Excitation without Proof Mass
135 The composite structure of sample D, with width 9 mm, length 1 cm, and total thickness
of 196 m was weighed at 0.11g. This gives an average density of 6240 kgm
3
. Sample C and D series were found to have a thickness of 208
m and 192 m respectively. Both samples D and C series were assumed to have the same density. Since the natural
frequency structure is inverse proportional to the length of the cantilever structure, therefore the Youn
g’s modulus of the structure can be estimated by using equation 3-8
. The calculated Young’s modulus of sample C and D series are 3.78 10
10
Nm
2
and 1.17 10
10
Nm
2
respectively.
The total
Q
-factor,
Q
T
of the structure can be determined experimentally by exciting the free-standing structures over a range of frequencies close to the fundamental resonant
frequency to determine the value of the full bandwidth at half maximum electrical output power, then substituting this value into equation 6-7. Figure 6-6 shows that the
calculated values for
Q
T
of the samples lie in the range 120 to 215, with the largest value associated with sample D3, which is roughly a square shape. Shorter or longer
cantilever lengths do not appear to exhibit the same
Q
-factor as those having a square structure. This is because shorter or longer cantilever structures suffer losses at different
rates and with different dominant factors. The energy dissipation losses at the support are dominant for a shorter structure [108], while air damping losses become dominant
for longer cantilever structure [114]. With the measured Q-factor value, the total damping ratio for the samples was calculated to be in the range of 0.002 to 0.005.
Figure 6-6:
Q
T
as a function of cantilever length.
136