Thermoelastic damping in FGM nano-electromechanical system in axial vibration based on Eringen nonlocal theory
The thermo-elastic damping is a dominant source of internal damping in microelectromechanical systems (MEMS) and nano-electromechanical systems (NEMS). The internal damping cannot neither be controlled nor minimized unless either mechanical or geometrical properties are changed. Therefore, a novel F...
Збережено в:
| Дата: | 2017 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут механіки ім. С.П. Тимошенка НАН України
2017
|
| Назва видання: | Прикладная механика |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/174142 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Thermoelastic damping in FGM nano-electromechanical system in axial vibration based on Eringen nonlocal theory / Z. Rahimi, S. Rashahmadi // Прикладная механика. — 2017. — Т. 53, № 6. — С. 133-141. — Бібліогр.: 23 назв. — англ |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-174142 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1741422021-01-06T01:26:05Z Thermoelastic damping in FGM nano-electromechanical system in axial vibration based on Eringen nonlocal theory Rahimi, Z. Rashahmadi, S. The thermo-elastic damping is a dominant source of internal damping in microelectromechanical systems (MEMS) and nano-electromechanical systems (NEMS). The internal damping cannot neither be controlled nor minimized unless either mechanical or geometrical properties are changed. Therefore, a novel FGMNEM system with a controllable thermo-elastic damping of axial vibration based on Eringen nonlocal theory is considered. The effects of different parameter like the gradient index, nonlocal parameter, length of nanobeam and ambient temperature on the thermo-elastic damping quality factor are presented. It is shown that the thermo-elastic damping can be controlled by changing different parameter. Термопружне демпфування є головним джерелом внутрішнього демпфування в мікроелектромеханічних і наноелектромеханічних системах. Внутрішнє демпфування не може бути ні кероване, ні мінімізоване, поки механічні і геометричні властивості матеріалу не є змінними. Тому розглянуто осьові коливання балки з нового матеріалу – функціонально градієнтного матеріалу з врахуванням наноелектромеханічних явищ – в рамках нелокальної теорії Ерінгена. Розглянуто вплив різних параметрів – показника градієнтості, параметрів нелокальності, довжини нанобалки, температури навколишнього середовища – на коефіцієнт термопружного демпфування. Показано, що термопружне демпфування може бути кероване зміною цих параметрів. 2017 Article Thermoelastic damping in FGM nano-electromechanical system in axial vibration based on Eringen nonlocal theory / Z. Rahimi, S. Rashahmadi // Прикладная механика. — 2017. — Т. 53, № 6. — С. 133-141. — Бібліогр.: 23 назв. — англ 0032-8243 http://dspace.nbuv.gov.ua/handle/123456789/174142 en Прикладная механика Інститут механіки ім. С.П. Тимошенка НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
The thermo-elastic damping is a dominant source of internal damping in microelectromechanical systems (MEMS) and nano-electromechanical systems (NEMS). The internal damping cannot neither be controlled nor minimized unless either mechanical or geometrical properties are changed. Therefore, a novel FGMNEM system with a controllable thermo-elastic damping of axial vibration based on Eringen nonlocal theory is considered. The effects of different parameter like the gradient index, nonlocal parameter, length of nanobeam and ambient temperature on the thermo-elastic damping quality factor are presented. It is shown that the thermo-elastic damping can be controlled by changing different parameter. |
| format |
Article |
| author |
Rahimi, Z. Rashahmadi, S. |
| spellingShingle |
Rahimi, Z. Rashahmadi, S. Thermoelastic damping in FGM nano-electromechanical system in axial vibration based on Eringen nonlocal theory Прикладная механика |
| author_facet |
