Explicit solution format for complex-valued natural frequency of beam with R -shunted piezoelectric laminate transducer

Analysis and design of resistive shunt circuits for piezoelectric damping of beam structures is often based on a representation in terms of the single target vibration mode of the beam, neglecting spill-over effects from the out-of-bandwidth or residual vibration modes. In this article, a solution format is derived for the complex-valued natural frequency of the beam with a shunted piezoelectric laminate transducer, where the influence from the residual modes is taken into account by a quasi-static representation. This explicit solution format contains system parameters that directly represent the authority of the transducer and the spill-over from residual modes, and it recovers the short- and open-circuit frequencies as limit solutions. Furthermore, the frequency solution format provides the basis for design expressions for the optimal resistance and the corresponding attainable damping of the beam. The accuracy of the explicit frequency solution format is verified by comparison with numerical results. It is found that the complex-valued natural frequency of the first vibration mode of a beam with a piezoelectric laminate transducer shunted to a resistance is estimated with sufficient accuracy for engineering design purposes.