Page 29 - Mechanical design of microresonators _ modeling and applications

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Design at Resonance of Mechanical Microsystems

28 Chapter One

T
T
{X } M {X }= 0 {X } K {X }= 0 (1.89)
j
i
j
i
for i 실 j. When i = j, the generalized stiffness and mass matrices can be
defined as
T
T
M ={X } M {X } K ={X } K {X } (1.90)
i
i
i
i
i
i
The generalized mass and stiffness matrices of Eqs. (1.90) can be uti-
lized to quantify the participation of various modes in defining the free
vibrations of a multiple-DOF system as the superposition of normal
modes in the form:
{X}=෸ c {X } (1.91)
i
i
i
where c i is a participation factor corresponding to the ith mode. Pre-
T
multiplication in Eq. (1.91) by {X } [M] gives the mass participation
i
factor as
{X } T M {X}
i
c i M = (1.92)
M
i
Similarly, the stiffness participation factor is
{X } T K {X}
i
c K = (1.93)
i K
i
2
It can be shown — see Thomson, for instance — that by using the
modal matrix, defined as
P = {X }{X }. . .{X }. . .{X } (1.94)
n
1
2
i
where {X i } is the ith eigenvector, the equation of motion can be
decoupled, as it can be written in the form:
..
M m {x}+ K m {x}={0} (1.95)
where the transformed (modal) mass and stiffness matrices
T T
M m = P M P K m = P K P (1.96)
are both of diagonal form.
For an n-DOF dynamic system, the mass matrix [M] and the stiffness
matrix [K] are symmetric and generally fully populated. Such a system
is also known as fully coupled. Systems where the mass matrix is in

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