199081003816100381610038161003816119909=119886119909=0 + 11987211990911991012057512059311991010038161003816100381610038161003816119909=119886119909=0) 119889119910119889119905

int11990500int1198860(1198961199100119910 119908120575119908 + 1198701199100119910119909120593119909120575120593119909 + 1198701199100119910 120593119910120575120593119910)10038161003816100381610038161003816119910=0 119889119909119889119905 + int1199050

0int1198860(119896119910119887119910 119908120575119908 + 119870119910119887119910119909120593119909120575120593119909 + 119870119910119887119910 120593119910120575120593119910)10038161003816100381610038161003816119910=119887 119889119909119889119905

+ int11990500int1198870(1198961199090119909 119908120575119908 + 1198701199090119909 120593119909120575120593119909 + 1198701199090119909119910120593119910120575120593119910)10038161003816100381610038161003816119909=0 119889119910119889119905 + int1199050

0int1198870(119896119909119886119909 119908120575119908 + 119870119909119886119909 120593119909120575120593119909 + 119870119909119886119909119910120593119910120575120593119910)10038161003816100381610038161003816119909=119886 119889119910119889119905


0int1198860int1198870119902 (119909 119910) 120575119908119889119909119889119910119889119905 = 0

(17)



int11990500int1198860int1198870

((((



+ int11990500int1198860((119896119910119887119910 119908 + 119876119910) 120575119908 + (119870119910119887119910119909120593119909 +119872119909119910) 120575120593119909 + (119870119910119887119910 120593119910 +119872119910) 120575120593119910)10038161003816100381610038161003816119910=119887 119889119909119889119905


+ int11990500int1198870((119896119909119886119909 119908 + 119876119909) 120575119908 + (119870119909119886119909 120593119909 +119872119909) 120575120593119909 + (119870119909119886119909119910120593119910 +119872119909119910) 120575120593119910)10038161003816100381610038161003816119909=119886 119889119910119889119905 = 0

(18)



(19)




([[[[11987111 11987112 1198711311987121 11987122 1198712311987131 11987132 11987133

]]]] minus 1205962 [[[[11987211 0 00 11987222 00 0 11987233

]]]])[[[[120593119909120593119910119908]]]]

= [[[[00119902]]]]

(21)


where


(22)


[[[[[11987111989511 11987111989512 1198711198951311987111989521 11987111989522 1198711198952311987111989531 11987111989532 11987111989533

]]]]][[[[120593119909120593119910119908]]]] = [[[[

000]]]] (119895 = 1199090 119909119886 1199100 119910b)

(23)



(24)





119898=0

119873sum119899=0

119860119898119899 cos 120582119886119898119909 cos 120582119887119899119910+ 2sum119897=1

119872sum119898=0

119887119897119898120577119897119887 (119910) cos120582119886119898119909+ 2sum119897=1

119873sum119899=0

119886119897119899120577119897119886 (119909) cos 120582119887119899119910120593119910 (119909 119910) = 119872sum

119898=0

119873sum119899=0

119861119898119899 cos 120582119886119898119909 cos 120582119887119899119910+ 2sum119897=1

119872sum119898=0

119889119897119898120577119897119887 (119910) cos 120582119886119898119909+ 2sum119897=1

119873sum119899=0

119888119897119899120577119897119886 (119909) cos 120582119887119899119910119908 (119909 119910) = 119872sum

119898=0

119873sum119899=0

119862119898119899 cos 120582119886119898119909 cos 120582119887119899119910+ 2sum119897=1

119872sum119898=0

119891119897119898120577119897119887 (119910) cos 120582119886119898119909+ 2sum119897=1

119873sum119899=0

119890119897119899120577119897119886 (119909) cos120582119887119899119910

(25)


ln c

lm d

ln e



(26)


(27)

where




(28)



where

L119894 = [[[11987111H119894 11987112H119894 11987113H11989411987121H119894 11987122H119894 11987123H11989411987131H119894 11987132H119894 11987133H119894]]] (119894 = 119909119910 119909 119910)

M119894 = [[[11987211H119894 0 00 11987222H119894 00 0 11987233H119894]]] (119894 = 119909119910 119909 119910)

Θ119909119910 = [A B C]119879 Θ119909 = [a c e]119879 Θ119910 = [b d f]119879

F = [0 0 q]119879

(30)


L119895119909119910Θ119909119910 + L119895119909Θ119909 + L119895119910Θ119910 = 0 (119895 = 1199090 1199100 119909119886 119910119887) (31)

where

L119895119894 = [[[[11987111989511H119894 11987111989512H119894 11987111989513H11989411987111989521H119894 11987111989522H119894 11987111989523H11989411987111989531H119894 11987111989532H119894 11987111989533H119894

]]]] (119894 = 119909119910 119909 119910 119895 = 1199090 1199100 119909119886 119910119887)

(32)



(33)


L119909119910Θ119909119910 + [L119909 L119910] [Θ119909Θ119910

]minus 1205962 (M119909119910Θ119909119910 + [M119909 M119910] [Θ119909

Θ119910

]) = F

(34)

where

L119909119910 = int1198860int1198870

C119909119910L119909119910119889119909119889119910M119909119910 = int119886

0int1198870

C119909119910M119909119910119889119909119889119910L119909 = int119886

0int1198870

C119909119910L119909119889119909119889119910M119909 = int119886

0int1198870

C119909119910M119909119889119909119889119910L119910 = int119886

0int1198870

C119909119910L119910119889119909119889119910M119910 = int119886

0int1198870

C119909119910M119910119889119909119889119910

(35)


L119895119909119910Θ119909119910 + L119895119909Θ119909 + L119895119910Θ119910 = 0 (119895 = 1199090 1199100 1199091 1199101) (36)

where

L1199090119909119910 = int1198870

C119910L1199090119909119910119889119910

L119909119886119909119910 = int1198870

C119910L119909119886119909119910119889119910

L1199090119909 = int1198870

C119910L1199090119909 119889119910

L119909119886119909 = int1198870

C119910L119909119886119909 119889119910

L1199090119910 = int1198870

C119910L1199090119910 119889119910

L119909119886119910 = int1198870

C119910L119909119886119910 119889119910

L1199100119909119910 = int1198860

C119909L1199100119909119910119889119909

L119910119887119909119910 = int1198860

C119909L119910119887119909119910119889119909




L1199100119909 = int1198860

C119909L1199100119909 119889119909

L119910119887119909 = int1198860

C119909L119910119887119909 119889119909

L1199100119910 = int1198860

C119909L1199100119910 119889119909

L119910119887119910 = int1198860

C119909L119910119887119910 119889119909


[Θ119909Θ119910

] = [[[[[[[[L1199090119909 L1199090119910L119909119886119909 L119909119886119910L1199100119909 L1199100119910L119910119886119909 L119910119887119910

]]]]]]]]

minus1 [[[[[[[[L1199090119909119910L119909119886119909119910L1199100119909119910L119910119887119909119910

]]]]]]]]Θ119909119910 (38)


where


]]]]]]]]

minus1 [[[[[[[[L1199090119909119910L119909119886119909119910L1199100119909119910L119910119887119909119910

]]]]]]]]


]]]]]]]]

minus1 [[[[[[[[L1199090119909119910L119909119886119909119910L1199100119909119910L119910119887119909119910

]]]]]]]]

(40)









Clamped (C) 119908 = 120593119909 = 120593119910 = 0 1015 1015 1015


119910 = constantFree (F) 119876119909 = 119872119910 = 119872119909119910 = 0 0 0 0

Clamped (C) 119908 = 120593119910 = 120593119909 = 0 1015 1015 1015










Table4Com

paris

onof

them

axim


pics

quarep

latesh

avingvario

usbo

undary

cond

ition

swith

different

load

functio

ns

Boun

dary

cond

ition

sMetho

dUniform

pressure

PointL

oad(0505

)[(01)

(01)][(010

9)(0109)]

[(0208)(02

08)][(030

7)(0307)]

[(0406)(04

06)]CCCC

Present

5602E

-07

5438E

-07

4530E

-07

2790E

-07

8901E-08

2522E

-10

FEM

5602E

-07

5438E

-07

4530E

-07

2789E

-07

8896E

-08

2563E

-10

CFCF

Present

1278E-06

9716E-07

7275E-07

4116

E-07

1237E-07

340

1E-10

FEM

1278E-06

9716E-07

7273E-07

4115

E-07

1236E-07

3431E-10

CCCF

Present

1301E-09

8452E

-07

6152E

-07

3502E

-07

1066

E-07

2950E

-10

FEM

1302E-09

8452E

-07

6150E

-07

3501E-07

1065E-07

2981E-10

CSCS

Present

840

6E-07

7930E-07

6272E

-07

3677E

-07

1129E-07

3157E

-10

FEM

840

6E-07

7930E-07

6272E

-07

3677E

-07

1129E-07

3174E

-10

SSSS

Present

1737E-06

1584E-06

1175E-06

644

8E-07

1874E-07

506

4E-10

FEM

1737E-06

1585E-06

1740

E-06

644

7E-07

1873E-07

5076E

-10

CSFF

Present

3084E

-05

1858E-05

9955E-06

4267E

-06

1043E-06

260

4E-09

FEM

3084E

-05

1858E-05

9955E-06

4267E

-06

1043E-06

2618E

-09


CCCC[(0208) (0208)]

CFCF [(0406) (0406)]

CCCF [(01) (01)]


(a) (b)










4 Conclusions




00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus9

40 times 10minus9

60 times 10minus9

80 times 10minus9


(a)

00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus9

40 times 10minus9

60 times 10minus9

80 times 10minus9


(b)


00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus6

40 times 10minus6

60 times 10minus6

80 times 10minus6


(a)

00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

10 times 10minus6

20 times 10minus6

30 times 10minus6


(b)


CCCC CCCS CFCF CFFF





0 20 40 60 80 100

004

005

006

007

frequ

ency

par

amet

ers Ω


CCCC


CFCF

0008

0010

0012

0014

0016

frequ

ency

par

amet

ers Ω


CFFF

00016

00020

00024

frequ

ency

par

amet

ers Ω


CSCF

0012

0016

0020

frequ

ency

par

amet

ers Ω


Data Availability




Acknowledgments




2

0 008325 008325 008729 008729 006713 006713 006364 006365 006781 006781025 007600 007600 007950 007950 006145 006145 005829 005829 006205 0062051 006541 006541 006790 006790 005346 005346 005080 005080 005391 0053915 005524 005524 005695 005695 004568 004568 004349 004349 004600 004600infin 004263 004237 004443 004443 003417 003417 003239 003240 003451 003452

1

0 014378 014377 016713 016712 007537 007537 006290 006290 008062 008062025 012974 012973 014927 014926 006890 006890 005761 005761 007351 0073501 010725 010725 011955 011955 005968 005968 005021 005021 006308 0063085 008720 008720 009479 009479 005078 005078 004301 004300 005322 005322infin 007318 007318 008507 008507 003836 003836 003202 003202 004104 004104

05

0 035045 035037 041996 041980 010065 010065 006217 006216 013484 013483025 030709 030703 036112 036102 009170 009170 005695 005695 012160 0121591 023262 023260 026091 026088 007851 007851 004970 004970 010066 0100665 017691 017690 019258 019257 006610 006610 004262 004262 008226 008226infin 017921 017837 021375 021374 005123 005123 003164 003164 006863 006863

0 20 40 60 80 100

006

007

frequ

ency

par

amet

ers Ω


CCCC


CFCF

0014

0015

0016

frequ

ency

par

amet

ers Ω


CFFF

00024

00026

frequ

ency

par

amet

ers Ω


CSCF

0017

0016

0018

0019

frequ

ency

par

amet

ers Ω




005

0 003129 003130 004076 004076 001024 001024 000719 000719 001249 0012492025 002899 002899 003664 003664 000981 000981 000692 000692 001185 00118511 002667 002667 003250 003250 000948 000948 000674 000674 001132 00113165 002677 002677 003239 003239 000963 000963 000685 000685 001146 001146infin 002689 002689 003502 003502 000880 000880 000618 000618 001073 0010734

01

0 011639 011638 014580 014580 004001 004001 002835 002835 004817 004817025 010561 010561 012781 012780 003810 003810 002717 002717 004532 00453151 009734 009733 011453 011453 003679 003679 002641 002641 004327 00432675 009646 009646 011234 011234 003718 003718 002677 002677 004352 0043517infin 010001 010001 012528 012528 003438 003438 002426 002436 004139 0041391

02

0 037876 037870 043939 043930 014871 014871 010795 010795 017323 0173218025 036117 036113 041624 041617 014354 014354 010436 010436 016671 016671 033549 033547 037962 037958 013851 013851 010127 010127 015937 01593665 032783 032781 036695 036692 013888 013887 010200 010200 015878 0158779infin 032545 032544 037755 037753 012779 012778 009276 009276 014885 0148845


005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001541 001175 000360 000355 000429 000139 000057 0002641 001291 001027 000329 000333 000389 000129 000054 0002465 001111 000893 000290 000296 000342 000114 000048 00021810 001107 000868 000274 000274 000324 000107 000044 000203

01

0 006897 018772 001632 001608 001941 000630 000259 00120005 005652 015857 001406 001405 001663 000547 000227 0010431 004862 014291 001291 001320 001519 000509 000214 0009745 003953 011409 001113 001166 001303 000446 000190 00085610 003929 011157 001054 001082 001239 000417 000175 000798

02

0 023286 018772 006198 006260 007269 002439 001026 00465705 019360 015857 005338 005472 006242 002121 000899 0040551 017136 014291 004927 005147 005740 001983 000848 0037985 013242 011409 004126 004505 004776 001714 000749 00329510 013167 011157 003935 004191 004569 001606 000694 003081



005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001606 001246 000388 000386 000461 000150 000062 0002871 001588 001235 000386 000385 000458 000150 000062 0002865 001598 001249 000393 000392 000465 000153 000063 00029110 001481 001179 000378 000383 000447 000148 000062 000283

01

0 006897 005286 001632 001608 001941 000630 000259 00120005 006029 004770 001524 001532 001801 000595 000247 0011351 005888 004690 001511 001526 001782 000591 000246 0011295 005858 004702 001529 001553 001801 000601 000251 00114810 005748 004611 001498 001521 001765 000588 000246 001124

02

0 023288 018773 006198 006260 007269 002439 001026 00465705 021550 017558 005870 005988 006871 002324 000983 0044421 020752 017074 005782 005952 006754 002304 000979 0044075 020352 016912 005810 006038 006770 002332 000996 00446410 020672 017013 005770 005935 006737 002300 000978 004398



References






































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia












120583 (119911) = 3119870 minus 21198666119870 + 2119866 (3)








z

x

y

b

a

Metal

z

h

Ceramic

y

q(xy)

1

2

3

4

y1

y2

y2

y1

x1

x2


p=10

p=5

p=2

p=1

p=05

p=02p=01

minus05

minus04

minus03

minus02

minus01

00

01

02

03

04

05

zh

02 04 06 08 1000Vc



120576119909 = 119911120597120593119909120597119909 = 119911120581119909120576119910 = 119911120597120593119910120597119910 = 119911120581119910120574119909119910 = 119911(120597120593119910120597119909 + 120597120593119909120597119910 ) = 119911120581119909119910

(6a)

120574119909119911 = 120597119908120597119909 + 120593119909120574119910119911 = 120597119908120597119910 + 120593119910 (6b)


120590119909120590119910120591119909119910120591119909119911120591119910119911

= [[[[[[[[[

11987611 (119911) 11987612 (119911) 0 0 011987612 (119911) 11987611 (119911) 0 0 00 0 11987666 (119911) 0 00 0 0 11987666 (119911) 00 0 0 0 11987666 (119911)]]]]]]]]]120576119909120576119910120574119909119910120574119909119911120574119910119911

(7)


(8)


119872119909 = 11986311 120597120593119909120597119909 + 11986312 120597120593119910120597119910 119872119910 = 11986312 120597120593119909120597119909 + 11986311 120597120593119910120597119910 119872119909119910 = 11986366 (120597120593119910120597119909 + 120597120593119909120597119910 )


119876119909 = 11986066 (120597119908120597119909 + 120593119909) 119876119910 = 11986066 (120597119908120597119910 + 120593119910)

(9)


(11986066 11986311 11986322 11986312) = intℎ2minusℎ2

(12058111987666 (119911) 11987611 (119911)sdot 1199112 11987611 (119911) 1199112 11987612 (119911) 1199112) 119889119911 (10)


+ 120574119910119911119876119910) 119889119909119889119910 = 12 int1198860 int1198870 (119872119909 120597120593119909120597119909+119872119910 120597120593119910120597119910 +119872119909119910 (120597120593119909120597119910 + 120597120593119910120597119909 )+ (120597119908120597119909 + 120593119909)119876119909 + (120597119908120597119910 + 120593119910)119876119910)119889119909119889119910

(11)

119879 = 12 int1198860 int1198870 [1198680 (120597119908120597119905 )2+ 1198682((120597120593119909120597119905 )2 + (120597120593119910120597119905 )2)]119889119909119889119910

[1198680 1198681] = intℎ2minusℎ2

120588 (1 1199112) 119889119911(12)


119881 = 12 int1198860 (1198961199100119910 1199082 + 11987011991001199101199091205932119909 + 1198701199100119910 1205932119910)10038161003816100381610038161003816119910=0 119889119909+ 12 int1198860 (119896119910119887119910 1199082 + 1198701199101198871199101199091205932119909 + 119870119910119887119910 1205932119910)10038161003816100381610038161003816119910=119887 119889119909+ 12 int1198870 (1198961199090119909 1199082 + 1198701199090119909 1205932119909 + 11987011990901199091199101205932119910)10038161003816100381610038161003816119909=0 119889119910+ 12 int1198870 (119896119909119886119909 1199082 + 119870119909119886119909 1205932119909 + 1198701199091198861199091199101205932119910)10038161003816100381610038161003816119909=119886 119889119910

(13)


119860119902 (119909 119910)119908 (119909 119910) 119889119860 (14)




int11990500(120575119879 + 120575119882 minus 120575119880 minus 120575119881) 119889119905 = 0 (16)



int11990500int1198860(11987211990911991012057512059311990910038161003816100381610038161003816119910=119887119910=0 + 11987611991012057511990810038161003816100381610038161003816119910=119887119910=0 + 11987211991012057512059311991010038161003816100381610038161003816119910=119887119910=0) 119889119909119889119905 + int1199050

0int1198870(1198721199091205751205931199091003816100381610038161003816119909=119886119909=0 + 1198761199091205751199081003816100381610038161003816119909=119886119909=0 + 11987211990911991012057512059311991010038161003816100381610038161003816119909=119886119909=0) 119889119910119889119905

int11990500int1198860(1198961199100119910 119908120575119908 + 1198701199100119910119909120593119909120575120593119909 + 1198701199100119910 120593119910120575120593119910)10038161003816100381610038161003816119910=0 119889119909119889119905 + int1199050

0int1198860(119896119910119887119910 119908120575119908 + 119870119910119887119910119909120593119909120575120593119909 + 119870119910119887119910 120593119910120575120593119910)10038161003816100381610038161003816119910=119887 119889119909119889119905

+ int11990500int1198870(1198961199090119909 119908120575119908 + 1198701199090119909 120593119909120575120593119909 + 1198701199090119909119910120593119910120575120593119910)10038161003816100381610038161003816119909=0 119889119910119889119905 + int1199050

0int1198870(119896119909119886119909 119908120575119908 + 119870119909119886119909 120593119909120575120593119909 + 119870119909119886119909119910120593119910120575120593119910)10038161003816100381610038161003816119909=119886 119889119910119889119905


0int1198860int1198870119902 (119909 119910) 120575119908119889119909119889119910119889119905 = 0

(17)



int11990500int1198860int1198870

((((



+ int11990500int1198860((119896119910119887119910 119908 + 119876119910) 120575119908 + (119870119910119887119910119909120593119909 +119872119909119910) 120575120593119909 + (119870119910119887119910 120593119910 +119872119910) 120575120593119910)10038161003816100381610038161003816119910=119887 119889119909119889119905


+ int11990500int1198870((119896119909119886119909 119908 + 119876119909) 120575119908 + (119870119909119886119909 120593119909 +119872119909) 120575120593119909 + (119870119909119886119909119910120593119910 +119872119909119910) 120575120593119910)10038161003816100381610038161003816119909=119886 119889119910119889119905 = 0

(18)



(19)




([[[[11987111 11987112 1198711311987121 11987122 1198712311987131 11987132 11987133

]]]] minus 1205962 [[[[11987211 0 00 11987222 00 0 11987233

]]]])[[[[120593119909120593119910119908]]]]

= [[[[00119902]]]]

(21)


where


(22)


[[[[[11987111989511 11987111989512 1198711198951311987111989521 11987111989522 1198711198952311987111989531 11987111989532 11987111989533

]]]]][[[[120593119909120593119910119908]]]] = [[[[

000]]]] (119895 = 1199090 119909119886 1199100 119910b)

