Mechanical Properties of Cantilever Structure

131 Air damping is related to air pressure, P air and gas constant. At a constant air pressure, the Q -factor related to air damping is inversely proportional to the square of length of the cantilever, given by [112], 6-4 The surface loss which is mainly caused by surface stress [109], becomes dominant when the surface-to-volume ratio increases thickness length or width, and it is proportional to thickness and given by [108], 6-5 The rate of energy dissipation due to heat conduction produced when a beam is in oscillation, is inversely proportional to the product of the resonant frequency and the square of the cantilever thickness [109], 6-6 The total Q -factor of the structures can be determined from an experiment by dividing the measured fundamental natural frequency of the structure, f , by the full width at half maximum electric output power,  f , according to: 6-7 The Q -factor is used to determine the performance of the free-standing structures. A higher value indicates a lower rate of energy dissipation relative to the oscillation frequency. For this reason, cantilevers with thin, narrow and long structures are required to design sensitive and low loss devices. The Q -factor can be used to estimate the damping ratio for free-standing structures, provided that the damping is smaller than 0.05, where the relation is, 6-8 2 l h Q a ir  h Q sur fa ce  1 2 h f Q r ther mo  f f Q T    vibration of cycle per lost Energy energy vibration Stored 2  T T Q 2 1   132 Damping ratio is an important parameter used to calculate piezoelectric properties, which will be discussed in section 6.4. The mechanical properties of a cantilever change with the addition of a proof mass at the tip of the beam. The sensitivity of the cantilever is inversely proportional to the additional mass as given by [113], 6-9 where, m is the mass of the composite cantilever and M m is the additional mass proof mass.

6.3 Experimental Procedure

The samples were characterised on a shaker table operated in sinusoidal vibration over a range of different frequencies close to the resonant frequency of the unimorph cantilever beam. The acceleration level was maintained at a constant level by using a feedback system as shown in Figure 6-3 a. The accelerometer in the shaker measures the actual value of frequency and acceleration level and is fedback into the control processor. A processed signal is then generated and amplified to drive the shaker to produce the desired acceleration level at a given frequency. The output voltage power from the device is driven into a programmable resistance load and subsequently converted to a digital signal and is measured with a National Instrument Sequence Test Programme. In a further experiment, tungsten proof masses of density 19.25 gcm 3 were attached at the free-standing cantilever samples, in order to investigate the Q -factor, coupling factor, the efficiency of energy conversion and the maximum stress that the structure can withstand before it fails to perform accordingly. Four different dimensions of tungsten blocks with same thickness of 1 mm were used to investigate the mechanical and electrical performance of the piezoelectric cantilever. The proof masses are denoted as M1, M2, M3 and M4, with lengths and widths as shown in Figure 6-4. In order to increase the total mass for the experiment, identical proof masses were stacked on top of each other and adhered with double-sided tapes. The tape is thin, in comparison to the thicknesses of the proof mass and the cantilever sample and does not m M m Q  133 significantly contribute to the total mass. Furthermore, the experiments are conducted at a relatively low frequency ≤ 500 Hz and low acceleration level ≤ 10 ms 2 , therefore the damping effect of the tape could be ignored. Figure 6-3: a Diagram of a sequence test system and b a shaker table where the device is being tested. Figure 6-4: Schematic diagram of four different proof masses M1 – M4 shaded with the same thickness of 1 mm distributed on the tip of a cantilever. Planar view of cantilevers M1 M3 9 mm 2.5 mm 5 mm M2 9 mm 4.5 mm M4 2.5 mm 2.5 mm 4.5 mm Device under test Shaker PC Function Generator Analog-Digital Shaker DUT Sensor Signal Conditioner Amplifier Programmable Resistance Substituter a b