three-dimensional vibration analysis of crystal … · keywords: three-dimensional vibration,...

III European Conference on Computational MechanicsSolids, Structures and Coupled Problems in Engineering

C.A. Mota Soares et.al. (eds.)Lisbon, Portugal, 5–8 June 2006

THREE-DIMENSIONAL VIBRATION ANALYSIS OF CRYSTALPLATES VIA RITZ METHOD

Qian LI1 and VaiPan IU2

1Department of Civil and Environmental Engineering, University of MacauMacao SAR, PRC

e-mail: [email protected]

2 Department of Civil and Environmental Engineering, University of MacauMacao SAR, PRC

e-mail: [email protected]

Keywords: Three-dimensional Vibration, Crystal Plate, Chebyshev Polynomial, Ritz Method.

Abstract. Three-dimensional vibration of rectangular Y-cut crystalplate has been investigatedin this paper. The three displacements components of plate are expanded in three directionsby series of Chebyshev polynomial multiplied by the boundaryfunctionR which makes ex-pansions satisfy the essential boundary conditions along the edges. The eigenvalue matrix fornatural vibration frequencies is obtained for rectangularcrystal plate by Ritz method and thensolved by computer program. The result of an infinite plate excited by thickness-shear defor-mation parallel to one edge is proved by exact solutions. Thenatural free vibration frequenciesof rectangular Y-cut crystal plate of are compared with thoseby finite element method. Theagreements are close. Besides, convergence study demonstrates the high efficiency. Finally, thefree vibration mode shapes of clamped square Y-cut crystal are plotted.

1

Qian LI and VaiPan IU

1 INTRODUCTION

Exact solutions of three-dimensional equations of plates always have been pursued by manyresearchers for 3-D vibration analysis on the basis of linear, small strain elasticity theory whichdoes not rely on any hypothesis. Such analysis not only provides realistic results but also bringsout physical insights, which can not be predicted by the two-dimensional analysis. Attemptshave been made for 3-D vibration analysis of rectangular plates with general boundary condi-tions in the recent decades.

Early efforts on numerical vibration analysis through 3-D elasticity equations include thework of Cheung and Chakrabarti [1] who used the finite layer method to study the vibration ofthick rectangular plates with general boundary conditions. Fromme and Leissa [2], Hutchinsonand Zillimer [3] used the series solution method to analyze the free vibration of a completelyfree parallelepiped. Malik and Bert [4] and Liew and Teo [5] used the differential quadrature(DQ) method to analyze the vibration characteristics of rectangular plates, and so on. There aretoo many researchers pursuing the 3-D solutions to be mentioned here due to the limited paperlength.

Mathematically, Ritz method is a good approach to deduce the 3-D governing equationsthrough energy equations. Various authors have reported onapplications of Ritz method to 3-Dvibration analysis of plates. Leissa and Zhang [6] used simple algebraic polynomials and Liew[7, 8, 9] used orthogonal polynomials as admissible functions in Ritz method to analyze plates.Cheung and Zhou [10, 11] analyzed the 3-D vibration of rectangular and triangular plates andtori with circular cross-section by using Chebyshev polynomials as the admissible functions.

The choice of polynomials or series as admissible functionsis vital to approach the differentmethods. For example, on the road to pursue the two-dimensional equations of plates from 3-Dtheory of elasticity, different series or polynomials wereemployed. Mindlin first introducedhis approximate two-dimensional equations for elastic plates [12] based on the series expan-sion methods of Cauchy [13] and Poisson [14] and variational method of Kirchhoff [15]. Leeand Nikodem[16] also deduced the two-dimensional equations for elastic plates by employ-ing the same general procedure as Mindlin, but using a trigonometric series expansion. Thistrigonometric function series expansion later was improved by Lee [17] for these series lackthe proportional term. Lee added an additional proportional term, which contributed to annulshear correction factors, in his new 2-D theory. Chebyshev polynomial series were utilized byCheung and Zhou [10, 11] as mentioned before. These series have good properties of that it isa set of complete and orthogonal series in the interval[−1, 1].

