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AIP Advances ARTICLE scitation.org/journal/adv
Analysis of thermoelastic damping limited quality
factor and critical dimensions of circular plate
resonators based on axisymmetric
and non-axisymmetric vibrations
Cite as: AIP Advances 11, 035108 (2021); doi: 10.1063/5.0033087
Submitted: 15 October 2020 •Accepted: 30 January 2021 •
Published Online: 3 March 2021
Resmi R,1,a) V. Suresh Babu,2,b) and M. R. Baiju3,c)
AFFILIATIONS
1University of Kerala, LBS Institute of Technology for Women, Thiruvananthapuram, Kerala 695012, India
2APJ Abdul Kalam Technological University, Govt. Engineering College, Wayanad, Kerala 670644, India
3University of Kerala, Kerala Public Service Commission, Kerala 695004, India
a)Author to whom correspondence should be addressed: resmilbs@gmail.com and resmi@lbsitw.ac.in
b)vsbsreeragam@gmail.com and vsb@cet.ac.in
c)mrbaiju@gmail.com
ABSTRACT
Thermoelastic damping effects are very important intrinsic losses in microelectromechanical system/nanoelectromechanical system based
sensors and filters, which limit the maximum achievable quality factor. Thermoelasticity arises due to coupling between the temperature
field and elastic field of the material and its interaction within the material structure. The impacts of axisymmetric and non-axisymmetric
vibrations, plate dimensions, material parameters, boundary conditions, mode switching, and temperature on thermoelastic damping limited
quality factors (QTED) and critical thickness (hc) were analyzed, and the conditions for an enhanced quality factor were optimized in this
work. The analytical models of circular plate resonators have been developed in terms of material performance indices for axisymmetric and
non-axisymmetric vibrations. QTED and hc were analyzed based on two boundary conditions: simply supported and clamped–clamped. In
order to obtain maximum QTED, micro-circular plates with diamond as the structural material operating at a lower temperature and with
non-axisymmetric vibrations are proposed in this paper.
©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license
(http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0033087
., s
I. INTRODUCTION
The quality factor is one of the important design parameters of
a micro-/nano-resonator for many applications, which is the ratio
of the amount of energy stored in the system to the amount of
energy dissipated by the system.1Thermal currents generated due to
the compression/decompression in elastic media cause thermoelas-
tic damping.2,3 Because of the reliability of batch fabrications, low
power consumption, small size, high sensitivity, fast response,4–6
and many other potential applications in engineering, microelec-
tromechanical system (MEMS) based devices are widely used as sen-
sors, filters, modulators, and actuators.7–9 To evaluate thermoelastic
damping in vibrating beams, Zener2,10 derived an analytical solution
for energy dissipation, which is expressed as
Q−1
zener =Δˆ
E
2π
Emax =Eα2
TT0
Cv
ωτ
1 + (ωτ)2, (1)
where Δˆ
E,ˆ
Emax, E, αT,T0,Cv,ω, and kare the energy dissipated
per cycle of vibration due to irreversible heat conduction, the maxi-
mum stored elastic energy in the structure during vibration, Young’s
modulus, the coefficient of thermal expansion, the equilibrium tem-
perature of the beam, the specific heat per unit volume, the angular
frequency of the vibration, and the thermal conductivity, respec-
tively. The characteristic time constant for thermal relaxation with
AIP Advances 11, 035108 (2021); doi: 10.1063/5.0033087 11, 035108-1
© Author(s) 2021
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thickness his expressed as
τ=h2Cvπ2k. (2)
To study the effect of different geometrical parameters, Lifshitz
and Roukes11 derived an analytical expression for the thermoelastic
damping limited quality factor in microbeams by improving Zener’s
work, which is expressed as
Q−1
LR =Eα2
TT0
Cv
6
ξ21−1
ξsinh ξ+ sin ξ
cos ξ+ cosh ξ, (3)
where
ξ=hωCv2k.
In many resonators and sensors, the circular plate is considered
a commonly used structure with a wide range of applications in IR
detection and imaging, in chemical and biological sensing, as highly
sensitive temperature sensors, and as interferometric gravitational-
wave detectors. Sun and Saka12 investigated the thermoelastic damp-
ing (TED) of circular resonators in general vibration modes in the
context of the coupled theory of thermoelasticity. Hao13 analyzed
TED of in-plane vibration in a circular plate resonator by using the
thermal-energy method. Sun and Tohmyoh14 further studied TED
of axisymmetric flexural vibration in circular plate resonators.
By using the GTE theory of the dual phase lagging model,
Guo15 analyzed the thermoelastic dissipation in circular micro-plate
resonators. Li16 compared the thermoelastic damping in rectan-
gular and circular microplate resonators of previous models with
that of the FEM model. Salajeghe17 studied the nonlinear analy-
sis of TED in axisymmetric vibration of micro-circular thin-plate
resonators. Hayati18 analyzed axisymmetric problems by using the
mesh-free element-free Galerkin method. Farahani19 extended the
radial point interpolation method (RPIM) to the elasto-static anal-
ysis of circular plates. By using the finite element method, Hin-
ton20 analyzed the dynamic transient of axisymmetric circular plates.
Salehi21 presented non-axisymmetric circular viscoelastic plates by
using dynamic relaxation (DR) non-linear shear deformable govern-
ing equations. Using the finite element method, Pardoen22 discussed
the asymmetric vibration and stability of circular and annular plates.
Without using the reduced integration technique, Chen23 developed
a general approach axisymmetric finite element analysis by consti-
tuting a non-conforming displacement function. For an axisymmet-
ric finite element formulation, Clayton24 reported the integration of
the element stiffness matrix. Using the first-order shear deforma-
tion Mindlin plate theory, Reddy25 studied the axisymmetric bend-
ing and stretching of functionally graded solid and annular circular
plates. Kwak26 studied the effect of axisymmetric vibration of cir-
cular plates in contact with fluid. Ma27 employed the third-order
shear deformation plate theory to solve the axisymmetric bend-
ing and buckling problems of functionally graded circular plates.
Using a nonlocal plate theory, Duan28 studied the axisymmetric
bending of micro-/nano-scale circular plates. To find accurate nat-
ural frequencies and mode shapes for the flexural vibrations of
thick free circular plates, Hutchinson29 used a series of solutions of
the general three-dimensional equations of linear elasticity. Based
on the unconstrained third-order shear deformation plate theory
(UTST), Sahraee and Saidi30 studied the axisymmetric bending and
buckling of perfect functionally graded solid circular plates. Reddy
and Huang31 presented conventional and mixed finite element for-
mulations of axisymmetric circular plates.
Singh32 used the Rayleigh–Ritz method to study the axisym-
metric transverse vibration of a circular plate with two differ-
ent linear variations in thickness. Gallego Juárez33 analyzed the
axisymmetric vibration of circular plates with a single circular step.
Eisenberger34 found the exact axisymmetric vibration frequencies
of circular and annular variable thickness plates using the exact
element method. Under random excitation, the nonlinear axisym-
metric behavior of circular isotropic plates with a linearly vary-
ing thickness was investigated by Do˘
gan.35 Lamacchia36 studied
the circumferentially distributed bending moments of a thin annu-
lar plate under non-linear deformation. Using the Zhemochkin
method, Bosakov37 solved the non-axisymmetric contact problem
for a circular plate. The analytical development for thermoelas-
tic damping was illustrated numerically for silicon-like material by
Chugh and Pratap.38 Ma39 analyzed the impact of radial preten-
sion on thermoelastic damping of micromechanical circular plate
resonators.
For microelectromechanical system (MEMS)/nanoelectro-
mechanical system (NEMS) structures, it is desired to design and
fabricate systems with as little loss of mechanical energy as pos-
sible. Thermoelastic damping is an intrinsic energy dissipation
mechanism in micro-/nano-mechanical resonators, which limits the
maximum achievable quality factor and thus causes performance
deterioration. It is necessary to accurately analyze and control ther-
moelastic damping in circular plate resonators to get enhanced qual-
ity factors and better sensitivity. In this work, to curtail energy
dissipation, the impacts of types of vibration such as axisymmet-
ric and non-axisymmetric modes, plate dimensions, mode switch-
ing, boundary conditions, and temperature on the thermoelastic
damping limited quality factor (QTED) and critical thickness (hc) are
investigated.
Thermoelastic coupled equations of a thin circular plate res-
onator intended for axisymmetric and non-axisymmetric vibrations
were derived in terms of material performance index parameters
shown in Sec. II. In Sec. III, expressions for thermoelastic damping
limited quality factors for both axisymmetric and non-axisymmetric
vibrations are formulated. QTED and hc were simulated numeri-
cally using MATLAB 2015, and investigations have been carried
out. Section IV deals with the impacts of axisymmetric and non-
axisymmetric modes of vibrations, plate dimensions, mode switch-
ing, boundary conditions, and temperature variations on the ther-
moelastic damping limited quality factor and critical thickness of
the circular plates. The analyses of resonance frequency associated
with axisymmetric and non-axisymmetric vibrations have also been
carried out in Sec. IV.
II. FORMULATION OF BASIC EQUATIONS
OF CIRCULAR PLATE RESONATORS BASED
ON AXISYMMETRIC AND NON-AXISYMMETRIC
VIBRATIONS
In this work, a homogeneous, isotropic circular plate resonator
of radius “R” is considered with thermoelastic vibrations. The cylin-
drical coordinate system (r,θ, z) is selected to analyze the vibra-
tions; the origin of the coordinate system is placed at the center
of the plate, and the r–θplane corresponds to the mid-plane of
AIP Advances 11, 035108 (2021); doi: 10.1063/5.0033087 11, 035108-2
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the plate. The axial, circumferential, and radial directions are defined
as w(r,θ,z,t), v(r,θ,z,t), and u(r,θ,z,t), respectively, and the tem-
perature field is defined as T(r,θ,z,t). While taking the initial con-
ditions, the resonator is considered as unstressed and unstrained in
equilibrium by keeping the environment temperature at T0.14
For analysis, it is assumed that the plane remains neutral dur-
ing bending, and points of the plate are considered normal to the
middle surface before and after bending. The deformation along
the middle surface is neglected. The governing equations of the
circular plate resonator are formulated based on Kirchhoff–Love
(KL) plate theory. According to KL plate theory, normal stress
σzz and the deformation along the middle surface are neglected,
i.e., σzz = 0 and εzz = 0. The rectilinear element to the middle
surface remains perpendicular to the strained surface both before
and after deformation, and the elongation is also neglected, i.e., εrz
=εθz= 0.12 The structure and the coordinate system are illustrated
in Fig. 1.
Transverse deflection of the plate under forced harmonic
vibration is
w(r,θ,z,t)=W(r,θ,t)eiωt, (4)
where W(r,θ,t) is the displacement function.
Due to thermoelastic damping,
Θ=T−T0=Θ0(r,θ,z)eiωt. (5)
The stress components for the thin plate are as follows:
Normal stress:
σrr(r,θ,t)=ˆ
σr(r,θ)eiωt,
σθθ(r,θ,t)=ˆ
σθ(r,θ)eiωt,
σzz(r,θ,t)=0.
(6)
Shear stress:
ψrθ(r,θ)=ˆ
ψrθ(r,θ)eiωt,
ψrz(r,θ)=ψθz(r,θ)=0. (7)
FIG. 1. Schematic drawing of a circular plate and the coordinate system.
Here,
ˆ
σr(r,θ)=−C1zWrr
r+vWθθ
r2+C2
EαTΘ0,
ˆ
σθ(r,θ)=−C1zWθθ
r2+vWrr
r+C2
EαTΘ0,
ˆ
ψrθ(r,θ)=−C2zW,r
r,
C1=E
mn
,
C2=E
m
,
where two new material dependent parameters mand nwere derived
from Poisson’s ratio and are given in Table III.
The strain components are as follows:
Normal strain:
εr(r,θ,t)=ˆ
εr(r,θ)eiωt,
εθ(r,θ,t)=ˆ
εθ(r,θ)eiωt,
εz(r,θ,t)=−v
E[σθθ(r,θ,t)+σrr (r,θ,t)]+αTΘ0
=(m−n)
2E[σθθ(r,θ,t)+σrr (r,θ,t)]+αTΘ0. (8)
Shear strain:
γrθ(r,θ, t)=ˆ
γrθ(r,θ)eiωt,
γrz=γθz=0, (9)
where
ˆ
εr(r,θ)=−zW,rr ,
ˆ
εθ(r,θ)=−zW,r
r+W,θθ
r2,
ˆ
γrθ(r,θ)=−2
rzW,rθ.
Consider the thermal conduction equation
Cv˙
Θ=k∇2Θ−C3αTT0˙
ε, (10)
where
C3=E
p
,
ε=εrr +εθθ +εzz.
pis also derived from Poisson’s ratio, as shown in Table III.
A. Axisymmetric vibrations
i. εzz =0,
ii. ε=εrr +εθθ.
Hence, Eq. (10) becomes
Cv˙
Θ=kΘzz −C3αTT0˙
εθ, (11)
kΘzz =iωCvΘ0−iωC3αTT0, (12)
AIP Advances 11, 035108 (2021); doi: 10.1063/5.0033087 11, 035108-3
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TABLE I. Values of qkl (k = 1, 2, 3 and l= 0, 1, 2, 3).
