Longitudinal-transverse coupled vibrations of variable-height asymmetric beams: Modeling, analysis, and case study

Abstract

In general, the cross-sectional centroids of the variable-height asymmetric beam change in the height direction. The Timoshenko beam theory does not consider the influence of the varying cross-sectional centroids and cannot accurately reveal the vibration of variable-height asymmetric beams. Considering that the cross-sectional centroids are dislocated, we propose a novel dynamic model for the variable-height asymmetric beam element. The proposed dynamic model is based on the rigid-plane assumption and can reflect its main mechanical characteristics. The non-collinear axial forces produce the complementary moment. The rotational radiuses of the two ends of longitudinal fibers are not equal. After retaining the main mechanical characteristics, the variable-height element is roughly treated as a constant-height element to simplify the governing differential equations. A bonded-type piezoelectric transducer with variable-height asymmetric beams is taken as an example to verify the developed dynamic model. The theoretical and experimental resonant frequencies are 39338 Hz and 38704 Hz, respectively, with a difference of 634 Hz and a relative error of 1.638%. The calculated displacement vibration shapes are consistent with the experimental results. Finally, the modeling differences between variable-height symmetric and asymmetric beams are clarified from three aspects.