Identiﬁcation of Vibration Modes of Quartz Crystal Plates

Identification of Vibration Modes of Quartz CrystalPlates with Proportion of Strain and KineticEnergiesQi HuangPiezoelectric Device Laboratory, School of Mechanical Engineering and Mechanics, Ningbo University,818Fenghua Road, Ningbo, Zhejiang 315211, China.Department of Architectural Engineering, Huzhou Vocational & Technical College, 299 Xuefu Road, Wuxing Dis-trict, Huzhou, Zhejiang 313000, China.

Rongxing WuDepartment of Architectural Engineering, Ningbo Polytechnic, 1069 Xinda Road, Beilun District, Ningbo, Zhe-jiang 315800, China.

Lihong Wang, Longtao Xie, Jianke Du, Tingfeng Ma and Ji Wang†

Piezoelectric Device Laboratory, School of Mechanical Engineering and Mechanics, Ningbo University,818Fenghua Road, Ningbo, Zhejiang 315211, China.† Corresponding author

(Received 5 November 2019; accepted 18 May 2020)

For the design of quartz crystal resonators, finding and determining the vibration modes have always been veryimportant and cumbersome. Vibration modes are usually identified through plotting displacement patterns of eachcoupled modes and making comparisons. Over the years, there is not much improvement in the identificationprocedure while tremendous efforts have been made in refining the equations of the Mindlin plate theory to obtainmore accurate results, such as the adoption of the Finite Element Method (FEM) by implementing the high-orderMindlin plate equations for efficient analysis. However, due to the old fashioned mode identification method, theFEM application is still inadequate and cannot be fully automated for this purpose. To have this situation improved,a method using the proportions of strain and kinetic energies to characterize the energy level of each vibration modeis proposed. With solutions of displacements, the energy distribution of each vibration mode is calculated and themode with the highest energy concentration at a specific frequency is designated as the dominant mode. The resultshave been validated with the traditional approach by plotting mode shapes at each frequency. Clearly, this energyapproach will be advantageous with the FEM analysis for vibration mode identification automatically.

1. INTRODUCTION

In the design of a quartz crystal resonator, it is impor-tant to make full and accurate analysis of its vibrations andto identify the functioning mode through the detailed resultslike frequency and mode shapes.1–4 The precise identifica-tion of vibration modes is essential because the optimal func-tioning mode of the resonator in the designated vibration fre-quency should be dominant and needs to be well isolated fromso-called spurious modes of negative effects. Consequently,the functioning mode is extremely important to be identified,which is the thickness-shear mode usually, to immunize fromcouplings to other modes, or the notorious spurious modes. Asthere is no suitable way to eliminate the infinite couplings ofvibration modes in a finite quartz crystal plate, it is practicalto find the least coupled thickness-shear vibrations to enhancethe performance characteristics of a quartz crystal resonator.In other words, the challenge is to determine the optimal pa-rameters of the resonator structure, avoid the strong couplingof modes, and let the primary working mode be enhanced.3 In

a typical resonator, the primary working mode mainly refers tothe thickness-shear mode and the strongly coupled one is theflexural mode.2 For the accurate analysis of vibrations within aquartz crystal plate in the vicinity of the thickness-shear mode,the most effective method we have chosen is based on theMindlin plate theory which can easily include more vibrationmodes.

The strong couplings happen just between the neighbor-ing vibration modes, so only a few strongly coupled vibrationmodes are needed for the analysis to obtain the accurate re-sults such as frequency and displacements which can be usedfor mode identification. Usually the analysis is done withthe straight-crested waves for applications, and currently theMindlin plate equations have been implemented in the finiteelement analysis for more precise considerations of structuralcomplications. The identification of vibration modes, as fromthe beginning, is done with the plotting of mode shapes whichshow the dominance of specific mode at each frequency bypicking out the largest displacement.4, 5

392 https://doi.org/10.20855/ijav.2020.25.31671 (pp. 392–407) International Journal of Acoustics and Vibration, Vol. 25, No. 3, 2020

Q. Huang, et al.: IDENTIFICATION OF VIBRATION MODES OF QUARTZ CRYSTAL PLATES WITH PROPORTION OF STRAIN AND KINETIC. . .

Initially, the analysis of plate vibrations was presented as athree-dimensional problem6 which inevitably demanded com-plicated calculations in solving the differential equations ofmotion. Fortunately, with the Mindlin plate theory, we cananalyze the vibrations from a two-dimensional perspectiveand this greatly simplifies the solution procedure. Accord-ingly, accurate solutions from the Mindlin plate equations withanalytical and numerical procedures have made considerableprogress recently corresponding with methods to enhance themode identification, and in conjunction with the finite elementmethod, it is highly regarded as a major improvement.7 As aresult, the mode shapes can be plotted and identified with thedisplacements.

In the early beginning, the Mindlin plate theory was pre-sented with only equations of zeroth- and first-order mechan-ical displacements,8 then Mindlin and Tiersten9 expanded thetwo-dimensional equations of the plate under forced vibrationsto higher-order equations, and introduced the electric field con-siderations. Later Mindlin and Spencer2 presented the nor-malized wavenumber and dispersion relations of the infiniteplate. Furthermore, equations of AT-cut quartz crystal rectan-gular plates in free vibrations were derived and the frequencyspectra were calculated.2 The normalized dispersion relationsand frequency spectra, as effective representations of the vibra-tion analysis results, were widely used for the optimal selectionof plate parameters. Of course, other researchers also proposedsimilar or novel two-dimensional theory in resonator vibrationanalysis, like Lee, et al.10 and Sekimoto, et al.11

In recent years, applications of Mindlin plate theory havebeen further improved. Based on the high-order Mindlin platetheory, Yong and Zhang12 and Wang, et al.13 developed the fi-nite element method in analyzing high-frequency vibrations ofquartz crystal resonators. Then Yong, et al.14 studied vibrationmodes with the first- and third-order plate equations and foundout that the higher-order theory results are more accurate thanlower-order ones for some vibration modes. Therefore, moresolutions of vibration analysis prefer higher-order plate theoryand finite element methods regarding design tool for develop-ment. However, using such solutions usually means solvinga very large matrix problem in order to obtain the eigenval-ues as vibration frequencies. In the traditional linear algebraproblems, this means a lot of computer memory and com-puting time which greatly affect the computational efficiency.Lately, Wang, et al.15 have developed a finite element pro-gram with parallel computing capabilities on Linux clusters toquickly handle sparse matrices and algebraic operations withsoftware components from public domains and this effectivelyimproved the computational efficiency of high frequency vi-bration problems. Subsequently, they also implemented non-linear finite element analysis with the program. 16, 17

Currently, the Mindlin plate equations with truncation andcorrections for parallel finite element analysis are up to thefifth-order or even higher which also takes into account thepotential and overtone displacement variations.18 At thesame time, the analysis of nonlinear vibrations is also in-cluded.16, 17, 19 Generally speaking, with the higher-order platetheory, considerations of nonlinearity, and other factors to im-prove the accuracy and parallel computing technology to im-prove the computation efficiency, we can effectively obtain the

vibration solutions with great accuracy. For the design of thequartz crystal resonators, we still need the exact mode from themassive calculation results by plotting the mode shapes. Thereis a lack of efficiency because it requires manual interventionswhich will affect the automation of the parallel computing.

