Finding the thermoelastic damping in a vibrating body, for the most general case, involves the simultaneous solving of the three equations for displacements and one equation for temperature (called the heat equation). Since these are a set of coupled nonlinear partial differential equations there is considerable difficulty in solving them, especially for finite geometries. This paper presents a single degree of freedom (SDOF) model that explores the possibility of estimating thermoelastic damping in a body, vibrating in a particular mode, using only its geometry and material properties, without solving the heat equation. In doing so, the model incorporates the notion of “modal temperatures,” akin to modal displacements and modal frequencies. The procedure for deriving the equations that determine the thermoelastic damping for an arbitrary system, based on the model, is presented. The procedure is implemented for the specific case of a rectangular cantilever beam vibrating in its first mode and the resulting equations solved to obtain the damping behavior. The damping characteristics obtained for the rectangular cantilever beam, using the model, is compared with results previously published in the literature. The results show good qualitative agreement with Zener’s well known approximation. The good qualitative agreement between the predictions of the model and Zener’s approximation suggests that the model captures the essence of thermoelastic damping in vibrating bodies. The ability of this model to provide a good qualitative picture of thermoelastic damping suggests that other forms of dissipation might also be amenable for description using such simple models.

1.
Nowacki
,
W.
, 1986,
Thermoelasticity
,
2nd ed.
,
Pergamon Press
,
New York
.
2.
Nowick
,
A. S.
, and
Berry
,
B. S.
, 1972,
Anelastic Relaxation in Crystalline Solids
,
Academic
,
New York
.
3.
Zener
,
C.
, 1937, “
Internal Friction in Solids. I. Theory of Internal Friction in Reeds
,”
Phys. Rev.
0031-899X,
52
, pp.
230
235
.
4.
Zener
,
C.
, 1938, “
Internal Friction in Solids. II. General Theory of Thermoelastic Internal Friction
,”
Phys. Rev.
0031-899X,
53
, pp.
90
99
.
5.
Zener
,
C.
,
Otis
,
W.
, and
Nuckolls
,
R.
, 1938, “
Internal Friction in Solids. III. Experimental Demonstration of Thermoelastic Internal Friction
,”
Phys. Rev.
0031-899X,
53
, pp.
100
101
.
6.
Alblas
,
J. B.
, 1961, “
On the General Theory of Thermo-Elastic Friction
,”
Appl. Sci. Res., Sect. A
0365-7132,
10
, pp.
349
362
.
7.
Alblas
,
J. B.
, 1981, “
A Note on the General Theory of Thermoelastic Damping
,”
J. Therm. Stresses
0149-5739,
4
, pp.
333
355
.
8.
Chadwick
,
P.
, 1962, “
On the Propagation of Thermoelastic Disturbances in Thin Plates and Rods
,”
J. Mech. Phys. Solids
0022-5096,
10
, pp.
99
109
.
9.
Lord
,
H. W.
, and
Shulman
,
Y.
, 1967, “
A Generalized Dynamical Theory of Thermoelasticity
,”
J. Mech. Phys. Solids
0022-5096,
15
, pp.
299
309
.
10.
Nguyen
,
C. T. C.
, 2004, “
Vibrating RF Mems for Next Generation Wireless Applications
,”
Customs Integrated Circuits Conferencee, 2004
,
Proceedings of the IEEE 2004
, 3–6, Oct. 2004,
IEEE
,
Piscataway, NJ
, pp.
257
274
.
11.
Cleland
,
A. N.
, and
Roukes
,
M. L.
, 1998, “
A Nanometer-Scale Mechanical Electrometer
,”
Nature (London)
0028-0836,
392
, pp.
160
162
.
12.
Tilman
,
H. A. C.
,
Elwenspoek
,
M.
, and
Fluitman
,
J. H. J.
, 1992, “
Micro Resonant Force Gauges
,”
Sens. Actuators, A
0924-4247,
30
, pp.
35
53
.
13.
Zook
,
J. D.
,
Burns
,
D. W.
,
Guckel
,
H.
,
Sniegowski
,
J. J.
,
Engelstad
,
R. L.
, and
Feng
,
Z.
, 1992, “
Characteristics of Polysilicon Resonant Microbeams
,”
Sens. Actuators, A
0924-4247,
35
, pp.
51
59
.
14.
Lifshitz
,
R.
, and
Roukes
,
M. L.
, 2000, “
Thermoelastic Damping in Micro- and Nanomechanical Systems
,”
Phys. Rev. B
0163-1829,
61
, pp.
5600
5609
.
15.
D. M.
Photiadis
,
A. B. H. H.
,
Liu
,
X.
,
Bucaro
,
J. A.
, and
Marcus
,
M. H.
, 2002, “
Thermoelastic Loss Observed in a High q Mechanical Oscillator
,”
Physica B
0921-4526,
316
, pp.
408
410
.
16.
Houston
,
B. H.
,
Photiadis
,
D. M.
,
Marcus
,
M. H.
,
Bucaro
,
J. A.
,
Liu
,
X.
, and
Vignola
,
J. F.
, 2002, “
Thermoelastic Loss in Microscale Oscillators
,”
Appl. Phys. Lett.
0003-6951,
80
, pp.
1300
1302
.
17.
Houston
,
B. H.
,
Photiadis
,
D. M.
,
Vignola
,
J. F.
,
Marcus
,
M. H.
,
Liu
,
X.
,
Czaplewski
,
D.
,
Sekaric
,
L.
,
Butler
,
J.
,
Pehrsson
,
P.
, and
Bucaro
,
J. A.
, 2004, “
Loss Due to Transverse Thermoelastic Currents in Microscale Resonators
,”
Mater. Sci. Eng., A
0921-5093,
370
, pp.
407
411
.
18.
Nayfeh
,
A. H.
, and
Younis
,
M. I.
, 2004, “
Modeling and Simulations of Thermoelastic Damping in Microplates
,”
J. Micromech. Microeng.
0960-1317,
14
, pp.
1711
1717
.
19.
Norris
,
A. N.
, and
Photiadis
,
D. M.
, 2005, “
Thermoelastic Relaxation in Elastic Structures with Applications to Thin Plates
,”
Q. J. Mech. Appl. Math.
0033-5614,
58
, pp.
143
163
.
20.
Landau
,
L. D.
, and
Lifshitz
,
E. M.
, 1959,
Theory of Elasticity
,
Pergamon Press
,
London
.
You do not currently have access to this content.