322
8 Thermoelastic Vibrations of Timoshenko Microbeams
stress theory (F = 0), we employ an approximate method. Namely, using modes
(8.79) as the vibration modes for F = 0 one may also approximately compute
fundamental frequency F = 0 via the Rayleigh method. The eigenfrequency is
defined by comparison of the formulas governing kinetic and potential energies of
the vibrating beam. As following from (8.62), under the lack of temperature field
Re [S ω ] = Re [D ω ] = 1 and the values of kinetic and potential energies of the linear
beams carrying out vibrations on the fundamental frequencies ω F =0 are defined by
the following formulas:
K =
ω
2
F =0
2
1
0
w
2
(x) +
1
P
ψ
2
(x)
dx,
(8.93)
U L =
1
2
⎧
⎨
⎩
1
0
d ψ
d x
2
dx+ H
1
0
d w
d x
+ ψ
2
dx + F
1
0
d
2 w
d x 2 −
dψ
dx
2
dx
⎫
⎬
⎭
,
(8.94)
where functions w(x), ψ (x) are defined by (8.89) and (c − c) beam boundary conditions. The exact value of frequency ω F =0 , owing to the Rayleigh relation, is as
follows:
ω
2
F =0 =
⎧
⎨
⎩
1
0
d ψ
d x
2
+H
1
0
d w
d x
+ ψ
2
dx +
+F
1
0
d
2 w
d x 2 −
dψ
dx
2
dx
⎫
⎬
⎭
/
1
0
w
2
(x) +
1
P
ψ
2
(x)
dx .
(8.95)
If we substitute w 0 (x), ψ 0 (x) from (8.79) for F = 0, instead of w(x), ψ (x), we
obtain
ω
2
F =0
ω
2
F=0
≈ 1 +
F
1
0
d
2 w 0
d x 2 −
dψ 0
dx
2
dx
1
0
d ψ 0
d x
2
dx + H
1
0
d w 0
d x
+ ψ 0
2 dx
.
(8.96)
In other words, the eigenfrequencies for F = 0 are computed directly from the
classical case for F = 0. In reference [105], a similar approach has been employed to
find the approximate value ω
2
F =0 using the Bernoulli hypothesis ∂ w 0 /∂ x = −ψ 0 .
If we neglect Poisson’s effect, (8.96) yields
ω
2
F =0 ≈ (1 + 4F) ω
2
F=0 .
(8.97)