Based on the fact that the vibration energies are stored ineach vibration mode, the energy method can be used for vibra-tion analysis, as shown in modal strain energy approach.20 Forpiezoelectric plate vibrations, the energy calculations were of-ten used to describe the energy trapping and harvest effect.21, 22

Through the research of Yong, et al.,23 the ratios of the trappedenergies can be calculated to evaluate the quality of resonators.

Previously, the use of energy methods in vibrations has beencommon in structural dynamics and the calculated energy hasbeen used for the identification of vibration modes in a com-plex structure.24 The statistical energy analysis (SEA) is amature method for the efficient calculation of the response ofstructural vibration.25 In a different approach from the tradi-tional solutions of mechanical vibrations, the statistical methoduses the energy of each subsystem as the basic parameters tosolve the vibration problem.26

Besides SEA, there are also other methods which utilize theenergy contributions of vibration modes to make more accurateapproximations to vibrations27 such as the detection of domi-nant modes of shell structures under external excitations likewind or ground motion28, 29 and using dominating energy dis-tribution to characterize the vibration signals.30 In this paper,we focus on the high frequency vibration analysis of the quartzcrystal resonator and target locating the pure thickness-shearmode from the coupled vibrations with energy formulation ateach frequency with multiple vibration modes or degree-of-freedoms (DOFs).

To serve the purpose of identifying the dominant mode of aquartz crystal resonator, the strain and kinetic energies of vi-brations with multiple DOFs are presented with formulationsby vibration solutions at each frequency. Using the strain en-ergy of each DOF to distinguish the dominant mode of a gearpair determined effective as described by Wang, et al.31 Unlikeusing the time-averaged kinetic energy,28–30 in this research,the strain and kinetic energies in each vibration cycle are cal-culated because the quartz crystal resonator exhibits perfectlysteady vibrations. As a result, both the kinetic and strain en-ergy distributions in a vibration cycle are used to identify thedominant functioning mode with the highest energy concentra-tion.

To demonstrate the use of the energy-based vibration modeidentification method, we chose the well-known Mindlin first-order equations of a rectangular plate for calculations. First,we adopted the first-order plate equations and calculated thedisplacement expressions of vibration modes. Then, we es-tablished the theoretical formulation of the kinetic and strainenergy densities and integrated them over the plate for totalenergy. Next, we divided the total energy according to specificmode types and calculated the energy distributions of each vi-bration mode. We calculated the energies in three differentmaterial types, which were two different sets of mode cou-plings with an AT-cut and a full set of mode couplings with anSC-cut quartz crystal plate. Finally, we successfully observedthe functioning thickness-shear mode from the highest concen-

International Journal of Acoustics and Vibration, Vol. 25, No. 3, 2020 393


Figure 1. The simple model of a plate for the crystal resonator.

trations of kinetic and strain energies. The results were alsoconfirmed by mode shapes at each resonant frequency by thetraditional mode plotting method.32 This energy approach canbe effective and easy to use in the vibration mode identifica-tion and is capable of improving the features of finite elementanalysis.

2. THE FIRST-ORDER EQUATIONS OF THEMINDLIN PLATE THEORY

For the analysis of thickness-shear vibrations of a quartzcrystal plate, the first-order Mindlin plate equations with lim-ited couplings of modes are adequate for accurate results.1–5

We chose a rectangular plate with large length to thickness ra-tio for the calculations, and the analytical solutions based onthe straight-crested wave assumptions are useful in the designand optimization of quartz crystal resonators.

As known in fundamental thickness-shear vibrations ofquartz crystal resonators, there are six modes to be analyzedbased on the Mindlin first-order plate theory, which are theextension (E) u(0)1 , flexural (F) u(0)2 , face-shear (FS) u(0)3 ,thickness-shear (TSh) u(1)1 , thickness-stretch (TSt) u(1)2 , andthickness-twist (TT) u(1)3 .8 However, in different materialtypes the coupling modes are different with the number ofmodes in the solutions. The simple model of a plate for thecrystal resonator we used here is a rectangular AT-cut quartzcrystal plate with four free sides as shown in Fig. 1 with 2b,2a and 2c representing its thickness, length, and width, respec-tively.

To demonstrate our energy approach, we utilized the equa-tions without electrodes,10, 33, 34 without external forces,35 andwith temperature at 25C to avoid frequency-temperature ef-fect.36, 37 Furthermore, only the straight-crested waves travel-ing along the x1 direction are considered, making the equationsgreatly simplified.

With the given plate model and boundary conditions inFig. 1, we have the stress-displacement equations for quartzcrystal plate with the full set of elastic constants and modifica-

tions as follows8, 38

T(0)j = 2bκ

(0)j

[cj1u

(0)1,1 + cj2κ

(0)2 u

(1)2 + cj3u

(0)3,3

+cj4κ(0)4

(u(0)2,3 + u

(1)3

)+ cj5

(u(0)3,1 + u

(0)1,3

)+cj6κ

(0)6

(u(0)2,1 + u

(1)1

)]; (1a)

T(1)j =

2b3

3κ(1)j

[cj1u

(1)1,1 + c13u

(1)3,3 + c14κ

(1)4 u

(1)2,3

+c15

(u(1)3,1 + c15u

(1)1,3

)+ c16κ

(1)6 u

(1)2,1

]; (1b)

T(1)2 = 0, cij = cij −

ci2c2jc22

, i, j = 1, 2, 3, 4, 5, 6; (1c)

where T (n)j (n = 0, 1; j = 1, 2, . . . 6) are the nth-order stresses,

and cij are the elastic constants of quartz crystal. To correctthe discrepancy due to the truncation of higher-order flexuralmode, elastic constants cij have been modified.1 The constantκ(n)j are the correction factors to compensate the truncation

and their value are2, 6, 8, 38–40

κ(0)p =π

2√

3, κ(1)p = 1, p = 2, 4, 6; (2a)

κ(0)q = 1, κ(1)q = 1, q = 1, 3, 5. (2b)

The corresponding two-dimensional equations of motion of thesix vibration modes are8

∂T(0)1

∂x1+∂T

(0)5

∂x3= 2bρu

(0)1 ; (3a)

∂T(0)6

∂x1+∂T

(0)4

∂x3= 2bρu

(0)2 ; (3b)

∂T(0)5

∂x1+∂T

(0)3

∂x3= 2bρu

(0)3 ; (3c)

∂T(1)1

∂x1+∂T

(1)5

∂x3− T (0)

6 =2b3

3ρu

(1)1 ; (3d)

∂T(1)6

∂x1+∂T

(1)4

∂x3− T (0)

2 =2b3

3ρu

(1)2 ; (3e)

∂T(1)5

∂x1+∂T

(1)3

∂x3− T (0)

4 =2b3

3ρu

(1)3 ; (3f)

where ρ is the density of quartz crystal.The six displacements of straight-crested waves travelling

along the x1 direction are defined as1, 8

u(0)j = Aj sin ξx1e

iωt; (4a)

u(1)j =

Aj+3

bcos ξx1e

iωt, j = 1, 2, 3; (4b)

394 International Journal of Acoustics and Vibration, Vol. 25, No. 3, 2020


Figure 2. Dispersion relations of coupled thickness-shear and flexural vibra-tions of an AT-cut quartz crystal plate.

Figure 3. Dispersion relations of coupled thickness-shear, flexural, and face-shear vibrations of an AT-cut quartz crystal plate.

where Ai (i = 1, 2, 3, 4, 5, 6) , ξ, ω are the amplitudes,wavenumber, and vibration frequency, respectively.

