Page 228 - An Introduction to Microelectromechanical Systems Engineering

Microelectromechanical Resonators 207

To visualize this complex effect, let us imagine two physically separate but iden-
tical simple resonators consisting of a mass and a spring. These resonators can freely
oscillate at the natural frequency determined by the mass and the spring constant.
Adding a weak and compliant flexure or spring between the two masses (see
Figure 7.13) restricts the allowed oscillations of this two-body system. The two
masses can move either in phase or out of phase with respect to each other; these are
the two oscillation modes of the system. When the motions are in phase, there is no
relative displacement between the two masses and, consequently, no restoring force
from the weak flexure. The oscillation frequency of this first mode is then equal to
the natural frequency of a single resonator. When the two masses move out of phase
with respect to each other, however, their displacements are in opposite directions
at any instant of time. This motion produces the largest relative displacement across
the coupling flexure, thereby resulting in a restoring force, which, according to
Newton’s second law, provides a higher oscillation frequency. The physical cou-
pling of the two masses effectively split the two overlapping resonant frequencies (of
the two identical resonators) into two distinct frequencies, with a frequency separa-
tion dependent on the stiffness of the coupling flexure. In physics, it is said that the
coupling lifts the degeneracy of the oscillation modes. For a very compliant coupling
spring, the two split frequencies are sufficiently close to each other that they effec-
tively form a narrow passband. Increasing the number of coupled oscillators in a

Stiff spring with Weak flexure with Stiff spring with
spring constant k 1 spring constant k 2 spring constant k 1

Mass m Mass m

In phase Out of phase

Amplitude

f f
r1 r2 Frequency
1 k 1 k +2 k
f = 2π m 1 f = 2π 1 m 2
r2
r1
Figure 7.13 Illustration of two identical resonators, each with a mass and spring, coupled by a
weak and compliant intermediate flexure. The system has two resonant oscillation modes, for in-
phase and out-of-phase motion, resulting in a bandpass characteristic.