Abstract
This paper considers the RL shunt damping of vibration with a piezoelectric transducer of a structure with a variable natural frequency. The inductive shunt damping is notorious for not being robust when the natural frequency of the electrical circuit does not match the natural frequency of the structure. In the proposed implementation, the shunted piezoelectric transducer is supplemented with a small additional one (with open electrodes) measuring the mechanical extension of the structure at the location of the transducer. The adaptation strategy uses the property that, at resonance, the electric charge in the shunted transducer is in quadrature of phase with the mechanical strain at the location of the transducer (i.e. the voltage in the transducer with open electrodes). A Phase Shift to Voltage Converter, inspired from the Phase Locked Loop technique (PLL), is built to evaluate the phase shift between these two signals and to adapt the (synthetic) inductor L via a voltage controlled resistor, involving a photoresistive optoisolator (photoresistor). The proposed strategy is supported by simulations and experimental results.
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Notes
- 1.
In practice, the transducer is mounted with a prestressing structure to prevent the splitting of the PZT slices under traction. This can be represented by a linear spring \(K_1\) mounted in parallel to the piezoelectric stack. By considering the prestressing spring \(K_1\), one can find readily the same form of the piezoelectric constitutive equations with the effective properties: \(K_a^\star =K_a+K_1\), \(C^\star =C(1-k^2+\nu k^2)\), \(d_{33}^{\star }=\nu d_{33}\) and \(k^{\star 2}= k^2 \nu /(1-k^2+\nu k^2)\), where \(\nu \) is the fraction of strain energy in the transducer defined as: \(\nu = K_a / (K_1+K_a)\), see [12].
- 2.
The definition of \(k^2\) has been used, where \(k^2=\frac{n^2d_{33}^2K_a}{C}\).
- 3.
These values are representatives of typical generalized electromechanical coupling factors \(K_i\) met in practice.
- 4.
We used a pure integral controller since we aim at cancelling the static error between the measured and the reference phases.
- 5.
Because the reference PZT is mounted with opposite polarization, a phase shift of \(180^{\circ }\) is introduced in the frequency response of \(V_L/V_{ref}\).
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Acknowledgements
This research is supported by the Wallonia Region of Belgium through the Mecatech M4 Project. The comments of the reviewers are gratefully acknowledged.
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Mokrani, B., Burda, I., Preumont, A. (2017). Adaptive Inductor for Vibration Damping in Presence of Uncertainty. In: Araujo, A., Mota Soares, C. (eds) Smart Structures and Materials. Computational Methods in Applied Sciences, vol 43. Springer, Cham. https://doi.org/10.1007/978-3-319-44507-6_7
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DOI: https://doi.org/10.1007/978-3-319-44507-6_7
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