In this paper, there are three mode coupling combinations asshown below,3, 14u(0)2 , u

(1)1 , AT - cut, two modes;

u(0)2 , u

(0)3 , u

(1)1 , AT - cut, three modes;

u(0)1 , u

(0)2 , u

(0)3 , u

(1)1 , u

(1)2 , u

(1)3 , SC - cut, six modes.

(5)

By substituting Eq. (5) according to Eq. (4) into Eqs. (1)and (3), we can obtain the dispersion relations as functions ofthe normalized frequency Ω and wavenumber Z as

M (Ω, Z) = 0; (6)

where

Ω =ω

ω0, ω0 =

π

2b

√c66ρ, Z =

ξπ2b

,

and M stands for the dispersion relation matrix of differentmode types and we give the detail later in the Appendix.

Figure 4. Dispersion relations of coupled thickness-shear, thickness-stretch,thickness-twist, extension, flexural, and face-shear vibrations of an SC-cutquartz crystal plate.

By solving Eq. (6), we can obtain the dispersion relationsof a quartz crystal plate as shown in Figs. 2-4. The dispersionrelations validated the equations and also revealed the charac-teristics of waves in the plate.

With the above results of normalized wavenumbers at eachfrequency, we can rewrite the displacements of Eq. (4) as

u(0)j =

6∑r=1

[αjrβrA44 sin

(πZr

2

x1b

)]eiωt; (7a)

u(1)j =

6∑r=1

[α(j+3)rβr

A44

bcos

(πZr

2

x1b

)]eiωt, j = 1, 2, 3;

(7b)

where Zr (r = 1, 2, 3, 4, 5, 6) are the the normalizedwavenumbers, A44 stands for the amplitude A4 at thefourth wavenumber and αjr are the corresponding amplituderatios defined as:

αjr =AjrA4r

, j, r =

2, 4, AT - cut, two modes;2, 3, 4, AT - cut, three modes;1, 2, 3, 4, 5, 6, SC - cut, six modes;

(8a)

βr =A4r

A44, r =

2, 4, AT - cut, two modes;2, 3, 4, AT - cut, three modes;1, 2, 3, 4, 5, 6, SC - cut, six modes;

(8b)

where Ajr is the amplitude Aj at the rth wavenumber.Using the traction-free boundary conditions of the plate

shown in Fig. 1, the corresponding stress boundary conditionsat x1 = ±a are:

T(0)5 = T

(0)6 = T

(0)1 = T

(1)5 = T

(1)6 = T

(1)1 = 0. (9)

By substituting Eq. (7) to Eq. (1) then to Eq. (9), we cansolve the frequency equations

|N (βr, Zr)| = 0; (10)



where N stands for the coefficient matrix of frequency equa-tion of the coupled modes and we give the details in the Ap-pendix.

Let the coefficient determinant of Eq. (10) vanishing, wecan obtain the frequency spectra as frequency versus length tothickness ratio. Then, from the dispersion relations, we canget a set of wavenumbers Zr at a specific frequency Ω. Bysubstituting Zr back into Eq. (10), we then obtain βr.

With displacements solved in Eq. (7) and for the calculationof strain energies later, we also give the expressions of nontriv-ial strains with displacements of the x1-propagating straight-crested waves as:

S(0)1 = u

(0)1,1; S

(0)4 = u

(1)3 ; S

(0)2 = u

(1)2 ;

S(0)5 = u

(0)3,1; S

(0)3 = 0; S

(0)6 = u

(0)2,1 + u

(1)1 ; (11a)

S(1)1 = u

(1)1,1; S

(1)4 = 0; S

(1)2 = 2u

(2)2 ;

S(1)5 = u

(1)3,1; S

(1)3 = 0; S

(1)6 = u

(1)2,1. (11b)

All the derivation above is primarily based on the first-orderMindlin plate theory.3, 8 After all the displacements and strainsin Eqs. (7) and (8) are obtained, the calculation of the strainand kinetic energies of the plate can be carried out without anydifficulty.

To obtain the dominant vibration modes, it needs the vibra-tion mode shapes from Eq. (7) to find out the largest amplitudefor each frequency in the frequency spectra. With this process,all the curves in the spectra will be labeled by the dominant vi-bration mode and the optimal selection of length, or the lengthto thickness ratio, can also be made from frequency spectra.Clearly, it is a lengthy and tedious procedure requiring the cal-culation and plotting of vibration modes with a visual check byan experienced researcher. In case of numerical analysis with alarger set of data and solutions, the plotting of vibration modesis the last choice of the vibration mode identification procedurewe can consider.

3. THE CALCULATION OF ENERGIES OFVIBRATIONS OF A PLATE

To enable a formulation and calculation with displacementsolutions for the vibration mode identification, it is naturalto turn to the energy of vibrations for a possible solution.The idea is that the largest vibration mode should be the onewith the largest proportion of the strain energy or kinetic en-ergy. Then the identification of vibration modes can be madeby checking the energy proportion from vibration solutions.Clearly, it is simpler to calculate and compare the energy pro-portion in both analytical and numerical solutions for this pur-pose. To this objective, we first need to obtain the energy for-mulation with vibration solutions.

3.1. The Strain EnergyFirst, the strain energy is the energy stored in the plate when

it undergoes deformation in vibrations. By analyzing strain en-ergy of the plate, the relations of stresses, strains, and displace-ments will be merged, implying a simple result for the evalua-tion of the state of vibrations. We start with the strain energy

density which can be used to represent the energy in the platethrough integration. With the first-order Mindlin plate theoryof selected modes, the strain energy density is8

U =1

2cpq∑m

∑n

BmnS(m)p S(n)

q

=1

2cpq

(B00S

(0)p S(0)

q +B11S(1)p S(1)

q

); (12)

where Bmn (m,n = 0, 1) and S(n)p (p = 1, 2, 3, 4, 5, 6;n =

0, 1) are the integral constants and strains of the nth-order asgiven in Eq. (12), respectively. The energy varies with differentcases of mode coupling as we show in the following.

3.1.1. AT-cut quartz crystal plates with two vibrationmodes

As the case of two modes u(0)2 and u(1)1 , Eq. (11) is simpli-fied with zero displacements and the Eq. (12) becomes:

U =1

2c66B00S

(0)6 S

(0)6 +

1

2c11B

11S(1)1 S

(1)1

+1

2c22B

11S(1)2 S

(1)2 + c12B11S

(1)1 S

(1)2

=1

2B00c66 ·

(u(0)2,1

)2+

1

2B00c66 ·

(u(1)1

)2+

1

2B11c11 ·

(u(1)1,1

)2+B00c66 · u(0)2,1u

(1)1 . (13)

Next, we classify the energy densities according to the vibra-tion modes in Eq. (13) as follows:

UTSh =1

2B00c66

(u(1)1

)2+

1

2B11c11

(u(1)1,1

)2; (14a)

UF =1

2B00c66

(u(0)2,1

)2; (14b)

UF - TSh = B00c66u(0)2,1u

(1)1 ; (14c)

where UTSh and UF are strain energy densities of singlethickness-shear and flexural mode, while UF - TSh is of the cou-pled ones.

