Chapter 8
Thermoelastic Vibrations of Timoshenko
Microbeams (Modified Couple Stress
Theory)
8.1 Introduction
Thermoelastic vibrations of the Timoshenko microbeams based on the modified couple stress theory are studied. In particular, the dependence of the quality factor of nonlinear microbeam resonators under thermoelastic damping for Timoshenko beams
with regard to geometric nonlinearity is analyzed. The constructed mathematical
model is based on the modified couple stress theory which implies prediction of sizedependent effects in microbeam resonators. The Hamilton principle yields coupled
nonlinear thermoelastic PDEs governing dynamics of the Timoshenko microbeams
for both plane stresses and plane deformations. Nonlinear thermoelastic vibrations
are investigated analytically and numerically and quality factors of the resonators
versus geometric and material microbeam properties are estimated. Results are presented for gold microbeams for different ambient temperatures and different beam
thicknesses, and they are compared with results yielded by the classical theory of
elasticity in linear/nonlinear cases [1]. The most important conclusions of our study
are summarized in the following three points.
1. Results reported in Figs. 8.4, 8.6 and 8.5, 8.7 with size-dependent behaviour and in
the case of linear vibrations exhibit increase of the eigenfrequencies and increase
of the quality factor of the beam resonator.
2. The occurrence of nonlinearity in both tested boundary (simple-simple) and
(clamped-clamped) conditions implies a significant increase in the frequency
of the fundamental vibration mode as well as the quality factor of the resonator
(Figs. 8.8, 8.9, 8.10 and 8.11).
3. The analysis of the obtained results shows that in the case of the thermoelastic
damping, it is necessary to take into account the nonlinear behaviour as well as
size-dependent effects of the microbeams. Both mentioned features are crucial
for the quality factor of the beam resonators.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
J. Awrejcewicz et al., Mathematical Modelling and Numerical Analysis of Size-Dependent
Structural Members in Temperature Fields, Advanced Structured Materials 142,
https://doi.org/10.1007/978-3-030-55993-9_8
295
Thermoelastic Vibrations of Timoshenko
Microbeams (Modified Couple Stress
Theory)
8.1 Introduction
Thermoelastic vibrations of the Timoshenko microbeams based on the modified couple stress theory are studied. In particular, the dependence of the quality factor of nonlinear microbeam resonators under thermoelastic damping for Timoshenko beams
with regard to geometric nonlinearity is analyzed. The constructed mathematical
model is based on the modified couple stress theory which implies prediction of sizedependent effects in microbeam resonators. The Hamilton principle yields coupled
nonlinear thermoelastic PDEs governing dynamics of the Timoshenko microbeams
for both plane stresses and plane deformations. Nonlinear thermoelastic vibrations
are investigated analytically and numerically and quality factors of the resonators
versus geometric and material microbeam properties are estimated. Results are presented for gold microbeams for different ambient temperatures and different beam
thicknesses, and they are compared with results yielded by the classical theory of
elasticity in linear/nonlinear cases [1]. The most important conclusions of our study
are summarized in the following three points.
1. Results reported in Figs. 8.4, 8.6 and 8.5, 8.7 with size-dependent behaviour and in
the case of linear vibrations exhibit increase of the eigenfrequencies and increase
of the quality factor of the beam resonator.
2. The occurrence of nonlinearity in both tested boundary (simple-simple) and
(clamped-clamped) conditions implies a significant increase in the frequency
of the fundamental vibration mode as well as the quality factor of the resonator
(Figs. 8.8, 8.9, 8.10 and 8.11).
3. The analysis of the obtained results shows that in the case of the thermoelastic
damping, it is necessary to take into account the nonlinear behaviour as well as
size-dependent effects of the microbeams. Both mentioned features are crucial
for the quality factor of the beam resonators.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
J. Awrejcewicz et al., Mathematical Modelling and Numerical Analysis of Size-Dependent
Structural Members in Temperature Fields, Advanced Structured Materials 142,
https://doi.org/10.1007/978-3-030-55993-9_8
295