CC SS
l q1lq2lq3lq1lq2lq3l
0 9.85 39.43 88.73 5.5460 30.19 74.56
1 22.18 61.62 120.780 15.405 49.91 104.14
2 39.43 88.73 157.753 30.195 74.56 138.65
3 61.62 120.78 199.6569 49.914 104.14 178.08
where ωis the vibration frequency of the circular plate and is
expressed as
ω=ω01+r′(1 + f(ωkl)), (13)
where f(ωkl) is the term introduced for indicating the coupled
thermoelastic field. The thermoelastic coupling term for frequency
f(ωkl) is not included in classical thermoelasticity theories intro-
duced by Zener as well as by Lifshitz and Roukes.
ω0is the frequency of vibration in the absence of thermoelastic
damping (TED) and is given by
ω0=qklhC4
r2,
C4=C1
12ρ,
r′=Eα2
TT0
Cv
.
(14)
r′is the term associated mainly with temperature dependency. For
indicating the different modes of operation, kand lare denoted with
k=1,2,...and l=1,2,....
qkl values for simply supported (SS) and clamped–clamped
(CC) boundary conditions are given in Table I.
B. Non-axisymmetric vibrations
i. εzz ≠0,
ii. ϵ=εrr +εθθ +εzz, and
iii. εzz =C2
E[−vεrr +εθθ +nαTΘ0]=C2
E(m−n)εrr
2+εθθ +nαTΘ0.
Thus, Eq. (7) becomes
(Cv+C5αTT0)˙
Θ=kΘ,zz −C2αTT0˙
εθ, (15)
where
C5=C3
n
m
.
Since C5αTT0≪Cv, Eq. (15) becomes
Cv˙
Θ=kΘ,zz −C2αTT0˙
εθ. (16)
III. THERMOELASTIC DAMPING LIMITED QUALITY
FACTOR OF CIRCULAR PLATES
The boundary condition nΘz=0atz=±h/2 is applied.
The magnitude of thermoelastic damping for the circular
plate is
Θ0(r,Θ,z)=C3
E
r′
αT
z∇2˙
W
z−sin(gz)
gcosgh
2
, (17)
where
g=(1−i)ω
2χ,
Q−1
TED =1
2π
Δ˜
E
˜
Emax
.
(18)
The energy dissipation per cycle (Δ˜
E) is expressed as
Δ˜
E=Q−1
LRn
pY, (19)
where
Y=πh3C2
4∫2π
0∫r
0φ1rdrdθ,
φ1=W,rr +W,r
r+W,θθ2
,
and the maximum stored elastic energy (˜
Emax) is
˜
Emax =h3
2C2
4∫2π
0∫r
0{φ1−φ2−φ3}rdrdθ, (20)
where
φ2=2mW,rrW,r
r+W,θθ
r2,
φ3=W,rθ
r−W,θθ
r22
.
Thus, the thermoelastic damping for axisymmetric vibration in
terms of material indices is expressed as
Q−1
TED1=Q−1
LRn
p. (21)
By following the derivation of axisymmetry, we obtain
Q−1
TED =1
2π
Δˆ
E
˜
Emax
, (22)
where
Δˆ
E=pC5Δ˜
EQ−1
LR.
The thermoelastic damping for non-axisymmetric vibration in
terms of material indices can be expressed as
Q−1
TED2=Q−1
LRn
m. (23)
Table II gives the thermal properties such as thermal conductivity
(k), specific heat at constant pressure (Cp), the coefficient of linear
thermal expansion (α), thermal diffusivity (χ), and specific heat at
constant volume (Cv) and mechanical properties such as Young’s
AIP Advances 11, 035108 (2021); doi: 10.1063/5.0033087 11, 035108-4
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TABLE II. Thermal and mechanical properties of five structural materials at
T0= 298 K. Reproduced with permission from Resmi R, M. R. Baiju, and V. Suresh
Babu, AIP Conf. Proc. 2166, 020029 (2019). Copyright 2019 AIP Publishing LLC.
PolySi Diamond Si GaAs SiC
E(GPa) 150 800 130 85.9 415
v0.226 0.069 0.28 0.31 0.192
ρ(kg/m3) 2 328 3 515 2 230 5 316 3 200
k(W/m/K) 40 100 90 52 70
Cp[(J/kg)/K] 713 510 699 550 937.5
α(10−6) 2.3 1.2 2.59 5.73 3.0
χ(cm2/s) 0.241 0.558 0.577 0.178 0.233
Cv[(J/kg)/K] 1 659 864 1 792 650 1 558 770 2 923 800 3 000 000
modulus (E), Poisson’s ratio (v), and density (ρ) of polySi, dia-
mond, Si, GaAs, and SiC, which were used as different structural
materials.40
Table III shows that the material dependent performance
indices and the ratio of the parameters can be taken as a scaling
factor for expressing energy dissipation in terms of the classical
elasticity theory. The expression for the thermoelastic damping lim-
ited quality factor of the circular plate resonator can be derived in
TABLE III. Material dependent performance index parameters derived from Poisson’s
ratio of the structural material for a circular plate.
Parameters PolySi Diamond Si GaAs SiC
ma0.774 0.931 0.72 0.69 0.808
nb1.226 1.069 1.28 1.31 1.192
pc0.548 0.862 0.44 0.38 0.616
a(1 −v).
b(1 + v).
c(1 −2v).
terms of material performance indices and Lifshitz and Rouke’s (LR)
expressions, as shown in Eqs. (21) and (23).
IV. RESULTS AND DISCUSSIONS
In this section, the relationships between energy dissipa-
tion, material performance parameters, plate dimensions, types of
vibrations (axisymmetric and non-axisymmetric), vibrating modes,
boundary conditions, and temperature for a circular plate res-
onator were examined. Besides, the influences of axisymmetric and
non-axisymmetric vibrations, material parameters, vibration modes,
boundary conditions, and plate dimensions on critical thickness (hc)
TABLE IV. QTED and hc of circular plate resonators with axisymmetric vibrations for both boundary conditions [simply
supported (SS) and clamped–clamped (CC)]; R/h = 50; T0= 298 K; mode (k,l) with k= 1 and l= 1.
