Piezoelectric Cantilever beam

To fully validate the derived double-mass model, experiments with a piezoelectric beam was conducted. The beam was made from a 5A4E 2-layer series poled piezoelectric sheet¹ from Piezo Systems, Inc. Two identical iron cubes are glued on the beam. The experimental set-up is shown in Figure 5.11. The PZT beam is fixed by a rigid aluminium clamp, which also serves as conductors for the electrodes on each side of the beam. Nylon bolts are used to secure the clamp while keeping it electrically insulated. Since the main interest is to validate the frequencies, mode shapes and the voltage output, the electrodes of the PZT beam are not segmented for power optimization. The instruments involved are the same as the ones used by the previous experiments, with the exception that LabVIEW^TM with differential input and output modules from National Instruments^TM. are used to drive the shaker and log measurement results.The parameters and material properties involved in the experiment can be found in Table B.4 of Appendix B.

To measure the frequency response, firstly a swept-sine input is used to quickly identify the two resonant frequencies. To assure a steady state response, the beam

11 Part number T220-A4-503X.

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A

B

C

D

Figure 5.11: Experimental set-up. (A) Clamp. (B) Double-mass PZT beam. (C) Ac-celerometer. (D) Electromagnetic shaker.

is then excited sinusoidally with a fixed frequency for at least 2 seconds. The excitation frequency is incremented in steps of 10 Hz in the non-resonant region and 1 Hz around the resonant peaks, and the velocity of each mass and the voltage output are measured. Figures 5.12 and 5.13 show the velocity FRF of M_t1 and M_t2, respectively. As can be seen, for both masses the overall velocity response

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Figure 5.12: Velocity FRF of the first mass Mt1. The lower frequency peak is the 1st mode and the higher frequency peak is the 2nd mode.

0 100 200 300 400 500

Figure 5.13: Velocity FRF of the second mass Mt2. The lower frequency peak is the 1st mode and the higher frequency peak is the 2nd mode.

and resonant frequencies closely match the experimental measurements. It is in-teresting to see that M_t2 has very little amplitude at the second mode, this is due to its proximity to the stationary location of the beam as indicated in Figure 5.15. One major source of uncertainty is the mechanical damping ratios. It was observed that mechanical damping could vary noticeably for different excitation frequencies and amplitudes. Applying logarithmic decrement technique, the

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age mechanical damping ratios around the first and second resonant frequencies were determined. The exact position where the velocity was measured also con-tributed to the errors. This is because the laser spot that measures the velocity is approximately 2 mm in diameter, which results in a ±1 mm uncertainty. If the beam’s gradient is relatively large at the measuring position, the error can be significant. Slight adjustments have been made to the model to compensate for these measurement uncertainties. In Figure 5.14, excellent agreement is also found between the measured voltage output and the model. cantilevers that are not in resonance contribute very little to the power output. This is also true for multi-modal harvesters. In voltage validation, the low frequency peak has a higher value

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Figure 5.14: Voltage FRF, electrode segmentation was not implemented hence charge cancellation did occur.

than experiment and the high frequency peak a lower value than experiment. One of the major source of errors is the capacitance, which has a direct effect on the voltage output. The capacitance in the model is calculated based on the geometry parameters and properties provided by the PZT manufacturer. The already small capacitance (typically around 10 to 20 nF) can be changed dramatically due to the beam’s condition and external factors. The electrode on each side of the beam, whose area is proportional to the capacitance, is a conductive layer of nickel with a thickness of only 0.5 µm. The actual area could be reduced if the nickel layer is wiped off sporadically during handling. Moreover, the total capacitance can be altered by the set-up of the experiment. In the presented set-up, the aluminium clamp and nylon bolts add extra capacitance to the circuit. These factors have not been taken into account in the estimation of the capacitance. Nonetheless, the model is able to produce satisfactory results that are valuable for design purposes.

With the same procedure carried out before, the mode shapes were constructed by the velocity measurement along the piezoelectric beam. The first and second mode shapes shown in Figure 5.15 are normalized by the velocity FRF of the free end.

Although the relative velocity of the fixed end is zero, the non-zero velocities at the fixed end in Figure 5.15 correspond to the absolute velocity FRF of the vibrating 82

Stationary location

Stationary location

Normalizedvelocity

Beam length (mm)

Figure 5.15: Normalised mode shapes of 1st and 2nd mode.

base. The uncertainty in laser positions mentioned earlier also contributed to the small inconsistency of the mode shape validation. Yet, the calculated mode shapes closely resemble the actual measurements.

5.4 Conclusions

While vibration-based energy harvesting technologies have become ever promis-ing, narrow bandwidth is still the major challenge. Adding more proof masses is a cost effective and reliable way to enhance the usable bandwidth of a harvester.

In this chapter, a continuum-based model for a double-mass bimorph cantilever is presented. By applying Hamilton’s principle, the equations of motion together with all necessary boundary and transition conditions are derived. Undamped nat-ural frequencies and the mass normalized eigenfunctions are determined through modal analysis. The steady state solution for harmonic base excitation is obtained.

Both series and parallel connection for the bimorph are incorporated in the model through the definitions of some geometric variables. In addition, electrode seg-mentation for power output enhancement is incorporated into the model. As an incremental investigation, the derived model was firstly experimentally validated on a stainless steel beam with two masses. The results confirmed that the eigen-functions can accurately determined the natural frequencies and amplitude of the beam. The model was then validated experimentally on a 2-layer piezoelectric beam with two proof masses. In the experiment, the resonant frequencies, velocity frequency response, voltage output and the mode shape of the first two vibration modes are measured and compared with modelling results. In all quantities

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pared, the model showed excellent agreement with the actual measurements. High credibility means this model can be used to quickly evaluate the performance of a double-mass energy harvester, and design the two resonant frequencies and the power output to desired specifications. Depending on the required bandwidth, a broadband energy harvester can be implemented by an array of double-mass PZT beams for more compactness compared to its single-mass counterpart. The de-velopment of this model is an important step forward to solving the optimization problem of double-mass PZT harvesters, and to deriving a generalised model for a PZT beam with any number of proof masses.

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Resonance Tuning by Impedance