Finally, the total strain energies of the quartz crystal plateare obtained by integrating the energy densities of Eq. (14) in



the entire plate in one vibration cycle as:

UX =

∫ a

−a

∫ c

−c

∫ T

0

UXdx1dx3dt, X = TSh, F, F - TSh;

(15a)

UTSh =2πc

ωc66

2∑i=1

A24iα

24iVc2 +

2∑i=1,j 6=i

A4iA4jα4iα4jVcc

+π3c

6ωc11

2∑i=1

A24iα

24iZ

2i Vs2

+

2∑i=1,j 6=i

A4iA4jα4iα4jZiZjVss

; (15b)

UF =π3c

ωc66

2∑i=1

A24iα

22iZ

2i V c2

+

2∑i=1,j 6=i

A4iA4jα2iα2jZiZjVcc

; (15c)

UF - TSh =4π3c

ωc66

2∑i=1

A4iA4iα2iα4iZiV c2

+

2∑i=1,j 6=i

A4iA4jα2iα4jZiVcc

; (15d)

U=

∫ a

−a

∫ c

−c

∫ T

0

Udx1dx3dt = UTSh + UF + UF - TSh, T =2π

ω;

(15e)

where T is the period of vibration, U is the total strain energy,UTSh and UF are the strain energy of thickness-shear and flexu-ral modes, and UF - TSh is the coupled energy, respectively. Andthe coefficients Vc2, Vcc, Vs2, and Vss in Eq. (15) are given as:

Vc2 =1

b

∫ a

−acos2

(πZi

2

x1b

)dx1 =

a

b+

1

πZisin(abπZi

);

(16a)

Vcc =1

b

∫ a

−acos

(πZi

2

x1b

)cos

(πZj

2

x1b

)dx1

=2

π

[sin(abπ2 (Zi − Zj)

)Zi − Zj

+sin(abπ2 (Zi + Zj)

)Zi + Zj

];

(16b)

Vs2 =1

b

∫ a

−asin2

(πZi

2

x1b

)dx1 =

a

b− 1

πZisin(abπZi

);

(16c)

Vss =1

b

∫ a

−asin

(πZi

2

x1b

)sin

(πZj

2

x1b

)dx1

=2

π

[sin(abπ2 (Zi − Zj)

)Zi − Zj

−sin(abπ2 (Zi + Zj)

)Zi + Zj

].

(16d)

3.1.2. AT-cut quartz crystal plate with three vibrationmodes

With the same procedure, we can obtain the strain energydensity of the case of three modes as:

U =1

2c55B00S

(0)5 S

(0)5 +

1

2c66B00S

(0)6 S

(0)6 + c56B00S

(0)5 S

(0)6

+1

2c11B

11S(1)1 S

(1)1 +

1

2c22B

11S(1)2 S

(1)2 + c12B11S

(1)1 S

(1)2

=1

2B00c66 ·

(u(0)2,1

)2+

1

2B00c55 ·

(u(0)3,1

)2+

1

2B00c66 ·

(u(1)1

)2+

1

2B11c11 ·

(u(1)1,1

)2+B00c66 · u(0)2,1u

(1)1

+B00c65 · u(0)2,1u(0)3,1 +B00c65 · u(1)1 u

(0)3,1 (17)

Next, we classify the energy densities according to the vi-bration modes in Eq. (17) as follows:

UTSh =1

2B00c66

(u(1)1

)2+

1

2B11c11

(u(1)1,1

)2; (18a)

UF =1

2B00c66

(u(0)2,1

)2; (18b)

UFS =1

2B00c55

(u(0)3,1

)2; (18c)

UF - TSh =B00c66u(0)2,1u

(1)1 ; (18d)

UF - FS =B00c65u(0)2,1u

(0)3,1; (18e)

UTSh - FS =B00c65u(1)1 u

(0)3,1; (18f)

where UX (X = TSh, F, FS, F - TSh, F - FS, TSh - FS) are thestrain energies of different modes. Since for AT-cut quartzcrystal c65 is much smaller, it is clear that the coupled ener-gies should be much smaller in comparison with the energiesof single modes. Thus, these coupled energies are neglected infollowing calculations.

Finally, the total strain energies of the quartz crystal plateare obtained by integrating the energy densities of Eq. (18) inthe entire plate in one vibration cycle as:

UX =

∫ a

−a

∫ c

−c

∫ T

0

UXdx1dx3dt,

X = TSh, F, FS, F - TSh, F - FS, TSh - FS; (19a)

UTSh =2πc

ωc66

3∑i=1

A24iα

24iVc2

+

3∑i=1,j 6=i

A4iA4jα4iα4jVcc

+π3c

6ωc11

3∑i=1

A24iα

24iZ

2i Vs2



+

3∑i=1,j 6=i


; (19b)

UF =π3c

ωc66

3∑i=1

A24iα

22iZ

2i V c2

+

3∑i=1,j 6=i


; (19c)

UFS =π3c

ωc55

3∑i=1

A24iα

23iZ

2i V c2

+

3∑i=1,j 6=i


; (19d)

UF - TSh =4π3c

ωc66

4∑i=2


+

4∑i=2,j 6=i

A4iA4jα2iα4jZiVcc

; (19e)

UF - FS =2π4c

ωc65

3∑i=1

A4iA4iα3iα2iZ2i V c2

+

3∑i=1,j 6=i

A4iA4jα3iα2jZ2i Vcc

; (19f)

UTSh - FS =4π3c

ωc65

3∑i=1


+

3∑i=1,j 6=i

A4iA4jα4iα3jZiVcc

; (19g)

U =

∫ a

−a

∫ c

−c

∫ T

0

Udx1dx3dt

=UTSh + UF + UFS + UF - TSh + UF - FS + UTSh - FS;(19h)

where U is the total strain energy, UTSh, UF, and UFS arethe strain energy of thickness-shear, flexural and face-shearmodes, and UF - TSh, UF - FS, and UTSh - FS are the coupled en-ergies, respectively.

3.1.3. SC-cut quartz crystal plates with six vibrationmodes

The SC-cut quartz crystal is one of the most complicatedmaterial types in vibration analysis with the first-order Mindlinplate equations as the elastic constant matrix is full. Due to thematerial characteristics of this special orientation, all the sixvibration modes are coupled and need to be calculated in theenergy solutions. First, we can obtain the strain energy densityof six modes as:

U =1

2

∑p=1,2,4,5,6

∑q=1,2,4,5,6

cpqB00S(0)p S(0)

q

+1

2

∑p=1,2,5,6

∑q=1,2,5,6

cpqB11S(1)p S(1)

q

=1

2B00

[c11

(u(0)1,1

)2+ c22

(u(1)2

)2+ c44

(u(1)3

)2+c55

(u(0)3,1

)2+ c66

(u(0)2,1

)2+ c66

(u(1)1

)2]+B00

[c66u

(0)2,1u

(1)1 + c12u

(0)1,1u

(1)2 + c14u

(0)1,1u

(1)3

+ c15u(0)1,1u

(0)3,1 +c16u

(0)1,1

(u(0)2,1 + u

(1)1

)+ c24u

(1)2 u

(1)3

]+B00

[c25u

(1)2 u

(0)3,1 + c26u

(1)2

(u(0)2,1 + u

(1)1

)+ c45u

(1)3 u

(0)3,1

+c46u(1)3

(u(0)2,1 + u

(1)1

)+ c56u

(0)3,1

(u(0)2,1 + u

(1)1

)]+

1

2B11

[c11

(u(1)1,1

)2+ c55

(u(1)3,1

)2+ c66

(u(1)2,1

)2]+B11

[c16u

(1)2,1u

(1)1 + c15u

(1)1,1u

(1)3,1 + c65u

(1)2,1u

(1)3,1

].