(23)



(24)





119898=0

119873sum119899=0

119860119898119899 cos 120582119886119898119909 cos 120582119887119899119910+ 2sum119897=1

119872sum119898=0

119887119897119898120577119897119887 (119910) cos120582119886119898119909+ 2sum119897=1

119873sum119899=0

119886119897119899120577119897119886 (119909) cos 120582119887119899119910120593119910 (119909 119910) = 119872sum

119898=0

119873sum119899=0

119861119898119899 cos 120582119886119898119909 cos 120582119887119899119910+ 2sum119897=1

119872sum119898=0

119889119897119898120577119897119887 (119910) cos 120582119886119898119909+ 2sum119897=1

119873sum119899=0

119888119897119899120577119897119886 (119909) cos 120582119887119899119910119908 (119909 119910) = 119872sum

119898=0

119873sum119899=0

119862119898119899 cos 120582119886119898119909 cos 120582119887119899119910+ 2sum119897=1

119872sum119898=0

119891119897119898120577119897119887 (119910) cos 120582119886119898119909+ 2sum119897=1

119873sum119899=0

119890119897119899120577119897119886 (119909) cos120582119887119899119910

(25)


ln c

lm d

ln e



(26)


(27)

where




(28)



where

L119894 = [[[11987111H119894 11987112H119894 11987113H11989411987121H119894 11987122H119894 11987123H11989411987131H119894 11987132H119894 11987133H119894]]] (119894 = 119909119910 119909 119910)

M119894 = [[[11987211H119894 0 00 11987222H119894 00 0 11987233H119894]]] (119894 = 119909119910 119909 119910)

Θ119909119910 = [A B C]119879 Θ119909 = [a c e]119879 Θ119910 = [b d f]119879

F = [0 0 q]119879

(30)


L119895119909119910Θ119909119910 + L119895119909Θ119909 + L119895119910Θ119910 = 0 (119895 = 1199090 1199100 119909119886 119910119887) (31)

where

L119895119894 = [[[[11987111989511H119894 11987111989512H119894 11987111989513H11989411987111989521H119894 11987111989522H119894 11987111989523H11989411987111989531H119894 11987111989532H119894 11987111989533H119894

]]]] (119894 = 119909119910 119909 119910 119895 = 1199090 1199100 119909119886 119910119887)

(32)



(33)


L119909119910Θ119909119910 + [L119909 L119910] [Θ119909Θ119910

]minus 1205962 (M119909119910Θ119909119910 + [M119909 M119910] [Θ119909

Θ119910

]) = F

(34)

where

L119909119910 = int1198860int1198870

C119909119910L119909119910119889119909119889119910M119909119910 = int119886

0int1198870

C119909119910M119909119910119889119909119889119910L119909 = int119886

0int1198870

C119909119910L119909119889119909119889119910M119909 = int119886

0int1198870

C119909119910M119909119889119909119889119910L119910 = int119886

0int1198870

C119909119910L119910119889119909119889119910M119910 = int119886

0int1198870

C119909119910M119910119889119909119889119910

(35)


L119895119909119910Θ119909119910 + L119895119909Θ119909 + L119895119910Θ119910 = 0 (119895 = 1199090 1199100 1199091 1199101) (36)

where

L1199090119909119910 = int1198870

C119910L1199090119909119910119889119910

L119909119886119909119910 = int1198870

C119910L119909119886119909119910119889119910

L1199090119909 = int1198870

C119910L1199090119909 119889119910

L119909119886119909 = int1198870

C119910L119909119886119909 119889119910

L1199090119910 = int1198870

C119910L1199090119910 119889119910

L119909119886119910 = int1198870

C119910L119909119886119910 119889119910

L1199100119909119910 = int1198860

C119909L1199100119909119910119889119909

L119910119887119909119910 = int1198860

C119909L119910119887119909119910119889119909




L1199100119909 = int1198860

C119909L1199100119909 119889119909

L119910119887119909 = int1198860

C119909L119910119887119909 119889119909

L1199100119910 = int1198860

C119909L1199100119910 119889119909

L119910119887119910 = int1198860

C119909L119910119887119910 119889119909


[Θ119909Θ119910

] = [[[[[[[[L1199090119909 L1199090119910L119909119886119909 L119909119886119910L1199100119909 L1199100119910L119910119886119909 L119910119887119910

]]]]]]]]

minus1 [[[[[[[[L1199090119909119910L119909119886119909119910L1199100119909119910L119910119887119909119910

]]]]]]]]Θ119909119910 (38)


where


]]]]]]]]

minus1 [[[[[[[[L1199090119909119910L119909119886119909119910L1199100119909119910L119910119887119909119910

]]]]]]]]


]]]]]]]]

minus1 [[[[[[[[L1199090119909119910L119909119886119909119910L1199100119909119910L119910119887119909119910

]]]]]]]]

(40)









Clamped (C) 119908 = 120593119909 = 120593119910 = 0 1015 1015 1015


119910 = constantFree (F) 119876119909 = 119872119910 = 119872119909119910 = 0 0 0 0

Clamped (C) 119908 = 120593119910 = 120593119909 = 0 1015 1015 1015










Table4Com

paris

onof

them

axim


pics

quarep

latesh

avingvario

usbo

undary

cond

ition

swith

different

load

functio

ns

Boun

dary

cond

ition

sMetho

dUniform

pressure

PointL

oad(0505

)[(01)

(01)][(010

9)(0109)]

[(0208)(02

08)][(030

7)(0307)]

[(0406)(04

06)]CCCC

Present

5602E

-07

5438E

-07

4530E

-07

2790E

-07

8901E-08

2522E

-10

FEM

5602E

-07

5438E

-07

4530E

-07

2789E

-07

8896E

-08

2563E

-10

CFCF

Present

1278E-06

9716E-07

7275E-07

4116

E-07

1237E-07

340

1E-10

FEM

1278E-06

9716E-07

7273E-07

4115

E-07

1236E-07

3431E-10

CCCF

Present

1301E-09

8452E

-07

6152E

-07

3502E

-07

1066

E-07

2950E

-10

FEM

1302E-09

8452E

-07

6150E

-07

3501E-07

1065E-07

2981E-10

CSCS

Present

840

6E-07

7930E-07

6272E

-07

3677E

-07

1129E-07

3157E

-10

FEM

840

6E-07

7930E-07

6272E

-07

3677E

-07

1129E-07

3174E

-10

SSSS

Present

1737E-06

1584E-06

1175E-06

644

8E-07

1874E-07

506

4E-10

FEM

1737E-06

1585E-06

1740

E-06

644

7E-07

1873E-07

5076E

-10

CSFF

Present

3084E

-05

1858E-05

9955E-06

4267E

-06

1043E-06

260

4E-09

FEM

3084E

-05

1858E-05

9955E-06

4267E

-06

1043E-06

2618E

-09


CCCC[(0208) (0208)]

CFCF [(0406) (0406)]

CCCF [(01) (01)]


(a) (b)










4 Conclusions




00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus9

40 times 10minus9

60 times 10minus9

80 times 10minus9


(a)

00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus9

40 times 10minus9

60 times 10minus9

80 times 10minus9


(b)


00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus6

40 times 10minus6

60 times 10minus6

80 times 10minus6


(a)

00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

10 times 10minus6

20 times 10minus6

30 times 10minus6


(b)


CCCC CCCS CFCF CFFF





0 20 40 60 80 100

004

005

006

007

frequ

ency

par

amet

ers Ω


CCCC


CFCF

0008

0010

0012

0014

0016

frequ

ency

par

amet

ers Ω


CFFF

00016

00020

00024

frequ

ency

par

amet

ers Ω


CSCF

0012

0016

0020

frequ

ency

par

amet

ers Ω


Data Availability




Acknowledgments




2

0 008325 008325 008729 008729 006713 006713 006364 006365 006781 006781025 007600 007600 007950 007950 006145 006145 005829 005829 006205 0062051 006541 006541 006790 006790 005346 005346 005080 005080 005391 0053915 005524 005524 005695 005695 004568 004568 004349 004349 004600 004600infin 004263 004237 004443 004443 003417 003417 003239 003240 003451 003452

1

0 014378 014377 016713 016712 007537 007537 006290 006290 008062 008062025 012974 012973 014927 014926 006890 006890 005761 005761 007351 0073501 010725 010725 011955 011955 005968 005968 005021 005021 006308 0063085 008720 008720 009479 009479 005078 005078 004301 004300 005322 005322infin 007318 007318 008507 008507 003836 003836 003202 003202 004104 004104

05

0 035045 035037 041996 041980 010065 010065 006217 006216 013484 013483025 030709 030703 036112 036102 009170 009170 005695 005695 012160 0121591 023262 023260 026091 026088 007851 007851 004970 004970 010066 0100665 017691 017690 019258 019257 006610 006610 004262 004262 008226 008226infin 017921 017837 021375 021374 005123 005123 003164 003164 006863 006863

0 20 40 60 80 100

006

007

frequ

ency

par

amet

ers Ω


CCCC


CFCF

0014

0015

0016

frequ

ency

par

amet

ers Ω


CFFF

00024

00026

frequ

ency

par

amet

ers Ω


CSCF

0017

0016

0018

0019

frequ

ency

par

amet

ers Ω




005

0 003129 003130 004076 004076 001024 001024 000719 000719 001249 0012492025 002899 002899 003664 003664 000981 000981 000692 000692 001185 00118511 002667 002667 003250 003250 000948 000948 000674 000674 001132 00113165 002677 002677 003239 003239 000963 000963 000685 000685 001146 001146infin 002689 002689 003502 003502 000880 000880 000618 000618 001073 0010734

01

0 011639 011638 014580 014580 004001 004001 002835 002835 004817 004817025 010561 010561 012781 012780 003810 003810 002717 002717 004532 00453151 009734 009733 011453 011453 003679 003679 002641 002641 004327 00432675 009646 009646 011234 011234 003718 003718 002677 002677 004352 0043517infin 010001 010001 012528 012528 003438 003438 002426 002436 004139 0041391

02

0 037876 037870 043939 043930 014871 014871 010795 010795 017323 0173218025 036117 036113 041624 041617 014354 014354 010436 010436 016671 016671 033549 033547 037962 037958 013851 013851 010127 010127 015937 01593665 032783 032781 036695 036692 013888 013887 010200 010200 015878 0158779infin 032545 032544 037755 037753 012779 012778 009276 009276 014885 0148845


005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001541 001175 000360 000355 000429 000139 000057 0002641 001291 001027 000329 000333 000389 000129 000054 0002465 001111 000893 000290 000296 000342 000114 000048 00021810 001107 000868 000274 000274 000324 000107 000044 000203

01

0 006897 018772 001632 001608 001941 000630 000259 00120005 005652 015857 001406 001405 001663 000547 000227 0010431 004862 014291 001291 001320 001519 000509 000214 0009745 003953 011409 001113 001166 001303 000446 000190 00085610 003929 011157 001054 001082 001239 000417 000175 000798

02

0 023286 018772 006198 006260 007269 002439 001026 00465705 019360 015857 005338 005472 006242 002121 000899 0040551 017136 014291 004927 005147 005740 001983 000848 0037985 013242 011409 004126 004505 004776 001714 000749 00329510 013167 011157 003935 004191 004569 001606 000694 003081



005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001606 001246 000388 000386 000461 000150 000062 0002871 001588 001235 000386 000385 000458 000150 000062 0002865 001598 001249 000393 000392 000465 000153 000063 00029110 001481 001179 000378 000383 000447 000148 000062 000283

01

0 006897 005286 001632 001608 001941 000630 000259 00120005 006029 004770 001524 001532 001801 000595 000247 0011351 005888 004690 001511 001526 001782 000591 000246 0011295 005858 004702 001529 001553 001801 000601 000251 00114810 005748 004611 001498 001521 001765 000588 000246 001124

02

0 023288 018773 006198 006260 007269 002439 001026 00465705 021550 017558 005870 005988 006871 002324 000983 0044421 020752 017074 005782 005952 006754 002304 000979 0044075 020352 016912 005810 006038 006770 002332 000996 00446410 020672 017013 005770 005935 006737 002300 000978 004398



References






































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



z

x

y

b

a

Metal

z

h

Ceramic

y

q(xy)

1

2

3

4

y1

y2

y2

y1

x1

x2


p=10

p=5

p=2

p=1

p=05

p=02p=01

minus05

minus04

minus03

minus02

minus01

00

01

02

03

04

05

zh

02 04 06 08 1000Vc



120576119909 = 119911120597120593119909120597119909 = 119911120581119909120576119910 = 119911120597120593119910120597119910 = 119911120581119910120574119909119910 = 119911(120597120593119910120597119909 + 120597120593119909120597119910 ) = 119911120581119909119910

(6a)

120574119909119911 = 120597119908120597119909 + 120593119909120574119910119911 = 120597119908120597119910 + 120593119910 (6b)


120590119909120590119910120591119909119910120591119909119911120591119910119911

= [[[[[[[[[

11987611 (119911) 11987612 (119911) 0 0 011987612 (119911) 11987611 (119911) 0 0 00 0 11987666 (119911) 0 00 0 0 11987666 (119911) 00 0 0 0 11987666 (119911)]]]]]]]]]120576119909120576119910120574119909119910120574119909119911120574119910119911

(7)


(8)


119872119909 = 11986311 120597120593119909120597119909 + 11986312 120597120593119910120597119910 119872119910 = 11986312 120597120593119909120597119909 + 11986311 120597120593119910120597119910 119872119909119910 = 11986366 (120597120593119910120597119909 + 120597120593119909120597119910 )


119876119909 = 11986066 (120597119908120597119909 + 120593119909) 119876119910 = 11986066 (120597119908120597119910 + 120593119910)

(9)


(11986066 11986311 11986322 11986312) = intℎ2minusℎ2

(12058111987666 (119911) 11987611 (119911)sdot 1199112 11987611 (119911) 1199112 11987612 (119911) 1199112) 119889119911 (10)


+ 120574119910119911119876119910) 119889119909119889119910 = 12 int1198860 int1198870 (119872119909 120597120593119909120597119909+119872119910 120597120593119910120597119910 +119872119909119910 (120597120593119909120597119910 + 120597120593119910120597119909 )+ (120597119908120597119909 + 120593119909)119876119909 + (120597119908120597119910 + 120593119910)119876119910)119889119909119889119910

(11)

119879 = 12 int1198860 int1198870 [1198680 (120597119908120597119905 )2+ 1198682((120597120593119909120597119905 )2 + (120597120593119910120597119905 )2)]119889119909119889119910

[1198680 1198681] = intℎ2minusℎ2

120588 (1 1199112) 119889119911(12)


119881 = 12 int1198860 (1198961199100119910 1199082 + 11987011991001199101199091205932119909 + 1198701199100119910 1205932119910)10038161003816100381610038161003816119910=0 119889119909+ 12 int1198860 (119896119910119887119910 1199082 + 1198701199101198871199101199091205932119909 + 119870119910119887119910 1205932119910)10038161003816100381610038161003816119910=119887 119889119909+ 12 int1198870 (1198961199090119909 1199082 + 1198701199090119909 1205932119909 + 11987011990901199091199101205932119910)10038161003816100381610038161003816119909=0 119889119910+ 12 int1198870 (119896119909119886119909 1199082 + 119870119909119886119909 1205932119909 + 1198701199091198861199091199101205932119910)10038161003816100381610038161003816119909=119886 119889119910

(13)


119860119902 (119909 119910)119908 (119909 119910) 119889119860 (14)




int11990500(120575119879 + 120575119882 minus 120575119880 minus 120575119881) 119889119905 = 0 (16)



int11990500int1198860(11987211990911991012057512059311990910038161003816100381610038161003816119910=119887119910=0 + 11987611991012057511990810038161003816100381610038161003816119910=119887119910=0 + 11987211991012057512059311991010038161003816100381610038161003816119910=119887119910=0) 119889119909119889119905 + int1199050

0int1198870(1198721199091205751205931199091003816100381610038161003816119909=119886119909=0 + 1198761199091205751199081003816100381610038161003816119909=119886119909=0 + 11987211990911991012057512059311991010038161003816100381610038161003816119909=119886119909=0) 119889119910119889119905

int11990500int1198860(1198961199100119910 119908120575119908 + 1198701199100119910119909120593119909120575120593119909 + 1198701199100119910 120593119910120575120593119910)10038161003816100381610038161003816119910=0 119889119909119889119905 + int1199050

0int1198860(119896119910119887119910 119908120575119908 + 119870119910119887119910119909120593119909120575120593119909 + 119870119910119887119910 120593119910120575120593119910)10038161003816100381610038161003816119910=119887 119889119909119889119905

+ int11990500int1198870(1198961199090119909 119908120575119908 + 1198701199090119909 120593119909120575120593119909 + 1198701199090119909119910120593119910120575120593119910)10038161003816100381610038161003816119909=0 119889119910119889119905 + int1199050

0int1198870(119896119909119886119909 119908120575119908 + 119870119909119886119909 120593119909120575120593119909 + 119870119909119886119909119910120593119910120575120593119910)10038161003816100381610038161003816119909=119886 119889119910119889119905


0int1198860int1198870119902 (119909 119910) 120575119908119889119909119889119910119889119905 = 0

(17)



int11990500int1198860int1198870

((((



+ int11990500int1198860((119896119910119887119910 119908 + 119876119910) 120575119908 + (119870119910119887119910119909120593119909 +119872119909119910) 120575120593119909 + (119870119910119887119910 120593119910 +119872119910) 120575120593119910)10038161003816100381610038161003816119910=119887 119889119909119889119905


+ int11990500int1198870((119896119909119886119909 119908 + 119876119909) 120575119908 + (119870119909119886119909 120593119909 +119872119909) 120575120593119909 + (119870119909119886119909119910120593119910 +119872119909119910) 120575120593119910)10038161003816100381610038161003816119909=119886 119889119910119889119905 = 0

(18)



(19)




([[[[11987111 11987112 1198711311987121 11987122 1198712311987131 11987132 11987133

]]]] minus 1205962 [[[[11987211 0 00 11987222 00 0 11987233

]]]])[[[[120593119909120593119910119908]]]]

= [[[[00119902]]]]

(21)


where


(22)


[[[[[11987111989511 11987111989512 1198711198951311987111989521 11987111989522 1198711198952311987111989531 11987111989532 11987111989533

]]]]][[[[120593119909120593119910119908]]]] = [[[[

000]]]] (119895 = 1199090 119909119886 1199100 119910b)

(23)



(24)





119898=0

119873sum119899=0

119860119898119899 cos 120582119886119898119909 cos 120582119887119899119910+ 2sum119897=1

119872sum119898=0

119887119897119898120577119897119887 (119910) cos120582119886119898119909+ 2sum119897=1

119873sum119899=0

119886119897119899120577119897119886 (119909) cos 120582119887119899119910120593119910 (119909 119910) = 119872sum

119898=0

119873sum119899=0

119861119898119899 cos 120582119886119898119909 cos 120582119887119899119910+ 2sum119897=1

119872sum119898=0

119889119897119898120577119897119887 (119910) cos 120582119886119898119909+ 2sum119897=1

119873sum119899=0

119888119897119899120577119897119886 (119909) cos 120582119887119899119910119908 (119909 119910) = 119872sum

119898=0

119873sum119899=0

119862119898119899 cos 120582119886119898119909 cos 120582119887119899119910+ 2sum119897=1

119872sum119898=0

119891119897119898120577119897119887 (119910) cos 120582119886119898119909+ 2sum119897=1

119873sum119899=0

119890119897119899120577119897119886 (119909) cos120582119887119899119910

(25)


ln c

lm d

ln e



(26)


(27)

where




(28)



where

L119894 = [[[11987111H119894 11987112H119894 11987113H11989411987121H119894 11987122H119894 11987123H11989411987131H119894 11987132H119894 11987133H119894]]] (119894 = 119909119910 119909 119910)

M119894 = [[[11987211H119894 0 00 11987222H119894 00 0 11987233H119894]]] (119894 = 119909119910 119909 119910)

Θ119909119910 = [A B C]119879 Θ119909 = [a c e]119879 Θ119910 = [b d f]119879

F = [0 0 q]119879

(30)


L119895119909119910Θ119909119910 + L119895119909Θ119909 + L119895119910Θ119910 = 0 (119895 = 1199090 1199100 119909119886 119910119887) (31)

where

L119895119894 = [[[[11987111989511H119894 11987111989512H119894 11987111989513H11989411987111989521H119894 11987111989522H119894 11987111989523H11989411987111989531H119894 11987111989532H119894 11987111989533H119894

]]]] (119894 = 119909119910 119909 119910 119895 = 1199090 1199100 119909119886 119910119887)

(32)



(33)