In this paper, Chebyshev polynomial series are used to expandthe displacements of plate inthe application of Ritz method to analyze 3-D vibration of crystal plates. The numerical resultsof an infinite plate excited by thickness-shear deformationparallel to one edge is verified byexact solutions. The natural free vibration frequencies ofrectangular Y-cut crystal plates arecompared with those finite element method. Convergence rate is also studied. And finally, thefree vibration mode shapes of clamped square Y-cut crystal plate are plotted.

2 DERIVATION IN RITZ METHOD PROCEDURE

A plate is referred to anx1, x2, x3 system of rectangular coordinates. The geometry of therectangular crystal plate is shown in Figure 1. The plate hasa length2a, a width 2b, and auniform thickness2h. The faces of the plate are the planesx2 = ±h, and axes are parallel tothe edges of the plate. The corresponding displacement components at generic points areu1,u2, andu3 in thex1, x2, andx3 directions, respectively.

2


Figure 1: Geometry of rectangular crystal plate with uniform thickness.

The linear elastic strain energyV for the rectangular plate can be written in integral form as

V =1

2

∫ b

−b

∫ h

−h

∫ a

−acijklSijSkldx1dx2dx3 (1)

wherecijkl is the elastic stiffness coefficient;Sij =1

2

(

∂ui

∂xj

+∂uj

∂xi

)

is strain component; the

subscriptsi, j, k, l = 1, 2, 3.The kinetic energyK of the plate can be written as

K =1

2ρ∫ b

−b

∫ h

−h

∫ a

−a

(

∂u1

∂t

)2

+

(

∂u2

∂t

)2

+

(

∂u3

∂t

)2

dx1dx2dx3 (2)

whereρ is the material mass density per unit volume;ui (i = 1, 2, 3) is the displacementcomponent of plate.

The maximum energy functionΠ of the plate is

Π = Vmax − Kmax (3)

Vmax andKmax mentioned above are obtained from equations (1) and (2), respectively.In the case of a plate undergoing free vibration, its periodic displacement components can

be expressed in terms of the displacement amplitude functions:

ui = Uieiωt (i = 1, 2, 3) (4)

whereω denotes the natural frequency of the plate andi =√−1 . The amplitude functionsUi

are expressed as

U1 = U01 +

∑

l

Alαl U2 = U02 +

∑

m

Bmβm U3 = U03 +

∑

n

Cnγn (5)

whereU0i (i = 1, 2, 3) is specified function;Al, Bm, andCn are unknown constants to be

determined;αl, βm, andγn are suitably chosen coordinate functions. By applying Ritz method,substituting (4) into (3) and minimizing the functionΠ with respect to the coefficients of (5),

∂Π

∂Al

= 0∂Π

∂Bm

= 0∂Π

∂Cn

= 0 (6)

where the subscriptsl,m, n = 1, 2, 3, . . ., leads to the eigenvalue equations.

3


3 THICKNESS-SHEAR VIBRATION OF AN INFINITE PLATE

The exact solution of the three-dimensional equations for thickness-shear vibration parallelto an axis (i.e.x1 axis) is obtained by setting

u1 = (A sin Px2 + B cos Px2)eiωt (7)

u2 = 0

u3 = 0

The exact solution of equations of motion gives the circularfrequency of first modeω =π

2h

√

c66

ρ.

The corresponding displacement componentsui in (4) are reduced to

u1 =

(

∑

l

AlPl(η)

)

eiωt u2 = u3 = 0 (8)

whereη =x2

h∈ [−1, 1]; Pl(η), (l = 1, 2, 3, . . .) is the one-dimensionallth Chebyshev polyno-

mial [18]:Pl(η) = cos[(l − 1) arccos(η)] (9)

Table 1 shows the comparison of the first eight frequency parameters of thickness-shearvibration of infinite Y-cut crystal plate obtained from Ritz method with analytical solutions.

To facilitate comparison, the non-dimensional frequency parameterω is introduced asω =ω

ω.

