Page 93 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH02_92_08/30/05

Basic Members: Lumped- and Distributed-Parameter Modeling and Design

92 Chapter Two
Long microcantilevers and bridges. The differential equation that gov-
erns the free vibrations of a long variable-cross-section beam (Euler-
Bernoulli model) is

4
2
˜ EI (x)u (x, t) ˜ ȡA(x)u (x, t)
y z z
+ =0 (2.157)
˜x 4 ˜t 2
When the cross section is constant, Eq. (2.157) simplifies to
2
4
˜ u (x, t) ˜ u (x, t)
z
z
EI + ȡA =0 (2.158)
y 4 2
˜x ˜t
The method of separation of variables is usually employed in solving
the problem of free vibrations which assumes that
u (x, t) = U (x)T (t) (2.159)
z
z
By using the notation u (x) for U (x), the following differential equation
z
z
can be derived from Eqs. (2.158) and (2.159):
4
d u (x) íȕ u (x) =0 (2.160)
z
4
dx 4 z
4
where ȕ = ȡAȦ 2 (2.161)
EI y
6
The general solution to Eq. (2.161) is, as shown by Thomson, for in-
stance, of the form:
u (x) = Acosh (ȕx) + B sinh (ȕx) + C cos (ȕx) + D sin (ȕx) (2.162)
z

For a cantilever of length l and with fixed-free ends, the following
equation is obtained by imposing the zero slope and zero deflection
conditions at the fixed end:

1+cosh (ȕl) cos (ȕl) =0 (2.163)

By solving the transcendental Eq. (2.163), the following solutions are
obtained, which describe the first three modes:

{ 1.8751
ȕl = 4.6941 (2.164)
7.85476

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