Mode B.C. Material PolySi Diamond Si GaAs SiC
I
SS
QTED 6306.761 8463.817 4144.562 2028.398 2799.866
hc (μm) 16.201 20.601 40.401 23.401 11.201
CC
QTED 6307.159 8463.817 4144.562 2028.398 2799.866
hc (μm) 11.2 14.2 28 16.2 7.8
II
SS
QTED 6307.157 8463.817 4144.562 2028.398 2800.258
hc (μm) 8.401 10.401 20.602 12.001 5.601
CC
QTED 6307.159 8463.817 4144.562 2030.498 2799.944
hc (μm) 6.401 8.001 15.8 9.201 4.401
III
SS
QTED 6306.761 8464.534 4144.562 2028.398 2800.101
hc (μm) 5.001 6.4 12.401 7.201 3.401
CC
QTED 6307.556 8464.534 4144.734 2028.48 2799.866
hc (μm) 4.001 5.201 10.001 5.801 2.801
IV
SS
QTED 6307.159 8464.534 4144.734 2028.439 2801.984
hc (μm) 3.401 4.201 8.401 4.801 2.401
CC
QTED 6307.159 8464.534 4144.562 2028.645 2801.199
hc (μm) 2.801 3.601 7.001 4.001 2.001
AIP Advances 11, 035108 (2021); doi: 10.1063/5.0033087 11, 035108-5
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were also explored. Resonance frequencies associated with axisym-
metric and non-axisymmetric vibrations for two boundary condi-
tions were also simulated and investigated. Five different structural
materials (diamond, Si, polySi, GaAs, SiC) were selected for circu-
lar microplates and simulated numerically using MATLAB 2015.
Properties of the material are listed in Table II.40
The thermoelastic damping limited quality factor (QTED), crit-
ical thickness (hc), and resonance frequency of a circular plate res-
onator were simulated numerically and analyzed in the subsequent
sections. The diameter of the circular plate was taken as 100 μm, and
aspect ratio, R/h, was fixed at 50. The initial equilibrium temperature
was taken as T0= 298 K.
A. Factors affecting QTED and hc
The thermoelastic damping limited quality factor and critical
thickness of circular plate resonators were analyzed for different
conditions such as axisymmetric and non-axisymmetric vibrations,
geometry adaptations, i.e., plate dimension optimizations, bound-
ary conditions, mode switching, and temperature variations. The
numerically simulated results using MATLAB 2015 for the ther-
moelastic limited quality factor and critical thickness from Eqs. (21)
and (23) were shown in Table IV (for axisymmetric vibrations) and
Table V (for non-axisymmetric vibrations).
The circular microplate was simulated for five different struc-
tural materials and two boundary conditions: simply supported and
clamped–clamped. Tables IV and Vshow the simulation results for
circular microplates vibrating in the first mode.
It is found that as the thickness increases, the thermoelastic
damping increases first and then decreases, and there is a critical
thickness, denoted as hc, with the maximum value of thermoelas-
tic damping at which QTED is minimum. Tables IV and Valso show
the critical thickness (hc) of circular plate resonators with the above-
mentioned stipulated conditions. Table IV shows hc of circular plate
resonators in axisymmetric vibrations, and Table V exhibits that of
the inplane mode, i.e., with non-axisymmetric vibrations.
The improvement in QTED was brought by employing non-
axisymmetric vibrations, which provide better QTEDs than axisym-
metric vibrations. Analysis also shows that slight improvement in
QTED is achieved by increasing the mode number and selecting the
clamped–clamped boundary conditions.
1. Effect of axisymmetric and non-axisymmetric
vibrations
The normal strain in the z direction is represented by εzz, and in
the case of in plane vibrations, cubical dilations are also considered,
i.e., εzz ≠o. In the governing equations of the circular microplate,
TABLE V. QTED and hc of circular plate resonators with non-axisymmetric vibrations for both boundary conditions [simply
supported (SS) and clamped–clamped (CC)]; R/h = 50; T0= 298 K; mode (k,l) with k= 1, 2 and l= 1, 2.
Mode B.C. Material PolySi Diamond Si GaAs SiC
I
SS
QTED 8907.892 9141.603 6781.96 3683.106 3692.555
hc (μm) 16.201 20.601 40.401 23.401 11.201
CC
QTED 8907.892 9141.603 6781.96 3683.106 3692.555
hc (μm) 11.2 14.2 28 16.2 7.8
II
SS
QTED 8908.686 9141.603 6781.96 3683.106 3693.095
hc (μm) 8.401 10.401 20.601 12.001 5.601
CC
QTED 8907.892 9141.603 6781.96 3683.241 3692.555
hc (μm) 6.401 8.001 15.801 9.201 4.401
III
SS
QTED 8156.607 9141.603 6781.96 3683.106 3692.825
hc (μm) 5.001 6.401 12.401 7.201 3.401
CC
QTED 8908.686 9142.439 6782.42 3683.241 3695.39
hc (μm) 4.001 5.201 10.001 5.801 2.801
IV
SS
QTED 8908.686 9141.603 6781.96 3683.241 3695.39
hc (μm) 3.401 4.201 8.401 4.801 2.401
CC
QTED 8907.892 9141.603 6781.96 3683.648 3695.39
hc (μm) 2.801 3.601 7.001 4.001 2.001
AIP Advances 11, 035108 (2021); doi: 10.1063/5.0033087 11, 035108-6
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TABLE VI. QTED and hc of circular plate resonators with axisymmetric and non-
axisymmetric vibrations for clamped–clamped boundary conditions; R/h = 50;
T0= 298 K; mode (k,l) with k= 1 and l= 1.
Axisymmetric Non-axisymmetric
Material QTED hc (μm) QTED hc (μm)
PolySi 6307.159 11.2 8907.892 11.2
Diamond 8463.817 14.2 9141.603 14.2
Si 4144.562 28 6781.96 28
GaAs 2028.398 16.2 3683.106 16.2
SiC 2799.866 7.8 3692.555 7.8
εzz ≠ois assigned for nonaxisymmetric vibrations and εzz =ocorre-
sponds to the axisymmetric vibrations. Table VI shows the compar-
ison of the thermoelastic damping limited quality factor and critical
thickness of a circular microplate with both vibration modes. The
comparison of axisymmetric and non-axisymmetric vibrations is
carried out for a clamped–clamped beam vibrating in the first mode
with five different structural materials, as shown in Table VI.
The energy dissipation in axisymmetric vibrations is always
greater than that of the non-axisymmetric case, i.e., the QTED of
non-axisymmetric vibrations is higher than that of axisymmetric
vibrations. The relation is applicable to all five materials, and Q−1
TED
is maximum for GaAs and minimum for diamond. The sequence in
which QTED diminishes is diamond >polySi >Si >SiC >GaAs, and
the order in which critical thickness reduces is Si >GaAs >diamond
>polySi >SiC. The material order in which QTED and hc diminish
is the same for both axisymmetric and nonaxisymmetric vibrations,
as shown in Table VI. The enhancement in QTED is achieved by
selecting non-axisymmetric vibrations for circular plates. For both
axisymmetric and non-axisymmetric vibrations, the critical thick-
ness (hc) and the order in which hcdrops off are the same, i.e., hc(AS)
≈hc(NAS).