Good agreements have been observed for both symmetric and anti-symmetric modes.

Modes Method ω1 ω2 ω3 ω4 ω5 ω6 ω7 ω8

Anti-sym Analytical 1.0 3.0 5.0 7.0 9.0 11.0 13.0 15.0Present 1.000 3.000 5.000 7.000 9.000 11.000 13.000 15.001

Symmetric Analytical 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0Present 2.000 4.000 6.000 8.000 10.000 11.998 14.000 16.000

Table 1: Comparison of the first eight frequency parameters of thickness-shear vibration of infinite Y-cut crystalplates for antisymmetric and symmetric modes.

4 FREE VIBRATION OF RECTANGULAR CRYSTAL PLATES

The plate in Figure 1 is considered for 3-D vibration analysis in this section. The amplitudefunctionsUi have to be expressed in triplicate summations in three directions. For coding andderivational convenience, the following non-dimensionalparameters are introduced first

ξ =x1

aη =

x2

hζ =

x3

b(10)

whereξ, η, ζ ∈ [−1, 1]. Each of the mechanical displacement amplitude functionsU1(ξ, η, ζ),U2(ξ, η, ζ), andU3(ξ, η, ζ) in (Eq. 4) is written in the form of the triplicate series of Chebyshevpolynomial:

U1(ξ, η, ζ) = Ru1

∞∑

i=0

∞∑

j=0

∞∑

k=0

AijkPi(ξ)Pj(η)Pk(ζ) (11)

4


U2(ξ, η, ζ) = Ru2

∞∑

l=0

∞∑

m=0

∞∑

n=0

BlmnPl(ξ)Pm(η)Pn(ζ)

U3(ξ, η, ζ) = Ru3

∞∑

p=0

∞∑

q=0

∞∑

r=0

CpqrPp(ξ)Pq(η)Pr(ζ)

whereRδ, (δ = u1, u2, u3) are R-functions [19] and served here to make it sure thatUi satisfythe essential boundary conditions along the edges. In this paper,

Ru1= 1 − ζ2 (12)

Ru2= (1 − ξ2)(1 − ζ2)

Ru3= 1 − ξ2

for simply supported condition;

Ru1= (1 − ξ2)(1 − ζ2) (13)

Ru2= (1 − ξ2)(1 − ζ2)

Ru3= (1 − ξ2)(1 − ζ2)

for clamped condition.Minimizing the maximum energy functionΠ with respect to the coefficientsAijk, Blmn, and

Cpqr yields the eigenvalue matrix:

K11 K12 K13

KT

12 K22 K23

KT

13 KT

23 K33

− ω2

M11 0 00 M22 00 0 M33

A

B

C

= 0 (14)

whereA, B, andC are the column vectors containing unknown coefficients,Aijk, Blmn, andCpqr, respectively; the superscript “T” denotes the transpose action. The elements of the stiffnesssubmatricesK and mass submatricesM are listed below

K11 =bh

ac1111∆

1,1ξ,u1iu1i∆

0,0η,jj∆

0,0ζ,u1ku1k +

ah

4bc1313∆

0,0ξ,u1iu1i∆

0,0η,jj∆

1,1ζ,u1ku1k (15a)

+a

4c1312∆

0,0ξ,u1iu1i∆

1,0η,jj∆

0,1ζ,u1ku1k +

a

4c1312∆

0,0ξ,u1iu1i∆

0,1η,jj∆

1,0ζ,u1ku1k

+ab

4hc1212∆

0,0ξ,u1iu1i∆

1,1η,jj∆

0,0ζ,u1ku1k

K12 = bc1122∆1,0ξ,u1iu2l∆

0,1η,jm∆0,0

ζ,u1ku2n +h

2c1123∆

1,0ξ,u1iu2l∆

0,0η,jm∆0,1

ζ,u1ku2n (15b)