Rahimi, Z. Rashahmadi, S. |
| author_sort |
Rahimi, Z. |
| title |
Thermoelastic damping in FGM nano-electromechanical system in axial vibration based on Eringen nonlocal theory |
| title_short |
Thermoelastic damping in FGM nano-electromechanical system in axial vibration based on Eringen nonlocal theory |
| title_full |
Thermoelastic damping in FGM nano-electromechanical system in axial vibration based on Eringen nonlocal theory |
| title_fullStr |
Thermoelastic damping in FGM nano-electromechanical system in axial vibration based on Eringen nonlocal theory |
| title_full_unstemmed |
Thermoelastic damping in FGM nano-electromechanical system in axial vibration based on Eringen nonlocal theory |
| title_sort |
thermoelastic damping in fgm nano-electromechanical system in axial vibration based on eringen nonlocal theory |
| publisher |
Інститут механіки ім. С.П. Тимошенка НАН України |
| publishDate |
2017 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/174142 |
| citation_txt |
Thermoelastic damping in FGM nano-electromechanical system in axial vibration based on Eringen nonlocal theory / Z. Rahimi, S. Rashahmadi // Прикладная механика. — 2017. — Т. 53, № 6. — С. 133-141. — Бібліогр.: 23 назв. — англ |
| series |
Прикладная механика |
| work_keys_str_mv |
AT rahimiz thermoelasticdampinginfgmnanoelectromechanicalsysteminaxialvibrationbasedoneringennonlocaltheory AT rashahmadis thermoelasticdampinginfgmnanoelectromechanicalsysteminaxialvibrationbasedoneringennonlocaltheory |
| first_indexed |
2025-07-15T11:01:17Z |
| last_indexed |
2025-07-15T11:01:17Z |
| _version_ |
1837710471666335744 |
| fulltext |
2017 ПРИКЛАДНАЯ МЕХАНИКА Том 53, № 6
ISSN0032–8243. Прикл. механика, 2017, 53, № 6 133
Z . R a h i m i , S . R a s h a h m a d i 1
THERMOELASTIC DAMPING IN FGM NANO-ELECTROMECHANICAL
SYSTEM IN AXIAL VIBRATION BASED ON ERINGEN NONLOCAL THEORY
1 Mechanical Engineering Department, Urmia University,
Urmia, Iran; e-mail: s.rashahmadi@urmia.ac.ir
Abstract. The thermo-elastic damping is a dominant source of internal damping in mi-
cro-electromechanical systems (MEMS) and nano-electromechanical systems (NEMS). The
internal damping cannot neither be controlled nor minimized unless either mechanical or
geometrical properties are changed. Therefore, a novel FGMNEM system with a controlla-
ble thermo-elastic damping of axial vibration based on Eringen nonlocal theory is consid-
ered. The effects of different parameter like the gradient index, nonlocal parameter, length
of nanobeam and ambient temperature on the thermo-elastic damping quality factor are pre-
sented. It is shown that the thermo-elastic damping can be controlled by changing different
parameter.
Key words: FGM nanobeam, Eringen nonlocal theory, axial vibrations, thermoelastic
damping.
1. Introduction.
Nowadays MEMS and NEMS industries are playing important roles in scientific and
engineering communities. There are a lot of advantages that make MEMS and NEMS com-
mercialization attractive. Ink jet printer heads, micropumps, airbag accelerometers and mi-
cro sensors are a few examples of devices which these systems have successfully replaced
more conventional systems [1].
There are different kind of loss mechanism in MEMS and NEMS which classified into
the two categories: extrinsic and intrinsic losses. [2]. internal damping cannot neither be
controlled norminimized unless either mechanical or geometrical properties are changed
[3]TED is a dominant source of intrinsic damping in MEMS. [4] and NEMS. [5] working
under vacuum condition. [6]. It has been identified as one of important loss mechanisms in
NEMS and MEMS [7]. From the other hand development of low-power, high performance
MEM and NEM systems are of great importance [8] so TED is a very active research ap-
proach [7]. Lifshitz and Roukes [9] derived an analytical expression for the QF of TED in-
micro-beams and studied the effect of different geometrical parameters. Saeedi vahdat et al.
