Analytical Modelling and Non-linear Characterisation of Piezoelectric Materials for Actuation and Vibration Control of Beams
The use of piezoelectric materials for actuation, and vibration suppression of thin beams, is the subject of study in this doctoral thesis. The initial focus is set on reducing beam vibrations with resistively shunted piezoelectric patches, where the converted electrical energy is dissipated by the resistor to give an additional damping. The amount of additional damping achieved depends on the value of shunted resistor, the dimensions of the piezoelectric, and its location on the substructure. Hence, the resistively shunted piezoelectric-beam was modelled to determine the optimal values, and to examine its dynamics. A multi-modal model was derived based on the Euler-Bernoulli beam theory, and a reduced non-dimensionalized transfer function was obtained from the multi-modal model. The presented model was derived from assumptions which aptly describe the dynamics of the resistively shunted piezoelectric-beam. The aptness of the presented model in representing the system, over the existing models, was evident from the comparison of the analytical predictions with the existing experimental data. With the model derived, the second part of the work deals with determining the value of resistance which would yield maximum amplitude attenuation (referred as the optimal resistance value). A method for obtaining the optimal resistance value from the analytical model, based on the presence of a fixed-point in the amplitude response, exists in the literature. But, this method cannot be used on the presented analytical model, as it includes the base-damping of the structure. Hence, a different approach was adopted to determine the optimal resistance from the analytical model. Analytical results were also validated with experimental results from a cantilever piezoelectric-beam. The amplitude plots of the first, second, and third modes of the piezoelectric-beam exhibited a softening e ect, indicating a non-linear behaviour of the piezoelectric patches. Hence, a non-linear constitutive equation was required to describe the behaviour of the piezoelectric patches. In the third part of the work, a two-step experimental procedure was devised to construct the non-linear constitutive equation of the piezoelectric actuators. In the first step, the piezoelectric patches were short circuited and a family of displacement curves were obtained for the first, second and third modes of the piezoelectric-beam by base excitation. The pro le of backbone curves from these plots were used to identify the type of non-linear terms required to describe the mechanical domain. In the second step, voltage excitation was used to obtain a similar set of displacement curves. A comparison of the profile of the backbone curves, of the displacement frequency response plot, from the voltage excited data with those from the base excited data, lead to the identification of the non-linear electromechanical coupling term. The constitutive equation, which accounts for the non-linear nature, of the piezoelectric actuator contains (apart from the linear terms) a quadratic strain term, a cubic strain term, and a term with the product of cubic strain and electric field.
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