+h

4c1312∆

0,1ξ,u1iu2l∆

0,0η,jm∆1,0

ζ,u1ku2n +b

4c1212∆

0,1ξ,u1iu2l∆

1,0η,jm∆0,0

ζ,u1ku2n

K13 = hc1133∆1,0ξ,u1iu3p∆

0,0η,jq∆

0,1ζ,u1ku3r +

b

2c1123∆

1,0ξ,u1iu3p∆

0,1η,jq∆

0,0ζ,u1ku3r (15c)

+h

4c1313∆

0,1ξ,u1iu3p∆

0,0η,jq∆

1,0ζ,u1ku3r +

b

4c1312∆

0,1ξ,u1iu3p∆

1,0η,jq∆

0,0ζ,u1ku3r

K22 =ab

hc2222∆

0,0ξ,u2lu2l∆

1,1η,mm∆0,0

ζ,u2nu2n +a

2c2223∆

0,0ξ,u2lu2l∆

0,1η,mm∆1,0

ζ,u2nu2n (15d)

+a

2c2223∆

0,0ξ,u2lu2l∆

1,0η,mm∆0,1

ζ,u2nu2n +ah

4bc2323∆

0,0ξ,u2lu2l∆

0,0η,mm∆1,1

ζ,u2nu2n

5


+bh

4ac1212∆

1,1ξ,u2lu2l∆

0,0η,mm∆0,0

ζ,u2nu2n

K23 = ac2233∆0,0ξ,u2lu3p∆

1,0η,mq∆

0,1ζ,u2nu3r +

ab

2hc2223∆

0,0ξ,u2lu3p∆

1,1η,mq∆

0,0ζ,u2nu3r (15e)

+a

4c2323∆

0,0ξ,u2lu3p∆

0,1η,mq∆

1,0ζ,u2nu3r +

bh

4ac1312∆

1,1ξ,u2lu3p∆

0,0η,mq∆

0,0ζ,u2nu3r

K33 =ah

bc3333∆

0,0ξ,u3pu3p∆

0,0η,qq∆

1,1ζ,u3ru3r +

ab

4hc2323∆

0,0ξ,u3pu3p∆

1,1η,qq∆

0,0ζ,u3ru3r (15f)

+bh

4ac1313∆

1,1ξ,u3pu3p∆

0,0η,qq∆

0,0ζ,u3ru3r

M11 = ρabh∆0,0ξ,u1iu1i∆

0,0η,jj∆

0,0ζ,u1ku1k (16a)

M22 = ρabh∆0,0ξ,u2lu2l∆

0,0η,mm∆0,0

ζ,u2nu2n (16b)

M33 = ρabh∆0,0ξ,u3pu3p∆

0,0η,qq∆

0,0ζ,u3ru3r (16c)

For the describable convenience, the operators∆ξ, ∆η, and∆ζ are used in above equationsand they are defined as

∆s,sξ,σϕσϕ =

∫ 1

−1

ds[f (1)σ (ξ)Pϕ(ξ)]

dξs· ds[f

(1)σ (ξ)Pϕ(ξ)]

dξsdξ (17a)

∆s,sη,ϕϕ =

∫ 1

−1

dsPϕ(η)

dηs· dsPϕ(η)

dηsdη (17b)

∆s,sζ,σϕσϕ =

∫ 1

−1

ds[f (2)σ (ζ)Pϕ(ζ)]

dζs· ds[f

(2)σ (ζ)Pϕ(ζ)]

dζ sdζ (17c)

where

s, s = 0, 1, (18)

σ, σ = u1, u2, u3,

ϕ, ϕ = i, j, k, l,m, n, p, q, r

Solving the eigenvalue equation (14) yields the frequencyω. The mode shapes correspondingto each eigenvalue may be obtained by back substitution of the eigenvalues one by one in theusual manner.