[4] study the effects of axial and residual stresses on thermoelastic damping in capacitive
micro-beam. Lepage et al. [10] studied thermoelastic effects in a vibrating beam acceler-
ometer that modelled using finite elements. Nayfeh and Younis [11] presented ananalytical
expression for the QF of micro-plates of general shapes and boundary conditions due to
TED. Rezazadeh et al. [6] studied thermoelastic damping in a micro-beam resonator using
modified couple stress theory and also thermoelastic damping in capacitive micro-beam
resonators using hyperbolic heat conduction model [12].
As can be seen there are a lot of work has been done on the TED of transverse vibration
but investigations about longitudinal vibrations are few in comparison with it. These vibra-
tions are quite different [13]for example, the natural frequencies in transversal vibration are
much lower than that of longitudinal vibration [13]and probably can achieve high QF. [14].
TED of the longitudinal vibration of micro beams is studied by maroofi et al. [14] and they
showed increasing of the ambient temperature and length of micro beam increases the QF.
134
In the present work, by considering the coupled equations of motion and heat conduc-
tivity, TED of nano FGM beam’s axial vibration based on nonlocal theory has been pre-
sented and has been proposed a novel FGMNEM system, with a controllable thermo-elastic
damping of axial vibration.
The Material properties of the FGM beam vary continuously along the beam thickness
according to the power law distribution. The coupled equations is solved by Galerkin
method for a clamped-clamped boundary condition and finally, the effects of different pa-
rameters like gradient index, nonlocal parameter, length of nano beam and ambient tempera-
ture has been presented on the QF.
2. Formulation.
Consider functionally graded clamped nanobeam of length ,L width b, and thickness h.
Material properties of the beam vary continuously along the beam thickness are functions of
z according to the power law distribution [15]:
2
( ) ( ) ,
2
m
l u u
z h
p z p p p
h
where subscripts u and l refer to material properties of the upper and lower surfaces respec-
tively. m is a non-negative number that dictates the material variation profile through the
thickness of the beam. The general strain field results from both mechanical and thermal
effects [16, 17]
,M T
ij ij ije e e (1)
where M
ije and T
ije are mechanical and thermal strain as bellow:
0
1
; ( )( ) ,
( ) ( )
M T
ij ij kk ij ij ije e z T T
E z E z
(2)
where is passion’s ratio, E is young modulus, is coefficient of the linear thermal expan-
sion and 0T is equal to the ambient temperature so the stress-strain relation is as follow [16]:
02 (3 2 ) ( )( ) .ij kk ij ij ije e z T T (3)
For both plane strain and plane stress condition we have [18]:
0[ ] ( ).ef M T ef
xx xx xx xxE e e E e T T (4)
In plane stress condition ( ) ( )efE z z and in plane strain condition ( ) ( ) / (1 )efE z z
and where 25 , / (1 )efb h E E otherwise .efE E
2.1 Nonlocal theory. According to Eringen [19 – 21], the stress field at a point x in an
elastic continuum not only depends on the strain field at the point but also on strains at all
other points of the body. Eringen attributed this fact to the atomic theory of lattice dynamics
and experimental observations on phonon dispersion. Thus, the nonlocal stress tensor com-
ponents σij at point x are expressed as:
( ) ( , ) ( ) ,ij ij
v
x k x x t x dx (5)
where tij(x) are the components of the classical macroscopic stress tensor at point x and the
kernel function ( , )k x x represents the nonlocal modulus, x x being the distance (in
Euclidean norm) and τ is a material constant that depends on internal and external character-
istic lengths (such as the lattice spacing and wavelength, respectively). It is possible (see
Eringen [20]) to represent the integral constitutive relations in an equivalent differential
form as:
135
2
2
02
1 ( ( ) ),ef
xx xxE e e a
x
(6)
where a is an internal characteristic length and 0e is a constant. Now based on the Eringen
nonlocal theory Eq. (4) changes to following form.