L119909119910Θ119909119910 + [L119909 L119910] [Θ119909Θ119910

]minus 1205962 (M119909119910Θ119909119910 + [M119909 M119910] [Θ119909

Θ119910

]) = F

(34)

where

L119909119910 = int1198860int1198870

C119909119910L119909119910119889119909119889119910M119909119910 = int119886

0int1198870

C119909119910M119909119910119889119909119889119910L119909 = int119886

0int1198870

C119909119910L119909119889119909119889119910M119909 = int119886

0int1198870

C119909119910M119909119889119909119889119910L119910 = int119886

0int1198870

C119909119910L119910119889119909119889119910M119910 = int119886

0int1198870

C119909119910M119910119889119909119889119910

(35)


L119895119909119910Θ119909119910 + L119895119909Θ119909 + L119895119910Θ119910 = 0 (119895 = 1199090 1199100 1199091 1199101) (36)

where

L1199090119909119910 = int1198870

C119910L1199090119909119910119889119910

L119909119886119909119910 = int1198870

C119910L119909119886119909119910119889119910

L1199090119909 = int1198870

C119910L1199090119909 119889119910

L119909119886119909 = int1198870

C119910L119909119886119909 119889119910

L1199090119910 = int1198870

C119910L1199090119910 119889119910

L119909119886119910 = int1198870

C119910L119909119886119910 119889119910

L1199100119909119910 = int1198860

C119909L1199100119909119910119889119909

L119910119887119909119910 = int1198860

C119909L119910119887119909119910119889119909




L1199100119909 = int1198860

C119909L1199100119909 119889119909

L119910119887119909 = int1198860

C119909L119910119887119909 119889119909

L1199100119910 = int1198860

C119909L1199100119910 119889119909

L119910119887119910 = int1198860

C119909L119910119887119910 119889119909


[Θ119909Θ119910

] = [[[[[[[[L1199090119909 L1199090119910L119909119886119909 L119909119886119910L1199100119909 L1199100119910L119910119886119909 L119910119887119910

]]]]]]]]

minus1 [[[[[[[[L1199090119909119910L119909119886119909119910L1199100119909119910L119910119887119909119910

]]]]]]]]Θ119909119910 (38)


where


]]]]]]]]

minus1 [[[[[[[[L1199090119909119910L119909119886119909119910L1199100119909119910L119910119887119909119910

]]]]]]]]


]]]]]]]]

minus1 [[[[[[[[L1199090119909119910L119909119886119909119910L1199100119909119910L119910119887119909119910

]]]]]]]]

(40)









Clamped (C) 119908 = 120593119909 = 120593119910 = 0 1015 1015 1015


119910 = constantFree (F) 119876119909 = 119872119910 = 119872119909119910 = 0 0 0 0

Clamped (C) 119908 = 120593119910 = 120593119909 = 0 1015 1015 1015










Table4Com

paris

onof

them

axim


pics

quarep

latesh

avingvario

usbo

undary

cond

ition

swith

different

load

functio

ns

Boun

dary

cond

ition

sMetho

dUniform

pressure

PointL

oad(0505

)[(01)

(01)][(010

9)(0109)]

[(0208)(02

08)][(030

7)(0307)]

[(0406)(04

06)]CCCC

Present

5602E

-07

5438E

-07

4530E

-07

2790E

-07

8901E-08

2522E

-10

FEM

5602E

-07

5438E

-07

4530E

-07

2789E

-07

8896E

-08

2563E

-10

CFCF

Present

1278E-06

9716E-07

7275E-07

4116

E-07

1237E-07

340

1E-10

FEM

1278E-06

9716E-07

7273E-07

4115

E-07

1236E-07

3431E-10

CCCF

Present

1301E-09

8452E

-07

6152E

-07

3502E

-07

1066

E-07

2950E

-10

FEM

1302E-09

8452E

-07

6150E

-07

3501E-07

1065E-07

2981E-10

CSCS

Present

840

6E-07

7930E-07

6272E

-07

3677E

-07

1129E-07

3157E

-10

FEM

840

6E-07

7930E-07

6272E

-07

3677E

-07

1129E-07

3174E

-10

SSSS

Present

1737E-06

1584E-06

1175E-06

644

8E-07

1874E-07

506

4E-10

FEM

1737E-06

1585E-06

1740

E-06

644

7E-07

1873E-07

5076E

-10

CSFF

Present

3084E

-05

1858E-05

9955E-06

4267E

-06

1043E-06

260

4E-09

FEM

3084E

-05

1858E-05

9955E-06

4267E

-06

1043E-06

2618E

-09


CCCC[(0208) (0208)]

CFCF [(0406) (0406)]

CCCF [(01) (01)]


(a) (b)










4 Conclusions




00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus9

40 times 10minus9

60 times 10minus9

80 times 10minus9


(a)

00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus9

40 times 10minus9

60 times 10minus9

80 times 10minus9


(b)


00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus6

40 times 10minus6

60 times 10minus6

80 times 10minus6


(a)

00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

10 times 10minus6

20 times 10minus6

30 times 10minus6


(b)


CCCC CCCS CFCF CFFF





0 20 40 60 80 100

004

005

006

007

frequ

ency

par

amet

ers Ω


CCCC


CFCF

0008

0010

0012

0014

0016

frequ

ency

par

amet

ers Ω


CFFF

00016

00020

00024

frequ

ency

par

amet

ers Ω


CSCF

0012

0016

0020

frequ

ency

par

amet

ers Ω


Data Availability




Acknowledgments




2

0 008325 008325 008729 008729 006713 006713 006364 006365 006781 006781025 007600 007600 007950 007950 006145 006145 005829 005829 006205 0062051 006541 006541 006790 006790 005346 005346 005080 005080 005391 0053915 005524 005524 005695 005695 004568 004568 004349 004349 004600 004600infin 004263 004237 004443 004443 003417 003417 003239 003240 003451 003452

1

0 014378 014377 016713 016712 007537 007537 006290 006290 008062 008062025 012974 012973 014927 014926 006890 006890 005761 005761 007351 0073501 010725 010725 011955 011955 005968 005968 005021 005021 006308 0063085 008720 008720 009479 009479 005078 005078 004301 004300 005322 005322infin 007318 007318 008507 008507 003836 003836 003202 003202 004104 004104

05

0 035045 035037 041996 041980 010065 010065 006217 006216 013484 013483025 030709 030703 036112 036102 009170 009170 005695 005695 012160 0121591 023262 023260 026091 026088 007851 007851 004970 004970 010066 0100665 017691 017690 019258 019257 006610 006610 004262 004262 008226 008226infin 017921 017837 021375 021374 005123 005123 003164 003164 006863 006863

0 20 40 60 80 100

006

007

frequ

ency

par

amet

ers Ω


CCCC


CFCF

0014

0015

0016

frequ

ency

par

amet

ers Ω


CFFF

00024

00026

frequ

ency

par

amet

ers Ω


CSCF

0017

0016

0018

0019

frequ

ency

par

amet

ers Ω




005

0 003129 003130 004076 004076 001024 001024 000719 000719 001249 0012492025 002899 002899 003664 003664 000981 000981 000692 000692 001185 00118511 002667 002667 003250 003250 000948 000948 000674 000674 001132 00113165 002677 002677 003239 003239 000963 000963 000685 000685 001146 001146infin 002689 002689 003502 003502 000880 000880 000618 000618 001073 0010734

01

0 011639 011638 014580 014580 004001 004001 002835 002835 004817 004817025 010561 010561 012781 012780 003810 003810 002717 002717 004532 00453151 009734 009733 011453 011453 003679 003679 002641 002641 004327 00432675 009646 009646 011234 011234 003718 003718 002677 002677 004352 0043517infin 010001 010001 012528 012528 003438 003438 002426 002436 004139 0041391

02

0 037876 037870 043939 043930 014871 014871 010795 010795 017323 0173218025 036117 036113 041624 041617 014354 014354 010436 010436 016671 016671 033549 033547 037962 037958 013851 013851 010127 010127 015937 01593665 032783 032781 036695 036692 013888 013887 010200 010200 015878 0158779infin 032545 032544 037755 037753 012779 012778 009276 009276 014885 0148845


005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001541 001175 000360 000355 000429 000139 000057 0002641 001291 001027 000329 000333 000389 000129 000054 0002465 001111 000893 000290 000296 000342 000114 000048 00021810 001107 000868 000274 000274 000324 000107 000044 000203

01

0 006897 018772 001632 001608 001941 000630 000259 00120005 005652 015857 001406 001405 001663 000547 000227 0010431 004862 014291 001291 001320 001519 000509 000214 0009745 003953 011409 001113 001166 001303 000446 000190 00085610 003929 011157 001054 001082 001239 000417 000175 000798

02

0 023286 018772 006198 006260 007269 002439 001026 00465705 019360 015857 005338 005472 006242 002121 000899 0040551 017136 014291 004927 005147 005740 001983 000848 0037985 013242 011409 004126 004505 004776 001714 000749 00329510 013167 011157 003935 004191 004569 001606 000694 003081



005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001606 001246 000388 000386 000461 000150 000062 0002871 001588 001235 000386 000385 000458 000150 000062 0002865 001598 001249 000393 000392 000465 000153 000063 00029110 001481 001179 000378 000383 000447 000148 000062 000283

01

0 006897 005286 001632 001608 001941 000630 000259 00120005 006029 004770 001524 001532 001801 000595 000247 0011351 005888 004690 001511 001526 001782 000591 000246 0011295 005858 004702 001529 001553 001801 000601 000251 00114810 005748 004611 001498 001521 001765 000588 000246 001124

02

0 023288 018773 006198 006260 007269 002439 001026 00465705 021550 017558 005870 005988 006871 002324 000983 0044421 020752 017074 005782 005952 006754 002304 000979 0044075 020352 016912 005810 006038 006770 002332 000996 00446410 020672 017013 005770 005935 006737 002300 000978 004398



References






































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



119876119909 = 11986066 (120597119908120597119909 + 120593119909) 119876119910 = 11986066 (120597119908120597119910 + 120593119910)

(9)


(11986066 11986311 11986322 11986312) = intℎ2minusℎ2

(12058111987666 (119911) 11987611 (119911)sdot 1199112 11987611 (119911) 1199112 11987612 (119911) 1199112) 119889119911 (10)


+ 120574119910119911119876119910) 119889119909119889119910 = 12 int1198860 int1198870 (119872119909 120597120593119909120597119909+119872119910 120597120593119910120597119910 +119872119909119910 (120597120593119909120597119910 + 120597120593119910120597119909 )+ (120597119908120597119909 + 120593119909)119876119909 + (120597119908120597119910 + 120593119910)119876119910)119889119909119889119910

(11)

119879 = 12 int1198860 int1198870 [1198680 (120597119908120597119905 )2+ 1198682((120597120593119909120597119905 )2 + (120597120593119910120597119905 )2)]119889119909119889119910

[1198680 1198681] = intℎ2minusℎ2

120588 (1 1199112) 119889119911(12)


119881 = 12 int1198860 (1198961199100119910 1199082 + 11987011991001199101199091205932119909 + 1198701199100119910 1205932119910)10038161003816100381610038161003816119910=0 119889119909+ 12 int1198860 (119896119910119887119910 1199082 + 1198701199101198871199101199091205932119909 + 119870119910119887119910 1205932119910)10038161003816100381610038161003816119910=119887 119889119909+ 12 int1198870 (1198961199090119909 1199082 + 1198701199090119909 1205932119909 + 11987011990901199091199101205932119910)10038161003816100381610038161003816119909=0 119889119910+ 12 int1198870 (119896119909119886119909 1199082 + 119870119909119886119909 1205932119909 + 1198701199091198861199091199101205932119910)10038161003816100381610038161003816119909=119886 119889119910

(13)


119860119902 (119909 119910)119908 (119909 119910) 119889119860 (14)




int11990500(120575119879 + 120575119882 minus 120575119880 minus 120575119881) 119889119905 = 0 (16)



int11990500int1198860(11987211990911991012057512059311990910038161003816100381610038161003816119910=119887119910=0 + 11987611991012057511990810038161003816100381610038161003816119910=119887119910=0 + 11987211991012057512059311991010038161003816100381610038161003816119910=119887119910=0) 119889119909119889119905 + int1199050

0int1198870(1198721199091205751205931199091003816100381610038161003816119909=119886119909=0 + 1198761199091205751199081003816100381610038161003816119909=119886119909=0 + 11987211990911991012057512059311991010038161003816100381610038161003816119909=119886119909=0) 119889119910119889119905

int11990500int1198860(1198961199100119910 119908120575119908 + 1198701199100119910119909120593119909120575120593119909 + 1198701199100119910 120593119910120575120593119910)10038161003816100381610038161003816119910=0 119889119909119889119905 + int1199050

0int1198860(119896119910119887119910 119908120575119908 + 119870119910119887119910119909120593119909120575120593119909 + 119870119910119887119910 120593119910120575120593119910)10038161003816100381610038161003816119910=119887 119889119909119889119905

+ int11990500int1198870(1198961199090119909 119908120575119908 + 1198701199090119909 120593119909120575120593119909 + 1198701199090119909119910120593119910120575120593119910)10038161003816100381610038161003816119909=0 119889119910119889119905 + int1199050

0int1198870(119896119909119886119909 119908120575119908 + 119870119909119886119909 120593119909120575120593119909 + 119870119909119886119909119910120593119910120575120593119910)10038161003816100381610038161003816119909=119886 119889119910119889119905


0int1198860int1198870119902 (119909 119910) 120575119908119889119909119889119910119889119905 = 0

(17)



int11990500int1198860int1198870

((((



+ int11990500int1198860((119896119910119887119910 119908 + 119876119910) 120575119908 + (119870119910119887119910119909120593119909 +119872119909119910) 120575120593119909 + (119870119910119887119910 120593119910 +119872119910) 120575120593119910)10038161003816100381610038161003816119910=119887 119889119909119889119905


+ int11990500int1198870((119896119909119886119909 119908 + 119876119909) 120575119908 + (119870119909119886119909 120593119909 +119872119909) 120575120593119909 + (119870119909119886119909119910120593119910 +119872119909119910) 120575120593119910)10038161003816100381610038161003816119909=119886 119889119910119889119905 = 0

(18)



(19)




([[[[11987111 11987112 1198711311987121 11987122 1198712311987131 11987132 11987133

]]]] minus 1205962 [[[[11987211 0 00 11987222 00 0 11987233

]]]])[[[[120593119909120593119910119908]]]]

= [[[[00119902]]]]

(21)


where


(22)


[[[[[11987111989511 11987111989512 1198711198951311987111989521 11987111989522 1198711198952311987111989531 11987111989532 11987111989533

]]]]][[[[120593119909120593119910119908]]]] = [[[[

000]]]] (119895 = 1199090 119909119886 1199100 119910b)

(23)



(24)





119898=0

119873sum119899=0

119860119898119899 cos 120582119886119898119909 cos 120582119887119899119910+ 2sum119897=1

119872sum119898=0

119887119897119898120577119897119887 (119910) cos120582119886119898119909+ 2sum119897=1

119873sum119899=0

119886119897119899120577119897119886 (119909) cos 120582119887119899119910120593119910 (119909 119910) = 119872sum

119898=0

119873sum119899=0

119861119898119899 cos 120582119886119898119909 cos 120582119887119899119910+ 2sum119897=1

119872sum119898=0

119889119897119898120577119897119887 (119910) cos 120582119886119898119909+ 2sum119897=1

119873sum119899=0

119888119897119899120577119897119886 (119909) cos 120582119887119899119910119908 (119909 119910) = 119872sum

119898=0

119873sum119899=0

119862119898119899 cos 120582119886119898119909 cos 120582119887119899119910+ 2sum119897=1

119872sum119898=0

119891119897119898120577119897119887 (119910) cos 120582119886119898119909+ 2sum119897=1

119873sum119899=0

119890119897119899120577119897119886 (119909) cos120582119887119899119910

(25)


ln c

lm d

ln e



(26)


(27)

where




(28)



where

L119894 = [[[11987111H119894 11987112H119894 11987113H11989411987121H119894 11987122H119894 11987123H11989411987131H119894 11987132H119894 11987133H119894]]] (119894 = 119909119910 119909 119910)

M119894 = [[[11987211H119894 0 00 11987222H119894 00 0 11987233H119894]]] (119894 = 119909119910 119909 119910)

Θ119909119910 = [A B C]119879 Θ119909 = [a c e]119879 Θ119910 = [b d f]119879

F = [0 0 q]119879

(30)


L119895119909119910Θ119909119910 + L119895119909Θ119909 + L119895119910Θ119910 = 0 (119895 = 1199090 1199100 119909119886 119910119887) (31)

where

L119895119894 = [[[[11987111989511H119894 11987111989512H119894 11987111989513H11989411987111989521H119894 11987111989522H119894 11987111989523H11989411987111989531H119894 11987111989532H119894 11987111989533H119894

]]]] (119894 = 119909119910 119909 119910 119895 = 1199090 1199100 119909119886 119910119887)

(32)



(33)


L119909119910Θ119909119910 + [L119909 L119910] [Θ119909Θ119910

]minus 1205962 (M119909119910Θ119909119910 + [M119909 M119910] [Θ119909

Θ119910

]) = F

(34)

where

L119909119910 = int1198860int1198870

C119909119910L119909119910119889119909119889119910M119909119910 = int119886

0int1198870

C119909119910M119909119910119889119909119889119910L119909 = int119886

0int1198870

C119909119910L119909119889119909119889119910M119909 = int119886

0int1198870

C119909119910M119909119889119909119889119910L119910 = int119886

0int1198870

C119909119910L119910119889119909119889119910M119910 = int119886

0int1198870

C119909119910M119910119889119909119889119910

(35)


L119895119909119910Θ119909119910 + L119895119909Θ119909 + L119895119910Θ119910 = 0 (119895 = 1199090 1199100 1199091 1199101) (36)

where

L1199090119909119910 = int1198870

C119910L1199090119909119910119889119910

L119909119886119909119910 = int1198870

C119910L119909119886119909119910119889119910

L1199090119909 = int1198870

C119910L1199090119909 119889119910

L119909119886119909 = int1198870

C119910L119909119886119909 119889119910

L1199090119910 = int1198870

C119910L1199090119910 119889119910

L119909119886119910 = int1198870

C119910L119909119886119910 119889119910

L1199100119909119910 = int1198860

C119909L1199100119909119910119889119909

L119910119887119909119910 = int1198860

C119909L119910119887119909119910119889119909




L1199100119909 = int1198860

C119909L1199100119909 119889119909

L119910119887119909 = int1198860

C119909L119910119887119909 119889119909

L1199100119910 = int1198860

C119909L1199100119910 119889119909

L119910119887119910 = int1198860

C119909L119910119887119910 119889119909


[Θ119909Θ119910

] = [[[[[[[[L1199090119909 L1199090119910L119909119886119909 L119909119886119910L1199100119909 L1199100119910L119910119886119909 L119910119887119910

]]]]]]]]

minus1 [[[[[[[[L1199090119909119910L119909119886119909119910L1199100119909119910L119910119887119909119910

]]]]]]]]Θ119909119910 (38)


where


]]]]]]]]

minus1 [[[[[[[[L1199090119909119910L119909119886119909119910L1199100119909119910L119910119887119909119910

]]]]]]]]


]]]]]]]]

minus1 [[[[[[[[L1199090119909119910L119909119886119909119910L1199100119909119910L119910119887119909119910

]]]]]]]]

(40)









Clamped (C) 119908 = 120593119909 = 120593119910 = 0 1015 1015 1015


119910 = constantFree (F) 119876119909 = 119872119910 = 119872119909119910 = 0 0 0 0

Clamped (C) 119908 = 120593119910 = 120593119909 = 0 1015 1015 1015










Table4Com

paris

onof

them

axim


pics

quarep

latesh

avingvario

usbo

undary

cond

ition

swith

different

load

functio

ns

Boun

dary

cond

ition

sMetho

dUniform

pressure

PointL

oad(0505

)[(01)

(01)][(010

9)(0109)]

[(0208)(02

08)][(030

7)(0307)]

[(0406)(04

06)]CCCC

Present

5602E

-07

5438E

-07

4530E

-07

2790E

-07

8901E-08

2522E

-10

FEM

5602E

-07

5438E

-07

4530E

-07

2789E

-07

8896E

-08

2563E

-10

CFCF

Present

1278E-06

9716E-07

7275E-07

4116

E-07

1237E-07

340

1E-10

FEM

1278E-06

9716E-07

7273E-07

4115

E-07

1236E-07

3431E-10

CCCF

Present

1301E-09

8452E

-07

6152E

-07

3502E

-07

1066

E-07

2950E

-10

FEM

1302E-09

8452E

-07

6150E

-07

3501E-07

1065E-07

2981E-10

CSCS

Present

840

6E-07

7930E-07

6272E

-07

3677E

-07

1129E-07

3157E

-10

FEM

840

6E-07

7930E-07

6272E

-07

3677E

-07

1129E-07

3174E

-10

SSSS

Present

1737E-06

1584E-06

1175E-06

644

8E-07

1874E-07

506

4E-10

FEM

1737E-06

1585E-06

1740

E-06

644

7E-07

1873E-07

5076E

-10

CSFF

Present

3084E

-05

1858E-05

9955E-06

4267E

-06

1043E-06

260

4E-09

FEM

3084E

-05

1858E-05

9955E-06

4267E

-06

1043E-06

2618E

-09


CCCC[(0208) (0208)]