The following examples of four clamped edges and four simplysupported edges plates arecarried out under the procedures described before. The trial rectangular Y-cut crystal plates areof three different edge lengths (a = 25mm, b = 25, 15, 10mm) andtwo different thicknesses (t =0.1, 0.5mm). In the computation of present method,11×6×11 terms of admissible functions foreach displacement function were used. The finite element models in ABAQUS analysis weremeshed by shell and solid element types according to their thickness t = 0.1mm and 0.5mm,respectively. The first 20 frequencies are plotted in Figure2. The dot-lines are the results fromABAQUS and others are the results obtained from present method. The numbers in the legendsdenote the geometry properties of the plate: length, width,and thickness, respectively. It also isshown that the results from present method and ABAQUS have a good agreement.

6


0 5 10 15 200

2

4

6

8

10

12x 10

5

Modes

Fre

quen

cy (

Hz)

25×25×0.125×15×0.125×10×0.125×25×0.525×15×0.525×10×0.5

(a)

0 5 10 15 200

1

2

3

4

5

6

7

8

9x 10

5

Modes

Fre

quen

cy (

Hz)

25×25×0.125×15×0.125×10×0.125×25×0.525×15×0.525×10×0.5

(b)

Figure 2: Comparison of the first 20 natural vibration frequencies of rectangular crystal plate: (a) clamped plate(b) simply supported plate.

5 CONVERGENCE STUDY

Theoretically, the Ritz method could provide the solutions as accurate as we want. In thisstudy, the efficiency of solutions depends greatly on the choice of displacement componentsamplitude functionsUi. The natural frequencies obtained by Ritz method converge asupperbounds to the exact values and these upper bound estimates can be improved by increasing thenumber of terms of admissible functions in the computation.It is a balance problem of costand accuracy. In fact, a practical limit to the number of terms used always exists because ofthe limited speed, the capacity and the numerical accuracy of computers. Zhou studied theconvergence rate, the numerical stability and accuracy of Ritz method for isotropic plate in hispaper [11]. The frequencies monotonically decrease and approach the exact values with theincrease in the number of terms of admissible functions.

In present study, square Y-cut quartz plates with four clamped edges are taken as an examplefor the simple convergence studies. The dimension of the plate is25 × 25 × 0.807mm. Bech-mann’s values [20] for Y-cut quartz referred to the principal axis of crystal symmetry, in unitsof 1010N/m2, arec11 = c22 = 8.674, c12 = 0.699, c13 = c23 = 1.191, c14 = c56 = −c24 =−1.791, c33 = 10.72, c44 = c55 = 5.794, c66 = 3.988. All the computations were performed indouble precision and the integrals were evaluated analytically in (17).

0 5 10 15 200

1

2

3

4

5

6

7x 10

5

Modes

Fre

quen

cy (

Hz)

8×3×88×4×88×5×89×3×99×4×99×5×910×3×1010×4×1010×5×10

Figure 3: Convergence of frequencies of square Y-cut crystal plate with four clamped edges.

Figure 3 shows the convergence of first 20 frequencies of square crystal plate. The numbersin legend refer to the number of terms in displacement amplitude functions expansions alongx1,

7


x2, andx3 directions, respectively. The number of terms in sides is needed greater than that inthickness for the length and width dimensions are larger than thickness’s one. In general, moreterms are needed with the increasing dimension of one direction. From this figure, it is seenthat the convergence rate is very rapid. The frequencies monotonically decrease and approachcertain values with the increase in the number of terms of admissible functions. The few termscan obtain the reasonable frequencies of first few modes, andthe small number of admissiblefunctions terms mainly leads to the deviation of high mode frequencies.

6 SHAPES OF MODES

−1

0

1

−1

0

1−1.5

−1

−0.5

0

0.5

ξζ

Re

lativ

e d

isp

lace

me

nts

−1.5 −1 −0.5 0 0.5−1

−0.5

0

0.5

1

Relative displacements

Th

ickn

ess

co

ord

ina

te, η u

1u

2u

3

(a) 1st mode (flexural,ω = 273.06 kHz)

−1

0

1

−1

0

1−0.5

0

0.5

ξζ

Re

lativ

e d

isp

lace

me

nts

−0.4 −0.2 0 0.2 0.4 0.6−1

−0.5

0

0.5

1


Th

ickn

ess

co

ord

ina

te, η u

1u

2u

3

(b) 2nd mode (flexural,ω = 425.47 kHz)

