Equivalent circuit for end-loaded piezoelectric bimorph actuators

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Abstract
Following on Mason's equivalent circuit for a piezoelectric bar with a length-expansion mode, we have designed an equivalent circuit for a piezoelectric bimorph. A transformer with a transformation ratio with the dimension ‘‘length'’ allows us to construct a moment from a force and an arm. The total moment is transformed back into a force at the tip of the bimorph by a reduced transformer with the dimension of ‘‘inverse length''. A mass m attached to the tip of the bimorph can be represented as an inductor of inductance L = m, which can be placed in series with the inductance which represents the mass of the bimorph. The velocity of the mass can be calculated as the current through the inductor, and the displacement of the mass expressed as the time integral of the velocity. The dynamic behavior of the mass is calculated as a function of the frequency of excitation with an electric voltage V. The displacement is observed under a microscope as a function of frequency, and shows a close agreement between the theoretical and experimental curves. Attaching a mass at the tip of the bimorph increases the total mass as well as the total inductance of the equivalent circuit. In this manner the change in resonance frequency can easily be predicted. Experimentally the resonance frequency shifts towards lower values with increasing masses, according to the theoretical predictions. This model has been adapted to a ‘‘one-and-a-half-morph'’ in which a piezoelectric bar is glued to a non piezoelectric bar. We selected a non piezoelectric material with the same elastic properties and specific mass as the piezoelectric material and loaded this assembly with masses at the tip of the bimorph, and calculated and measured its displacement as a function of frequency. Once again there was a close agreement between theory and experiment.

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