CFCF [(0406) (0406)]

CCCF [(01) (01)]


(a) (b)










4 Conclusions




00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus9

40 times 10minus9

60 times 10minus9

80 times 10minus9


(a)

00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus9

40 times 10minus9

60 times 10minus9

80 times 10minus9


(b)


00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus6

40 times 10minus6

60 times 10minus6

80 times 10minus6


(a)

00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

10 times 10minus6

20 times 10minus6

30 times 10minus6


(b)


CCCC CCCS CFCF CFFF





0 20 40 60 80 100

004

005

006

007

frequ

ency

par

amet

ers Ω


CCCC


CFCF

0008

0010

0012

0014

0016

frequ

ency

par

amet

ers Ω


CFFF

00016

00020

00024

frequ

ency

par

amet

ers Ω


CSCF

0012

0016

0020

frequ

ency

par

amet

ers Ω


Data Availability




Acknowledgments




2

0 008325 008325 008729 008729 006713 006713 006364 006365 006781 006781025 007600 007600 007950 007950 006145 006145 005829 005829 006205 0062051 006541 006541 006790 006790 005346 005346 005080 005080 005391 0053915 005524 005524 005695 005695 004568 004568 004349 004349 004600 004600infin 004263 004237 004443 004443 003417 003417 003239 003240 003451 003452

1

0 014378 014377 016713 016712 007537 007537 006290 006290 008062 008062025 012974 012973 014927 014926 006890 006890 005761 005761 007351 0073501 010725 010725 011955 011955 005968 005968 005021 005021 006308 0063085 008720 008720 009479 009479 005078 005078 004301 004300 005322 005322infin 007318 007318 008507 008507 003836 003836 003202 003202 004104 004104

05

0 035045 035037 041996 041980 010065 010065 006217 006216 013484 013483025 030709 030703 036112 036102 009170 009170 005695 005695 012160 0121591 023262 023260 026091 026088 007851 007851 004970 004970 010066 0100665 017691 017690 019258 019257 006610 006610 004262 004262 008226 008226infin 017921 017837 021375 021374 005123 005123 003164 003164 006863 006863

0 20 40 60 80 100

006

007

frequ

ency

par

amet

ers Ω


CCCC


CFCF

0014

0015

0016

frequ

ency

par

amet

ers Ω


CFFF

00024

00026

frequ

ency

par

amet

ers Ω


CSCF

0017

0016

0018

0019

frequ

ency

par

amet

ers Ω




005

0 003129 003130 004076 004076 001024 001024 000719 000719 001249 0012492025 002899 002899 003664 003664 000981 000981 000692 000692 001185 00118511 002667 002667 003250 003250 000948 000948 000674 000674 001132 00113165 002677 002677 003239 003239 000963 000963 000685 000685 001146 001146infin 002689 002689 003502 003502 000880 000880 000618 000618 001073 0010734

01

0 011639 011638 014580 014580 004001 004001 002835 002835 004817 004817025 010561 010561 012781 012780 003810 003810 002717 002717 004532 00453151 009734 009733 011453 011453 003679 003679 002641 002641 004327 00432675 009646 009646 011234 011234 003718 003718 002677 002677 004352 0043517infin 010001 010001 012528 012528 003438 003438 002426 002436 004139 0041391

02

0 037876 037870 043939 043930 014871 014871 010795 010795 017323 0173218025 036117 036113 041624 041617 014354 014354 010436 010436 016671 016671 033549 033547 037962 037958 013851 013851 010127 010127 015937 01593665 032783 032781 036695 036692 013888 013887 010200 010200 015878 0158779infin 032545 032544 037755 037753 012779 012778 009276 009276 014885 0148845


005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001541 001175 000360 000355 000429 000139 000057 0002641 001291 001027 000329 000333 000389 000129 000054 0002465 001111 000893 000290 000296 000342 000114 000048 00021810 001107 000868 000274 000274 000324 000107 000044 000203

01

0 006897 018772 001632 001608 001941 000630 000259 00120005 005652 015857 001406 001405 001663 000547 000227 0010431 004862 014291 001291 001320 001519 000509 000214 0009745 003953 011409 001113 001166 001303 000446 000190 00085610 003929 011157 001054 001082 001239 000417 000175 000798

02

0 023286 018772 006198 006260 007269 002439 001026 00465705 019360 015857 005338 005472 006242 002121 000899 0040551 017136 014291 004927 005147 005740 001983 000848 0037985 013242 011409 004126 004505 004776 001714 000749 00329510 013167 011157 003935 004191 004569 001606 000694 003081



005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001606 001246 000388 000386 000461 000150 000062 0002871 001588 001235 000386 000385 000458 000150 000062 0002865 001598 001249 000393 000392 000465 000153 000063 00029110 001481 001179 000378 000383 000447 000148 000062 000283

01

0 006897 005286 001632 001608 001941 000630 000259 00120005 006029 004770 001524 001532 001801 000595 000247 0011351 005888 004690 001511 001526 001782 000591 000246 0011295 005858 004702 001529 001553 001801 000601 000251 00114810 005748 004611 001498 001521 001765 000588 000246 001124

02

0 023288 018773 006198 006260 007269 002439 001026 00465705 021550 017558 005870 005988 006871 002324 000983 0044421 020752 017074 005782 005952 006754 002304 000979 0044075 020352 016912 005810 006038 006770 002332 000996 00446410 020672 017013 005770 005935 006737 002300 000978 004398



References






































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




int11990500int1198860int1198870

((((



+ int11990500int1198860((119896119910119887119910 119908 + 119876119910) 120575119908 + (119870119910119887119910119909120593119909 +119872119909119910) 120575120593119909 + (119870119910119887119910 120593119910 +119872119910) 120575120593119910)10038161003816100381610038161003816119910=119887 119889119909119889119905


+ int11990500int1198870((119896119909119886119909 119908 + 119876119909) 120575119908 + (119870119909119886119909 120593119909 +119872119909) 120575120593119909 + (119870119909119886119909119910120593119910 +119872119909119910) 120575120593119910)10038161003816100381610038161003816119909=119886 119889119910119889119905 = 0

(18)



(19)




([[[[11987111 11987112 1198711311987121 11987122 1198712311987131 11987132 11987133

]]]] minus 1205962 [[[[11987211 0 00 11987222 00 0 11987233

]]]])[[[[120593119909120593119910119908]]]]

= [[[[00119902]]]]

(21)


where


(22)


[[[[[11987111989511 11987111989512 1198711198951311987111989521 11987111989522 1198711198952311987111989531 11987111989532 11987111989533

]]]]][[[[120593119909120593119910119908]]]] = [[[[

000]]]] (119895 = 1199090 119909119886 1199100 119910b)

(23)



(24)





119898=0

119873sum119899=0

119860119898119899 cos 120582119886119898119909 cos 120582119887119899119910+ 2sum119897=1

119872sum119898=0

119887119897119898120577119897119887 (119910) cos120582119886119898119909+ 2sum119897=1

119873sum119899=0

119886119897119899120577119897119886 (119909) cos 120582119887119899119910120593119910 (119909 119910) = 119872sum

119898=0

119873sum119899=0

119861119898119899 cos 120582119886119898119909 cos 120582119887119899119910+ 2sum119897=1

119872sum119898=0

119889119897119898120577119897119887 (119910) cos 120582119886119898119909+ 2sum119897=1

119873sum119899=0

119888119897119899120577119897119886 (119909) cos 120582119887119899119910119908 (119909 119910) = 119872sum

119898=0

119873sum119899=0

119862119898119899 cos 120582119886119898119909 cos 120582119887119899119910+ 2sum119897=1

119872sum119898=0

119891119897119898120577119897119887 (119910) cos 120582119886119898119909+ 2sum119897=1

119873sum119899=0

119890119897119899120577119897119886 (119909) cos120582119887119899119910

(25)


ln c

lm d

ln e



(26)


(27)

where




(28)



where

L119894 = [[[11987111H119894 11987112H119894 11987113H11989411987121H119894 11987122H119894 11987123H11989411987131H119894 11987132H119894 11987133H119894]]] (119894 = 119909119910 119909 119910)

M119894 = [[[11987211H119894 0 00 11987222H119894 00 0 11987233H119894]]] (119894 = 119909119910 119909 119910)

Θ119909119910 = [A B C]119879 Θ119909 = [a c e]119879 Θ119910 = [b d f]119879

F = [0 0 q]119879

(30)


L119895119909119910Θ119909119910 + L119895119909Θ119909 + L119895119910Θ119910 = 0 (119895 = 1199090 1199100 119909119886 119910119887) (31)

where

L119895119894 = [[[[11987111989511H119894 11987111989512H119894 11987111989513H11989411987111989521H119894 11987111989522H119894 11987111989523H11989411987111989531H119894 11987111989532H119894 11987111989533H119894

]]]] (119894 = 119909119910 119909 119910 119895 = 1199090 1199100 119909119886 119910119887)

(32)



(33)


L119909119910Θ119909119910 + [L119909 L119910] [Θ119909Θ119910

]minus 1205962 (M119909119910Θ119909119910 + [M119909 M119910] [Θ119909

Θ119910

]) = F

(34)

where

L119909119910 = int1198860int1198870

C119909119910L119909119910119889119909119889119910M119909119910 = int119886

0int1198870

C119909119910M119909119910119889119909119889119910L119909 = int119886

0int1198870

C119909119910L119909119889119909119889119910M119909 = int119886

0int1198870

C119909119910M119909119889119909119889119910L119910 = int119886

0int1198870

C119909119910L119910119889119909119889119910M119910 = int119886

0int1198870

C119909119910M119910119889119909119889119910

(35)


L119895119909119910Θ119909119910 + L119895119909Θ119909 + L119895119910Θ119910 = 0 (119895 = 1199090 1199100 1199091 1199101) (36)

where

L1199090119909119910 = int1198870

C119910L1199090119909119910119889119910

L119909119886119909119910 = int1198870

C119910L119909119886119909119910119889119910

L1199090119909 = int1198870

C119910L1199090119909 119889119910

L119909119886119909 = int1198870

C119910L119909119886119909 119889119910

L1199090119910 = int1198870

C119910L1199090119910 119889119910

L119909119886119910 = int1198870

C119910L119909119886119910 119889119910

L1199100119909119910 = int1198860

C119909L1199100119909119910119889119909

L119910119887119909119910 = int1198860

C119909L119910119887119909119910119889119909




L1199100119909 = int1198860

C119909L1199100119909 119889119909

L119910119887119909 = int1198860

C119909L119910119887119909 119889119909

L1199100119910 = int1198860

C119909L1199100119910 119889119909

L119910119887119910 = int1198860

C119909L119910119887119910 119889119909


[Θ119909Θ119910

] = [[[[[[[[L1199090119909 L1199090119910L119909119886119909 L119909119886119910L1199100119909 L1199100119910L119910119886119909 L119910119887119910

]]]]]]]]

minus1 [[[[[[[[L1199090119909119910L119909119886119909119910L1199100119909119910L119910119887119909119910

]]]]]]]]Θ119909119910 (38)


where


]]]]]]]]

minus1 [[[[[[[[L1199090119909119910L119909119886119909119910L1199100119909119910L119910119887119909119910

]]]]]]]]


]]]]]]]]

minus1 [[[[[[[[L1199090119909119910L119909119886119909119910L1199100119909119910L119910119887119909119910

]]]]]]]]

(40)









Clamped (C) 119908 = 120593119909 = 120593119910 = 0 1015 1015 1015


119910 = constantFree (F) 119876119909 = 119872119910 = 119872119909119910 = 0 0 0 0

Clamped (C) 119908 = 120593119910 = 120593119909 = 0 1015 1015 1015










Table4Com

paris

onof

them

axim


pics

quarep

latesh

avingvario

usbo

undary

cond

ition

swith

different

load

functio

ns

Boun

dary

cond

ition

sMetho

dUniform

pressure

PointL

oad(0505

)[(01)

(01)][(010

9)(0109)]

[(0208)(02

08)][(030

7)(0307)]

[(0406)(04

06)]CCCC

Present

5602E

-07

5438E

-07

4530E

-07

2790E

-07

8901E-08

2522E

-10

FEM

5602E

-07

5438E

-07

4530E

-07

2789E

-07

8896E

-08

2563E

-10

CFCF

Present

1278E-06

9716E-07

7275E-07

4116

E-07

1237E-07

340

1E-10

FEM

1278E-06

9716E-07

7273E-07

4115

E-07

1236E-07

3431E-10

CCCF

Present

1301E-09

8452E

-07

6152E

-07

3502E

-07

1066

E-07

2950E

-10

FEM

1302E-09

8452E

-07

6150E

-07

3501E-07

1065E-07

2981E-10

CSCS

Present

840

6E-07

7930E-07

6272E

-07

3677E

-07

1129E-07

3157E

-10

FEM

840

6E-07

7930E-07

6272E

-07

3677E

-07

1129E-07

3174E

-10

SSSS

Present

1737E-06

1584E-06

1175E-06

644

8E-07

1874E-07

506

4E-10

FEM

1737E-06

1585E-06

1740

E-06

644

7E-07

1873E-07

5076E

-10

CSFF

Present

3084E

-05

1858E-05

9955E-06

4267E

-06

1043E-06

260

4E-09

FEM

3084E

-05

1858E-05

9955E-06

4267E

-06

1043E-06

2618E

-09


CCCC[(0208) (0208)]

CFCF [(0406) (0406)]

CCCF [(01) (01)]


(a) (b)










4 Conclusions




00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus9

40 times 10minus9

60 times 10minus9

80 times 10minus9


(a)

00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus9

40 times 10minus9

60 times 10minus9

80 times 10minus9


(b)


00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus6

40 times 10minus6

60 times 10minus6

80 times 10minus6


(a)

00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

10 times 10minus6

20 times 10minus6

30 times 10minus6


(b)


CCCC CCCS CFCF CFFF





0 20 40 60 80 100

004

005

006

007

frequ

ency

par

amet

ers Ω


CCCC


CFCF

0008

0010

0012

0014

0016

frequ

ency

par

amet

ers Ω


CFFF

00016

00020

00024

frequ

ency

par

amet

ers Ω


CSCF

0012

0016

0020

frequ

ency

par

amet

ers Ω


Data Availability




Acknowledgments




2

0 008325 008325 008729 008729 006713 006713 006364 006365 006781 006781025 007600 007600 007950 007950 006145 006145 005829 005829 006205 0062051 006541 006541 006790 006790 005346 005346 005080 005080 005391 0053915 005524 005524 005695 005695 004568 004568 004349 004349 004600 004600infin 004263 004237 004443 004443 003417 003417 003239 003240 003451 003452

1

0 014378 014377 016713 016712 007537 007537 006290 006290 008062 008062025 012974 012973 014927 014926 006890 006890 005761 005761 007351 0073501 010725 010725 011955 011955 005968 005968 005021 005021 006308 0063085 008720 008720 009479 009479 005078 005078 004301 004300 005322 005322infin 007318 007318 008507 008507 003836 003836 003202 003202 004104 004104

05

0 035045 035037 041996 041980 010065 010065 006217 006216 013484 013483025 030709 030703 036112 036102 009170 009170 005695 005695 012160 0121591 023262 023260 026091 026088 007851 007851 004970 004970 010066 0100665 017691 017690 019258 019257 006610 006610 004262 004262 008226 008226infin 017921 017837 021375 021374 005123 005123 003164 003164 006863 006863

0 20 40 60 80 100

006

007

frequ

ency

par

amet

ers Ω


CCCC


CFCF

0014

0015

0016

frequ

ency

par

amet

ers Ω


CFFF

00024

00026

frequ

ency

par

amet

ers Ω


CSCF

0017

0016

0018

0019

frequ

ency

par

amet

ers Ω




005

0 003129 003130 004076 004076 001024 001024 000719 000719 001249 0012492025 002899 002899 003664 003664 000981 000981 000692 000692 001185 00118511 002667 002667 003250 003250 000948 000948 000674 000674 001132 00113165 002677 002677 003239 003239 000963 000963 000685 000685 001146 001146infin 002689 002689 003502 003502 000880 000880 000618 000618 001073 0010734

01

0 011639 011638 014580 014580 004001 004001 002835 002835 004817 004817025 010561 010561 012781 012780 003810 003810 002717 002717 004532 00453151 009734 009733 011453 011453 003679 003679 002641 002641 004327 00432675 009646 009646 011234 011234 003718 003718 002677 002677 004352 0043517infin 010001 010001 012528 012528 003438 003438 002426 002436 004139 0041391

02

0 037876 037870 043939 043930 014871 014871 010795 010795 017323 0173218025 036117 036113 041624 041617 014354 014354 010436 010436 016671 016671 033549 033547 037962 037958 013851 013851 010127 010127 015937 01593665 032783 032781 036695 036692 013888 013887 010200 010200 015878 0158779infin 032545 032544 037755 037753 012779 012778 009276 009276 014885 0148845


005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001541 001175 000360 000355 000429 000139 000057 0002641 001291 001027 000329 000333 000389 000129 000054 0002465 001111 000893 000290 000296 000342 000114 000048 00021810 001107 000868 000274 000274 000324 000107 000044 000203

01

0 006897 018772 001632 001608 001941 000630 000259 00120005 005652 015857 001406 001405 001663 000547 000227 0010431 004862 014291 001291 001320 001519 000509 000214 0009745 003953 011409 001113 001166 001303 000446 000190 00085610 003929 011157 001054 001082 001239 000417 000175 000798

02

0 023286 018772 006198 006260 007269 002439 001026 00465705 019360 015857 005338 005472 006242 002121 000899 0040551 017136 014291 004927 005147 005740 001983 000848 0037985 013242 011409 004126 004505 004776 001714 000749 00329510 013167 011157 003935 004191 004569 001606 000694 003081



005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001606 001246 000388 000386 000461 000150 000062 0002871 001588 001235 000386 000385 000458 000150 000062 0002865 001598 001249 000393 000392 000465 000153 000063 00029110 001481 001179 000378 000383 000447 000148 000062 000283

01

0 006897 005286 001632 001608 001941 000630 000259 00120005 006029 004770 001524 001532 001801 000595 000247 0011351 005888 004690 001511 001526 001782 000591 000246 0011295 005858 004702 001529 001553 001801 000601 000251 00114810 005748 004611 001498 001521 001765 000588 000246 001124

02

0 023288 018773 006198 006260 007269 002439 001026 00465705 021550 017558 005870 005988 006871 002324 000983 0044421 020752 017074 005782 005952 006754 002304 000979 0044075 020352 016912 005810 006038 006770 002332 000996 00446410 020672 017013 005770 005935 006737 002300 000978 004398



References






































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



where


(22)


[[[[[11987111989511 11987111989512 1198711198951311987111989521 11987111989522 1198711198952311987111989531 11987111989532 11987111989533

]]]]][[[[120593119909120593119910119908]]]] = [[[[

000]]]] (119895 = 1199090 119909119886 1199100 119910b)

(23)



(24)





119898=0

119873sum119899=0

119860119898119899 cos 120582119886119898119909 cos 120582119887119899119910+ 2sum119897=1

119872sum119898=0

119887119897119898120577119897119887 (119910) cos120582119886119898119909+ 2sum119897=1

119873sum119899=0

119886119897119899120577119897119886 (119909) cos 120582119887119899119910120593119910 (119909 119910) = 119872sum

119898=0

119873sum119899=0

119861119898119899 cos 120582119886119898119909 cos 120582119887119899119910+ 2sum119897=1

119872sum119898=0

119889119897119898120577119897119887 (119910) cos 120582119886119898119909+ 2sum119897=1

119873sum119899=0

119888119897119899120577119897119886 (119909) cos 120582119887119899119910119908 (119909 119910) = 119872sum

119898=0

119873sum119899=0

119862119898119899 cos 120582119886119898119909 cos 120582119887119899119910+ 2sum119897=1

119872sum119898=0

119891119897119898120577119897119887 (119910) cos 120582119886119898119909+ 2sum119897=1

119873sum119899=0

119890119897119899120577119897119886 (119909) cos120582119887119899119910

(25)


ln c

lm d

ln e



(26)


(27)

where




(28)



where

L119894 = [[[11987111H119894 11987112H119894 11987113H11989411987121H119894 11987122H119894 11987123H11989411987131H119894 11987132H119894 11987133H119894]]] (119894 = 119909119910 119909 119910)

M119894 = [[[11987211H119894 0 00 11987222H119894 00 0 11987233H119894]]] (119894 = 119909119910 119909 119910)

Θ119909119910 = [A B C]119879 Θ119909 = [a c e]119879 Θ119910 = [b d f]119879

F = [0 0 q]119879

(30)


