Page 113 - MEMS Mechanical Sensors

102 Mechanical Transduction Techniques

(a) (b)
Figure 5.11 Examples of two balanced resonators: (a) DETF and (b) TBTF.

1/Q is of fundamental importance since it not only affects the Q-factor of the
s
resonator, but provides a key determinant of resonator performance. A dynamically
balanced resonator design that minimizes 1/Q provides many benefits [10]:
s
• High resonator Q-factor and therefore good resolution of frequency;
• A high degree of immunity to environmental vibrations;
• Immunity to interference from surrounding structural resonances;
• Improved long-term performance since the influence of the surrounding struc-
ture on the resonator is minimized.

The Q-factor of a resonator is ultimately limited by the energy loss mechanisms
within the resonator material. This is illustrated by the fact that even if the external
damping mechanisms 1/Q and 1/Q are removed, the amplitude of its vibrations will
a s
still decay with time. There are several internal loss mechanisms by which vibrations
can be attenuated. These include the movement of dislocations and scattering by
impurities, phonon interaction, and the thermoelastic effect.

5.5.2.2 Nonlinear Behavior and Hysteresis
Nonlinear behavior becomes apparent at higher vibration amplitudes when the
resonator’s restoring force becomes a nonlinear function of its displacement. This
effect is present in all resonant structures. In the case of a flexurally vibrating fixed-
fixed beam, the transverse deflection results in a stretching of its neutral axis. A ten-
sile force is effectively applied and the resonant frequency increases. This is known
as the hard spring effect. The magnitude of this effect depends upon the boundary
conditions of the beam. If the beam is not clamped firmly, the nonlinear relationship
can exhibit the soft spring effect whereby the resonant frequency falls with increas-
ing amplitude. The nature of the effect and its magnitude also depends upon the
geometry of the resonator.
The equation of motion for an oscillating force applied to an undamped structure
is given by (5.23) where m is the mass of the system, F is the applied driving force, ω is
the frequency, y is the displacement, and s(y) is the nonlinear function [11].
my &&+ s () y = F cos ω t (5.23)
0