2
02
1 [ ] ( )( ).ef M T ef ef
xx xx xx xxE e e E e E z T T
x
(7)
The equation for the axial motion of the beam in the absence of external force can be
obtained as. [16]:
2
1 2
( , )
,
N u x t
I
x t
(8)
where ( , )u x t is the axial displacement, ( )z is the mass per unit length, N is the axial
force per unit length and I is as bellow,
/2
1
/2
; ( ) ,
h
xx
A h
N dA I b z dz
(9)
where A is the cross-sectional area of the beam. Integrating Eq. (6) with respect to area
gives the following relation:
2
1 2 02
1 ( ),xxN B e B T T
x
(10)
where
/2
1
/2
( )
h
ef
h
B b E z dz
and
/2
2
/2
( ) (z)
h
ef
h
B b E z dz
.
Using Eqs. (7) – (9), the equation of axial vibration can be found in terms of displace-
ment
2 4 2
1 1 1 2 02 2 2 2
( , ) ( , ) ( , )
( ),
u x t u x t u x t
I I B B T T
xt t x x
(11)
where 0 Eq. (10) reduces to the classic form.
In the other hand the heat conduction equation without any thermal source is. [16]:
, 0
( )
( ) ( ) ( ) ,
1 2ii ii
E z
k z c z z T e
(12)
where k and c are thermal conductively and specific heat constant at a constant volume. Eq. (11)
in the form of displacement obtained as:
2 2
02
( )
( ) ( ) ( ) ( ) .
1 2
E z u
k z z c z z T
t x tx
(13)
By using the following non-dimensional parameters:
ˆ ;
u
u
l
ˆ ;
x
x
l
0
ˆ ;
T
0
ˆ ;
t
t
t
2
2
0
( ) (1 )(1 )
.
( )
z L
t
E z
The dimensional-less form of coupled Eqs. (10), (12) obtains as bellow:
2 2 4
1 2 3 42 2 2 2
ˆˆ ˆ ˆ
= ;
ˆ ˆˆˆ ˆ
u u u
D D D D
xx t t x
(14a)
136
2 2
1 2 32
ˆ ˆ ˆ
,
ˆ ˆˆˆ
u
C C C
t x tx
(14b)
where D and C coefficients are as:
2 01 1
1 2 3 1 42 2
0 0
; ; ; ;
B TB L I
D D D I D
L L t t L
0 0 0
1 2 32
0 0
( ) ( ) ( ) ( )
; ; .
(1 2 )
efk z T z c z T E z T
C C C
L t t
3. Numerical solution.
To solve the coupled equation Galerkin method has been used. Based on the Galerkin
method:
1 1 1 1
ˆˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ( , ) ( ) ( ) ( ) ( ); ( , ) ( ) ( ) ( ) ( ),
N M
n n n n m m m m
n n m m
u x t x a t x a t x t x b t x b t
(15)
where ˆ( )n x , ˆ( )m x are the suitable shape functions and ˆ( )na t , ˆ( )mb t are time dependent
coefficients. The shape functions satisfy the both end clamped boundary conditions of our
problem so the form of they is as bellow:
ˆ ˆ ˆ ˆ( ) sin( ); ( ) sin( ).n mx n x x m x (16)
Substituting Eq. (15) into Eq. (14) and multiplying Eq. (14a) into ˆ( )i x and Eq. (15)
into ˆ( )j x and integrating outcome from 0 to 1 and also considering first mode shape of
displacement and second one of thermo leads to:
3 4 1 1 1 2 2 1 2 2 2 3 1
ˆ ˆ ˆ ˆ ˆ ˆ (- ) ( ) ( ) ( ) 0; ( ) ( ) ( ) 0,D D a t D a t D b t C b t C b t C a t (17)
By considering
ˆ
1
ˆ( ) ta t ae and
ˆ
2
ˆ( ) tb t be Eq. (17) changes to the following form:
2
3 4 1 2 1 2 3(( ) ) 0; ( ) 0.D D s D a D b C C s b C sa (18)
So the natural frequencies of the system obtain by solving Eq. (18). The D and C coef-
ficients are as bellow:
1 1
1 1 2 2
0 0
1 1
3 3 4 4
0 0
ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) ; ( ) ( ) ;
ˆ ˆ ˆ ˆ ˆ ˆ ( ) ( ) ; ( ) ( ) ;
n i m i
n i n i
D D x x dx D D x x dx
D D x x dx D D x x dx
1 1 1
1 1 2 2 3 3
0 0 0
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) ; ( ) ( ) ; ( ) ( ) .m j m j n jC C x x dx C C x x dx C C x x dx