L119895119909119910Θ119909119910 + L119895119909Θ119909 + L119895119910Θ119910 = 0 (119895 = 1199090 1199100 119909119886 119910119887) (31)

where

L119895119894 = [[[[11987111989511H119894 11987111989512H119894 11987111989513H11989411987111989521H119894 11987111989522H119894 11987111989523H11989411987111989531H119894 11987111989532H119894 11987111989533H119894

]]]] (119894 = 119909119910 119909 119910 119895 = 1199090 1199100 119909119886 119910119887)

(32)



(33)


L119909119910Θ119909119910 + [L119909 L119910] [Θ119909Θ119910

]minus 1205962 (M119909119910Θ119909119910 + [M119909 M119910] [Θ119909

Θ119910

]) = F

(34)

where

L119909119910 = int1198860int1198870

C119909119910L119909119910119889119909119889119910M119909119910 = int119886

0int1198870

C119909119910M119909119910119889119909119889119910L119909 = int119886

0int1198870

C119909119910L119909119889119909119889119910M119909 = int119886

0int1198870

C119909119910M119909119889119909119889119910L119910 = int119886

0int1198870

C119909119910L119910119889119909119889119910M119910 = int119886

0int1198870

C119909119910M119910119889119909119889119910

(35)


L119895119909119910Θ119909119910 + L119895119909Θ119909 + L119895119910Θ119910 = 0 (119895 = 1199090 1199100 1199091 1199101) (36)

where

L1199090119909119910 = int1198870

C119910L1199090119909119910119889119910

L119909119886119909119910 = int1198870

C119910L119909119886119909119910119889119910

L1199090119909 = int1198870

C119910L1199090119909 119889119910

L119909119886119909 = int1198870

C119910L119909119886119909 119889119910

L1199090119910 = int1198870

C119910L1199090119910 119889119910

L119909119886119910 = int1198870

C119910L119909119886119910 119889119910

L1199100119909119910 = int1198860

C119909L1199100119909119910119889119909

L119910119887119909119910 = int1198860

C119909L119910119887119909119910119889119909




L1199100119909 = int1198860

C119909L1199100119909 119889119909

L119910119887119909 = int1198860

C119909L119910119887119909 119889119909

L1199100119910 = int1198860

C119909L1199100119910 119889119909

L119910119887119910 = int1198860

C119909L119910119887119910 119889119909


[Θ119909Θ119910

] = [[[[[[[[L1199090119909 L1199090119910L119909119886119909 L119909119886119910L1199100119909 L1199100119910L119910119886119909 L119910119887119910

]]]]]]]]

minus1 [[[[[[[[L1199090119909119910L119909119886119909119910L1199100119909119910L119910119887119909119910

]]]]]]]]Θ119909119910 (38)


where


]]]]]]]]

minus1 [[[[[[[[L1199090119909119910L119909119886119909119910L1199100119909119910L119910119887119909119910

]]]]]]]]


]]]]]]]]

minus1 [[[[[[[[L1199090119909119910L119909119886119909119910L1199100119909119910L119910119887119909119910

]]]]]]]]

(40)









Clamped (C) 119908 = 120593119909 = 120593119910 = 0 1015 1015 1015


119910 = constantFree (F) 119876119909 = 119872119910 = 119872119909119910 = 0 0 0 0

Clamped (C) 119908 = 120593119910 = 120593119909 = 0 1015 1015 1015










Table4Com

paris

onof

them

axim


pics

quarep

latesh

avingvario

usbo

undary

cond

ition

swith

different

load

functio

ns

Boun

dary

cond

ition

sMetho

dUniform

pressure

PointL

oad(0505

)[(01)

(01)][(010

9)(0109)]

[(0208)(02

08)][(030

7)(0307)]

[(0406)(04

06)]CCCC

Present

5602E

-07

5438E

-07

4530E

-07

2790E

-07

8901E-08

2522E

-10

FEM

5602E

-07

5438E

-07

4530E

-07

2789E

-07

8896E

-08

2563E

-10

CFCF

Present

1278E-06

9716E-07

7275E-07

4116

E-07

1237E-07

340

1E-10

FEM

1278E-06

9716E-07

7273E-07

4115

E-07

1236E-07

3431E-10

CCCF

Present

1301E-09

8452E

-07

6152E

-07

3502E

-07

1066

E-07

2950E

-10

FEM

1302E-09

8452E

-07

6150E

-07

3501E-07

1065E-07

2981E-10

CSCS

Present

840

6E-07

7930E-07

6272E

-07

3677E

-07

1129E-07

3157E

-10

FEM

840

6E-07

7930E-07

6272E

-07

3677E

-07

1129E-07

3174E

-10

SSSS

Present

1737E-06

1584E-06

1175E-06

644

8E-07

1874E-07

506

4E-10

FEM

1737E-06

1585E-06

1740

E-06

644

7E-07

1873E-07

5076E

-10

CSFF

Present

3084E

-05

1858E-05

9955E-06

4267E

-06

1043E-06

260

4E-09

FEM

3084E

-05

1858E-05

9955E-06

4267E

-06

1043E-06

2618E

-09


CCCC[(0208) (0208)]

CFCF [(0406) (0406)]

CCCF [(01) (01)]


(a) (b)










4 Conclusions




00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus9

40 times 10minus9

60 times 10minus9

80 times 10minus9


(a)

00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus9

40 times 10minus9

60 times 10minus9

80 times 10minus9


(b)


00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus6

40 times 10minus6

60 times 10minus6

80 times 10minus6


(a)

00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

10 times 10minus6

20 times 10minus6

30 times 10minus6


(b)


CCCC CCCS CFCF CFFF





0 20 40 60 80 100

004

005

006

007

frequ

ency

par

amet

ers Ω


CCCC


CFCF

0008

0010

0012

0014

0016

frequ

ency

par

amet

ers Ω


CFFF

00016

00020

00024

frequ

ency

par

amet

ers Ω


CSCF

0012

0016

0020

frequ

ency

par

amet

ers Ω


Data Availability




Acknowledgments




2

0 008325 008325 008729 008729 006713 006713 006364 006365 006781 006781025 007600 007600 007950 007950 006145 006145 005829 005829 006205 0062051 006541 006541 006790 006790 005346 005346 005080 005080 005391 0053915 005524 005524 005695 005695 004568 004568 004349 004349 004600 004600infin 004263 004237 004443 004443 003417 003417 003239 003240 003451 003452

1

0 014378 014377 016713 016712 007537 007537 006290 006290 008062 008062025 012974 012973 014927 014926 006890 006890 005761 005761 007351 0073501 010725 010725 011955 011955 005968 005968 005021 005021 006308 0063085 008720 008720 009479 009479 005078 005078 004301 004300 005322 005322infin 007318 007318 008507 008507 003836 003836 003202 003202 004104 004104

05

0 035045 035037 041996 041980 010065 010065 006217 006216 013484 013483025 030709 030703 036112 036102 009170 009170 005695 005695 012160 0121591 023262 023260 026091 026088 007851 007851 004970 004970 010066 0100665 017691 017690 019258 019257 006610 006610 004262 004262 008226 008226infin 017921 017837 021375 021374 005123 005123 003164 003164 006863 006863

0 20 40 60 80 100

006

007

frequ

ency

par

amet

ers Ω


CCCC


CFCF

0014

0015

0016

frequ

ency

par

amet

ers Ω


CFFF

00024

00026

frequ

ency

par

amet

ers Ω


CSCF

0017

0016

0018

0019

frequ

ency

par

amet

ers Ω




005

0 003129 003130 004076 004076 001024 001024 000719 000719 001249 0012492025 002899 002899 003664 003664 000981 000981 000692 000692 001185 00118511 002667 002667 003250 003250 000948 000948 000674 000674 001132 00113165 002677 002677 003239 003239 000963 000963 000685 000685 001146 001146infin 002689 002689 003502 003502 000880 000880 000618 000618 001073 0010734

01

0 011639 011638 014580 014580 004001 004001 002835 002835 004817 004817025 010561 010561 012781 012780 003810 003810 002717 002717 004532 00453151 009734 009733 011453 011453 003679 003679 002641 002641 004327 00432675 009646 009646 011234 011234 003718 003718 002677 002677 004352 0043517infin 010001 010001 012528 012528 003438 003438 002426 002436 004139 0041391

02

0 037876 037870 043939 043930 014871 014871 010795 010795 017323 0173218025 036117 036113 041624 041617 014354 014354 010436 010436 016671 016671 033549 033547 037962 037958 013851 013851 010127 010127 015937 01593665 032783 032781 036695 036692 013888 013887 010200 010200 015878 0158779infin 032545 032544 037755 037753 012779 012778 009276 009276 014885 0148845


005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001541 001175 000360 000355 000429 000139 000057 0002641 001291 001027 000329 000333 000389 000129 000054 0002465 001111 000893 000290 000296 000342 000114 000048 00021810 001107 000868 000274 000274 000324 000107 000044 000203

01

0 006897 018772 001632 001608 001941 000630 000259 00120005 005652 015857 001406 001405 001663 000547 000227 0010431 004862 014291 001291 001320 001519 000509 000214 0009745 003953 011409 001113 001166 001303 000446 000190 00085610 003929 011157 001054 001082 001239 000417 000175 000798

02

0 023286 018772 006198 006260 007269 002439 001026 00465705 019360 015857 005338 005472 006242 002121 000899 0040551 017136 014291 004927 005147 005740 001983 000848 0037985 013242 011409 004126 004505 004776 001714 000749 00329510 013167 011157 003935 004191 004569 001606 000694 003081



005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001606 001246 000388 000386 000461 000150 000062 0002871 001588 001235 000386 000385 000458 000150 000062 0002865 001598 001249 000393 000392 000465 000153 000063 00029110 001481 001179 000378 000383 000447 000148 000062 000283

01

0 006897 005286 001632 001608 001941 000630 000259 00120005 006029 004770 001524 001532 001801 000595 000247 0011351 005888 004690 001511 001526 001782 000591 000246 0011295 005858 004702 001529 001553 001801 000601 000251 00114810 005748 004611 001498 001521 001765 000588 000246 001124

02

0 023288 018773 006198 006260 007269 002439 001026 00465705 021550 017558 005870 005988 006871 002324 000983 0044421 020752 017074 005782 005952 006754 002304 000979 0044075 020352 016912 005810 006038 006770 002332 000996 00446410 020672 017013 005770 005935 006737 002300 000978 004398



References






































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





119898=0

119873sum119899=0

119860119898119899 cos 120582119886119898119909 cos 120582119887119899119910+ 2sum119897=1

119872sum119898=0

119887119897119898120577119897119887 (119910) cos120582119886119898119909+ 2sum119897=1

119873sum119899=0

119886119897119899120577119897119886 (119909) cos 120582119887119899119910120593119910 (119909 119910) = 119872sum

119898=0

119873sum119899=0

119861119898119899 cos 120582119886119898119909 cos 120582119887119899119910+ 2sum119897=1

119872sum119898=0

119889119897119898120577119897119887 (119910) cos 120582119886119898119909+ 2sum119897=1

119873sum119899=0

119888119897119899120577119897119886 (119909) cos 120582119887119899119910119908 (119909 119910) = 119872sum

119898=0

119873sum119899=0

119862119898119899 cos 120582119886119898119909 cos 120582119887119899119910+ 2sum119897=1

119872sum119898=0

119891119897119898120577119897119887 (119910) cos 120582119886119898119909+ 2sum119897=1

119873sum119899=0

119890119897119899120577119897119886 (119909) cos120582119887119899119910

(25)


ln c

lm d

ln e



(26)


(27)

where




(28)



where

L119894 = [[[11987111H119894 11987112H119894 11987113H11989411987121H119894 11987122H119894 11987123H11989411987131H119894 11987132H119894 11987133H119894]]] (119894 = 119909119910 119909 119910)

M119894 = [[[11987211H119894 0 00 11987222H119894 00 0 11987233H119894]]] (119894 = 119909119910 119909 119910)

Θ119909119910 = [A B C]119879 Θ119909 = [a c e]119879 Θ119910 = [b d f]119879

F = [0 0 q]119879

(30)


L119895119909119910Θ119909119910 + L119895119909Θ119909 + L119895119910Θ119910 = 0 (119895 = 1199090 1199100 119909119886 119910119887) (31)

where

L119895119894 = [[[[11987111989511H119894 11987111989512H119894 11987111989513H11989411987111989521H119894 11987111989522H119894 11987111989523H11989411987111989531H119894 11987111989532H119894 11987111989533H119894

]]]] (119894 = 119909119910 119909 119910 119895 = 1199090 1199100 119909119886 119910119887)

(32)



(33)


L119909119910Θ119909119910 + [L119909 L119910] [Θ119909Θ119910

]minus 1205962 (M119909119910Θ119909119910 + [M119909 M119910] [Θ119909

Θ119910

]) = F

(34)

where

L119909119910 = int1198860int1198870

C119909119910L119909119910119889119909119889119910M119909119910 = int119886

0int1198870

C119909119910M119909119910119889119909119889119910L119909 = int119886

0int1198870

C119909119910L119909119889119909119889119910M119909 = int119886

0int1198870

C119909119910M119909119889119909119889119910L119910 = int119886

0int1198870

C119909119910L119910119889119909119889119910M119910 = int119886

0int1198870

C119909119910M119910119889119909119889119910

(35)


L119895119909119910Θ119909119910 + L119895119909Θ119909 + L119895119910Θ119910 = 0 (119895 = 1199090 1199100 1199091 1199101) (36)

where

L1199090119909119910 = int1198870

C119910L1199090119909119910119889119910

L119909119886119909119910 = int1198870

C119910L119909119886119909119910119889119910

L1199090119909 = int1198870

C119910L1199090119909 119889119910

L119909119886119909 = int1198870

C119910L119909119886119909 119889119910

L1199090119910 = int1198870

C119910L1199090119910 119889119910

L119909119886119910 = int1198870

C119910L119909119886119910 119889119910

L1199100119909119910 = int1198860

C119909L1199100119909119910119889119909

L119910119887119909119910 = int1198860

C119909L119910119887119909119910119889119909




L1199100119909 = int1198860

C119909L1199100119909 119889119909

L119910119887119909 = int1198860

C119909L119910119887119909 119889119909

L1199100119910 = int1198860

C119909L1199100119910 119889119909

L119910119887119910 = int1198860

C119909L119910119887119910 119889119909


[Θ119909Θ119910

] = [[[[[[[[L1199090119909 L1199090119910L119909119886119909 L119909119886119910L1199100119909 L1199100119910L119910119886119909 L119910119887119910

]]]]]]]]

minus1 [[[[[[[[L1199090119909119910L119909119886119909119910L1199100119909119910L119910119887119909119910

]]]]]]]]Θ119909119910 (38)


where


]]]]]]]]

minus1 [[[[[[[[L1199090119909119910L119909119886119909119910L1199100119909119910L119910119887119909119910

]]]]]]]]


]]]]]]]]

minus1 [[[[[[[[L1199090119909119910L119909119886119909119910L1199100119909119910L119910119887119909119910

]]]]]]]]

(40)









Clamped (C) 119908 = 120593119909 = 120593119910 = 0 1015 1015 1015


119910 = constantFree (F) 119876119909 = 119872119910 = 119872119909119910 = 0 0 0 0

Clamped (C) 119908 = 120593119910 = 120593119909 = 0 1015 1015 1015










Table4Com

paris

onof

them

axim


pics

quarep

latesh

avingvario

usbo

undary

cond

ition

swith

different

load

functio

ns

Boun

dary

cond

ition

sMetho

dUniform

pressure

PointL

oad(0505

)[(01)

(01)][(010

9)(0109)]

[(0208)(02

08)][(030

7)(0307)]

[(0406)(04

06)]CCCC

Present

5602E

-07

5438E

-07

4530E

-07

2790E

-07

8901E-08

2522E

-10

FEM

5602E

-07

5438E

-07

4530E

-07

2789E

-07

8896E

-08

2563E

-10

CFCF

Present

1278E-06

9716E-07

7275E-07

4116

E-07

1237E-07

340

1E-10

FEM

1278E-06

9716E-07

7273E-07

4115

E-07

1236E-07

3431E-10

CCCF

Present

1301E-09

8452E

-07

6152E

-07

3502E

-07

1066

E-07

2950E

-10

FEM

1302E-09

8452E

-07

6150E

-07

3501E-07

1065E-07

2981E-10

CSCS

Present

840

6E-07

7930E-07

6272E

-07

3677E

-07

1129E-07

3157E

-10

FEM

840

6E-07

7930E-07

6272E

-07

3677E

-07

1129E-07

3174E

-10

SSSS

Present

1737E-06

1584E-06

1175E-06

644

8E-07

1874E-07

506

4E-10

FEM

1737E-06

1585E-06

1740

E-06

644

7E-07

1873E-07

5076E

-10

CSFF

Present

3084E

-05

1858E-05

9955E-06

4267E

-06

1043E-06

260

4E-09

FEM

3084E

-05

1858E-05

9955E-06

4267E

-06

1043E-06

2618E

-09


CCCC[(0208) (0208)]

CFCF [(0406) (0406)]

CCCF [(01) (01)]


(a) (b)










4 Conclusions




00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus9

40 times 10minus9

60 times 10minus9

80 times 10minus9


(a)

00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus9

40 times 10minus9

60 times 10minus9

80 times 10minus9


(b)


00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus6

40 times 10minus6

60 times 10minus6

80 times 10minus6


(a)

00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

10 times 10minus6

20 times 10minus6

30 times 10minus6


(b)


CCCC CCCS CFCF CFFF





0 20 40 60 80 100

004

005

006

007

frequ

ency

par

amet

ers Ω


CCCC


CFCF

0008

0010

0012

0014

0016

frequ

ency

par

amet

ers Ω


CFFF

00016

00020

00024

frequ

ency

par

amet

ers Ω


CSCF

0012

0016

0020

frequ

ency

par

amet

ers Ω


Data Availability




Acknowledgments




2

0 008325 008325 008729 008729 006713 006713 006364 006365 006781 006781025 007600 007600 007950 007950 006145 006145 005829 005829 006205 0062051 006541 006541 006790 006790 005346 005346 005080 005080 005391 0053915 005524 005524 005695 005695 004568 004568 004349 004349 004600 004600infin 004263 004237 004443 004443 003417 003417 003239 003240 003451 003452

1

0 014378 014377 016713 016712 007537 007537 006290 006290 008062 008062025 012974 012973 014927 014926 006890 006890 005761 005761 007351 0073501 010725 010725 011955 011955 005968 005968 005021 005021 006308 0063085 008720 008720 009479 009479 005078 005078 004301 004300 005322 005322infin 007318 007318 008507 008507 003836 003836 003202 003202 004104 004104

05

0 035045 035037 041996 041980 010065 010065 006217 006216 013484 013483025 030709 030703 036112 036102 009170 009170 005695 005695 012160 0121591 023262 023260 026091 026088 007851 007851 004970 004970 010066 0100665 017691 017690 019258 019257 006610 006610 004262 004262 008226 008226infin 017921 017837 021375 021374 005123 005123 003164 003164 006863 006863

0 20 40 60 80 100

006

007

frequ

ency

par

amet

ers Ω


CCCC


CFCF

0014

0015

0016

frequ

ency

par

amet

ers Ω


CFFF

00024

00026

frequ

ency

par

amet

ers Ω


CSCF

0017

0016

0018

0019

frequ

ency

par

amet

ers Ω




005

0 003129 003130 004076 004076 001024 001024 000719 000719 001249 0012492025 002899 002899 003664 003664 000981 000981 000692 000692 001185 00118511 002667 002667 003250 003250 000948 000948 000674 000674 001132 00113165 002677 002677 003239 003239 000963 000963 000685 000685 001146 001146infin 002689 002689 003502 003502 000880 000880 000618 000618 001073 0010734

01

0 011639 011638 014580 014580 004001 004001 002835 002835 004817 004817025 010561 010561 012781 012780 003810 003810 002717 002717 004532 00453151 009734 009733 011453 011453 003679 003679 002641 002641 004327 00432675 009646 009646 011234 011234 003718 003718 002677 002677 004352 0043517infin 010001 010001 012528 012528 003438 003438 002426 002436 004139 0041391

02

0 037876 037870 043939 043930 014871 014871 010795 010795 017323 0173218025 036117 036113 041624 041617 014354 014354 010436 010436 016671 016671 033549 033547 037962 037958 013851 013851 010127 010127 015937 01593665 032783 032781 036695 036692 013888 013887 010200 010200 015878 0158779infin 032545 032544 037755 037753 012779 012778 009276 009276 014885 0148845


005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001541 001175 000360 000355 000429 000139 000057 0002641 001291 001027 000329 000333 000389 000129 000054 0002465 001111 000893 000290 000296 000342 000114 000048 00021810 001107 000868 000274 000274 000324 000107 000044 000203

01

0 006897 018772 001632 001608 001941 000630 000259 00120005 005652 015857 001406 001405 001663 000547 000227 0010431 004862 014291 001291 001320 001519 000509 000214 0009745 003953 011409 001113 001166 001303 000446 000190 00085610 003929 011157 001054 001082 001239 000417 000175 000798