The material dependent performance indices m,n, and pare
derived from the attributes and Poisson’s ratio of the material and
affect the axisymmetric and non-axisymmetric vibrations and the
resulting quality factors. The different ratios of the material indices
shown in Eqs. (21) and (23) can be taken as a scaling factor for
the expression for the thermoelastic damping limited quality factor
connecting to the classical Lifshitz and Rouke’s expression.
Table VII shows the scaling factor of non-axisymmetric and
axisymmetric vibrations. The order of materials for the scaling factor
of axisymmetric and non-axisymmetric vibrations is diamond >SiC
TABLE VII. Scaling factor of circular plate resonators with axisymmetric and
non-axisymmetric vibrations for clamped–clamped boundary conditions; R/h = 50;
T0= 298 K; mode (k,l) with k= 1 and l= 1.
Material NAS AS Difference
PolySi 0.6313 0.4470 0.1843
Diamond 0.8709 0.8064 0.0645
Si 0.5625 0.3437 0.2188
GaAs 0.5267 0.2901 0.2366
SiC 0.6779 0.5168 0.1611
FIG. 2. Variation in thermoelastic energy dissipation (Q−1) with thickness for circu-
lar microplate resonators with different structural materials; the type of vibrations is
axisymmetric and non-axisymmetric vibrations; R/h = 50; T0= 298 K; mode (k,l)
with k=1andl= 1.
>polySi >Si >GaAs. The order of materials in which the difference
between the scaling factor of axisymmetric and non-axisymmetric
vibrations varies is GaAs >Si >polySi >SiC >diamond.
The maximum QTED (9142) is attained for the diamond based
circular plate with non-axisymmetric vibrations, and the minimum
QTED (2028) is attained for GaAs as the structural material with
axisymmetric vibrations.
Figure 2 shows the variation in thermoelastic energy dissi-
pation with thickness of a clamped–clamped circular microplate
vibrating in the first mode (1,1). The maximum energy dissipation
(4.93 ×10−4) is obtained for GaAs in the axisymmetric mode
when the critical thickness is 16.2 μm, and the minimum energy
dissipation (1.0939 ×10−4) is achieved for diamond in the non-
axisymmetric mode when the critical thickness is 14.2 μm.
2. Impacts of geometry (plate dimensions)
Thermoelastic damping (TED) is an important factor in circu-
lar plate resonator design and development, and the dependency of
the TED limited quality factor and proper geometry optimizations
of the plate dimensions were investigated by analyzing
(a) variation in Q−1with the radius, R, for fixed h to find the
critical radius, Rc,
(b) variation in Q−1with thickness h for a fixed radius, R, to find
the critical thickness, hc, and
(c) variation in Q−1with thickness, h, for a fixed aspect ratio, R/h,
to find the critical thickness, hc.
a. Impacts of fixed thickness (h) on the critical radius (Rc) of cir-
cular plate resonators. A diamond based clamped–clamped circular
plate with fixed thickness, h = 5 μm, 10 μm, and 15 μm, and a vary-
ing radius, R, operating at T0= 298 K, vibrating in the first mode, i.e.,
(k,l) with k= 1 and l= 1, was analyzed for finding the critical radius.
Figure 3 depicts the behavior of TED of a clamped diamond plate
AIP Advances 11, 035108 (2021); doi: 10.1063/5.0033087 11, 035108-7
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FIG. 3. Variation in thermoelastic damping energy Q−1with the radius (R) for a
clamped–clamped diamond circular microplate resonator with axisymmetric and
non-axisymmetric vibrations; h = 5 μm, 10 μm, 15 μm, T0= 298 K; mode (k,l) with
k=1andl= 1.
resonator and shows that the energy dissipation increases gradually
and peaks at some particular value of the radius known as the critical
radius, Rc. As the fixed thickness selected for the analysis increases,
the value of Rc also increases.
From Table VIII, the thermoelastic damping limited quality
factor (QTED) was found to be same for different critical radius
(Rc) values and fixed thickness (h) values, i.e., QTED is found to be
independent of plate dimensions. As the thickness of the circular
TABLE VIII. QTED and Rc of circular plate resonators with axisymmetric (AS)/non-
axisymmetric (NAS) vibrations for the clamped–clamped boundary condition; h = 5
μm and 15 μm; T0= 298 K; mode (k,l) with k= 1 and l= 1.
Material
h (μm) 5 15
AS/NAS QTED Rc (mm) QTED Rc (mm)
PolySi
AS 8463.817 0.1485 8463.817 0.7695
NAS 8907.892 0.1485 8907.892 0.7695
Diamond
AS 6306.761 0.1665 6306.761 0.8645
NAS 9141.603 0.1665 9141.603 0.8645
Si
AS 4144.562 0.1055 4144.562 0.5485
NAS 6781.96 0.1055 6781.96 0.5485
GaAs
AS 2028.398 0.1385 2028.398 0.7205
NAS 3683.106 0.1385 3683.106 0.7205
SiC
AS 2799.787 0.2005 2799.787 1.0425
NAS 3672.555 0.2005 3672.555 1.0425
microplate resonators increases, the critical radius at which TED is
maximum also increases. The maximum critical radius for a fixed
value of thickness (h) was obtained for SiC and the minimum, for Si.
The order in which the critical radius decreases is SiC >diamond >
polySi >GaAs >Si.
b. Impacts of fixed radius (R) on critical thickness (hc) of circu-
lar plate resonators. The thermoelastic damping increases first and
then decreases with an increasing value of thickness. The value of
thickness at which peaking of energy takes place is known as the crit-
ical thickness. Figure 4 depicts the variation in thermoelastic energy
with the thickness of the diamond circular microplate for different
values of the radius (R = 100 μm, 300 μm, and 500 μm.). As shown
in Fig. 4, there exists a critical thickness (hc) for which the maxi-
mum value of the damping factor occurs. As the fixed values of the
radius increases, the value of critical thickness also increases. The
value of hc is the same for both axisymmetric and nonaxisymmetric
vibrations.
From Fig. 4, it can be seen that the energy dissipation was
almost constant and found to be independent of plate dimen-
sions such as radius. The maximum (3.5717 ×10−4) and minimum
(2.7229 ×10−4) energy dissipation were obtained for axisymmet-
ric (hc= 9.19 μm at R = 500 μm) and non-axisymmetric modes (hc
= 3.15 μm at R = 100 μm), respectively.