(20)

Observing the Eq. (20), though there are many terms of cou-pled energy, but many elastic constants associated with themare smaller in comparison to the single mode terms. As a re-sult, the coupled energies are also neglected here, and only thelarger energy density terms are presented as:

UTSh =1

2B00c66

(u(1)1

)2+

1

2B11c11

(u(1)1,1

)2; (21a)

UTSt =1

2B00c22

(u(1)2

)2+

1

2B11c66

(u(1)2,1

)2; (21b)

UTT =1

2B00c44

(u(1)3

)2+

1

2B11c55

(u(1)3,1

)2; (21c)

UE =1

2B00c11

(u(0)1,1

)2; (21d)

UF =1

2B00c66

(u(0)2,1

)2; (21e)

UFS =1

2B00c55

(u(0)3,1

)2. (21f)

We also integrated the energy densities of Eq. (20) as:

UX =

∫ a

−a

∫ c

−c

∫ T

0

UXdx1dx3dt,

X = TSh,TSt, TT, E, F, FS; (22a)

UTSh =2πc

ωc66

6∑i=1

A24iα

24iVc2

+

6∑i=1,j 6=i

A4iA4jα4iα4jVcc

+π3c

6ωc11

6∑i=1

A24iα

24iZ

2i Vs2

+

6∑i=1,j 6=i


; (22b)



UTSt =2πc

ωc22

6∑i=1

A24iα

25iVc2

+

6∑i=1,j 6=i

A4iA4jα5iα5jVcc

+π3c

6ωc66

6∑i=1

A24iα

25iZ

2i Vs2

+

6∑i=1,j 6=i


; (22c)

UTT =2πc

ωc44

6∑i=1

A24iα

26iVc2

+

6∑i=1,j 6=i

A4iA4jα6iα6jVcc

+π3c

6ωc55

6∑i=1

A24iα

26iZ

2i Vs2

+

6∑i=1,j 6=i


; (22d)

UE =π3c

ωc11

6∑i=1

A24iα

21iZ

2i V c2

+

6∑i=1,j 6=i


; (22e)

UF =π3c

ωc66

6∑i=1

A24iα

22iZ

2i V c2

+

6∑i=1,j 6=i


; (22f)

UFS =π3c

ωc55

6∑i=1

A24iα

23iZ

2i V c2

+

6∑i=1,j 6=i


; (22g)

U =

∫ a

−a

∫ c

−c

∫ T

0

Udx1dx3dt

= UTSh + UTSt + UTT + UE + UF + UFS. (22h)

3.1.4. Energy percentages of each vibration modes

With all the expressions of strain energies of the plate ob-tained, we can calculate the energy proportions of each vibra-tion mode. In this study, in order to develop a procedure basedon the energy for mode identification, we focus not only on the

value of energy itself but also its distribution at different vibra-tion modes. In other words, we define the energy percentagesof each vibration mode as

PX =UX

U× 100%,

X =

TSh,F, AT - cut, two modes,TSh,F,FS, AT - cut, three modes,TSh,TT, TSt, E, F, FS, SC - cut, six modes;

(23)

where U is total of the strain energy of the plate which excludesthe coupling energy parts as:

U =

UTSh + UF,AT - cut, two modes,

UTSh + UF + UFS,AT - cut, three modes,

UTSh + UTSt + UTT + UE + UF + UFS,SC - cut, six modes;

while PX are the percentage of strain energy of the X mode.The strain energies of coupled modes are neglected due totheir insignificance. Such description will be enough for themeasurement of dominance of strain energies of each vibra-tion mode.

Since the kinetic energy in vibration also spread to all modesjust as the strain energy, and its equivalence to strain energyalso implies the possibility to use its proportions for vibra-tion mode identification. With the successful utilization of thestrain energy, we also want to explore the possibility and ap-plicability of kinetic energy in a similar manner.

3.2. The Kinetic EnergyWith known vibrations of the plate, we can calculate the ki-

netic energy and their proportions as we have demonstratedin the calculation of strain energy. We are expecting that theproportions of kinetic energy are consistent with the strain en-ergies for the determination of dominance of specific modeshapes.

The kinetic energy of the first-order Mindlin plate is8

K =1

2

∑m

∑n

ρBmnu(m)j u

(n)j

=1

2ρB00

3∑i=1

u(0)i u

(0)i +

1

2ρB11

3∑i=1

u(1)i u

(1)i . (24)

By separating the vibration displacements as for the calcu-lation of the strain energy, Eq. (24) can also be rewritten by thevibration modes as:

KTSh =1

2ρB11u

(1)1 u

(1)1 ; (25a)

KTSt =1

2ρB11u

(1)2 u

(1)2 ; (25b)

KTT =1

2ρB11u

(1)3 u

(1)3 ; (25c)

KE =1

2ρB00u

(0)1 u

(0)1 ; (25d)

KF =1

2ρB00u

(0)2 u

(0)2 ; (25e)

KFS =1

2ρB00u

(0)3 u

(0)3 ; (25f)



where u(n)j (n = 0, 1; j = 1, 2, 3, ) are the velocities of eachmode in vibrations. We can see from Eq. (25), in compari-son with the strain energy, the kinetic energy density equationsare not coupled. This is certainly advantageous over the strainenergies in their complexity for calculation.

With displacement solutions in Eq. (7), we also integrateEq. (25) over the plate for the three cases of couplings.

3.2.1. AT-cut quartz crystal plate with two vibrationmodes

For two modes only thickness-shear and flexural mode arecalculated here as:

KX =

∫ a

−a

∫ c

−c

∫ T

0

KXdx1dx3dt, X = TSh, F; (26a)

KTSh =4πb2ρc

3ω

2∑i=1

A24iα

24iVc2 +

2∑i=1,j 6=i

A4iA4jα4iα4jVcc

;

(26b)

KF =4πb2ρc

ω

2∑i=1

A24iα

22iVs2 +

2∑i=1,j 6=i

A4iA4jα2iα2jVss

;

(26c)

K =

∫ a

−a

∫ c

−c

∫ T

0

Kdx1dx3dt = KTSh +KF. (26d)

3.2.2. AT-cut quartz crystal plate with three vibrationmodes

The kinetic energy of thickness-shear, flexural, and face-shear mode are as:

KX =

∫ a

−a

∫ c

−c

∫ T

0

KXdx1dx3dt, X = TSh, F, FS; (27a)

KTSh =4πb2ρc

3ω

3∑i=1

A24iα

24iVc2 +

3∑i=1,j 6=i

A4iA4jα4iα4jVcc

;

(27b)

KF =4πb2ρc

ω

3∑i=1

A24iα

22iVs2 +

3∑i=1,j 6=i

A4iA4jα2iα2jVss

;

(27c)

KFS =4πb2ρc

ω

3∑i=1

A24iα

23iVs2 +

3∑i=1,j 6=i

A4iA4jα3iα3jVss

;

(27d)

K =

∫ a

−a

∫ c

−c

∫ T

0

Kdx1dx3dt = KTSh +KF +KFS. (27e)

3.2.3. SC-cut quartz crystal plate with six vibrationmodes

In an SC-cut plate all the kinetic energies of the six modesare considered as:

KX =

∫ a

−a

∫ c

−c

∫ T

0

KXdx1dx3dt, X = TSh,TSt, TT, E, F, FS;

(28a)

KTSh =4πb2ρc

3ω

6∑i=1

A24iα

24iVc2 +

6∑i=1,j 6=i

A4iA4jα4iα4jVcc

;

(28b)