02

0 023286 018772 006198 006260 007269 002439 001026 00465705 019360 015857 005338 005472 006242 002121 000899 0040551 017136 014291 004927 005147 005740 001983 000848 0037985 013242 011409 004126 004505 004776 001714 000749 00329510 013167 011157 003935 004191 004569 001606 000694 003081



005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001606 001246 000388 000386 000461 000150 000062 0002871 001588 001235 000386 000385 000458 000150 000062 0002865 001598 001249 000393 000392 000465 000153 000063 00029110 001481 001179 000378 000383 000447 000148 000062 000283

01

0 006897 005286 001632 001608 001941 000630 000259 00120005 006029 004770 001524 001532 001801 000595 000247 0011351 005888 004690 001511 001526 001782 000591 000246 0011295 005858 004702 001529 001553 001801 000601 000251 00114810 005748 004611 001498 001521 001765 000588 000246 001124

02

0 023288 018773 006198 006260 007269 002439 001026 00465705 021550 017558 005870 005988 006871 002324 000983 0044421 020752 017074 005782 005952 006754 002304 000979 0044075 020352 016912 005810 006038 006770 002332 000996 00446410 020672 017013 005770 005935 006737 002300 000978 004398



References






































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




(28)



where

L119894 = [[[11987111H119894 11987112H119894 11987113H11989411987121H119894 11987122H119894 11987123H11989411987131H119894 11987132H119894 11987133H119894]]] (119894 = 119909119910 119909 119910)

M119894 = [[[11987211H119894 0 00 11987222H119894 00 0 11987233H119894]]] (119894 = 119909119910 119909 119910)

Θ119909119910 = [A B C]119879 Θ119909 = [a c e]119879 Θ119910 = [b d f]119879

F = [0 0 q]119879

(30)


L119895119909119910Θ119909119910 + L119895119909Θ119909 + L119895119910Θ119910 = 0 (119895 = 1199090 1199100 119909119886 119910119887) (31)

where

L119895119894 = [[[[11987111989511H119894 11987111989512H119894 11987111989513H11989411987111989521H119894 11987111989522H119894 11987111989523H11989411987111989531H119894 11987111989532H119894 11987111989533H119894

]]]] (119894 = 119909119910 119909 119910 119895 = 1199090 1199100 119909119886 119910119887)

(32)



(33)


L119909119910Θ119909119910 + [L119909 L119910] [Θ119909Θ119910

]minus 1205962 (M119909119910Θ119909119910 + [M119909 M119910] [Θ119909

Θ119910

]) = F

(34)

where

L119909119910 = int1198860int1198870

C119909119910L119909119910119889119909119889119910M119909119910 = int119886

0int1198870

C119909119910M119909119910119889119909119889119910L119909 = int119886

0int1198870

C119909119910L119909119889119909119889119910M119909 = int119886

0int1198870

C119909119910M119909119889119909119889119910L119910 = int119886

0int1198870

C119909119910L119910119889119909119889119910M119910 = int119886

0int1198870

C119909119910M119910119889119909119889119910

(35)


L119895119909119910Θ119909119910 + L119895119909Θ119909 + L119895119910Θ119910 = 0 (119895 = 1199090 1199100 1199091 1199101) (36)

where

L1199090119909119910 = int1198870

C119910L1199090119909119910119889119910

L119909119886119909119910 = int1198870

C119910L119909119886119909119910119889119910

L1199090119909 = int1198870

C119910L1199090119909 119889119910

L119909119886119909 = int1198870

C119910L119909119886119909 119889119910

L1199090119910 = int1198870

C119910L1199090119910 119889119910

L119909119886119910 = int1198870

C119910L119909119886119910 119889119910

L1199100119909119910 = int1198860

C119909L1199100119909119910119889119909

L119910119887119909119910 = int1198860

C119909L119910119887119909119910119889119909




L1199100119909 = int1198860

C119909L1199100119909 119889119909

L119910119887119909 = int1198860

C119909L119910119887119909 119889119909

L1199100119910 = int1198860

C119909L1199100119910 119889119909

L119910119887119910 = int1198860

C119909L119910119887119910 119889119909


[Θ119909Θ119910

] = [[[[[[[[L1199090119909 L1199090119910L119909119886119909 L119909119886119910L1199100119909 L1199100119910L119910119886119909 L119910119887119910

]]]]]]]]

minus1 [[[[[[[[L1199090119909119910L119909119886119909119910L1199100119909119910L119910119887119909119910

]]]]]]]]Θ119909119910 (38)


where


]]]]]]]]

minus1 [[[[[[[[L1199090119909119910L119909119886119909119910L1199100119909119910L119910119887119909119910

]]]]]]]]


]]]]]]]]

minus1 [[[[[[[[L1199090119909119910L119909119886119909119910L1199100119909119910L119910119887119909119910

]]]]]]]]

(40)









Clamped (C) 119908 = 120593119909 = 120593119910 = 0 1015 1015 1015


119910 = constantFree (F) 119876119909 = 119872119910 = 119872119909119910 = 0 0 0 0

Clamped (C) 119908 = 120593119910 = 120593119909 = 0 1015 1015 1015










Table4Com

paris

onof

them

axim


pics

quarep

latesh

avingvario

usbo

undary

cond

ition

swith

different

load

functio

ns

Boun

dary

cond

ition

sMetho

dUniform

pressure

PointL

oad(0505

)[(01)

(01)][(010

9)(0109)]

[(0208)(02

08)][(030

7)(0307)]

[(0406)(04

06)]CCCC

Present

5602E

-07

5438E

-07

4530E

-07

2790E

-07

8901E-08

2522E

-10

FEM

5602E

-07

5438E

-07

4530E

-07

2789E

-07

8896E

-08

2563E

-10

CFCF

Present

1278E-06

9716E-07

7275E-07

4116

E-07

1237E-07

340

1E-10

FEM

1278E-06

9716E-07

7273E-07

4115

E-07

1236E-07

3431E-10

CCCF

Present

1301E-09

8452E

-07

6152E

-07

3502E

-07

1066

E-07

2950E

-10

FEM

1302E-09

8452E

-07

6150E

-07

3501E-07

1065E-07

2981E-10

CSCS

Present

840

6E-07

7930E-07

6272E

-07

3677E

-07

1129E-07

3157E

-10

FEM

840

6E-07

7930E-07

6272E

-07

3677E

-07

1129E-07

3174E

-10

SSSS

Present

1737E-06

1584E-06

1175E-06

644

8E-07

1874E-07

506

4E-10

FEM

1737E-06

1585E-06

1740

E-06

644

7E-07

1873E-07

5076E

-10

CSFF

Present

3084E

-05

1858E-05

9955E-06

4267E

-06

1043E-06

260

4E-09

FEM

3084E

-05

1858E-05

9955E-06

4267E

-06

1043E-06

2618E

-09


CCCC[(0208) (0208)]

CFCF [(0406) (0406)]

CCCF [(01) (01)]


(a) (b)










4 Conclusions




00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus9

40 times 10minus9

60 times 10minus9

80 times 10minus9


(a)

00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus9

40 times 10minus9

60 times 10minus9

80 times 10minus9


(b)


00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus6

40 times 10minus6

60 times 10minus6

80 times 10minus6


(a)

00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

10 times 10minus6

20 times 10minus6

30 times 10minus6


(b)


CCCC CCCS CFCF CFFF





0 20 40 60 80 100

004

005

006

007

frequ

ency

par

amet

ers Ω


CCCC


CFCF

0008

0010

0012

0014

0016

frequ

ency

par

amet

ers Ω


CFFF

00016

00020

00024

frequ

ency

par

amet

ers Ω


CSCF

0012

0016

0020

frequ

ency

par

amet

ers Ω


Data Availability




Acknowledgments




2

0 008325 008325 008729 008729 006713 006713 006364 006365 006781 006781025 007600 007600 007950 007950 006145 006145 005829 005829 006205 0062051 006541 006541 006790 006790 005346 005346 005080 005080 005391 0053915 005524 005524 005695 005695 004568 004568 004349 004349 004600 004600infin 004263 004237 004443 004443 003417 003417 003239 003240 003451 003452

1

0 014378 014377 016713 016712 007537 007537 006290 006290 008062 008062025 012974 012973 014927 014926 006890 006890 005761 005761 007351 0073501 010725 010725 011955 011955 005968 005968 005021 005021 006308 0063085 008720 008720 009479 009479 005078 005078 004301 004300 005322 005322infin 007318 007318 008507 008507 003836 003836 003202 003202 004104 004104

05

0 035045 035037 041996 041980 010065 010065 006217 006216 013484 013483025 030709 030703 036112 036102 009170 009170 005695 005695 012160 0121591 023262 023260 026091 026088 007851 007851 004970 004970 010066 0100665 017691 017690 019258 019257 006610 006610 004262 004262 008226 008226infin 017921 017837 021375 021374 005123 005123 003164 003164 006863 006863

0 20 40 60 80 100

006

007

frequ

ency

par

amet

ers Ω


CCCC


CFCF

0014

0015

0016

frequ

ency

par

amet

ers Ω


CFFF

00024

00026

frequ

ency

par

amet

ers Ω


CSCF

0017

0016

0018

0019

frequ

ency

par

amet

ers Ω




005

0 003129 003130 004076 004076 001024 001024 000719 000719 001249 0012492025 002899 002899 003664 003664 000981 000981 000692 000692 001185 00118511 002667 002667 003250 003250 000948 000948 000674 000674 001132 00113165 002677 002677 003239 003239 000963 000963 000685 000685 001146 001146infin 002689 002689 003502 003502 000880 000880 000618 000618 001073 0010734

01

0 011639 011638 014580 014580 004001 004001 002835 002835 004817 004817025 010561 010561 012781 012780 003810 003810 002717 002717 004532 00453151 009734 009733 011453 011453 003679 003679 002641 002641 004327 00432675 009646 009646 011234 011234 003718 003718 002677 002677 004352 0043517infin 010001 010001 012528 012528 003438 003438 002426 002436 004139 0041391

02

0 037876 037870 043939 043930 014871 014871 010795 010795 017323 0173218025 036117 036113 041624 041617 014354 014354 010436 010436 016671 016671 033549 033547 037962 037958 013851 013851 010127 010127 015937 01593665 032783 032781 036695 036692 013888 013887 010200 010200 015878 0158779infin 032545 032544 037755 037753 012779 012778 009276 009276 014885 0148845


005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001541 001175 000360 000355 000429 000139 000057 0002641 001291 001027 000329 000333 000389 000129 000054 0002465 001111 000893 000290 000296 000342 000114 000048 00021810 001107 000868 000274 000274 000324 000107 000044 000203

01

0 006897 018772 001632 001608 001941 000630 000259 00120005 005652 015857 001406 001405 001663 000547 000227 0010431 004862 014291 001291 001320 001519 000509 000214 0009745 003953 011409 001113 001166 001303 000446 000190 00085610 003929 011157 001054 001082 001239 000417 000175 000798

02

0 023286 018772 006198 006260 007269 002439 001026 00465705 019360 015857 005338 005472 006242 002121 000899 0040551 017136 014291 004927 005147 005740 001983 000848 0037985 013242 011409 004126 004505 004776 001714 000749 00329510 013167 011157 003935 004191 004569 001606 000694 003081



005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001606 001246 000388 000386 000461 000150 000062 0002871 001588 001235 000386 000385 000458 000150 000062 0002865 001598 001249 000393 000392 000465 000153 000063 00029110 001481 001179 000378 000383 000447 000148 000062 000283

01

0 006897 005286 001632 001608 001941 000630 000259 00120005 006029 004770 001524 001532 001801 000595 000247 0011351 005888 004690 001511 001526 001782 000591 000246 0011295 005858 004702 001529 001553 001801 000601 000251 00114810 005748 004611 001498 001521 001765 000588 000246 001124

02

0 023288 018773 006198 006260 007269 002439 001026 00465705 021550 017558 005870 005988 006871 002324 000983 0044421 020752 017074 005782 005952 006754 002304 000979 0044075 020352 016912 005810 006038 006770 002332 000996 00446410 020672 017013 005770 005935 006737 002300 000978 004398



References






































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





L1199100119909 = int1198860

C119909L1199100119909 119889119909

L119910119887119909 = int1198860

C119909L119910119887119909 119889119909

L1199100119910 = int1198860

C119909L1199100119910 119889119909

L119910119887119910 = int1198860

C119909L119910119887119910 119889119909


[Θ119909Θ119910

] = [[[[[[[[L1199090119909 L1199090119910L119909119886119909 L119909119886119910L1199100119909 L1199100119910L119910119886119909 L119910119887119910

]]]]]]]]

minus1 [[[[[[[[L1199090119909119910L119909119886119909119910L1199100119909119910L119910119887119909119910

]]]]]]]]Θ119909119910 (38)


where


]]]]]]]]

minus1 [[[[[[[[L1199090119909119910L119909119886119909119910L1199100119909119910L119910119887119909119910

]]]]]]]]


]]]]]]]]

minus1 [[[[[[[[L1199090119909119910L119909119886119909119910L1199100119909119910L119910119887119909119910

]]]]]]]]

(40)









Clamped (C) 119908 = 120593119909 = 120593119910 = 0 1015 1015 1015


119910 = constantFree (F) 119876119909 = 119872119910 = 119872119909119910 = 0 0 0 0

Clamped (C) 119908 = 120593119910 = 120593119909 = 0 1015 1015 1015










Table4Com

paris

onof

them

axim


pics

quarep

latesh

avingvario

usbo

undary

cond

ition

swith

different

load

functio

ns

Boun

dary

cond

ition

sMetho

dUniform

pressure

PointL

oad(0505

)[(01)

(01)][(010

9)(0109)]

[(0208)(02

08)][(030

7)(0307)]

[(0406)(04

06)]CCCC

Present

5602E

-07

5438E

-07

4530E

-07

2790E

-07

8901E-08

2522E

-10

FEM

5602E

-07

5438E

-07

4530E

-07

2789E

-07

8896E

-08

2563E

-10

CFCF

Present

1278E-06

9716E-07

7275E-07

4116

E-07

1237E-07

340

1E-10

FEM

1278E-06

9716E-07

7273E-07

4115

E-07

1236E-07

3431E-10

CCCF

Present

1301E-09

8452E

-07

6152E

-07

3502E

-07

1066

E-07

2950E

-10

FEM

1302E-09

8452E

-07

6150E

-07

3501E-07

1065E-07

2981E-10

CSCS

Present

840

6E-07

7930E-07

6272E

-07

3677E

-07

1129E-07

3157E

-10

FEM

840

6E-07

7930E-07

6272E

-07

3677E

-07

1129E-07

3174E

-10

SSSS

Present

1737E-06

1584E-06

1175E-06

644

8E-07

1874E-07

506

4E-10

FEM

1737E-06

1585E-06

1740

E-06

644

7E-07

1873E-07

5076E

-10

CSFF

Present

3084E

-05

1858E-05

9955E-06

4267E

-06

1043E-06

260

4E-09

FEM

3084E

-05

1858E-05

9955E-06

4267E

-06

1043E-06

2618E

-09


CCCC[(0208) (0208)]

CFCF [(0406) (0406)]

CCCF [(01) (01)]


(a) (b)










4 Conclusions




00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus9

40 times 10minus9

60 times 10minus9

80 times 10minus9


(a)

00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus9

40 times 10minus9

60 times 10minus9

80 times 10minus9


(b)


00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus6

40 times 10minus6

60 times 10minus6

80 times 10minus6


(a)

00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

10 times 10minus6

20 times 10minus6

30 times 10minus6


(b)


CCCC CCCS CFCF CFFF





0 20 40 60 80 100

004

005

006

007

frequ

ency

par

amet

ers Ω


CCCC


CFCF

0008

0010

0012

0014

0016

frequ

ency

par

amet

ers Ω


CFFF

00016

00020

00024

frequ

ency

par

amet

ers Ω


CSCF

0012

0016

0020

frequ

ency

par

amet

ers Ω


Data Availability




Acknowledgments




2

0 008325 008325 008729 008729 006713 006713 006364 006365 006781 006781025 007600 007600 007950 007950 006145 006145 005829 005829 006205 0062051 006541 006541 006790 006790 005346 005346 005080 005080 005391 0053915 005524 005524 005695 005695 004568 004568 004349 004349 004600 004600infin 004263 004237 004443 004443 003417 003417 003239 003240 003451 003452

1

0 014378 014377 016713 016712 007537 007537 006290 006290 008062 008062025 012974 012973 014927 014926 006890 006890 005761 005761 007351 0073501 010725 010725 011955 011955 005968 005968 005021 005021 006308 0063085 008720 008720 009479 009479 005078 005078 004301 004300 005322 005322infin 007318 007318 008507 008507 003836 003836 003202 003202 004104 004104

05

0 035045 035037 041996 041980 010065 010065 006217 006216 013484 013483025 030709 030703 036112 036102 009170 009170 005695 005695 012160 0121591 023262 023260 026091 026088 007851 007851 004970 004970 010066 0100665 017691 017690 019258 019257 006610 006610 004262 004262 008226 008226infin 017921 017837 021375 021374 005123 005123 003164 003164 006863 006863

0 20 40 60 80 100

006

007

frequ

ency

par

amet

ers Ω


CCCC


CFCF

0014

0015

0016

frequ

ency

par

amet

ers Ω


CFFF

00024

00026

frequ

ency

par

amet

ers Ω


CSCF

0017

0016

0018

0019

frequ

ency

par

amet

ers Ω




005

0 003129 003130 004076 004076 001024 001024 000719 000719 001249 0012492025 002899 002899 003664 003664 000981 000981 000692 000692 001185 00118511 002667 002667 003250 003250 000948 000948 000674 000674 001132 00113165 002677 002677 003239 003239 000963 000963 000685 000685 001146 001146infin 002689 002689 003502 003502 000880 000880 000618 000618 001073 0010734

01

0 011639 011638 014580 014580 004001 004001 002835 002835 004817 004817025 010561 010561 012781 012780 003810 003810 002717 002717 004532 00453151 009734 009733 011453 011453 003679 003679 002641 002641 004327 00432675 009646 009646 011234 011234 003718 003718 002677 002677 004352 0043517infin 010001 010001 012528 012528 003438 003438 002426 002436 004139 0041391

02

0 037876 037870 043939 043930 014871 014871 010795 010795 017323 0173218025 036117 036113 041624 041617 014354 014354 010436 010436 016671 016671 033549 033547 037962 037958 013851 013851 010127 010127 015937 01593665 032783 032781 036695 036692 013888 013887 010200 010200 015878 0158779infin 032545 032544 037755 037753 012779 012778 009276 009276 014885 0148845


005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001541 001175 000360 000355 000429 000139 000057 0002641 001291 001027 000329 000333 000389 000129 000054 0002465 001111 000893 000290 000296 000342 000114 000048 00021810 001107 000868 000274 000274 000324 000107 000044 000203

01

0 006897 018772 001632 001608 001941 000630 000259 00120005 005652 015857 001406 001405 001663 000547 000227 0010431 004862 014291 001291 001320 001519 000509 000214 0009745 003953 011409 001113 001166 001303 000446 000190 00085610 003929 011157 001054 001082 001239 000417 000175 000798

02

0 023286 018772 006198 006260 007269 002439 001026 00465705 019360 015857 005338 005472 006242 002121 000899 0040551 017136 014291 004927 005147 005740 001983 000848 0037985 013242 011409 004126 004505 004776 001714 000749 00329510 013167 011157 003935 004191 004569 001606 000694 003081



005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001606 001246 000388 000386 000461 000150 000062 0002871 001588 001235 000386 000385 000458 000150 000062 0002865 001598 001249 000393 000392 000465 000153 000063 00029110 001481 001179 000378 000383 000447 000148 000062 000283

01

0 006897 005286 001632 001608 001941 000630 000259 00120005 006029 004770 001524 001532 001801 000595 000247 0011351 005888 004690 001511 001526 001782 000591 000246 0011295 005858 004702 001529 001553 001801 000601 000251 00114810 005748 004611 001498 001521 001765 000588 000246 001124

02

0 023288 018773 006198 006260 007269 002439 001026 00465705 021550 017558 005870 005988 006871 002324 000983 0044421 020752 017074 005782 005952 006754 002304 000979 0044075 020352 016912 005810 006038 006770 002332 000996 00446410 020672 017013 005770 005935 006737 002300 000978 004398



References






































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





Clamped (C) 119908 = 120593119909 = 120593119910 = 0 1015 1015 1015


119910 = constantFree (F) 119876119909 = 119872119910 = 119872119909119910 = 0 0 0 0

Clamped (C) 119908 = 120593119910 = 120593119909 = 0 1015 1015 1015










Table4Com

paris

onof

them

axim


pics

quarep

latesh

avingvario

usbo

undary

cond

ition

swith

different

load

functio

ns

Boun

dary

cond

ition

sMetho

dUniform

pressure

PointL

oad(0505

)[(01)

(01)][(010

9)(0109)]

