Page 28 - Mechanical design of microresonators _ modeling and applications

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Design at Resonance of Mechanical Microsystems

Design at Resonance of Mechanical Microsystems 27

( i )
X 1 =1 ( i ) ( i )
( i ) X j
X 3 X n
( i )
( i )
X n-1
X 2
Figure 1.25 Mode shape i corresponding to an eigenvector of dimension n.
where [A] is the dynamic matrix defined as in Eq. (1.67). The eigenvalue
Ȝ i is connected to the resonant frequency as
Ȝ = Ȧ 2 (1.86)
i i

and the vector of Eq. (1.85) is the eigenvector corresponding to the
eigenvalue Ȝ . The matrix form Eq. (1.85) is formed of n homogeneous
i
algebraic equations under the assumption that the system’s dimension
is n. The unknowns in these equations are the n components of the
eigenvector {X }. Because the equations are homogeneous, only n ෹ 1
i
unknown eigenvector components can be determined, all in terms of one
eigenvector component, which can be chosen arbitrarily. Usually, the
arbitrary component is equal to 1 and the other components are less
than 1, and this form can be achieved by a normalization procedure.
The graphical representation of the components of one eigenvector is
illustrated in Fig. 1.25 and is known as the eigenmode or mode shape.
As a result, an n-DOF mechanical system has n eigenvalues, and for
each eigenvalue there is an eigenvector and the corresponding mode
shape.
The following relationship also holds true:

det ( A íȜ I ) I = ( A íȜ I )adj( A íȜ I ) (1.87)

where for a square nonsingular matrix [B], its adjoint is found by means
of

B ෹1 = adj B (1.88)
det B
For the ith mode, the determinant of Eq. (1.87) is zero; and therefore,
by comparing Eqs. (1.85) and (1.87), it follows that one column of the
adjoint matrix ([A] íȜ[I]) is the eigenvector corresponding to the ith
mode. In other words, finding a specific eigenvector can be done by for-
mulating the adjoint matrix for that mode.
It should be pointed out that normal modes are orthogonal with
respect to the stiffness and mass matrices, namely,

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