4. Results.
In this section the numerical results has been presented. According to the power-law
distribution of a material’s property, as an example, variation of the elastic modulus and
thermal conductivity through the thickness with different material-variation profile parame-
ter m is shown in Fig. 1. As can be noticed, the variation of a material property can be en-
hanced via manipulating the control parameter m. The FGM nano beam has been composed
of Nickel and silicon nitride. Its property varying through the thickness based on the power
137
low. Its bottom and upper surface is pure nickel and pure silicon nitride respectively. Mate-
rials properties have been presented in Table 1.
Fig. 1. Variation of the elastic modulus and thermal conductivity through the thickness
with different material-variation profile parameter m.
Geometrical and material properties.
Table 1
Material Nickel [14] Silicon nitride [22]
Density (kg/m3) 8900 2300
Thermal conductivity (w/mk) 92 30
Young modulus(Gpa) 210 310
Thermal expansion (1/k)× 10-6 13 3.4
Specific heat at constant volume (j/kgk) 438 680
Poisson’s ratio 0,31 0,27
According to the complex frequency approach, quality factor of thermo-elastic damping
(QTED) can be achieved as [6, 23]:
1 1 Re( )
.
2 2 Im( )TEDQ
(19)
To validate our results a comparison has been made with the results of Maroofi et al.
[14] in Table. 2 for silicon micro beam where 200μm, 169Gpa, 150 w / mk,L E k
3695 j / kgk, 2300 kg / m , 0,28c and 62,6 10 . As it can be seen there is a
good agreement between them.
A comparison between quality factor (QTED×106) in longitudinal vibration for silicon
beam.
Table 2
Temperature (T0) Maroofi et al. [47] present
300 K 2,861 3,107
400 k 2,146 2,331
In Fig. 2 the effects of gradient index ( 0 40)m for three value of nonlocal parameter
2( 0, 24 nm ) have been showed. As it can be seen increasing of gradient index increases
138
the quality factor in the nonlocal and classic theory. For gradient index between 0 – 10 the
variation of the TEDQ is large but as the gradient index increases the diagram will be linear.
Note that 0 presents classic theory and where 0m the beam is pure nickel and by
increasing gradient index the silicon bromide’s present increases. Fig. 3 shows the effects of
nonlocal parameter on the quality factor. As there is no value has been introduced for the
FGM beam so here the nonlocal parameter has been chosen between 0 – 4 nm2. As it is visi-
ble increasing of causes increase in the TED.Q Comparison of Fig. 3, a and Fig. 3, b
shows that variation of for 20 nmL has lower effects than 10nmL on the TED.Q This
is means that increasing of the length decreases the effects of nonlocal parameter on the TED.Q
Fig. 2. Effects of gradient index on the thermo-elastic damping quality factor where
010nm, 300 K.L T
Fig. 3. Effects of nonlocal parameter on the thermo-elastic damping quality factor a:
0 010 nm, 300 kb : 10 nm, 300 K.L T L T
Effects of length variation have been showed in Fig. 4, where length varies from 10 – 100
nm and gradient index is 0, 2, 4,10.m As expected decreasing of length causing decrease
in TED.Q As it can be seen increasing of the gradient index decreases this variation. In other
139
hand it can be found that length variation’s effects on the TEDQ decreases as the silicon-
nitride’s present of the FGM beam increases.