[(0208)(02

08)][(030

7)(0307)]

[(0406)(04

06)]CCCC

Present

5602E

-07

5438E

-07

4530E

-07

2790E

-07

8901E-08

2522E

-10

FEM

5602E

-07

5438E

-07

4530E

-07

2789E

-07

8896E

-08

2563E

-10

CFCF

Present

1278E-06

9716E-07

7275E-07

4116

E-07

1237E-07

340

1E-10

FEM

1278E-06

9716E-07

7273E-07

4115

E-07

1236E-07

3431E-10

CCCF

Present

1301E-09

8452E

-07

6152E

-07

3502E

-07

1066

E-07

2950E

-10

FEM

1302E-09

8452E

-07

6150E

-07

3501E-07

1065E-07

2981E-10

CSCS

Present

840

6E-07

7930E-07

6272E

-07

3677E

-07

1129E-07

3157E

-10

FEM

840

6E-07

7930E-07

6272E

-07

3677E

-07

1129E-07

3174E

-10

SSSS

Present

1737E-06

1584E-06

1175E-06

644

8E-07

1874E-07

506

4E-10

FEM

1737E-06

1585E-06

1740

E-06

644

7E-07

1873E-07

5076E

-10

CSFF

Present

3084E

-05

1858E-05

9955E-06

4267E

-06

1043E-06

260

4E-09

FEM

3084E

-05

1858E-05

9955E-06

4267E

-06

1043E-06

2618E

-09


CCCC[(0208) (0208)]

CFCF [(0406) (0406)]

CCCF [(01) (01)]


(a) (b)










4 Conclusions




00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus9

40 times 10minus9

60 times 10minus9

80 times 10minus9


(a)

00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus9

40 times 10minus9

60 times 10minus9

80 times 10minus9


(b)


00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus6

40 times 10minus6

60 times 10minus6

80 times 10minus6


(a)

00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

10 times 10minus6

20 times 10minus6

30 times 10minus6


(b)


CCCC CCCS CFCF CFFF





0 20 40 60 80 100

004

005

006

007

frequ

ency

par

amet

ers Ω


CCCC


CFCF

0008

0010

0012

0014

0016

frequ

ency

par

amet

ers Ω


CFFF

00016

00020

00024

frequ

ency

par

amet

ers Ω


CSCF

0012

0016

0020

frequ

ency

par

amet

ers Ω


Data Availability




Acknowledgments




2

0 008325 008325 008729 008729 006713 006713 006364 006365 006781 006781025 007600 007600 007950 007950 006145 006145 005829 005829 006205 0062051 006541 006541 006790 006790 005346 005346 005080 005080 005391 0053915 005524 005524 005695 005695 004568 004568 004349 004349 004600 004600infin 004263 004237 004443 004443 003417 003417 003239 003240 003451 003452

1

0 014378 014377 016713 016712 007537 007537 006290 006290 008062 008062025 012974 012973 014927 014926 006890 006890 005761 005761 007351 0073501 010725 010725 011955 011955 005968 005968 005021 005021 006308 0063085 008720 008720 009479 009479 005078 005078 004301 004300 005322 005322infin 007318 007318 008507 008507 003836 003836 003202 003202 004104 004104

05

0 035045 035037 041996 041980 010065 010065 006217 006216 013484 013483025 030709 030703 036112 036102 009170 009170 005695 005695 012160 0121591 023262 023260 026091 026088 007851 007851 004970 004970 010066 0100665 017691 017690 019258 019257 006610 006610 004262 004262 008226 008226infin 017921 017837 021375 021374 005123 005123 003164 003164 006863 006863

0 20 40 60 80 100

006

007

frequ

ency

par

amet

ers Ω


CCCC


CFCF

0014

0015

0016

frequ

ency

par

amet

ers Ω


CFFF

00024

00026

frequ

ency

par

amet

ers Ω


CSCF

0017

0016

0018

0019

frequ

ency

par

amet

ers Ω




005

0 003129 003130 004076 004076 001024 001024 000719 000719 001249 0012492025 002899 002899 003664 003664 000981 000981 000692 000692 001185 00118511 002667 002667 003250 003250 000948 000948 000674 000674 001132 00113165 002677 002677 003239 003239 000963 000963 000685 000685 001146 001146infin 002689 002689 003502 003502 000880 000880 000618 000618 001073 0010734

01

0 011639 011638 014580 014580 004001 004001 002835 002835 004817 004817025 010561 010561 012781 012780 003810 003810 002717 002717 004532 00453151 009734 009733 011453 011453 003679 003679 002641 002641 004327 00432675 009646 009646 011234 011234 003718 003718 002677 002677 004352 0043517infin 010001 010001 012528 012528 003438 003438 002426 002436 004139 0041391

02

0 037876 037870 043939 043930 014871 014871 010795 010795 017323 0173218025 036117 036113 041624 041617 014354 014354 010436 010436 016671 016671 033549 033547 037962 037958 013851 013851 010127 010127 015937 01593665 032783 032781 036695 036692 013888 013887 010200 010200 015878 0158779infin 032545 032544 037755 037753 012779 012778 009276 009276 014885 0148845


005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001541 001175 000360 000355 000429 000139 000057 0002641 001291 001027 000329 000333 000389 000129 000054 0002465 001111 000893 000290 000296 000342 000114 000048 00021810 001107 000868 000274 000274 000324 000107 000044 000203

01

0 006897 018772 001632 001608 001941 000630 000259 00120005 005652 015857 001406 001405 001663 000547 000227 0010431 004862 014291 001291 001320 001519 000509 000214 0009745 003953 011409 001113 001166 001303 000446 000190 00085610 003929 011157 001054 001082 001239 000417 000175 000798

02

0 023286 018772 006198 006260 007269 002439 001026 00465705 019360 015857 005338 005472 006242 002121 000899 0040551 017136 014291 004927 005147 005740 001983 000848 0037985 013242 011409 004126 004505 004776 001714 000749 00329510 013167 011157 003935 004191 004569 001606 000694 003081



005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001606 001246 000388 000386 000461 000150 000062 0002871 001588 001235 000386 000385 000458 000150 000062 0002865 001598 001249 000393 000392 000465 000153 000063 00029110 001481 001179 000378 000383 000447 000148 000062 000283

01

0 006897 005286 001632 001608 001941 000630 000259 00120005 006029 004770 001524 001532 001801 000595 000247 0011351 005888 004690 001511 001526 001782 000591 000246 0011295 005858 004702 001529 001553 001801 000601 000251 00114810 005748 004611 001498 001521 001765 000588 000246 001124

02

0 023288 018773 006198 006260 007269 002439 001026 00465705 021550 017558 005870 005988 006871 002324 000983 0044421 020752 017074 005782 005952 006754 002304 000979 0044075 020352 016912 005810 006038 006770 002332 000996 00446410 020672 017013 005770 005935 006737 002300 000978 004398



References






































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Table4Com

paris

onof

them

axim


pics

quarep

latesh

avingvario

usbo

undary

cond

ition

swith

different

load

functio

ns

Boun

dary

cond

ition

sMetho

dUniform

pressure

PointL

oad(0505

)[(01)

(01)][(010

9)(0109)]

[(0208)(02

08)][(030

7)(0307)]

[(0406)(04

06)]CCCC

Present

5602E

-07

5438E

-07

4530E

-07

2790E

-07

8901E-08

2522E

-10

FEM

5602E

-07

5438E

-07

4530E

-07

2789E

-07

8896E

-08

2563E

-10

CFCF

Present

1278E-06

9716E-07

7275E-07

4116

E-07

1237E-07

340

1E-10

FEM

1278E-06

9716E-07

7273E-07

4115

E-07

1236E-07

3431E-10

CCCF

Present

1301E-09

8452E

-07

6152E

-07

3502E

-07

1066

E-07

2950E

-10

FEM

1302E-09

8452E

-07

6150E

-07

3501E-07

1065E-07

2981E-10

CSCS

Present

840

6E-07

7930E-07

6272E

-07

3677E

-07

1129E-07

3157E

-10

FEM

840

6E-07

7930E-07

6272E

-07

3677E

-07

1129E-07

3174E

-10

SSSS

Present

1737E-06

1584E-06

1175E-06

644

8E-07

1874E-07

506

4E-10

FEM

1737E-06

1585E-06

1740

E-06

644

7E-07

1873E-07

5076E

-10

CSFF

Present

3084E

-05

1858E-05

9955E-06

4267E

-06

1043E-06

260

4E-09

FEM

3084E

-05

1858E-05

9955E-06

4267E

-06

1043E-06

2618E

-09


CCCC[(0208) (0208)]

CFCF [(0406) (0406)]

CCCF [(01) (01)]


(a) (b)










4 Conclusions




00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus9

40 times 10minus9

60 times 10minus9

80 times 10minus9


(a)

00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus9

40 times 10minus9

60 times 10minus9

80 times 10minus9


(b)


00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus6

40 times 10minus6

60 times 10minus6

80 times 10minus6


(a)

00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

10 times 10minus6

20 times 10minus6

30 times 10minus6


(b)


CCCC CCCS CFCF CFFF





0 20 40 60 80 100

004

005

006

007

frequ

ency

par

amet

ers Ω


CCCC


CFCF

0008

0010

0012

0014

0016

frequ

ency

par

amet

ers Ω


CFFF

00016

00020

00024

frequ

ency

par

amet

ers Ω


CSCF

0012

0016

0020

frequ

ency

par

amet

ers Ω


Data Availability




Acknowledgments




2

0 008325 008325 008729 008729 006713 006713 006364 006365 006781 006781025 007600 007600 007950 007950 006145 006145 005829 005829 006205 0062051 006541 006541 006790 006790 005346 005346 005080 005080 005391 0053915 005524 005524 005695 005695 004568 004568 004349 004349 004600 004600infin 004263 004237 004443 004443 003417 003417 003239 003240 003451 003452

1

0 014378 014377 016713 016712 007537 007537 006290 006290 008062 008062025 012974 012973 014927 014926 006890 006890 005761 005761 007351 0073501 010725 010725 011955 011955 005968 005968 005021 005021 006308 0063085 008720 008720 009479 009479 005078 005078 004301 004300 005322 005322infin 007318 007318 008507 008507 003836 003836 003202 003202 004104 004104

05

0 035045 035037 041996 041980 010065 010065 006217 006216 013484 013483025 030709 030703 036112 036102 009170 009170 005695 005695 012160 0121591 023262 023260 026091 026088 007851 007851 004970 004970 010066 0100665 017691 017690 019258 019257 006610 006610 004262 004262 008226 008226infin 017921 017837 021375 021374 005123 005123 003164 003164 006863 006863

0 20 40 60 80 100

006

007

frequ

ency

par

amet

ers Ω


CCCC


CFCF

0014

0015

0016

frequ

ency

par

amet

ers Ω


CFFF

00024

00026

frequ

ency

par

amet

ers Ω


CSCF

0017

0016

0018

0019

frequ

ency

par

amet

ers Ω




005

0 003129 003130 004076 004076 001024 001024 000719 000719 001249 0012492025 002899 002899 003664 003664 000981 000981 000692 000692 001185 00118511 002667 002667 003250 003250 000948 000948 000674 000674 001132 00113165 002677 002677 003239 003239 000963 000963 000685 000685 001146 001146infin 002689 002689 003502 003502 000880 000880 000618 000618 001073 0010734

01

0 011639 011638 014580 014580 004001 004001 002835 002835 004817 004817025 010561 010561 012781 012780 003810 003810 002717 002717 004532 00453151 009734 009733 011453 011453 003679 003679 002641 002641 004327 00432675 009646 009646 011234 011234 003718 003718 002677 002677 004352 0043517infin 010001 010001 012528 012528 003438 003438 002426 002436 004139 0041391

02

0 037876 037870 043939 043930 014871 014871 010795 010795 017323 0173218025 036117 036113 041624 041617 014354 014354 010436 010436 016671 016671 033549 033547 037962 037958 013851 013851 010127 010127 015937 01593665 032783 032781 036695 036692 013888 013887 010200 010200 015878 0158779infin 032545 032544 037755 037753 012779 012778 009276 009276 014885 0148845


005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001541 001175 000360 000355 000429 000139 000057 0002641 001291 001027 000329 000333 000389 000129 000054 0002465 001111 000893 000290 000296 000342 000114 000048 00021810 001107 000868 000274 000274 000324 000107 000044 000203

01

0 006897 018772 001632 001608 001941 000630 000259 00120005 005652 015857 001406 001405 001663 000547 000227 0010431 004862 014291 001291 001320 001519 000509 000214 0009745 003953 011409 001113 001166 001303 000446 000190 00085610 003929 011157 001054 001082 001239 000417 000175 000798

02

0 023286 018772 006198 006260 007269 002439 001026 00465705 019360 015857 005338 005472 006242 002121 000899 0040551 017136 014291 004927 005147 005740 001983 000848 0037985 013242 011409 004126 004505 004776 001714 000749 00329510 013167 011157 003935 004191 004569 001606 000694 003081



005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001606 001246 000388 000386 000461 000150 000062 0002871 001588 001235 000386 000385 000458 000150 000062 0002865 001598 001249 000393 000392 000465 000153 000063 00029110 001481 001179 000378 000383 000447 000148 000062 000283

01

0 006897 005286 001632 001608 001941 000630 000259 00120005 006029 004770 001524 001532 001801 000595 000247 0011351 005888 004690 001511 001526 001782 000591 000246 0011295 005858 004702 001529 001553 001801 000601 000251 00114810 005748 004611 001498 001521 001765 000588 000246 001124

02

0 023288 018773 006198 006260 007269 002439 001026 00465705 021550 017558 005870 005988 006871 002324 000983 0044421 020752 017074 005782 005952 006754 002304 000979 0044075 020352 016912 005810 006038 006770 002332 000996 00446410 020672 017013 005770 005935 006737 002300 000978 004398



References






































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



CCCC[(0208) (0208)]

CFCF [(0406) (0406)]

CCCF [(01) (01)]


(a) (b)










4 Conclusions




00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus9

40 times 10minus9

60 times 10minus9

80 times 10minus9


(a)

00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus9

40 times 10minus9

60 times 10minus9

80 times 10minus9


(b)


00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus6

40 times 10minus6

60 times 10minus6

80 times 10minus6


(a)

00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

10 times 10minus6

20 times 10minus6

30 times 10minus6


(b)


CCCC CCCS CFCF CFFF





0 20 40 60 80 100

004

005

006

007

frequ

ency

par

amet

ers Ω


CCCC


CFCF

0008

0010

0012

0014

0016

frequ

ency

par

amet

ers Ω


CFFF

00016

00020

00024

frequ

ency

par

amet

ers Ω


CSCF

0012

0016

0020

frequ

ency

par

amet

ers Ω


Data Availability




Acknowledgments




2

0 008325 008325 008729 008729 006713 006713 006364 006365 006781 006781025 007600 007600 007950 007950 006145 006145 005829 005829 006205 0062051 006541 006541 006790 006790 005346 005346 005080 005080 005391 0053915 005524 005524 005695 005695 004568 004568 004349 004349 004600 004600infin 004263 004237 004443 004443 003417 003417 003239 003240 003451 003452

1

0 014378 014377 016713 016712 007537 007537 006290 006290 008062 008062025 012974 012973 014927 014926 006890 006890 005761 005761 007351 0073501 010725 010725 011955 011955 005968 005968 005021 005021 006308 0063085 008720 008720 009479 009479 005078 005078 004301 004300 005322 005322infin 007318 007318 008507 008507 003836 003836 003202 003202 004104 004104

05

0 035045 035037 041996 041980 010065 010065 006217 006216 013484 013483025 030709 030703 036112 036102 009170 009170 005695 005695 012160 0121591 023262 023260 026091 026088 007851 007851 004970 004970 010066 0100665 017691 017690 019258 019257 006610 006610 004262 004262 008226 008226infin 017921 017837 021375 021374 005123 005123 003164 003164 006863 006863

0 20 40 60 80 100

006

007

frequ

ency

par

amet

ers Ω


CCCC


CFCF

0014

0015

0016

frequ

ency

par

amet

ers Ω


CFFF

00024

00026

frequ

ency

par

amet

ers Ω


CSCF

0017

0016

0018

0019

frequ

ency

par

amet

ers Ω




005

0 003129 003130 004076 004076 001024 001024 000719 000719 001249 0012492025 002899 002899 003664 003664 000981 000981 000692 000692 001185 00118511 002667 002667 003250 003250 000948 000948 000674 000674 001132 00113165 002677 002677 003239 003239 000963 000963 000685 000685 001146 001146infin 002689 002689 003502 003502 000880 000880 000618 000618 001073 0010734

01

0 011639 011638 014580 014580 004001 004001 002835 002835 004817 004817025 010561 010561 012781 012780 003810 003810 002717 002717 004532 00453151 009734 009733 011453 011453 003679 003679 002641 002641 004327 00432675 009646 009646 011234 011234 003718 003718 002677 002677 004352 0043517infin 010001 010001 012528 012528 003438 003438 002426 002436 004139 0041391

02

0 037876 037870 043939 043930 014871 014871 010795 010795 017323 0173218025 036117 036113 041624 041617 014354 014354 010436 010436 016671 016671 033549 033547 037962 037958 013851 013851 010127 010127 015937 01593665 032783 032781 036695 036692 013888 013887 010200 010200 015878 0158779infin 032545 032544 037755 037753 012779 012778 009276 009276 014885 0148845


005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001541 001175 000360 000355 000429 000139 000057 0002641 001291 001027 000329 000333 000389 000129 000054 0002465 001111 000893 000290 000296 000342 000114 000048 00021810 001107 000868 000274 000274 000324 000107 000044 000203

01

0 006897 018772 001632 001608 001941 000630 000259 00120005 005652 015857 001406 001405 001663 000547 000227 0010431 004862 014291 001291 001320 001519 000509 000214 0009745 003953 011409 001113 001166 001303 000446 000190 00085610 003929 011157 001054 001082 001239 000417 000175 000798

02

0 023286 018772 006198 006260 007269 002439 001026 00465705 019360 015857 005338 005472 006242 002121 000899 0040551 017136 014291 004927 005147 005740 001983 000848 0037985 013242 011409 004126 004505 004776 001714 000749 00329510 013167 011157 003935 004191 004569 001606 000694 003081



005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001606 001246 000388 000386 000461 000150 000062 0002871 001588 001235 000386 000385 000458 000150 000062 0002865 001598 001249 000393 000392 000465 000153 000063 00029110 001481 001179 000378 000383 000447 000148 000062 000283

01

0 006897 005286 001632 001608 001941 000630 000259 00120005 006029 004770 001524 001532 001801 000595 000247 0011351 005888 004690 001511 001526 001782 000591 000246 0011295 005858 004702 001529 001553 001801 000601 000251 00114810 005748 004611 001498 001521 001765 000588 000246 001124

02

0 023288 018773 006198 006260 007269 002439 001026 00465705 021550 017558 005870 005988 006871 002324 000983 0044421 020752 017074 005782 005952 006754 002304 000979 0044075 020352 016912 005810 006038 006770 002332 000996 00446410 020672 017013 005770 005935 006737 002300 000978 004398



References






































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia










4 Conclusions




00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus9

40 times 10minus9

60 times 10minus9

80 times 10minus9


(a)

00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus9

40 times 10minus9

60 times 10minus9

80 times 10minus9


(b)


00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus6

40 times 10minus6

60 times 10minus6

80 times 10minus6


(a)

00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

10 times 10minus6

20 times 10minus6

30 times 10minus6


(b)


CCCC CCCS CFCF CFFF





0 20 40 60 80 100

004

005

006

007

frequ

ency

par

amet

ers Ω


CCCC


CFCF

0008

0010

0012

0014

0016

frequ

ency

par

amet

ers Ω


CFFF

00016

00020

00024

frequ

ency

par

amet

ers Ω


CSCF

0012

0016

0020

frequ

ency

par

amet

ers Ω


Data Availability




Acknowledgments




2

0 008325 008325 008729 008729 006713 006713 006364 006365 006781 006781025 007600 007600 007950 007950 006145 006145 005829 005829 006205 0062051 006541 006541 006790 006790 005346 005346 005080 005080 005391 0053915 005524 005524 005695 005695 004568 004568 004349 004349 004600 004600infin 004263 004237 004443 004443 003417 003417 003239 003240 003451 003452

1

0 014378 014377 016713 016712 007537 007537 006290 006290 008062 008062025 012974 012973 014927 014926 006890 006890 005761 005761 007351 0073501 010725 010725 011955 011955 005968 005968 005021 005021 006308 0063085 008720 008720 009479 009479 005078 005078 004301 004300 005322 005322infin 007318 007318 008507 008507 003836 003836 003202 003202 004104 004104

05

0 035045 035037 041996 041980 010065 010065 006217 006216 013484 013483025 030709 030703 036112 036102 009170 009170 005695 005695 012160 0121591 023262 023260 026091 026088 007851 007851 004970 004970 010066 0100665 017691 017690 019258 019257 006610 006610 004262 004262 008226 008226infin 017921 017837 021375 021374 005123 005123 003164 003164 006863 006863

