8.5 Numerical and Analytical Results for a Rectangular Cross-Sectional Beam Resonator 317
W = A 1 chαx + A 2 shαx + A 3 cos βx + A 4 sin βx,
(8.71)
ψ = A 1 M 1 shαx + A 2 M 1 chαx + A 3 M 2 sin βx − A 4 M 2 cos βx.
(8.72)
Formula (8.66) yields
M 1 =
−H α
2
− ω
2
H α
,
M 2 =
H β
2
− ω
2
Hβ
.
(8.73)
The characteristic equation
q
4
+
a 3
a 2
q
2
−
a 4
a 2
= 0
(8.74)
has the following roots
q 1,2 = ±i
(a 3 +
a 3
2 + 4a 2 a 4
2a 2
)
1
2
, q 3,4 = ±
(−a 3 +
a 3
2 + 4a 2 a 4
2a 2
)
1
2
,
(8.75)
and α and β can be easily found (see (8.71) and (8.72)
(α, β) =
(∓a 3 +
a 3
2 + 4a 2 a 4
2a 2
)
1
2
(8.76)
or equivalently
α =
⎡
⎣ −
H
P
+ 1
ω 2 +
H
P
+ 1
2
ω 4 − 4H
ω 4
P
− H ω 2
⎤
⎦ /2H . (8.77)
β =
⎡
⎣
H
P
+ 1
ω 2 +
H
P
+ 1
2
ω 4 − 4H
ω 4
P
− H ω 2
⎤
⎦ /2H . (8.78)
Boundary conditions (8.46) for the clamped beam (c ˆ
ac) follow
w| x=0,1 = ψ| x=0,1 = 0.
Hence, the following eigenfunctions are derived: