Multimodal passive damping of a mechanical structure can be implemented by a coupling to a secondary structure exhibiting similar modal properties. When considering a piezoelectric coupling, the secondary structure is an electrical network. A suitable topology for such a network can be obtained by a finite difference formulation of the mechanical equations, followed by a direct electromechanical analogy. This procedure is applied to the Kirchhoff-Love theory in order to find the electrical analogue of a clamped plate. The passive electrical network is implemented with inductors, transformers and the inherent capacitance of the piezoelectric patches. The electrical resonances are tuned to approach those of several mechanical modes simultaneously. This yields a broadband reduction of the plate vibrations through the array of interconnected piezoelectric patches. The robustness of the control strategy is evaluated by introducing perturbations in the mechanical or electrical designs. A non-optimal tuning is considered by way of a uniform variation of the network inductance. Then, the effect of local or boundary modifications of the electromechanical system is observed experimentally. In the end, the use of an analogous electrical network appears as an efficient and robust solution for the multimodal control of a plate.
Considering piezoelectric damping, a resonant shunt can lead to a significant vibration reduction when tuned to the mechanical mode to control. However, limits appear when looking at practical applications in a low frequency range: the required inductance is often too high to be satisfied with standard passive components. Moreover, even if the inductor is eventually available, the internal resistance of the component generally exceeds the value which is required for a shunt optimization. Suitable inductors can be designed for applications requiring high inductance and low resistance values. Indeed, the permeance of a magnetic circuit can be significantly increased by the use of closed cores made of high permeability materials. In this paper, three designs are described and compared: an inductor from standard series and two handmade inductors involving a ferrite core and a nanocrystalline toroid. The components are successively integrated into a piezoelectric shunt dedicated to the vibration control of a cantilever beam. Depending on the frequency of the target mechanical mode to control, the benefits and the limits of the different inductors are observed. It is shown that custom designs can definitely extend to lower frequency the application of the passive resonant shunt strategy.
In damping devices involving piezoelectric elements, a single piezoelectric patch cannot consistently achieve multimodal control because of charge cancellation or vibration node location. In order to sense and control structural vibration on a prescribed frequency range, a solution consists in using an array of several piezoelectric patches being small compared to the smallest wavelength to control. Then, as an extension of the tuned mass damper strategy, a passive multimodal control requires to implement a damping system whose modes are as close as possible to those of the controlled structure. In this way, the electrical equivalent of the discretized mechanical structure represents the passive network that optimizes the energy transfer between the two media. For one-dimensional structures, a periodic distribution in several unit cells enables the use of the transfer matrix method applied on electromechanical state-vectors. The optimal passive networks are obtained for the propagation of longitudinal and transverse waves and a numerical implementation of the coupled behavior is performed. Compared to the more classical resonant shunts, the network topology induces promising multimodal damping and a reduction of the needed inductance. It is thus possible to create a completely passive electrical structure as it is demonstrated experimentally by using only purely passive components.