In Fig. 5 ambient temperature versus TEDQ is showed for nickel nano beam based on
classic and Eringen nonlocal theory ( 0, 0 and 0, 2)m m and the FGM nano
beam based on classic and Eringen nonlocal theory ( 2, 0 and 2, 2).m m It is
visible by increasing of ambient temperature the QTED decreases. Also it can be seen that by
increasing of T0 the effects of gradient index and nonlocal parameter on the TEDQ de-
creases. In table 3 TEDQ in the 0 200T and 500 K and also the difference of this two qual-
ity factor ( TED1Q – TED2Q ) has been showed. This table shows that by increasing of gradi-
ent index and nonlocal parameter the effects of ambient temperature’s variation on the
TEDQ increases.
Fig. 4. Length variation versus thermo-elastic damping quality factor (μ=0).
Fig. 5. Ambient temperature versus thermo-elastic damping quality factor ( 10 nm).L
140
TEDQ of 0 200 & 500T for different values of gradient index and nonlocal parameter.
Table 3
m 2(nm )
TED1Q
0( 200K)T
TED2Q
0( 500K)T
TED1Q – TED2Q
0 1456 582,5 873,5
2 1586 634,5 951,5 0
4 1706 682,5 1023,5
0 1818 727,1 1090,9
2 1950 779,8 1170,2 2
4 2074 829,4 1244,6
0 2116 846,2 1269,8
2 2253 901 1352 4
4 2383 953,1 1429,9
5. Conclusion.
Thermoelastic damping of axial vibration of the FGM nano beam has been presented
based on the Eringen nonlocal theory. It has been composed of Nickel and silicon nitride. Its
property varying through the thickness based on the power low. Its bottom and upper sur-
face is pure nickel and pure silicon nitride respectively. The effects of gradient index,
nonlocal parameter, Length of the beam and ambient temperature has been presented on the
thermo-elastic damping quality factor. In the other word a FGMNEM system, with a con-
trollable thermo-elastic damping of axial vibration based on Eringen nonlocal theory has
been presented and has been showed that increasing of gradient index (increasing of silicon
bromide’s present), nonlocal parameter and length of the nano beam, increases the quality
factor and increasing of ambient temperature decreases it. Also is showed that by increasing
of ambient temperature decreases the effects of gradient index and nonlocal parameter and
increasing of the gradient index decreases the effects of the nano beam’s length variation, on
the TED.Q The achieved results can be used as a design implement for the designers to con-
trol TED.
РЕЗЮМЕ . Термопружне демпфування є головним джерелом внутрішнього демпфування в мік-
роелектромеханічних і наноелектромеханічних системах. Внутрішнє демпфування не може бути ні
кероване, ні мінімізоване, поки механічні і геометричні властивості матеріалу не є змінними. Тому
розглянуто осьові коливання балки з нового матеріалу – функціонально градієнтного матеріалу з
врахуванням наноелектромеханічних явищ – в рамках нелокальної теорії Ерінгена. Розглянуто вплив
різних параметрів – показника градієнтості, параметрів нелокальності, довжини нанобалки, темпера-
тури навколишнього середовища – на коефіцієнт термопружного демпфування. Показано, що термо-
пружне демпфування може бути кероване зміною цих параметрів.
1. Lepage S. Stochastic finite element method for the modeling of thermoelastic damping in microresonators,
Ph. D. Dissertation, University of Liege, Departmentof Aerospace and Mechanics (2006).
2. Lepage S. Stochastic finite element method for the modeling of thermoelastic damping in micro-
resonators, Ph.D. Dissertation, University of Liege, Department of Aerospace and Mechanics, 1-38
(2006).
3. Younis M.I. Modeling and Simulation of Microelectromechanical Systems in Multi-Physics Fields. Proc.
Virginia Polytechnic Institute and State University, Blacksburg (2004).
