Piezoelectric units with self-tuning multi-resonant shunts for vibration absorption

This thesis is focused on a lightweight and modular control system formed by a piezoelectric patch connected to either a single-resonant or a multi-resonant self-tuning shunt, which can be used to mitigate the resonant response of one or multiple low-order flexural modes of a hosting structure. The aim of the study is to develop a self-contained unit, which can be bonded in batches on thin structures to decrease the low frequency flexural response generated by stationary stochastic disturbances. To this end, the study investigates the optimal tuning of both single-resonant and multi-resonant shunts with reference to a global and a local cost function. Two configurations of the single-resonant shunt are considered, which are formed by a resistance-inductance (RL) connected respectively in series and in parallel. Instead, a single configuration of the multi-resonant shunt is investigated, which is formed by an array of parallel branches encompassing a resistance-inductance-capacitance (RLC) connected in series. The global cost function, given by the minimisation of the hosting structure time-averaged total flexural kinetic energy, is used as a reference metric to assess the optimal tuning of the shunt. Instead, the local cost function, given by the maximisation of the time-averaged electric power absorbed either by the RL single-resonant shunt or by each RLC branch of the multi-resonant shunt, is employed for the practical implementation of the self-tuning shunt. The study shows that, with respect to the resistance and inductance shunt parameters, the two cost functions are characterised by mirror bell surfaces. Hence, the optimal shunt resistance and inductance values that would minimise the global cost function coincide with those that would maximise the local cost functions. As a result, both the single-resonant and multi-resonant shunts can be suitably tuned within the shunt itself by maximising the time-average electric power absorbed by the single-resonant shunt or by each branch of the multi-resonant shunt. The study also shows that, the tuning can be effectively implemented with a recursive two-paths tuning approach, whereby the inductance is first tuned along a constant-resistance path characterised by a bell shaped curve of the cost function and then the resistance is tuned along a constant-inductance path characterised by a bell shaped curve of the cost function too. This two-paths tuning sequence can be run recursively online such that the shunt can be adapted to variations of the electro-mechanical response of the hosting structure and piezoelectric transducer as well as to variations of the electric response of the shunt components, which can both occur in presence of temperature variations or other exogenous physical effects. Since the optimisations along the constant resistance and constant inductance paths are characterised by non-convex cost functions, the study proposes to employ the extremum seeking algorithm to find the optimal shunt parameters that would maximise the electric power absorption. This is a model-free gradient driven search algorithm, which asymptotically leads to the maximum of the non-convex bell-shaped paths. The algorithm is based on a periodic dithering signal that perturbs the inductance and resistance tuning signals such that the resulting electric power absorbed by the shunt equally shows such a periodic signal, which is either in phase or out-of-phase with the dithering signal depending the tuning is under or over estimating the shunt parameter with respect to the optimal one that maximises the power absorption. The study shows that this algorithm suitably leads to the optimal shunt values regardless the structure is excited by a stochastic disturbance such that the power cost function undergoes significant variations over time.