Page 30 - Mechanical design of microresonators _ modeling and applications

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Design at Resonance of Mechanical Microsystems

Design at Resonance of Mechanical Microsystems 29
u x k
u u y x
u x
y
m
m
k c x
k x
y
c k
y y
(a) (b)
Figure 1.26 Model of a 2-DOF gyroscope: (a) without damping; (b) with damping.

diagonal form are mass- or dynamically decoupled, whereas systems
where the stiffness matrix is in diagonal form are statically decoupled.
A system is fully decoupled when both the mass and the stiffness
matrices are in diagonal form. In many cases, the design effort is
directed at realizing one or both of the decoupling forms. Also note
that coupling and decoupling depend on the manner of selecting the
coordinates.

Example: A gyroscope (this subject is treated in greater detail in Chap. 5),
which is considered as a 2-DOF system under the assumption of small de-
formations, is studied now based on Fig. 1.26a.
The equations of the undamped motion about the two directions are
.. ..
mx + k x =0 my + k y =0 (1.97)
x
y
These equations can be collected into the matrix form:
0 m { } + k { 0} (1.98)
m 0 .. x k x 0 0
=
y .. 0 y

which shows that the system is decoupled both statically and dynamically.
Similarly, the damped motions about x and y are based on the sketch of
Fig. 1.26b and can be written as

0 m { } + 0 { } + k { y } { 0} (1.99)
m 0 .. x c x 0 x . k x 0 x 0
=
y .. c y y . 0 y

This system, too, is decoupled, both statically (because of the diagonal shape
of the stiffness matrix) and dynamically (because of the diagonal mass and
damping matrices).

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