8.4 Quality Factor of the Resonator with Thermoelastic Damping
313
Again, we find real and imaginary parts of D ω in the following way:
D ω =
⎛
⎜
⎜
⎝ 1 +
α T 0 Rω
2
G 4
G 3
2 + ω 2
⎞
⎟
⎟
⎠ − i
ω α T 0 R
G 4
G 3
G 4
G 3
2 + ω 2
,
Re [D ω ] =
⎛
⎜
⎜
⎝ 1 +
α T 0 Rω
2
G 4
G 3
2 + ω 2
⎞
⎟
⎟
⎠ ,
Im [D ω ] = −
ω α T 0 R
G 4
G 3
G 4
G 3
2 + ω 2
.
(8.58)
A thermoelastic dissipation can be quantified by a ratio of the loss of mechanical
energy per cycle of vibration to the general conserved energy of deformation, and
it can be estimated using a required amount of energy for each cycle to keep a
periodic harmonic motion [3, 82, 83]. The required amount of energy for an infinitely
small beam element d x located in point x over period T = 2π/ω is defined by the
following formula:
dx
2π/ω
0
P N
d
dt
1
2
∂ w
∂ x
2
dt + dx
2π/ω
0
M
d
dt
∂ ψ
∂ x
dt.
(8.59)
Substituting (8.51), (8.56) for N and M into (8.59), one obtains
[Energy loss] extension =
1
0
2π/ω
0
P N
d
dt
1
2
∂ w
∂ x
2
dtdx =
=
1
0
2π/ω
0
P Re [N ] Re
d
dt
1
2
∂ w
∂ x
2
dt dx =
= −π P Im [S ω ]
⎡
⎣
1
0
d w 0
d x
2
dx
⎤
⎦
2
,
(8.60)