In Table IX, the QTEDfor different radii seems to be equal. The
material order according to which hcdiminishes was Si >GaAs
>diamond >polySi >SiC.
c. Impacts of the aspect ratio (radius to thickness ratio—R/h)
on critical thickness (hc) of circular plate resonators. Figure 5 illus-
trates the variation in thermoelastic energy with thickness (h) of a
diamond based circular microplate for varying aspect ratios (R/h
= 40, 50, and 60) vibrating in the first mode at an operating tem-
perature T0= 298 K. From Fig. 5, it can be seen that as R/h varies,
FIG. 4. Variation in thermoelastic damping energy Q−1with thickness (h) for a
clamped–clamped diamond circular microplate resonator with axisymmetric and
non-axisymmetric vibrations; R = 100 μm, 300 μm, and 500 μm, T0= 298 K; mode
(k,l)withk=1andl=1.
AIP Advances 11, 035108 (2021); doi: 10.1063/5.0033087 11, 035108-8
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TABLE IX. QTED and hc of circular plate resonators with axisymmetric (AS)/non-
axisymmetric (NAS) vibrations for the clamped–clamped boundary condition;
R = 100 μm and 500 μm; T0= 298 K; mode (k,l) with k= 1 and l= 1.
Material
R (μm) 100 500
AS/NAS QTED hc (μm) QTED hc (μm)
PolySi
AS 6306.761 3.57 6306.761 10.41
NAS 8156.607 3.57 8156.607 10.41
Diamond
AS 8463.817 3.85 8463.817 11.25
NAS 9141.603 3.85 9141.603 11.25
Si
AS 4144.562 4.83 4144.562 14.09
NAS 6781.96 4.83 6781.96 14.09
GaAs
AS 2028.439 4.03 2028.398 11.77
NAS 3683.241 4.03 3683.106 11.77
SiC
AS 2799.866 3.15 2799.787 9.19
NAS 3672.555 3.15 3672.555 9.19
the energy dissipation is almost constant but the critical thickness
(hc) changes. When the aspect ratio, R/h, increases, critical thickness
also increases. The maximum (2.4128 ×10−4) and minimum (1.4745
×10−4) energy dissipation are obtained for axisymmetric (hc= 28.03
μm at R/h = 50) and non-axisymmetric (hc= 17.93 μm at R/h = 40)
vibrations, respectively.
Table X compares the QTED and hc of a circular microplate
for different R/h values vibrating in the first mode with the
FIG. 5. Variation in thermoelastic damping energy Q−1with thickness (h) for a
clamped–clamped diamond circular microplate resonator with axisymmetric and
non-axisymmetric vibrations; R/h = 40, 50, and 60, T0= 298 K; mode (k,l) with
k=1andl= 1.
TABLE X. QTED and hc of circular plate resonators with axisymmetric (AS)/non-
axisymmetric (NAS) vibrations for the clamped–clamped boundary condition;
R/h = 20 and 50; T0= 298 K; mode (k,l) with k= 1 and l= 1.
Material
R/h20 50
AS/NAS QTED hc (μm) hc (μm) QTED
PolySi
AS 6306.761 1.801 6307.159 11.201
NAS 8907.892 1.801 8907.892 11.201
Diamond
AS 8469.552 2.201 8463.817 14.201
NAS 9147.457 2.201 9141.603 14.201
Si
AS 4145.25 4.401 4144.562 28.001
NAS 6783.34 4.401 6781.96 28.001
GaAs
AS 2028.398 2.601 2028.398 16.201
NAS 3683.106 2.601 3683.106 16.201
SiC
AS 2801.356 1.201 2799.866 7.801
NAS 3674.444 1.201 3672.555 7.801
clamped–clamped boundary condition. From Table X, it can be seen
that the order in which QTED diminishes for different structural
materials when the circular microplate is having axisymmetric vibra-
tions was diamond >polySi >Si >SiC >GaAs. The same sequence in
QTED variation was obtained for non-axisymmetric vibrations, and
larger values were achieved for the quality factor. The material order
in which critical thickness varies was Si >GaAs >diamond >polySi
>SiC, as shown in Table X.
3. Impacts of mode switching
The impacts of mode switching are analyzed for a circular
plate resonator with diamond as the structural material. Among the
five different structural materials (diamond, polysilicon, GaAs, Si,
and SiC), diamond shows the minimum energy dissipation, as dis-
cussed in Sec. IV A 1. For analyzing the mode switching effects,
in this work, k= 1, 2 and l= 1, 2 values are selected so that the
first four vibration modes comes into play, i.e., (1,1), (1,2), (2,1),
and (2,2). The analysis is carried out for both boundary condi-
tions (simply supported and clamped–clamped) with the aspect ratio
R/h = 50, and the temperature is set at environmental temperature,
T0= 293 K.
Figure 6 illustrates the variation in energy dissipation Q−1
with thickness (h) for a diamond based circular microplate with
the simply supported boundary condition. The impact of mode
switching on a clamped circular plate with diamond as the struc-
tural material is illustrated in Fig. 7. The maximum energy dissipa-
tion (1.181 51 ×10−4) and minimum QTED (8463.8) are obtained
for axisymmetric (AS) vibrations of the microplate. The min-
imum Q−1(1.0939 ×10−4) and maximum QTED (9141.6) are
achieved for non-axisymmetric (NAS) conditions, which are shown
in Table XI.
AIP Advances 11, 035108 (2021); doi: 10.1063/5.0033087 11, 035108-9
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FIG. 6. Variation in energy dissipation Q−1vs thickness for a diamond based
circular plate resonator with vibrating modes (k,l) with k= 1, 2 and l= 1, 2; aspect
ratio R/h = 50; boundary condition – simply supported; operating temperature
T0= 298 K.
The QTED and hcof a diamond based clamped–clamped cir-
cular plate for different modes are presented in Tables XI and XII.
When the mode increases, the quality factor remains almost con-
stant for both axisymmetric and non-axisymmetric vibrations, as
depicted in Table XI. The maximum value of energy dissipation
is maintained at the same value for increasing mode numbers and
found to be independent of vibration modes but varies with the
operating temperature, as described in Sec. IV A 4.
FIG. 7. Variation in energy dissipation Q−1vs thickness for a diamond based cir-
cular plate resonator with vibrating modes (k,l) with k= 1, 2 and l= 1, 2; aspect
ratio R/h = 50; boundary condition – clamped–clamped; operating temperature T0
= 298 K
TABLE XI. QTED of circular plate resonators with axisymmetric and non-axisymmetric
vibrations for the clamped–clamped boundary condition; R/h = 50; T0= 298 K; modes
(k,l) with k= 1, 2 and l= 1, 2.
