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8 Thermoelastic Vibrations of Timoshenko Microbeams
8.2 Literature Review
Thermoelastic damping/dissipation (TED) occurs in all elastic materials subjected
to cycling deformations, and particularly in the case when the period of an exciting
cycle is close to the material relaxation [2–5]. For instance, in the case of vibrations of
an elastic beam, a major part of mechanical work is transformed into elastic energy,
but part of the work is converted into thermal energy. During bending, one side of
the structure is extended and cooled, while the other side is compressed and heated.
Temperature non-homogeneity described so far yields a transfer of heat from the more
heated part to the more cooled part of the structure. According to the second law of
thermodynamics, the latter heat transfer causes the increase of entropy and finally
the loss of mechanical energy in the form of heat. There are various mechanisms
of energy loss, such as thermoelastic damping [2, 6–8], energy loss in supports [9],
internal energy loss [10] and air damping [11].
In almost all macroscopic systems with basic damping sources, thermoelastic
coupling plays a secondary role because it is an order of magnitude lower than the
damping caused by external and internal viscous effects. However, in the case of
MEMS/NEMS devices and resonators, it plays a crucial role. It should be emphasized that micromechanical resonators with high quality factors are employed in a
wide range of MEMS applications, such as sensors [12–16], electric filters [17, 18],
accelerometers [19], giros [8], atomic force microscopes [20], MEMS type switches,
and various other mechanisms [21–23]. From a technical point of view, the evaluation of thermoelastic damping (TED) of those structural elements at the design stage
plays an important role in maintaining the required engineering characteristics. TED
belongs to fundamental sources of quantification damping in the microelectromechanical systems (MEMS) [24] and nanoelectromechanical systems (NEMS) [4,
25] working in the vacuum regime. Evoy et al. [26] and Duwel et al. [8] have shown
experimentally that TED is the dominant source of damping in MEMS and NEMS
devices.
Zener [2, 27, 28] belongs to the first who defined the occurrence of TED and
showed its crucial role in dissipation exhibited by bending resonators. Landau and
Lifshitz [29] estimated analytically the TED coefficients of elastic vibrations. The
second widely employed analytical model has been developed by Lifshitz and Roukes
[3]. In that model, a change in the resonance frequency associated with TED was
observed, and thus the classical Zener model was improved. The mentioned authors
have also derived an analytical expression for quality factor Q in microbeam resonators and studied the influence of numerous geometric parameters on their quality factors. Nayfeh and Younis [30] constructed the analytical expression for Q of
a microplate of arbitrary forms for different boundary conditions and studied the
occurrence of TED. Sun et al. [31–33], Sharma et al. [34] and Rezazadeh et al.
[35] investigated thermoelastic damping in capacity microbeam resonators using
the hyperbolic heat transfer model. Besides, Vahdat and Rezazadeh [24] investigated the effect of axial and residual stresses on thermoelastic damping in a capacity
microbeam resonator.
8 Thermoelastic Vibrations of Timoshenko Microbeams
8.2 Literature Review
Thermoelastic damping/dissipation (TED) occurs in all elastic materials subjected
to cycling deformations, and particularly in the case when the period of an exciting
cycle is close to the material relaxation [2–5]. For instance, in the case of vibrations of
an elastic beam, a major part of mechanical work is transformed into elastic energy,
but part of the work is converted into thermal energy. During bending, one side of
the structure is extended and cooled, while the other side is compressed and heated.
Temperature non-homogeneity described so far yields a transfer of heat from the more
heated part to the more cooled part of the structure. According to the second law of
thermodynamics, the latter heat transfer causes the increase of entropy and finally
the loss of mechanical energy in the form of heat. There are various mechanisms
of energy loss, such as thermoelastic damping [2, 6–8], energy loss in supports [9],
internal energy loss [10] and air damping [11].
In almost all macroscopic systems with basic damping sources, thermoelastic
coupling plays a secondary role because it is an order of magnitude lower than the
damping caused by external and internal viscous effects. However, in the case of
MEMS/NEMS devices and resonators, it plays a crucial role. It should be emphasized that micromechanical resonators with high quality factors are employed in a
wide range of MEMS applications, such as sensors [12–16], electric filters [17, 18],
accelerometers [19], giros [8], atomic force microscopes [20], MEMS type switches,
and various other mechanisms [21–23]. From a technical point of view, the evaluation of thermoelastic damping (TED) of those structural elements at the design stage
plays an important role in maintaining the required engineering characteristics. TED
belongs to fundamental sources of quantification damping in the microelectromechanical systems (MEMS) [24] and nanoelectromechanical systems (NEMS) [4,
25] working in the vacuum regime. Evoy et al. [26] and Duwel et al. [8] have shown
experimentally that TED is the dominant source of damping in MEMS and NEMS
devices.
Zener [2, 27, 28] belongs to the first who defined the occurrence of TED and
showed its crucial role in dissipation exhibited by bending resonators. Landau and
Lifshitz [29] estimated analytically the TED coefficients of elastic vibrations. The
second widely employed analytical model has been developed by Lifshitz and Roukes
[3]. In that model, a change in the resonance frequency associated with TED was
observed, and thus the classical Zener model was improved. The mentioned authors
have also derived an analytical expression for quality factor Q in microbeam resonators and studied the influence of numerous geometric parameters on their quality factors. Nayfeh and Younis [30] constructed the analytical expression for Q of
a microplate of arbitrary forms for different boundary conditions and studied the
occurrence of TED. Sun et al. [31–33], Sharma et al. [34] and Rezazadeh et al.
[35] investigated thermoelastic damping in capacity microbeam resonators using
the hyperbolic heat transfer model. Besides, Vahdat and Rezazadeh [24] investigated the effect of axial and residual stresses on thermoelastic damping in a capacity
microbeam resonator.