4. Vahdat A.S., Rezazadeh G. Effects of axial and residual stresses on thermoelastic damping in capacitive
micro-beam resonator. J. Franklin Institute, 348(4), 622 – 639 (2011).
5. Kim S.-B., Kim J.-H. Quality factors for the nano-mechanical tubes with thermoelastic damping and initial
stress. J. Sound Vibr., 330, 1393 – 1402 (2011).
141
6. Rezazadeh G. et al. Thermoelastic damping in a micro-beam resonator using modified couple stress the-
ory. Acta Mechanica, 223(6), 1137 – 1152 (2012).
7. Duwel A., Gorman J., Weinstein M., Borenstein J., Ward P. Experimental study of thermoelastic damping
in MEMS gyros. Sens. Actuators, A103 (1 – 2), 70 – 75 (2003).
8. Lu P., Lee H.P., Lu C., Chen H.B. Thermoelastic damping incylindrical shells with application to tubular
oscillator structures. Int. J. Mech. Sci., 50 (3), 501 – 512 (2008).
9. Lifshitz R., Roukes M.L. Thermoelastic damping in micro- and nano mechanical systems. Phys. Rev. B61,
5600–5609 (2000).
10. Lepage S. et al. Thermoelastic Damping in Vibrating Beam Accelerometer: A new thermoelastic finite
element approach. CANEUS 2006: MNT for Aerospace Applications. American Society of Mechanical
Engineers (2006).
11. Nayfeh H., Younis M.I. Modeling and simulations of thermoelastic damping in microplates. J. Micro-
mech. Microeng. 14, 1711 – 1717 (2004).
12. Rezazadeh G. et al. Study of thermoelastic damping in capacitive micro-beam resonators using hyper-
bolic heat conduction model. Sens. Transducers J, 108(9), 54 – 72 (2009).
13. Gorman D. Free in-plane vibration analysis of rectangular plates by the method of superposition, Journal
of Sound and Vibration, 272, 831 – 851 (2004).
14. Maroofi M.S., Najafi R., Shabani Rezazadeh G. Investigation of Thermoelastic Damping in the Longitu-
dinal Vibration of a Micro Beam. Int. J. of Mechanics, 28 (2), 314 – 320 (2015).
15. Nazemnezhad Reza, Shahrokh Hosseini-Hashemi Nonlocal nonlinear free vibration of functionally
graded nanobeams. Composite Structures, 110, 192 – 199 (2014).
16. Sadd M.H. Elasticity theory, applications, and numerics, Academic Press, (2009).
17. Timoshenko S., Woinowsky-Krieger S., Woinowsky S. Theory of plates and shells, McGraw-Hill, New
York, (1959).
18. Khanchehgardan A. et al. Thermo-elastic damping in nano-beam resonators based on nonlocal theory.
International Journal of Engineering-Transactions C: Aspects 26(12), 1505 (2013).
19. Eringen A.C. Nonlocal polar elastic continua, Int. J. Eng. Sci. 10, 1 – 16, (1972).
20. Eringen A.C. On differential equations of nonlocal elasticity and solutions of screw dislocation and sur-
face waves, J. Appl. Phys. 54, 4703 – 4710, (1983).
21. Eringen A.C. Nonlocal Continuum Field Theories, Springer-Verlag, New York, (2002).
22. Zamanzadeh Mohammadreza, et al. Static and dynamic stability modeling of a capacitive FGM micro-
beam in presence of temperature changes. Appl. Mathem. Modelling, 37(10), 6964 – 6978, (2013).
23. Lifshitz R., Roukes M.L. Thermoelastic damping in micro-and nanomechanical systems, Physical Review
B, 61, 5600 (2000).
From the Editorial Board: The article corresponds completely to submitted manuscript.
Поступила 22.01.2016 Утверждена в печать 30.05.2017
|