Material Mode (1,1) (1,2) (2,1) (2,2)
PolySi
AS 6307.159 6307.159 6307.556 6307.159
NAS 8907.892 8907.892 8908.686 8907.892
Diamond
AS 8463.817 8463.817 8464.534 8464.534
NAS 9141.603 9141.603 9141.439 9141.603
Si
AS 4144.562 4144.562 4144.734 4144.562
NAS 6781.396 6781.396 6782.42 6781.96
GaAs
AS 2028.398 2030.498 2028.48 2028.644
NAS 3683.106 3683.241 3683.241 3683.648
SiC
AS 2799.866 2799.944 2799.866 2801.199
NAS 3672.555 3672.555 3672.555 3674.309
For mode shifting, the order in which critical thickness hc
varies is (1,1) >(2,1) >(1,2) >(2,2) for both axisymmetric and
non-axisymmetric vibrations, as shown in Table XII. The critical
thickness decreases in the sequence of (1, 1), (2, 1), (1, 2), and
(2, 2) for both boundary conditions. Figures 6 and 7depict that as the
mode number increases, the dissipation curve shifts toward the left,
which means critical thickness at which peaking of energy occurs
gets reduced. From Table XII, as the mode number increases, i.e., as
kand lincrease, qkl increases; as a result, hc gets smaller for higher
vibration modes.
TABLE XII. hc of circular plate resonators with axisymmetric and non-axisymmetric
vibrations for the clamped–clamped boundary condition; R/h = 50; T0= 298 K; mode
(k,l) with k= 1, 2 and l= 1, 2.
Material Mode (1,1) (1,2) (2,1) (2,2)
PolySi
AS 11.2 6.401 4.001 2.801
NAS 11.202 6.401 4.001 2.801
Diamond
AS 14.2 8.001 5.201 3.601
NAS 14.2 8.001 5.201 3.601
Si
AS 28 15.801 10.001 7.001
NAS 28 15.801 10.001 7.001
GaAs
AS 16.2 9.201 5.801 4.001
NAS 16.2 9.201 5.801 4.001
SiC
AS 7.8 4.401 2.801 2.001
NAS 7.8 4.401 2.801 2.001
AIP Advances 11, 035108 (2021); doi: 10.1063/5.0033087 11, 035108-10
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4. Impacts of boundary conditions
The impacts of two boundary conditions—simply supported
and clamped–clamped—on a micro-circular plate with a fixed aspect
ratio R/h = 50 under an operating temperature of T0= 293 K were
analyzed in this section. Figure 8 shows the comparison of ther-
moelastic energy dissipation for different boundary conditions of
a diamond based circular plate oscillating under axisymmetric and
non-axisymmetric modes vibrating in the first mode (1,1).
Table XIII shows the thermoelastic damping limited quality
factor and critical thickness of circular microplates with all five
structural materials (polySi, diamond, Si, GaAs, and SiC). The QTED
and hc values of the circular plates vibrating under axisymmetric and
non-axisymmetric modes and with both boundary conditions are
shown in Table XIII. The energy dissipation varies with thickness,
as shown in Fig. 8, with peaking occuring at a particular value of
thickness known as critical thickness, hc. At critical dimensions, the
energy dissipation for both the boundary conditions are the same,
and when the plate size is larger than the critical size, Q−1of the
simply supported plate is larger than that of the clamped plate for
the same plate size. The clamped conditions exert more force on
the plate conditions than simply supported conditions, and hence,
forced vibrations are set up in the former case instead of free vibra-
tions in the latter one after the critical dimensions are attained. The
critical thickness for the clamped plate is smaller than that for the
simply supported plate, as illustrated in Fig. 8. As shown in Fig. 8,
the maximum energy dissipation (1.1815 ×10−4) is for axisymmet-
ric vibrations with a critical thickness of 20.601 μm, and minimum
energy dissipation (1.0939 ×10−4) is for non-axisymmetric vibra-
tions with a critical thickness of 14.25 μm. The maximum and min-
imum values of QTED for a diamond based circular plate are 9141.6
and 8464, respectively. From the analytical study, it is also proved
that QTED NAS >QTED AS, which shows the energy dissipation in
FIG. 8. Comparison of thermoelastic damping limited energy dissipation and criti-
cal thickness of circular microplates for two boundary conditions (simply supported
and clamped–clamped) as well as vibrations (axisymmetric and non-axisymmetric)
; aspect ratio R/h = 50; mode (1,1) i.e., k=l= 1.
TABLE XIII. QTED and hc of circular plate resonators with axisymmetric
and non-axisymmetric vibrations for both boundary conditions – simply supported and
clamped–clamped; R/h = 50; T0= 298 K; mode (k,l) with k= 1and l= 1.
SS CC
Material Mode QTED hc (μm) QTED hc (μm)
PolySi
AS 6306.761 16.201 6306.761 11.202
NAS 8907.892 16.201 8907.892 11.202
Diamond
AS 8463.817 20.601 8463.817 14.2
NAS 9141.603 20.601 9141.603 14.2
Si
AS 4144.562 40.401 4144.562 28
NAS 6781.96 40.401 6781.96 28
GaAs
AS 2028.398 23.401 2028.398 16.2
NAS 3683.106 23.401 3683.106 16.2
SiC
AS 2799.866 11.201 2799.866 7.8
NAS 3672.555 11.201 3672.555 7.8
circular plates with axisymmetric vibrations is more than that with
non-axisymmetric vibrations.
5. Effect of temperature
Generally, the thermal and mechanical properties of all mate-
rials are temperature dependent. At lower temperatures, the tem-
perature change associated with thermoelastic vibration is known
to be small,12,14 and all mechanical and thermal properties can be
treated as constants (T0= 293 K). The important thermal properties
are thermal conductivity (k), specific heat at constant pressure (Cp),
the coefficient of linear thermal expansion (α), thermal diffusivity
(χ), and specific heat at constant volume (Cv). The mechanical prop-
erties such as Young’s modulus (E), Poisson’s ratio (v), and density
(ρ) are also important in the analysis. PolySi, diamond, Si, GaAs, and
SiC were used as the different structural materials for the analysis of
QTED and hc in circular microplates. The temperature dependency of
elastic and thermal properties of Si was experimentally validated.41
The energy loss due to the temperature rise causes an increase in
thermoelastic damping, which results in low QTED. The thermoelas-
tic damping depends on temperature clearly, and it is a significant
loss mechanism at room temperature for micro-scale circular plate
resonators.
Figures 9 and 10 show the variations in thermoelastic energy
dissipation (Q−1) with temperature for the first four vibration
modes, i.e., for modes (m, n) with m = n = 1, 2. The solid lines repre-
sent the Q−1values for axisymmetric vibrations, and the dashed lines
represent those for a non-axisymmetric plate. The order in which the
thermoelastic energy dissipation varies for a circular microplate for
different modes is (1,1) >(2,1) >(1,2) >(2,2).