−1

0

1

−1

0

1−0.5

0

0.5

ξζ

Re

lativ

e d

isp

lace

me

nts

−0.4 −0.2 0 0.2 0.4 0.6−1

−0.5

0

0.5

1


Th

ickn

ess

co

ord

ina

te, η u

1u

2u

3

(c) 3rd mode (flexural,ω = 471.99 kHz)

−1

0

1

−1

0

1−0.4

−0.2

0

0.2

0.4

ξζ

Re

lativ

e d

isp

lace

me

nts

−0.4 −0.3 −0.2 −0.1 0 0.1−1

−0.5

0

0.5

1


Th

ickn

ess

co

ord

ina

te, η u

1u

2u

3

(d) 4th mode (flexural,ω = 579.21 kHz)

−1

0

1

−1

0

1−0.5

0

0.5

ξζ

Re

lativ

e d

isp

lace

me

nts

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1−1

−0.5

0

0.5

1


Th

ickn

ess

co

ord

ina

te, η u

1u

2u

3

(e) 5th mode (flexural,ω = 619.94 kHz)

−1

0

1

−1

0

1−1

−0.5

0

0.5

1

ξζ

Re

lativ

e d

isp

lace

me

nts

−1 −0.8 −0.6 −0.4 −0.2 0−1

−0.5

0

0.5

1


Th

ickn

ess

co

ord

ina

te, η u

1u

2u

3

(f) 6th mode (thickness extensional,ω = 721.96kHz)

−1

0

1

−1

0

1−1

−0.5

0

0.5

1

ξζ

Re

lativ

e d

isp

lace

me

nts

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1


Th

ickn

ess

co

ord

ina

te, η u

1u

2u

3

(g) 7th mode (flexural,ω = 745.41 kHz)

−1

0

1

−1

0

1−0.4

−0.2

0

0.2

0.4

ξζ

Re

lativ

e d

isp

lace

me

nts

−0.4 −0.2 0 0.2 0.4−1

−0.5

0

0.5

1


Th

ickn

ess

co

ord

ina

te, η u

1u

2u

3

(h) 8th mode (flexural,ω = 745.95 kHz)

Figure 4: Deflected shapes of the first eight modes for the plates with clamped edges.

8


The shapes of modes of vibration can be determined very easily by back substitution of theeigenvalues, one by one, in the usual manner. In this section, the free vibration analysis ofclamped square Y-cut crystal plate is carried out. The edge length and thickness of the plate are25mm and 5mm, respectively. The deflected shapes of first eight modes are plotted in Figure4 by substituting the eigenvectors back into (11). The left surfs in each subfigures show therelative displacement amplitudes ofu2 in the middle plane (η = 0); and right curves are theamplitudes ofu1, u2, andu3 of the midline (ξ = 0, ζ = 0) at the center of plate along thethickness respectively.

It is shown that the first five modes are first flexural mode and its overtones and the sixth isthe thickness extensional mode. Then it switches back to first flexural overtones. Because Y-cutcrystal is of trigonal withx3 the trigonal axis andx1 a diagonal axis, the free vibration modesare not symmetric even of the symmetric geometry and supportconditions.

7 CONCLUSIONS

In this paper, the three-dimensional eigenvalue equationsfor natural vibration of Y-cut crys-tal plates are deduced based on the linear, small strain 3-D elasticity theory via Ritz methodand displacement components expansion by Chebyshev polynomials multiplying boundaryR-functions. The results of thickness-shear vibration of an infinite plate are obtained and verifiedby exact solutions. Another example of free vibration characteristics of rectangular Y-cut crystalplates have been investigated and compared with the result from a finite element code ABAQUS.Both comparisons show the good accuracy. Besides, due to the constant and proportional termsin Chebyshev polynomial series, the convergence studies reveal the rapid convergence rate andhigh efficiency. Finally, the deflected shapes are plotted todemonstrate the vibration modes.

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