KTSt =4πb2ρc

3ω

6∑i=1

A24iα

25iVc2 +

6∑i=1,j 6=i

A4iA4jα5iα5jVcc

;

(28c)

KTT =4πb2ρc

3ω

6∑i=1

A24iα

26iVc2 +

6∑i=1,j 6=i

A4iA4jα6iα6jVcc

;

(28d)

KE =4πb2ρc

ω

6∑i=1

A24iα

21iVs2 +

6∑i=1,j 6=i

A4iA4jα1iα1jVss

;

(28e)

KF =4πb2ρc

ω

6∑i=1

A24iα

22iVs2 +

6∑i=1,j 6=i

A4iA4jα2iα2jVss

;

(28f)

KFS =4πb2ρc

ω

6∑i=1

A24iα

23iVs2 +

6∑i=1,j 6=i

A4iA4jα3iα3jVss

;

(28g)

K =

∫ a

−a

∫ c

−c

∫ T

0

Kdx1dx3dt

=KTSh +KTSt +KTT +KE +KF +KFS. (28h)

3.2.4. Energy percentages of each vibration modes

Following the same procedure, the distributions of kineticenergies are also calculated as:

PX =KX

K× 100%,

X =

TSh,F, AT - cut, two modes,TSh,F,FS, AT - cut, three modes,TSh,TT, TSt, E, F, FS, SC - cut, six modes;

(29)



Figure 5. Frequency spectra of two vibration modes and specific length tothickness ratio with 11 resonant frequencies.

Table 1. Strain/Kinetic(S/K) energy distributions of the crystal plate at eachresonance with two vibration modes.

SerialNumber

NormalizedFrequency

Flexural(S/K %)

Thickness-shear(S/K %)

1 0.9213 86.76/88.87 13.24/11.132 0.9764 85.59/84.17 14.41/15.833 1.0017 2.45/1.05 97.55/98.954 1.0141 25.04/12.46 74.96/87.545 1.0285 69.67/52.86 30.33/47.146 1.0461 47.71/28.26 52.29/71.747 1.0733 57.04/37.54 42.96/62.468 1.0953 67.3/49.13 32.7/50.879 1.1237 61.75/42.49 38.25/57.5110 1.1509 68.49/50.48 31.51/49.5211 1.1800 68.57/50.49 31.43/49.51

where K is the sum of the kinetic energy of all vibration modesas:

K =

KTSh +KF,AT - cut, two modes,

KTSh +KF +KFS,AT - cut, three modes,

KTSh +KTSt +KTT +KE +KF +KFS,SC - cut, six modes,

and PX are the proportions of kinetic energies of X mode.The clear advantage of the kinetic energies is that there is nocoupling and the calculation can be simpler.

4. NUMERICAL RESULTS

With formulations of energy proportions in terms of dis-placements, we presented a simple model for energy-basedmethod for the identification of vibration modes in a quartzcrystal plate in the vicinity of thickness-shear vibrations. Withsolutions of displacements and frequencies, the calculations ofboth strain and kinetic energies are straightforward, and theidentification of vibration modes are also done with the nu-merical value corresponding to each displacement.

Figure 6. Displacements of two modes at a normalized frequency of 1.0017.

Figure 7. Displacements of two modes at a normalized frequency of 1.0953.

4.1. AT-cut quartz crystal plate with twovibration modes

The results have been tabulated in Table 1 for an AT-cutquartz crystal plate with given aspect ratios and thickness forthe plate model shown in Fig. 1. The model plate has fourfree sides and we choose the length to thickness ratio (a/b) to32.9148 for demonstration purposes. With the given aspect ra-tio, 11 modes in the frequency spectra and the identificationsolutions will be applied to these points shown in Fig. 5.

We use Eqs. (23) and (29) for the calculation and the en-ergy distributions of each frequency and the results are givenin Table 1.

We also use the displacement data of some frequenciesabove and plot a few selected mode shapes below for com-parisons with the results of Table 1, as shown in Figs. 6 and 7.

From Table 1, it can be seen that the energy distribution hasa series of fluctuations and the results of strain energy are sim-ilar to the kinetic energy in a systematic trend with small de-viations which are mainly caused by the coupling parts of thestrain energy at some frequencies. The working frequency aresuccessfully identified as the third frequency with the normal-



Figure 8. Frequency spectra of three vibration modes and specific length tothickness ratio of 14 resonant frequencies.

Table 2. Strain/Kinetic(S/K) energy distributions of the crystal plate at eachresonance with three vibration modes.

SerialNumber

NormalizedFrequency

Flexural(S/K %)

Face-shear(S/K %)


1 0.9200 76.4/88.6 0.4/0.4 23.2/11.02 0.9745 72.0/78.9 3.9/6.4 24.1/14.63 0.9818 9.2/5.5 97.2/98.4 3.0/1.04 1.0017 2.0/1.6 0.0/0.0 99.4/98.45 1.0139 17.7/13.7 0.1/0.1 82.2/86.26 1.0276 53.9/53.4 0.3/0.3 45.7/46.27 1.0457 29.1/26.1 0.2/0.1 70.7/73.88 1.0723 39.3/36.0 13.5/13.3 50.5/50.79 1.0752 5.0/3.9 77.7/81.3 17.3/14.810 1.0946 49.9/48.1 0.6/0.7 49.5/51.211 1.1230 44.9/42.7 0.2/0.1 55.0/57.212 1.1499 51.5/49.7 0.8/0.9 47.4/49.313 1.1685 1.5/1.0 97.5/98.5 1.0/0.514 1.1791 51.7/50.0 0.4/0.3 47.9/49.7

ized frequency of 1.0017, and the thickness-shear vibrationsare considered as the dominant mode because of its highestenergy proportion with the percentage value of 98%, almost100%, implying this frequency represents the optimal func-tioning mode. The conclusion is also validated by the Fig. 10.With different material types, the effect of energy terms of cou-pled modes can be significant, as shown at some frequencies inTable 1. Consequently, few modes examined in this materialtype show that the influences of the coupled strain energy partsare big in some rows while the kinetic energy shows their per-centages are nearly half to half. Fig. 7 also supports the view-point that the strong coupling may cause deviations of strainenergy percentages. The kinetic energy solution is more reli-able.

4.2. AT-cut quartz crystal plate with threevibration modes

The results have been tabulated in Table 2 for an AT-cutquartz crystal plate with three modes and the model is same asthe case of two modes. We also chose the length to thicknessratio (a/b) to 32.9148 for demonstration purposes. With thegiven aspect ratio, 14 frequencies are in the frequency spec-tra and the identification techniques based on energy will beapplied to these points shown in Fig. 8.

Figure 9. Displacements of the three modes at a normalized frequency of1.0017.

Figure 10. Displacements of the three modes at a normalized frequency of1.0752.

Similarly, the energy distributions of each frequency are cal-culated and listed in the Table 2.

From Table 2, it is clear that the energy distribution has fluc-tuations and the results of strain energy are similar to the ki-netic energy in a systematic trend with only some minor de-viations which are mainly caused by the coupling parts of thestrain energy. To see the details, we take a look at the 4th fre-quency with the mode feature which is obviously simple withenergy distributions. At the 4th frequency where the normal-ized frequency is 1.0017, and the thickness-shear vibration isthe dominant mode because of its highest energy proportion at99%, almost 100%, implying this frequency can be consideredas the optimal functioning mode. For the 5th column, we cansee that the strain energy distribution is varying from the lowest1.0 percent to the highest 99.4 percent and we can quickly lo-cate the mode according to these values. Thus, it demonstratedthat the energy method can help us quickly and effectively im-plement the mode identification technique and find the optimalfunctioning mode.