As the mode number increases, Q−1is less than that of lower
modes, and QTED also increases when the resonator is vibrating at
AIP Advances 11, 035108 (2021); doi: 10.1063/5.0033087 11, 035108-11
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FIG. 9. Variation in energy dissipation Q−1vs temperature T for a diamond based
circular plate resonator with vibrating modes (k,l) where k= 1, 2 and l= 1, 2;
aspect ratio R/h = 50; boundary condition—simply supported.
higher modes. The impact of temperature increases the energy dissi-
pation due to thermoelastic damping, and the effect can be subsided
by vibrating at higher modes. The difference between the axisym-
metric and non-axisymmetric vibrations also reduces at higher
modes, as shown in Figs. 9 and 10. The impacts of temperature are
due to the temperature dependency of E, αT, and Cv, which can be
proved according to the expression of Q−1
TED, i.e., Q−1
max =0.494r′,
where r′is a temperature dependent parameter.
FIG. 10. Variation in energy dissipation Q−1vs temperature T for a diamond based
circular plate resonator with vibrating modes (k,l) where k= 1, 2 and l= 1, 2;
aspect ratio R/h = 50; boundary condition—clamped–clamped.
TABLE XIV. QTED and resonance frequency with axisymmetric vibrations of circular
plate resonators for clamped–clamped and simply supported boundary conditions;
R/h = 50; T0= 298 K; mode (k,l) with k= 1 and l= 1.
B.C. SS CC
Material QTED RF (kHz) QTED RF (kHz)
PolySi 6307.2 1088.9 6307.2 1429.2
Diamond 8463.8 1653 8463.8 2148.9
Si 4144.4 422.38 4144.6 550.69
GaAs 2028.4 381.73 2028.4 497.9
SiC 2800.3 2317.1 2799.9 2949
B. Impacts of resonance frequency on axisymmetric
and non-axisymmetric vibrations
Table XIV compares the resonance frequency of circular
microplate resonators with both boundary conditions—simply
supported and clamped–clamped. The resonance frequency of plate
resonators with axisymmetric vibrations are shown in Table XIV
and proved that the thermoelastic damping limited quality factor
and resonance frequency of resonators with the clamped–clamped
boundary condition are greater than those of the simply supported
boundary scenario. The order in which resonance frequency dimin-
ishes for various structural materials is SiC >diamond >polySi >Si
>GaAs under both the boundary conditions.
The resonance frequency corresponding to non-axisymmetric
vibrations of a circular microplate with the clamped–clamped
boundary condition is greater than that of the simply supported
boundary condition. The order of structural materials in which res-
onance frequency reduces is the same for the non-axisymmetric
condition.
FIG. 11. Variation in energy dissipation Q−1with resonance frequency for circular
plate resonators with axisymmetric and non-axisymmetric vibrations; mode (k,l)
where k, = 1 and l= 1; aspect ratio R/h = 50; boundary condition—clamped–
clamped; operating temperature T0= 298 K.
AIP Advances 11, 035108 (2021); doi: 10.1063/5.0033087 11, 035108-12
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TABLE XV. QTED and resonance frequency with non-axisymmetric vibrations of circu-
lar plate resonators for clamped–clamped and simply supported boundary conditions;
R/h = 50; T0= 298 K; mode (k,l) with k= 1 and l= 1.
B.C. SS CC
Material QTED RF (kHz) QTED RF (kHz)
PolySi 8908.7 1088.9 8907.9 1429.1
Diamond 9141.6 1653 9141.6 2148.9
Si 6782 422.38 6782 550.69
GaAs 3683.1 381.73 3683.2 497.9
SiC 3673.1 2317.1 3672.6 2948.9
From Fig. 11, it is clear that the maximum energy dissipation is
obtained for lower resonance frequency. As the resonance frequency
increases, Q−1
TED decreases. The maximum energy dissipation (4.93
×10−4) is obtained for GaAs in the axisymmetric condition at R.F
= 2.8275 ×105, and the minimum energy dissipation (1.0939 ×10−4)
is obtained for diamond in the non-axisymmetric condition at R.F
= 1.2107 ×106. The increasing order of energy dissipation with
respect to resonance frequency is GaAs <SiC <Si <polySi
<diamond.
Table XV compares the resonance frequency between differ-
ent boundary conditions and their corresponding QTED with non-
axisymmetric vibration.
V. CONCLUSION
In this article, the coupling equations of thermoelasticity were
solved on the basis of material performance index parameters for
non-axisymmetric and axisymmetric vibrations of a circular plate
resonator, and its thermoelastic damping limited quality factors
were investigated. Based on the numerical simulations of the equa-
tions obtained from the analytical models, it was found that the non-
axisymmetric vibrations provide the maximum thermoelastic damp-
ing limited quality factor, QTEDMAX. QTED was further maximized by
selecting diamond as the structural material. The sequence in which
QTED diminishes is diamond >polySi >Si >GaAs >SiC and was
found to be dependent on material performance index parameters.
The thermoelastic energy dissipation was found to be almost inde-
pendent of mode switching and boundary conditions under equi-
librium temperature. The effects of elevated temperature were also
analyzed, and they show that the energy dissipation depends on tem-
perature for all modes of vibrations as well as boundary conditions
and material dependency. The quality factor was found to be less
affected at low temperatures, and as temperature increases, energy
dissipation increases for both axisymmetric and non-axisymmetric
vibrations. The energy dissipation is more for the axisymmetric case,
for all modes and boundary conditions. As the vibrating mode num-
ber increases, the energy dissipation reduces, and the difference in
dissipated energy for axisymmetric and non-axisymmetric vibra-
tions was also reduced. Plate dimensions have no impact on the
energy dissipation and the quality factor. The critical thickness was
found to be dependent on boundary conditions (hcSS >hcCC),
the mode of vibrations, plate dimensions, and material parame-
ters. When the vibrating mode number increases, hc diminishes.
As the aspect ratio of the plate increases, hc also gets enhanced.
For the five different materials, the order in which hc varies is Si
>GaAs >diamond >polySi >SiC. As far as the resonance frequency
is concerned, it was observed that both axisymmetric and non-
axisymmetric vibrations have the same resonance frequency even
though energy dissipation is more for the axisymmetric case. The
resonance frequency of a circular plate resonator with the clamped–
clamped boundary condition was greater than that of the simply
supported case. The order in which resonance frequency increases is
the same for both axisymmetric and non-axisymmetric vibrations.
From our study, designers can fabricate circular plate resonator sen-
sors with optimized geometry and material and enhanced quality
factors to get better performance parameters such as sensitivity and
fast response.
DATA AVAILABILITY
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
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