Application of the energy finite element analysis to vibration of beams with stepped thickness and variable cross-section

Energy finite element analysis (EFEA) has been developed to compute the energy distribution of vibrating structures. The method adopts the energy density as the basic variable of differential equation. The energy density can be used to analyze the behavior of vibrating beams. Firstly, an EFEA equation is obtained from the classical displacement equation. In the applications of uniform and non-uniform beams, the EFEA results are compared with the analytical and FEA results. Secondly, a junction formulation solving the discontinuity problem of energy density at the junction of two beams with stepped thickness is proposed. The EFEA equation combined with junction formulation is used to solve the energy transmission problem of the coupling beams with stepped thickness and variable cross-section. The smoothed results of coupling beams are achieved, and the differences of energy density at the junctions are analyzed. The feasibility of the EFEA approach is validated by using several design examples under the various frequencies and structural damping loss factors.