Page 31 - Mechanical design of microresonators _ modeling and applications

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Design at Resonance of Mechanical Microsystems

30 Chapter One

Example: Study the coupling and/decoupling of the 3-DOF system shown in
Fig. 1.23, and find the transformed matrices which will decouple the system
in case it is coupled. Consider that m 1 = m 2 = m 3 = m and k 1 = k 2 = k 3
= k 4 = k.
As Eq. (1.79) shows, the system is statically coupled as the stiffness matrix
is not in diagonal form. The eigenvectors corresponding to the mechanical
system of Fig. 1.23 are assembled into the modal matrix of Eq. (1.93) as

෹1 1 1
P = 0 2 ෹ 2 (1.100)
1 1 1

As a result of the definition in Eq. (1.96), the transformed stiffness matrix is

1 0 0
K =4k 0 2 ෹ 2 0 (1.101)
m
0 0 2+ 2
which is in diagonal form. The mass matrix [Eq. (1.78)] is already in diagonal
form, and there is no need to calculate a transformed mass matrix. The de-
coupled dynamic equation of motion will therefore be
..
M {x}+ K {x}= {0} (1.102)
m

1.3.3 Lagrange’s equations
Lagrange’s method and equations can be utilized to derive the equa-
tions of motion of a mechanical system with (generally) multiple de-
grees of freedom by utilizing the energy forms that define the system’s
motion. Presented next are Lagrange’s equations for conservative sys-
tems as well as for nonconservative ones.
In conservative systems there are no energy gains or losses, as
3
1
2
indicated by Timoshenko, Thomson, or Rao, and Fig. 1.27 shows a
serial chain composed of only masses and springs. The system possesses
n degrees of freedom. The number n represents the minimum number
of parameters that can describe the state of the system at any given
time, and these parameters (depending on time) are called generalized
coordinates.
Lagrange’s equation for the ith mass (and degree of freedom) is
d ˜T ˜T ˜U
. í + =0 (1.103)
dt( ˜x ) ˜x ˜x
i i i
where T is the kinetic energy of the whole system and is calculated as

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