326
8 Thermoelastic Vibrations of Timoshenko Microbeams
Fig. 8.9 Beam with boundary conditions (s − s) for F = 0.260417: a ω 2
N L versus amplitude of
vibrations w; b quality factor Q N L versus beam vibration amplitude w
Fig. 8.10 Beam under boundary conditions (s − s) and for F = 0: a frequency ω 2
N L versus the
vibration amplitude w; b quality factor Q N L versus the beam vibration amplitude w
Dependence w of nonlinear frequencies ω
2
N L obtained approximately from formula (8.104) versus amplitude w is shown in Fig. 8.8a. Figure 8.8b presents quality
factor Q N L of the beam resonator under the lack of the size beam dependence (F = 0)
versus w for three values of beam length L of the clamped beam.
Analogous results for F = 0.260417 are given for different w in Fig. 8.9.
For the (s − s) beam, the fundamental linear frequency ω
2
L is estimated based on
Eq. (8.98) and the corresponding linear vibration mode takes the following form:
w 0 = sin π x , ψ 0 = M 2 cos π x,
(8.105)
where
M 2 =
Fπ
4
+ H π
2
− ω
2
/
−Fπ
3
+ H π
.
(8.106)
Substituting (8.105) into (8.104) for a = 1 one may find the nonlinear frequency
ω
2
N L for different values of the vibration amplitude W 0 . The quality factor of the beam
resonator is computed by formula (8.63) based on the obtained frequencies ω
2
N L and