8.2 Literature Review
299
al. [76] derived analytical solutions to estimate the quality factor of TED obtained
on the basis of MCST for the plane stress and plane deformation states of a linear
Euler-Bernoulli beam. They found that when the beam thickness is close to the material length parameter, the results obtained by MCST are different compared to the
results obtained using the classical beam theory.
In order to explain size dependence for microbeams, Taati et al. [77] combined
the MCST and hyperbolic model of heat transfer which differs from the classical
Fourier model. They showed that if one uses the hyperbolic heat transfer model then
MCST shows more deviations from results compared to the classical theory.
In reference [78], the size-dependent model was proposed using the time of thermal relaxation for temporal analysis of thermoelasticity of Timoshenko microbeams.
Besides, the size-dependent effects of a simply supported beam were studied under
constant impulse force where two beam ends were exposed to constant ambient
temperature.
Since eigenfrequencies of microbeam vibrations change when accounting for
MCST, it may be expected that TED forecast by MCST will also differ from the
results yielded by the classical theory of beams. Therefore, the influence of the size
length material parameter on quality factor Q of microbeam resonators still requires
a deeper analysis (in particular, when the characteristic size is compared with the
internal size parameter of material length).
The microbeam resonators often work in nonlinear regimes with large amplitude
while harvesting energy [79]. A key role in the studies of beam resonators plays
energy dissipation in micro- and nanoscales [2, 4, 5, 80, 81]. In particular, TED
belongs to one of the principal mechanism of energy dissipation for a large amount
of the micro/nanomaterial resonators [3–5, 82, 83]. However, as it has already been
mentioned, almost all published results deal with linearized regimes of vibrations of
the microbeam resonators taking into account only vibrations with small amplitude.
Nonlinearity in micro-resonators may occur due to various reasons including large
deflections (geometric nonlinearity or material nonlinearity). Geometric nonlinearity plays a key role in the microbeam resonator [84–86]. Though it was observed in
high-frequency MEMS/NEMS resonators [79, 87–89], to our knowledge there are
no reference studies on large deflection impact on thermoelastic dissipation. There
are a few papers devoted to the analysis and quantitative estimation of the influence
of thermoelastic damping in MEMS. Majority of the mentioned works deal with
small amplitude linear vibrations [30, 32, 90]. For instance, in the work [83], though
large static deflection generated by electrostatic forces under the action of constant
stress is studied, thermoelastic investigation is based on the linearized vibration of
small amplitude in the neighbourhood of static equilibrium with large deflection.
An analogous approach can be found in reference [51], where large static deflection
generated by electrostatic input affects the thermoelastic damping of linear small
amplitude vibrations around static equilibrium. A series of analogous examples are
reported in paper [41], where TEDs of resonators made from carbon tubes and vibrating around equilibrium were studied. Despite the thermoelastic dissipation analysis
within linear theory, there are several studies devoted to nonlinear dynamics under
large deflections of beam MEMS resonators [20, 91, 92]. Méndez et al. [92] studied
299
al. [76] derived analytical solutions to estimate the quality factor of TED obtained
on the basis of MCST for the plane stress and plane deformation states of a linear
Euler-Bernoulli beam. They found that when the beam thickness is close to the material length parameter, the results obtained by MCST are different compared to the
results obtained using the classical beam theory.
In order to explain size dependence for microbeams, Taati et al. [77] combined
the MCST and hyperbolic model of heat transfer which differs from the classical
Fourier model. They showed that if one uses the hyperbolic heat transfer model then
MCST shows more deviations from results compared to the classical theory.
In reference [78], the size-dependent model was proposed using the time of thermal relaxation for temporal analysis of thermoelasticity of Timoshenko microbeams.
Besides, the size-dependent effects of a simply supported beam were studied under
constant impulse force where two beam ends were exposed to constant ambient
temperature.
Since eigenfrequencies of microbeam vibrations change when accounting for
MCST, it may be expected that TED forecast by MCST will also differ from the
results yielded by the classical theory of beams. Therefore, the influence of the size
length material parameter on quality factor Q of microbeam resonators still requires
a deeper analysis (in particular, when the characteristic size is compared with the
internal size parameter of material length).
The microbeam resonators often work in nonlinear regimes with large amplitude
while harvesting energy [79]. A key role in the studies of beam resonators plays
energy dissipation in micro- and nanoscales [2, 4, 5, 80, 81]. In particular, TED
belongs to one of the principal mechanism of energy dissipation for a large amount
of the micro/nanomaterial resonators [3–5, 82, 83]. However, as it has already been
mentioned, almost all published results deal with linearized regimes of vibrations of
the microbeam resonators taking into account only vibrations with small amplitude.
Nonlinearity in micro-resonators may occur due to various reasons including large
deflections (geometric nonlinearity or material nonlinearity). Geometric nonlinearity plays a key role in the microbeam resonator [84–86]. Though it was observed in
high-frequency MEMS/NEMS resonators [79, 87–89], to our knowledge there are
no reference studies on large deflection impact on thermoelastic dissipation. There
are a few papers devoted to the analysis and quantitative estimation of the influence
of thermoelastic damping in MEMS. Majority of the mentioned works deal with
small amplitude linear vibrations [30, 32, 90]. For instance, in the work [83], though
large static deflection generated by electrostatic forces under the action of constant
stress is studied, thermoelastic investigation is based on the linearized vibration of
small amplitude in the neighbourhood of static equilibrium with large deflection.
An analogous approach can be found in reference [51], where large static deflection
generated by electrostatic input affects the thermoelastic damping of linear small
amplitude vibrations around static equilibrium. A series of analogous examples are
reported in paper [41], where TEDs of resonators made from carbon tubes and vibrating around equilibrium were studied. Despite the thermoelastic dissipation analysis
within linear theory, there are several studies devoted to nonlinear dynamics under
large deflections of beam MEMS resonators [20, 91, 92]. Méndez et al. [92] studied