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8 Thermoelastic Vibrations of Timoshenko Microbeams
w 0 =
W 0
a
chαx − cos βx + δ
shαx
M 1
+
sin βx
M 2
,
ψ 0 =
W 0
a
[M 1 shαx − M 2 sin βx + δ (chαx − cos βx)] ,
(8.79)
where
δ =
M 2 sin β − M 1 shα
(chα − cos β)
, M 1 =
−(H α
2
+ ω
2
)
H α
, M 2 =
(H β
2
− ω
2
)
Hβ
, (8.80)
and the frequency equation is as follows:
2 − 2chα · cos β +
M
2
2 − M
2
1
M 1 M 2
shα · sin β = 0.
(8.81)
Quantity a in (8.79) is equal to the maximum value of the eigenfunction in square
brackets of the first formula of (8.79), whereas W 0 is the value of non-dimensional
amplitude of beam deflection. Solving equation (8.81) and defining a, functions
w 0 (x) , ψ 0 (x) in (8.79) are substituted to (8.63) in order to get quality factor Q of
the resonator.
As a case study, we consider a beam made from gold (see Table 7.1).
Graphs of the fundamental vibration mode w 0 (x) , ψ 0 (x) for W 0 = a = 1.589
for the case of beam clamped on both sides (c ˆ
ac) for L/ h = 100 are shown in
Figs. 8.2 and 8.3.
Fig. 8.2 Fundamental mode
w 0 (x)
Fig. 8.3 Fundamental mode
ψ 0 (x)