For further validation, we took the displacement data of eachfrequency from the above model and plot mode shapes individ-



Figure 11. Frequency spectra of six vibration modes and specific length tothickness ratio of 12 resonant frequencies.

ually, as shown in the Figs. 9 and 10.The mode shapes above clearly show that they are consis-

tent with the findings in Table 2. Taking Fig. 9 as an example,the amplitude of thickness-shear mode has the clear dominanceand obviously it is the functioning mode of the thickness-shearvibrations, when its energy distribution reaches 99.4% instrain and 98.4% in kinetic energies, respectively. This cor-relation also applies to other modes and their mode shapes,proving the energy identification method is correct.

4.3. SC-cut quartz crystal plate with sixvibration modes

The results have been presented in Table 3 for the SC-cutquartz crystal plate and we also chose the length to thick-ness ratio (a/b) to 34.4097 for demonstration purposes. Mean-while for the SC-cut plate the frequency of working modes ofthickness-shear vibrations are lower than the AT-cut and wechose a different frequency range for the examination. Withthe given aspect ratio, we chose 12 modes in the frequencyspectra and the identification method will be applied to thesefrequency points as shown in Fig. 11.

By examining the Table 3, the domain modes are also clearlyidentified by both strain and kinetic energy solutions. Differ-ent from the AT-cut quartz crystal plates above, the concen-tration of energy of the functioning thickness-shear mode isabout 91% due to more couplings. Some deviations occurwhen the flexural mode couples with other modes like as inthe 4th or 11th frequencies. Other than this, predictions ofdominant modes can be trusted based on energy distributions.The further validation of mode shapes are given with Figs. 12and 13.

From Fig. 12, we find the prediction of mode shapes with ki-netic energy pointing to the flexural mode which is clearly thedominant mode. As we know, the SC-cut plate is more likelyto be affected by the spurious modes due to stronger couplings.The strain energy solutions are more sensitive to stronger modecouplings as we chose to neglect the coupled terms in strain en-ergies. We can see the kinetic energy is better in determiningthe feature of strong couplings of vibration modes. In general,

Figure 12. Displacements of the extension and flexural modes from six modesat a normalized frequency of 0.9266.

Figure 13. Displacements of the six modes at a normalized frequency of0.9408.

both methods based on energy are reliable in identifying thedominant vibration modes as shown in Fig. 13.

4.4. Energy analysis with experimental data

Since the energy method is accurate with the theoreticalanalysis, we used the experimental data by Yamashita et al.41

for comparison. The models of the experiment are the rectan-gle AT-cut quartz plates with length-to-thickness aspects from10.33 to 19.40 and width-to-thickness aspect as 3.78. We com-pared the same frequency spectra with mode identification asshown in Fig. 14.

The experimental data marked with “x” are the thickness-shear modes. The selected four points A, B, C, and D arenear the middle of the curves and the dominant thickness-shearmodes from the theoretical analysis.15 By taking the plate in-formation of the four points, we calculated energies of flexuraland thickness-shear modes and list them in Table 4.

In Table 4, the predictions from energy of the mode char-acters of four points are consistent with the experimental data.The energy values are closer to the 90% level. It shows the



Table 3. Strain/Kinetic(S/K) energy distributions of the crystal plate at each resonance with six vibration modes.

SerialNumber

NormalizedFrequency

Extension(S/K %)

Flexural(S/K %)

Face-shear(S/K %)

Thickness-stretch(S/K %)

Thickness-twist(S/K %)


1 0.8669 0.36/2.89 90.39/80.54 0.21/0.86 0.58/1.89 0.15/0.32 8.31/13.482 0.9052 11.18/12.88 3.43/0.55 78.88/83.05 0.15/0.11 6/3.3 0.37/0.113 0.9142 1.72/15.45 87.79/64.57 0.5/2.95 0.65/1.96 0.17/0.34 9.17/14.724 0.9266 21.56/69 65.01/15.27 3.55/10.15 0.5/0.5 0.24/0.22 9.13/4.875 0.9408 0/0.01 1.87/0.55 0/0 2.61/5.69 1.82/2.6 93.69/91.156 0.9504 0.09/0.27 20.21/6.58 0.05/0.07 2.29/5.6 1.94/3.18 75.42/84.297 0.9626 0.38/2.16 74.93/45.42 0.21/0.6 1.08/4 0.75/2.33 22.64/45.498 0.9717 11.2/12.63 0.94/0.2 78.44/81.03 0.22/0.22 8.07/5.29 1.13/0.639 0.9743 0.26/0.96 55.3/24.29 0.31/0.76 1.57/5.05 1.95/4.66 40.61/64.2810 0.9963 0.18/0.66 48.95/20.2 0.14/0.24 1.9/5.71 4.16/9.08 44.66/64.1211 1.0099 14.16/51.71 62.62/18.43 2.42/7.72 0.97/1.78 1.74/2.87 18.09/17.4912 1.0181 5.99/28.08 64.33/24.44 1.28/4.58 1.27/3.31 4.42/9.64 22.7/29.95

Figure 14. Frequency spectra of the experimental and theoretical data.

Table 4. Strain/Kinetic(S/K) energy distributions of the crystal plate at eachresonance with experimental data.

Expe-rimentPoints

NormalizedFrequency

Length /thicknessratio

Flexural(S/K %)


DominantMode

A 1.01158 12.0867 14.41/6.84 85.59/93.16 Thickness-shear

B 1.00833 13.5978 11.75/5.29 88.25/94.71 Thickness-shear

C 1.00609 15.4778 3.79/8.3 96.21/91.70 Thickness-shear

D 1.00545 17.0067 8.65/3.9 91.35/96.1 Thickness-shear

consistence of frequency spectra and mode properties of theMindlin plate theory and the predictions based on energies ofvibration modes.

5. CONCLUSIONS

The objective of this study was to identify the vibrationmodes from the coupled vibrations with the Mindlin plateequations at high frequency by means of the energy proportionfrom displacement solutions. The formulations of kinetic andstrain energy densities of the first-order Mindlin plate has beenintroduced and the total energy is obtained. The vibrations ofa simple rectangular AT-cut quartz crystal plate with four freeparallel sides were considered without including the electrodesand frequency-temperature effect. The energy distributions ofeach vibration mode were calculated with the given plate con-figurations. The comparison between the energy distributions

and the mode shape patterns confirmed that the proposed ap-proach actually identified the vibration modes accurately andwas able to single out the optimal working mode without usingmode shape plots. More importantly, the method was purelynumerical and is well suited with the FEM and other numericalsolutions. Particularly, using the kinetic energy for the energydistributions of each mode will be a simpler calculation.

Calculating the kinetic and strain energies of the plate withina given frequency range and obtaining the energy distributionof each vibration mode were demonstrated to be simple andeasy as exhibited in this study. The conclusion from the en-ergy distribution were consistent with mode shape plots. Al-though there were some small differences between kinetic andstrain energies with the strongly coupled vibration modes, suchmethod of mode identification was simple and reliable. In gen-eral, the energy based method for mode identification led tothe accurate characterization of vibration modes without dif-ference. Overall this was a convenient and efficient methodfor the identification of vibration modes with strongly coupledmodes and high frequency in structures of anisotropic materi-als and complex configurations. Furthermore, this energy ap-proach will be particularly preferential in finite element analy-sis because the energy calculation and characterization can beeasily done without utilizing complicated visualization func-tions and tools. In addition, it was easy to be automated withdata flow and the numerical procedure. It is expected that theenergy based approach for vibration mode identification canbe adopted to meet similar requirements in the finite elementanalysis of general structures.

