Boris Lossouarn

Multimodal vibration damping of a beam with a periodic array of piezoelectric patches connected to a passive electrical network

A multimodal damping strategy is implemented by coupling a beam to its analogue electrical network. This network comes from the direct electromechanical analogy applied to a transverse lattice of point masses that represents the discrete model of a beam. The mechanical and electrical structures are connected together through an array of piezoelectric patches. A discrete and a semi-continuous model are proposed to describe the piezoelectric coupling. Both are based on the transfer matrix formulation and consider a finite number of patches. It is shown that a simple coupling condition gives a network that approximates the modal properties of the beam. A multimodal tuned mass effect is then obtained and a wide-band damping is introduced by choosing a suitable positioning for resistors in the network. The strategy and the models are experimentally validated by coupling a free-free beam to a completely passive network. A multimodal vibration reduction is observed, which proves the efficiency of the control solution and its potential in term of practical implementation.

Multimodal coupling of periodic lattices and application to rod vibration damping with a piezoelectric network

An elastic lattice of point masses can be a suitable representation of a continuous rod for the study of longitudinal wave propagation. By extrapolating the classical tuned mass damping strategy, a multimodal tuned mass damper is introduced from the coupling of two lattices having the same modal properties. The aim of the study is then to implement this multimodal control on a rod coupled to an electrical network. The electromechanical analogy applied to a lattice gives the required network, and the energy conversion is performed with piezoelectric patches. The coupled problem is modeled by a novel semi-continuous transfer matrix formulation, which is experimentally validated by a setup involving a rod equipped with 20 pairs of piezoelectric patches. The broadband efficiency of the multimodal control is also experimentally proved with vibration reductions up to 25 dB on the four first resonances of the rod. Finally, the practical interest of the network is pointed out, as it limits the required inductance. This is confirmed by the present purely passive setup that only involves standard low value inductors.

Multimodal vibration damping of a plate by piezoelectric coupling to its analogous electrical network

Multimodal damping can be achieved by coupling a mechanical structure to an electrical network exhibiting similar modal properties. Focusing on a plate, a new topology for such an electrical analogue is found from a finite difference approximation of the Kirchhoff-Love theory and the use of the direct electromechanical analogy. Discrete models based on element dynamic stiffness matrices are proposed to simulate square plate unit cells coupled to their electrical analogues through two-dimensional piezoelectric transducers. A setup made of a clamped plate covered with an array of piezoelectric patches is built in order to validate the control strategy and the numerical models. The analogous electrical network is implemented with passive components as inductors, transformers and the inherent capacitance of the piezoelectric patches. The effect of the piezoelectric coupling on the dynamics of the clamped plate is significant as it creates the equivalent of a multimodal tuned mass damping. An adequate tuning of the network then yields a broadband vibration reduction. In the end, the use of an analogous electrical network appears as an efficient solution for the multimodal control of a plate.

Multimodal vibration damping through a periodic array of piezoelectric patches connected to a passive network

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.

Robustness of a multimodal piezoelectric damping involving the electrical analogue of a plate

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.

Comparison of passive inductor designs for piezoelectric shunt damping

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.

Multimodal Damping of a Plate with a Passive Piezoelectric Network

Multimodal vibration reduction can be obtained by coupling a mechanical structure to an electrical network approximating its modal properties. This strategy is applied to the control of a clamped plate with a 2D network of passive electrical components. The plate and the network are coupled through a periodic array of piezoelectric patches that enables the energy conversion. The network is experimentally implemented with inductors and transformers in order to create electrical resonances that present a spatial distribution approaching the first mode shapes of the plate. By tuning electrical resonances to mechanical ones, it becomes possible to introduce the equivalent of a tuned mass effect. This effect only occurs if the electrical and mechanical mode shapes are sufficiently close, which requires specific network topology and boundary conditions. Once the network is tuned, adding resistors introduces damping that can lead to a broadband vibration reduction of the plate. It is thus shown that a 2D control can be implemented with a purely passive solution, which could be of great interest for many embedded applications.

Hybrid passive-active modal networks for structural acoustic control (Conference Presentation)

Distributions of piezoelectric patches bonded to structures provide a means to alter or control, through active or passive means, the dynamic response of the host structure. Numerous active control schemes for such composite structures have been explored. Alternatively, for certain structures, a passive electrical network may be implemented which presents an electrical analog of the modal response of the structure, effectively providing a multi-modal, distributed passive tuned mass modal damper capability. Numerous tuned-mass damper design concepts (“tunings”) may be applied to such a passive network. Further, the distributed network analog, when coupled with active control concepts, permits a hybrid distributed passive-active modal control capability. This paper explores this hybrid distributed network control concept applied to a clamped rectangular plate. A unit-cell discrete representation of the plate leads to an electrical analog comprised of passive inductors, transformers and resistors. Addition of synthetic (or controlled) impedances at a limited set of points within the network permits dynamic adjustment of the frequency response of the system.

Design of a passive electrical analogue for piezoelectric damping of a plate

Vibrations of a mechanical structure can be reduced through a piezoelectric coupling to a passive electrical network exhibiting similar modal properties. For the control of a plate, the design of a two-dimensional analogous electrical network is considered. Depending on the mechanical boundary conditions, a finite difference formulation of the Kirchhoff–Love equation of motion shows that we need to ensure specific electrical connections along the edges of the analogous network. A numerical model involving an assembly of element matrices validates the electrical topology. Then, the passive electrical circuit is implemented with capacitors, inductors, and transformers, whose practical design is closely described. Focusing on the analogue of a clamped plate, experiments prove the ability of the proposed electrical network to approximate the behavior of the mechanical structure.

TRANSVERSE WAVE PROPAGATION IN A ONE-DIMENSIONAL STRUCTURE COUPLED TO ITS ELECTRICAL ANALOGUE: COMPARISON OF TRANSFER MATRIX MODELS

Wave propagation in mechanical structures can be controlled by a coupling to an electrical network exhibiting a similar dispersion relation. The energy transfer between the two media is maximized on a broad frequency range when choosing a network that is the electrical analogue of the structure to control. In terms of structural vibration, this strategy is equivalent to a multimodal tuned mass damping. Indeed, the control is implemented thanks to a multimodal electrical network, whose modes are close enough to those of the mechanical structure. The electromechanical coupling can be achieved by using an array of piezoelectric patches, which are small enough compared to the smallest wavelength to control. Then, when considering interconnected patches, wave propagation occurs simultaneously in the mechanical and electrical media. Wave propagation in one-dimensional periodic structures can be analyzed with the transfer matrix method. In this paper, the definition of an electromechanical unit cell gives a relation between state vectors involving both mechanical and electrical degrees of freedom. As an extension of a previous work focusing on longitudinal propagation, four transfer matrix models are defined in order to describe the piezoelectric coupling of a beam to its discrete electrical analogue. Indeed, the beam can be approximated either by its discrete equivalent, a fully homogenized model, a piecewise homogenized model or a finite element model. Offering an increasing complexity, those formulations are compared in order to determine their respective limits. Depending on the frequency range of interest, it then becomes possible to choose a suitable model for the analysis of structures involving a piezoelectric coupling to their electrical analogues.