8.4 Quality Factor of the Resonator with Thermoelastic Damping
311
∂ θ 2 (x, t)
∂ t
+
R
PG 3
∂
∂ t
∂ ψ
∂ x
+
G 4
G 3
θ 2 (x, t) = 0,
(8.44)
whereas Eqs. (8.38) and (8.39) take the following counterpart non-dimensional form:
H
∂ 2 w
∂ x 2 +
∂ ψ
∂ x
− F
∂ 4 w
∂ x 4 −
∂ 3 ψ
∂ x 3
+ P · N
∂ 2 w
∂ x 2 =
∂ 2 w
∂ t 2 ,
∂ 2 ψ
∂ x 2 − H
∂ w
∂ x
+ ψ
− F
∂ 3 w
∂ x 3 −
∂ 2 ψ
∂ x 2
− α T 0 PG 3
∂ θ 2
∂ x
=
1
P
∂ 2 ψ
∂ t 2 .
(8.45)
P · N
∂ w
∂ x
+ H
∂ w
∂ x
+ ψ
−F
∂ 3 w
∂ x 3 −
∂ 2 ψ
∂ x 2
x=0, 1
= 0
or
δ w| x=0, 1 = 0,
∂ 2 w
∂ x 2 −
∂ ψ
∂ x
x=0, 1
= 0
or
δ
∂ w
∂ x
− ψ
x=0, 1
= 0,
(8.46)
∂ ψ
∂ x
−
α T 0 AL 2
I
G 3 θ 2
x=0, 1
= 0
or
δ ψ| x=0, 1 = 0.
We consider simple harmonic vibrations in the form w (x, t) = w 0 (x) e
−iω t ,
ψ (x, t) = ψ 0 (x) e
−iω t
, where ω stands for circular frequency, and w 0 (x) , ψ 0 (x)
are approximated by a linear vibration mode. Observe that a large beam deflection
may cause the frequency to depend on the amplitude of vibrations. Therefore, owing
to relations (8.43), (8.44) and (8.42), other remaining functions have the following
forms:
θ 1 (x, t) = θ 1 (x) e
−2iω t
, θ 2 (x, t) = θ 2 (x) e
−iω t
,
N (x, t) = N (x) e
−2iω t
, M (x, t) = M (x) e
−iω t
.
(8.47)
Substituting (8.47) into (8.43) yields
− iωθ 1 (x) − 2iω
R
G 1
h
L
2
1
2
∂ w
∂ x
2
+
1
τ
G 2
G 1
θ 1 (x) = 0,
(8.48)
whereas integration along the beam length gives
1
τ
G 2
G 1
− iω
1
0
θ 1 (x) dx = iω
R
G 1
h
L
2
1
0
∂ w
∂ x
2
dx.
(8.49)
We have
1
0
θ 1 (x) dx =
iω R
G 1
1
τ
G 2
G 1
− iω
h
L
2
1
0
∂ w
∂ x
2
dx,
(8.50)