5.1. Acknowledgement

This work is supported by the National Natural ScienceFoundation of China (Grants 11672142 and 11772163) and theK. C. Wong Magana Fund through Ningbo University. Ad-ditional funding is from the National Key R&D Program ofChina (Grant 2017YFB1102900) and the Technology Innova-tion 2025 Program of the City of Ningbo (Grant 2019B10122).

APPENDIX A

Equations and parameters for the vibrationequations

The material parameters of quartz crystal and its AT- andSC-cut can be found in Refs.3, 5, 8, 11, 14



The matrix M and N in Eq. (6) and Eq. (10) are given forthree cases.

1) AT-cut quartz crystal plates with two vibration modes

The dispersion relation from Eq. (6) is given as:∣∣∣∣∣Z2π2 − 12Ω2 2Zπ

Zπ 2(c11c66Z2 − Ω2 + 1

)∣∣∣∣∣ = 0. (A1)

The frequency equation in Eq. (10) is given as:∣∣∣∣(α22πZ2

2 +1)

cos(πZ2

2ab

) (α24

πZ2

2 +1)

cos(πZ4

2ab

)Z2 sin

(πZ2

2ab

)Z4 sin

(πZ4

2ab

) ∣∣∣∣=0.

(A2)

2) AT-cut quartz crystal plates with three vibration modes

The dispersion relation from Eq. (6) is given as:∣∣∣∣∣∣∣π2

12Z2 − Ω2 c65

c66π

2√3Z2 πZ

6c56c66

π2√3Z2 c55

c66Z2 − Ω2 c56

c66Z√3

π2Z

c65c66

√3Z c11

c66Z2 − Ω2 + 1

∣∣∣∣∣∣∣ = 0.

(A3)

The frequency equation from Eq. (10) is given as:∣∣∣∣∣∣∣∣∣

[α32

πZ2

2 + κ6c56c55

(α22

πZ2

2 + 1)]

cos(πZ2

2ab

)[c65c66α32

πZ2

2 + κ6(α22

πZ2

2 + 1)]

cos(πZ2

2ab

)Z2 sin

(πZ2

2ab

) · · ·

· · ·

[α33

πZ3

2 + κ6c56c55

(α23

πZ3

2 + 1)]

cos(πZ3

2ab

)[c65c66α33

πZ3

2 + κ6(α23

πZ3

2 + 1)]

cos(πZ3

2ab

)Z3 sin

(πZ3

2ab

) · · ·

· · ·

[α34

πZ4

2 + κ6c56c55

(α24

πZ4

2 + 1)]

cos(πZ4

2ab

)[c65c66α34

πZ4

2 + κ6(α24

πZ4

2 + 1)]

cos(πZ4

2ab

)Z4 sin

(πZ4

2ab

)∣∣∣∣∣∣∣∣∣ = 0.

(A4)

3) SC-cut quartz crystal plates with six vibration modes

The dispersion relation from Eq. (6) is given as:

|Mij| = 0, i, j = 1, 2, . . . 6;

M11 = κ(0)1 c11Z

2 − c66Ω2,M12 = κ(0)1 κ

(0)6 c16Z

2,M13

= κ(0)1 c15Z

2;

M14 = κ(0)1 κ

(0)6 c16

2

πZ,M15 = κ

(0)1 κ

(0)2 c12

2

πZ,M16

= κ(0)1 κ

(0)4 c14

2

πZ;

M21 = κ(0)6 c61Z

2,M22 = κ(0)6 κ

(0)6 c66Z

2 − c66Ω2,M23

= κ(0)6 c65Z

2;

M24 = κ(0)6 κ

(0)6 c66

2

πZ,M25 = κ

(0)6 κ

(0)2 c62

2

πZ,M26

= κ(0)6 κ

(0)4 c64

2

πZ;

M31 = κ(0)5 c51Z

2,M32 = κ(0)6 κ

(0)5 c56Z

2,M33

= κ(0)5 c55Z

2 − c66Ω2;

M34 = κ(0)6 κ

(0)5 c56

2

πZ,M35 = κ

(0)5 κ

(0)2 c52

2

πZ,M36

= κ(0)5 κ

(0)4 c54

2

πZ;

M41 = κ(0)6 c61

π

2Z,M42 = κ

(0)6 κ

(0)6 c66

π

2Z,M43

= κ(0)6 c65

π

2Z;

M44 = κ(0)6 κ

(0)6 +

π2

12

(κ(1)1 c11Z

2 − Ω2)

;

M45 =c163κ(1)6 κ

(1)1

(π2Z)2

+ c62κ(0)2 κ

(0)6 ;

M46 =c153κ(1)1

(π2Z)2

+ c64κ(0)4 κ

(0)6 ;

M51 = κ(0)2 c21

π

2Z,M52 = κ

(0)2 κ

(0)6 c26

π

2Z,M53

= κ(0)2 c25

π

2Z;

M54 =c613κ(1)6

(π2Z)2

+ c26κ(0)2 κ

(0)6 ;

M55 =c663κ(1)6 κ

(1)6

(π2Z)2

+ c22κ(0)2 κ

(0)2 −

c663

(π2

)2Ω

2

;

M56 =c653κ(1)6

(π2Z)2

+ c24κ(0)4 κ

(0)2 ;

M61 = κ(0)4 c41

π

2Z,M62 = κ

(0)4 κ

(0)6 c46

π

2Z,M63 = κ

(0)4 c45

π

2Z;

M64 =c513κ(1)5

(π2Z)2

+ c46κ(0)4 κ

(0)6 ,M65

=c563κ(1)6 κ

(1)5

(π2Z)2

+ c42κ(0)2 κ

(0)4 ;

M66 =c553κ(1)5

(π2Z)2

+ c44κ(0)4 κ

(0)4 −

c663

(π2

)2Ω

2

.

(A5)

The frequency equation from Eq. (10) is given as:

|Njr| = 0, j, r = 1, 2, . . . 6;

N1r =[(c61α1r + c65α3r + κ

(0)6 c66α2r

) π2Zr+κ

(0)2 c62α5r

+κ(0)4 c64α6r + κ

(0)6 c66α4r

]cos(πZr

2

a

b

);

N2r =[(c51α1r + c55α3r + κ

(0)6 c56α2r

) π2Zr+κ

(0)2 c52α5r

+κ(0)4 c54α6r + κ

(0)6 c56α4r

]cos

(πZr

2

a

b

);

N3r =[(c11α1r + c15α3r + κ

(0)6 c16α2r

) π2Zr+κ

(0)2 c12α5r

+κ(0)4 c14α6r + κ

(0)6 c16α4r

]cos

(πZr

2

a

b

);

N4r =(c15α6r + c16κ

(1)6 α5r + c11α4r

)Zr sin

(πZr

2

a

b

);

N5r =(c55α6r + c56κ

(1)6 α5r + c51α4r

)Zr sin

(πZr

2

a

b

);



N6r =(c65α6r + c66κ

(1)6 α5r + c61α4r

)Zr sin

(πZr

2

a

b

).

(A6)

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Identiﬁcation of Vibration Modes of Quartz Crystal Plates