300
8 Thermoelastic Vibrations of Timoshenko Microbeams
large deflection nonlinearity at the resonant frequency localization and investigated
the velocity of damping of cantilever microbeams but omitted the effect of large
amplitude vibration on TED. In reference [93], the investigated thermoelastic damping with an account of nonlinear effects of the circle plate modelled with the use of
the von Kármán hypothesis was analyzed. Employing the Kantorovich method and
the methods of perturbation, a formula describing the resonator quality factor was
obtained.
The authors of reference [94] studied nonlinear effects under large deflections and
under thermoelastic dissipation of the microbeam resonators for adiabatic/isothermic
beam surfaces. The models of thermoelasticity and thermoelastic dissipation were
formulated for the case of large amplitude vibrations for the Euler-Bernoulli beam
without the size-dependent effects. Mendez et al. [92] carried out numerical modelling for MEMS cantilever, and a difference between linear and nonlinear approximations was illustrated and discussed. It was shown that for large displacements,
the beam becomes stiffer (frequency increase) and its vibrations are damped faster.
Those observations made it possible to improve the fabrication of high-frequency
MEMS resonators. In the work [95], the importance of nonlinear MEMS behaviour
with a strong coupling between mechanical and electric fields was demonstrated.
The overview of the state of the art of the MEMS/NEMS beam vibrations carried
out so far yields the following conclusions:
(i) thermoelastic damping of microbeam vibrations with large amplitudes plays a
crucial role in a proper fabrication of the resonators;
(ii) the influence of geometric nonlinearity on the values of frequency, damping and
on quality factors of the resonators is often omitted in the available literature;
(iii) both thermoelastic damping and large displacements may have an important
effect not only on the high-frequency resonators but also on the MEMS/NEMS
devices [39, 79, 88, 89];
(iv) in general, the thermoelastic PDEs governing the dynamics of microbeams were
obtained based on the classical theory of continuum. There are no sufficient
results following from the nonlinear model of coupled thermoelasticity of the
Timoshenko microbeams based on the MCST.
This chapter is aimed at filling the gaps that exist in the research of MEMS/NEMS
resonators. In this work, we employ the MCST to study quality factor of microbeam
resonators with an account of thermoelastic damping and geometric nonlinearity.
The associated mathematical model is constructed which, in contrary to the classical
theory, consists of the internal scale length material parameter allowing for a reliable prediction of the size-dependent effect in microbeam resonators. The Hamilton
principle yields coupled nonlinear thermoelastic equations of motion of the Timoshenko microbeams in the case of plane stress/deflection state. Based on the obtained
equations both analytical and numerical investigations of the nonlinear thermoelastic
vibrations of the microbeams are carried out and then the values of quality factors
of the resonators versus geometric and material microbeam properties are estimated.
The results are presented for gold microbeams at different ambient temperatures and
8 Thermoelastic Vibrations of Timoshenko Microbeams
large deflection nonlinearity at the resonant frequency localization and investigated
the velocity of damping of cantilever microbeams but omitted the effect of large
amplitude vibration on TED. In reference [93], the investigated thermoelastic damping with an account of nonlinear effects of the circle plate modelled with the use of
the von Kármán hypothesis was analyzed. Employing the Kantorovich method and
the methods of perturbation, a formula describing the resonator quality factor was
obtained.
The authors of reference [94] studied nonlinear effects under large deflections and
under thermoelastic dissipation of the microbeam resonators for adiabatic/isothermic
beam surfaces. The models of thermoelasticity and thermoelastic dissipation were
formulated for the case of large amplitude vibrations for the Euler-Bernoulli beam
without the size-dependent effects. Mendez et al. [92] carried out numerical modelling for MEMS cantilever, and a difference between linear and nonlinear approximations was illustrated and discussed. It was shown that for large displacements,
the beam becomes stiffer (frequency increase) and its vibrations are damped faster.
Those observations made it possible to improve the fabrication of high-frequency
MEMS resonators. In the work [95], the importance of nonlinear MEMS behaviour
with a strong coupling between mechanical and electric fields was demonstrated.
The overview of the state of the art of the MEMS/NEMS beam vibrations carried
out so far yields the following conclusions:
(i) thermoelastic damping of microbeam vibrations with large amplitudes plays a
crucial role in a proper fabrication of the resonators;
(ii) the influence of geometric nonlinearity on the values of frequency, damping and
on quality factors of the resonators is often omitted in the available literature;
(iii) both thermoelastic damping and large displacements may have an important
effect not only on the high-frequency resonators but also on the MEMS/NEMS
devices [39, 79, 88, 89];
(iv) in general, the thermoelastic PDEs governing the dynamics of microbeams were
obtained based on the classical theory of continuum. There are no sufficient
results following from the nonlinear model of coupled thermoelasticity of the
Timoshenko microbeams based on the MCST.
This chapter is aimed at filling the gaps that exist in the research of MEMS/NEMS
resonators. In this work, we employ the MCST to study quality factor of microbeam
resonators with an account of thermoelastic damping and geometric nonlinearity.
The associated mathematical model is constructed which, in contrary to the classical
theory, consists of the internal scale length material parameter allowing for a reliable prediction of the size-dependent effect in microbeam resonators. The Hamilton
principle yields coupled nonlinear thermoelastic equations of motion of the Timoshenko microbeams in the case of plane stress/deflection state. Based on the obtained
equations both analytical and numerical investigations of the nonlinear thermoelastic
vibrations of the microbeams are carried out and then the values of quality factors
of the resonators versus geometric and material microbeam properties are estimated.
The results are presented for gold microbeams at different ambient temperatures and