0 20 40 60 80 100

006

007

frequ

ency

par

amet

ers Ω


CCCC


CFCF

0014

0015

0016

frequ

ency

par

amet

ers Ω


CFFF

00024

00026

frequ

ency

par

amet

ers Ω


CSCF

0017

0016

0018

0019

frequ

ency

par

amet

ers Ω




005

0 003129 003130 004076 004076 001024 001024 000719 000719 001249 0012492025 002899 002899 003664 003664 000981 000981 000692 000692 001185 00118511 002667 002667 003250 003250 000948 000948 000674 000674 001132 00113165 002677 002677 003239 003239 000963 000963 000685 000685 001146 001146infin 002689 002689 003502 003502 000880 000880 000618 000618 001073 0010734

01

0 011639 011638 014580 014580 004001 004001 002835 002835 004817 004817025 010561 010561 012781 012780 003810 003810 002717 002717 004532 00453151 009734 009733 011453 011453 003679 003679 002641 002641 004327 00432675 009646 009646 011234 011234 003718 003718 002677 002677 004352 0043517infin 010001 010001 012528 012528 003438 003438 002426 002436 004139 0041391

02

0 037876 037870 043939 043930 014871 014871 010795 010795 017323 0173218025 036117 036113 041624 041617 014354 014354 010436 010436 016671 016671 033549 033547 037962 037958 013851 013851 010127 010127 015937 01593665 032783 032781 036695 036692 013888 013887 010200 010200 015878 0158779infin 032545 032544 037755 037753 012779 012778 009276 009276 014885 0148845


005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001541 001175 000360 000355 000429 000139 000057 0002641 001291 001027 000329 000333 000389 000129 000054 0002465 001111 000893 000290 000296 000342 000114 000048 00021810 001107 000868 000274 000274 000324 000107 000044 000203

01

0 006897 018772 001632 001608 001941 000630 000259 00120005 005652 015857 001406 001405 001663 000547 000227 0010431 004862 014291 001291 001320 001519 000509 000214 0009745 003953 011409 001113 001166 001303 000446 000190 00085610 003929 011157 001054 001082 001239 000417 000175 000798

02

0 023286 018772 006198 006260 007269 002439 001026 00465705 019360 015857 005338 005472 006242 002121 000899 0040551 017136 014291 004927 005147 005740 001983 000848 0037985 013242 011409 004126 004505 004776 001714 000749 00329510 013167 011157 003935 004191 004569 001606 000694 003081



005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001606 001246 000388 000386 000461 000150 000062 0002871 001588 001235 000386 000385 000458 000150 000062 0002865 001598 001249 000393 000392 000465 000153 000063 00029110 001481 001179 000378 000383 000447 000148 000062 000283

01

0 006897 005286 001632 001608 001941 000630 000259 00120005 006029 004770 001524 001532 001801 000595 000247 0011351 005888 004690 001511 001526 001782 000591 000246 0011295 005858 004702 001529 001553 001801 000601 000251 00114810 005748 004611 001498 001521 001765 000588 000246 001124

02

0 023288 018773 006198 006260 007269 002439 001026 00465705 021550 017558 005870 005988 006871 002324 000983 0044421 020752 017074 005782 005952 006754 002304 000979 0044075 020352 016912 005810 006038 006770 002332 000996 00446410 020672 017013 005770 005935 006737 002300 000978 004398



References






































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus9

40 times 10minus9

60 times 10minus9

80 times 10minus9


(a)

00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus9

40 times 10minus9

60 times 10minus9

80 times 10minus9


(b)


00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

20 times 10minus6

40 times 10minus6

60 times 10minus6

80 times 10minus6


(a)

00

Max

imum

defl

ectio

n W


CCCCSSSS

CFCFCSCS

10 times 10minus6

20 times 10minus6

30 times 10minus6


(b)


CCCC CCCS CFCF CFFF





0 20 40 60 80 100

004

005

006

007

frequ

ency

par

amet

ers Ω


CCCC


CFCF

0008

0010

0012

0014

0016

frequ

ency

par

amet

ers Ω


CFFF

00016

00020

00024

frequ

ency

par

amet

ers Ω


CSCF

0012

0016

0020

frequ

ency

par

amet

ers Ω


Data Availability




Acknowledgments




2

0 008325 008325 008729 008729 006713 006713 006364 006365 006781 006781025 007600 007600 007950 007950 006145 006145 005829 005829 006205 0062051 006541 006541 006790 006790 005346 005346 005080 005080 005391 0053915 005524 005524 005695 005695 004568 004568 004349 004349 004600 004600infin 004263 004237 004443 004443 003417 003417 003239 003240 003451 003452

1

0 014378 014377 016713 016712 007537 007537 006290 006290 008062 008062025 012974 012973 014927 014926 006890 006890 005761 005761 007351 0073501 010725 010725 011955 011955 005968 005968 005021 005021 006308 0063085 008720 008720 009479 009479 005078 005078 004301 004300 005322 005322infin 007318 007318 008507 008507 003836 003836 003202 003202 004104 004104

05

0 035045 035037 041996 041980 010065 010065 006217 006216 013484 013483025 030709 030703 036112 036102 009170 009170 005695 005695 012160 0121591 023262 023260 026091 026088 007851 007851 004970 004970 010066 0100665 017691 017690 019258 019257 006610 006610 004262 004262 008226 008226infin 017921 017837 021375 021374 005123 005123 003164 003164 006863 006863

0 20 40 60 80 100

006

007

frequ

ency

par

amet

ers Ω


CCCC


CFCF

0014

0015

0016

frequ

ency

par

amet

ers Ω


CFFF

00024

00026

frequ

ency

par

amet

ers Ω


CSCF

0017

0016

0018

0019

frequ

ency

par

amet

ers Ω




005

0 003129 003130 004076 004076 001024 001024 000719 000719 001249 0012492025 002899 002899 003664 003664 000981 000981 000692 000692 001185 00118511 002667 002667 003250 003250 000948 000948 000674 000674 001132 00113165 002677 002677 003239 003239 000963 000963 000685 000685 001146 001146infin 002689 002689 003502 003502 000880 000880 000618 000618 001073 0010734

01

0 011639 011638 014580 014580 004001 004001 002835 002835 004817 004817025 010561 010561 012781 012780 003810 003810 002717 002717 004532 00453151 009734 009733 011453 011453 003679 003679 002641 002641 004327 00432675 009646 009646 011234 011234 003718 003718 002677 002677 004352 0043517infin 010001 010001 012528 012528 003438 003438 002426 002436 004139 0041391

02

0 037876 037870 043939 043930 014871 014871 010795 010795 017323 0173218025 036117 036113 041624 041617 014354 014354 010436 010436 016671 016671 033549 033547 037962 037958 013851 013851 010127 010127 015937 01593665 032783 032781 036695 036692 013888 013887 010200 010200 015878 0158779infin 032545 032544 037755 037753 012779 012778 009276 009276 014885 0148845


005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001541 001175 000360 000355 000429 000139 000057 0002641 001291 001027 000329 000333 000389 000129 000054 0002465 001111 000893 000290 000296 000342 000114 000048 00021810 001107 000868 000274 000274 000324 000107 000044 000203

01

0 006897 018772 001632 001608 001941 000630 000259 00120005 005652 015857 001406 001405 001663 000547 000227 0010431 004862 014291 001291 001320 001519 000509 000214 0009745 003953 011409 001113 001166 001303 000446 000190 00085610 003929 011157 001054 001082 001239 000417 000175 000798

02

0 023286 018772 006198 006260 007269 002439 001026 00465705 019360 015857 005338 005472 006242 002121 000899 0040551 017136 014291 004927 005147 005740 001983 000848 0037985 013242 011409 004126 004505 004776 001714 000749 00329510 013167 011157 003935 004191 004569 001606 000694 003081



005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001606 001246 000388 000386 000461 000150 000062 0002871 001588 001235 000386 000385 000458 000150 000062 0002865 001598 001249 000393 000392 000465 000153 000063 00029110 001481 001179 000378 000383 000447 000148 000062 000283

01

0 006897 005286 001632 001608 001941 000630 000259 00120005 006029 004770 001524 001532 001801 000595 000247 0011351 005888 004690 001511 001526 001782 000591 000246 0011295 005858 004702 001529 001553 001801 000601 000251 00114810 005748 004611 001498 001521 001765 000588 000246 001124

02

0 023288 018773 006198 006260 007269 002439 001026 00465705 021550 017558 005870 005988 006871 002324 000983 0044421 020752 017074 005782 005952 006754 002304 000979 0044075 020352 016912 005810 006038 006770 002332 000996 00446410 020672 017013 005770 005935 006737 002300 000978 004398



References






































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





0 20 40 60 80 100

004

005

006

007

frequ

ency

par

amet

ers Ω


CCCC


CFCF

0008

0010

0012

0014

0016

frequ

ency

par

amet

ers Ω


CFFF

00016

00020

00024

frequ

ency

par

amet

ers Ω


CSCF

0012

0016

0020

frequ

ency

par

amet

ers Ω


Data Availability




Acknowledgments




2

0 008325 008325 008729 008729 006713 006713 006364 006365 006781 006781025 007600 007600 007950 007950 006145 006145 005829 005829 006205 0062051 006541 006541 006790 006790 005346 005346 005080 005080 005391 0053915 005524 005524 005695 005695 004568 004568 004349 004349 004600 004600infin 004263 004237 004443 004443 003417 003417 003239 003240 003451 003452

1

0 014378 014377 016713 016712 007537 007537 006290 006290 008062 008062025 012974 012973 014927 014926 006890 006890 005761 005761 007351 0073501 010725 010725 011955 011955 005968 005968 005021 005021 006308 0063085 008720 008720 009479 009479 005078 005078 004301 004300 005322 005322infin 007318 007318 008507 008507 003836 003836 003202 003202 004104 004104

05

0 035045 035037 041996 041980 010065 010065 006217 006216 013484 013483025 030709 030703 036112 036102 009170 009170 005695 005695 012160 0121591 023262 023260 026091 026088 007851 007851 004970 004970 010066 0100665 017691 017690 019258 019257 006610 006610 004262 004262 008226 008226infin 017921 017837 021375 021374 005123 005123 003164 003164 006863 006863

0 20 40 60 80 100

006

007

frequ

ency

par

amet

ers Ω


CCCC


CFCF

0014

0015

0016

frequ

ency

par

amet

ers Ω


CFFF

00024

00026

frequ

ency

par

amet

ers Ω


CSCF

0017

0016

0018

0019

frequ

ency

par

amet

ers Ω




005

0 003129 003130 004076 004076 001024 001024 000719 000719 001249 0012492025 002899 002899 003664 003664 000981 000981 000692 000692 001185 00118511 002667 002667 003250 003250 000948 000948 000674 000674 001132 00113165 002677 002677 003239 003239 000963 000963 000685 000685 001146 001146infin 002689 002689 003502 003502 000880 000880 000618 000618 001073 0010734

01

0 011639 011638 014580 014580 004001 004001 002835 002835 004817 004817025 010561 010561 012781 012780 003810 003810 002717 002717 004532 00453151 009734 009733 011453 011453 003679 003679 002641 002641 004327 00432675 009646 009646 011234 011234 003718 003718 002677 002677 004352 0043517infin 010001 010001 012528 012528 003438 003438 002426 002436 004139 0041391

02

0 037876 037870 043939 043930 014871 014871 010795 010795 017323 0173218025 036117 036113 041624 041617 014354 014354 010436 010436 016671 016671 033549 033547 037962 037958 013851 013851 010127 010127 015937 01593665 032783 032781 036695 036692 013888 013887 010200 010200 015878 0158779infin 032545 032544 037755 037753 012779 012778 009276 009276 014885 0148845


005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001541 001175 000360 000355 000429 000139 000057 0002641 001291 001027 000329 000333 000389 000129 000054 0002465 001111 000893 000290 000296 000342 000114 000048 00021810 001107 000868 000274 000274 000324 000107 000044 000203

01

0 006897 018772 001632 001608 001941 000630 000259 00120005 005652 015857 001406 001405 001663 000547 000227 0010431 004862 014291 001291 001320 001519 000509 000214 0009745 003953 011409 001113 001166 001303 000446 000190 00085610 003929 011157 001054 001082 001239 000417 000175 000798

02

0 023286 018772 006198 006260 007269 002439 001026 00465705 019360 015857 005338 005472 006242 002121 000899 0040551 017136 014291 004927 005147 005740 001983 000848 0037985 013242 011409 004126 004505 004776 001714 000749 00329510 013167 011157 003935 004191 004569 001606 000694 003081



005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001606 001246 000388 000386 000461 000150 000062 0002871 001588 001235 000386 000385 000458 000150 000062 0002865 001598 001249 000393 000392 000465 000153 000063 00029110 001481 001179 000378 000383 000447 000148 000062 000283

01

0 006897 005286 001632 001608 001941 000630 000259 00120005 006029 004770 001524 001532 001801 000595 000247 0011351 005888 004690 001511 001526 001782 000591 000246 0011295 005858 004702 001529 001553 001801 000601 000251 00114810 005748 004611 001498 001521 001765 000588 000246 001124

02

0 023288 018773 006198 006260 007269 002439 001026 00465705 021550 017558 005870 005988 006871 002324 000983 0044421 020752 017074 005782 005952 006754 002304 000979 0044075 020352 016912 005810 006038 006770 002332 000996 00446410 020672 017013 005770 005935 006737 002300 000978 004398



References






































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




2

0 008325 008325 008729 008729 006713 006713 006364 006365 006781 006781025 007600 007600 007950 007950 006145 006145 005829 005829 006205 0062051 006541 006541 006790 006790 005346 005346 005080 005080 005391 0053915 005524 005524 005695 005695 004568 004568 004349 004349 004600 004600infin 004263 004237 004443 004443 003417 003417 003239 003240 003451 003452

1

0 014378 014377 016713 016712 007537 007537 006290 006290 008062 008062025 012974 012973 014927 014926 006890 006890 005761 005761 007351 0073501 010725 010725 011955 011955 005968 005968 005021 005021 006308 0063085 008720 008720 009479 009479 005078 005078 004301 004300 005322 005322infin 007318 007318 008507 008507 003836 003836 003202 003202 004104 004104

05

0 035045 035037 041996 041980 010065 010065 006217 006216 013484 013483025 030709 030703 036112 036102 009170 009170 005695 005695 012160 0121591 023262 023260 026091 026088 007851 007851 004970 004970 010066 0100665 017691 017690 019258 019257 006610 006610 004262 004262 008226 008226infin 017921 017837 021375 021374 005123 005123 003164 003164 006863 006863

0 20 40 60 80 100

006

007

frequ

ency

par

amet

ers Ω


CCCC


CFCF

0014

0015

0016

frequ

ency

par

amet

ers Ω


CFFF

00024

00026

frequ

ency

par

amet

ers Ω


CSCF

0017

0016

0018

0019

frequ

ency

par

amet

ers Ω




005

0 003129 003130 004076 004076 001024 001024 000719 000719 001249 0012492025 002899 002899 003664 003664 000981 000981 000692 000692 001185 00118511 002667 002667 003250 003250 000948 000948 000674 000674 001132 00113165 002677 002677 003239 003239 000963 000963 000685 000685 001146 001146infin 002689 002689 003502 003502 000880 000880 000618 000618 001073 0010734

01

0 011639 011638 014580 014580 004001 004001 002835 002835 004817 004817025 010561 010561 012781 012780 003810 003810 002717 002717 004532 00453151 009734 009733 011453 011453 003679 003679 002641 002641 004327 00432675 009646 009646 011234 011234 003718 003718 002677 002677 004352 0043517infin 010001 010001 012528 012528 003438 003438 002426 002436 004139 0041391

02

0 037876 037870 043939 043930 014871 014871 010795 010795 017323 0173218025 036117 036113 041624 041617 014354 014354 010436 010436 016671 016671 033549 033547 037962 037958 013851 013851 010127 010127 015937 01593665 032783 032781 036695 036692 013888 013887 010200 010200 015878 0158779infin 032545 032544 037755 037753 012779 012778 009276 009276 014885 0148845


005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001541 001175 000360 000355 000429 000139 000057 0002641 001291 001027 000329 000333 000389 000129 000054 0002465 001111 000893 000290 000296 000342 000114 000048 00021810 001107 000868 000274 000274 000324 000107 000044 000203

01

0 006897 018772 001632 001608 001941 000630 000259 00120005 005652 015857 001406 001405 001663 000547 000227 0010431 004862 014291 001291 001320 001519 000509 000214 0009745 003953 011409 001113 001166 001303 000446 000190 00085610 003929 011157 001054 001082 001239 000417 000175 000798

02

0 023286 018772 006198 006260 007269 002439 001026 00465705 019360 015857 005338 005472 006242 002121 000899 0040551 017136 014291 004927 005147 005740 001983 000848 0037985 013242 011409 004126 004505 004776 001714 000749 00329510 013167 011157 003935 004191 004569 001606 000694 003081



005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001606 001246 000388 000386 000461 000150 000062 0002871 001588 001235 000386 000385 000458 000150 000062 0002865 001598 001249 000393 000392 000465 000153 000063 00029110 001481 001179 000378 000383 000447 000148 000062 000283

01

0 006897 005286 001632 001608 001941 000630 000259 00120005 006029 004770 001524 001532 001801 000595 000247 0011351 005888 004690 001511 001526 001782 000591 000246 0011295 005858 004702 001529 001553 001801 000601 000251 00114810 005748 004611 001498 001521 001765 000588 000246 001124

02

0 023288 018773 006198 006260 007269 002439 001026 00465705 021550 017558 005870 005988 006871 002324 000983 0044421 020752 017074 005782 005952 006754 002304 000979 0044075 020352 016912 005810 006038 006770 002332 000996 00446410 020672 017013 005770 005935 006737 002300 000978 004398



References






































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




005

0 003129 003130 004076 004076 001024 001024 000719 000719 001249 0012492025 002899 002899 003664 003664 000981 000981 000692 000692 001185 00118511 002667 002667 003250 003250 000948 000948 000674 000674 001132 00113165 002677 002677 003239 003239 000963 000963 000685 000685 001146 001146infin 002689 002689 003502 003502 000880 000880 000618 000618 001073 0010734

01

0 011639 011638 014580 014580 004001 004001 002835 002835 004817 004817025 010561 010561 012781 012780 003810 003810 002717 002717 004532 00453151 009734 009733 011453 011453 003679 003679 002641 002641 004327 00432675 009646 009646 011234 011234 003718 003718 002677 002677 004352 0043517infin 010001 010001 012528 012528 003438 003438 002426 002436 004139 0041391

02

0 037876 037870 043939 043930 014871 014871 010795 010795 017323 0173218025 036117 036113 041624 041617 014354 014354 010436 010436 016671 016671 033549 033547 037962 037958 013851 013851 010127 010127 015937 01593665 032783 032781 036695 036692 013888 013887 010200 010200 015878 0158779infin 032545 032544 037755 037753 012779 012778 009276 009276 014885 0148845


005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001541 001175 000360 000355 000429 000139 000057 0002641 001291 001027 000329 000333 000389 000129 000054 0002465 001111 000893 000290 000296 000342 000114 000048 00021810 001107 000868 000274 000274 000324 000107 000044 000203

01

0 006897 018772 001632 001608 001941 000630 000259 00120005 005652 015857 001406 001405 001663 000547 000227 0010431 004862 014291 001291 001320 001519 000509 000214 0009745 003953 011409 001113 001166 001303 000446 000190 00085610 003929 011157 001054 001082 001239 000417 000175 000798

02

0 023286 018772 006198 006260 007269 002439 001026 00465705 019360 015857 005338 005472 006242 002121 000899 0040551 017136 014291 004927 005147 005740 001983 000848 0037985 013242 011409 004126 004505 004776 001714 000749 00329510 013167 011157 003935 004191 004569 001606 000694 003081



005

0 001822 001370 000414 000405 000495 000160 000065 00030305 001606 001246 000388 000386 000461 000150 000062 0002871 001588 001235 000386 000385 000458 000150 000062 0002865 001598 001249 000393 000392 000465 000153 000063 00029110 001481 001179 000378 000383 000447 000148 000062 000283

01

0 006897 005286 001632 001608 001941 000630 000259 00120005 006029 004770 001524 001532 001801 000595 000247 0011351 005888 004690 001511 001526 001782 000591 000246 0011295 005858 004702 001529 001553 001801 000601 000251 00114810 005748 004611 001498 001521 001765 000588 000246 001124

02

0 023288 018773 006198 006260 007269 002439 001026 00465705 021550 017558 005870 005988 006871 002324 000983 0044421 020752 017074 005782 005952 006754 002304 000979 0044075 020352 016912 005810 006038 006770 002332 000996 00446410 020672 017013 005770 005935 006737 002300 000978 004398



References






































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




References






































































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia








































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


an accurate solution method for the static and vibration analysis...

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