The lumped model parameters approach for static and dynamic power-law beam problems

Abstract

It is important to estimate the natural frequencies of the structural elements in the design of mechanical or electromechanical structures. There is a wide use of single lumped-parameter spring-mass models in the industry for materials . Their behaviour is linear by Hooke’s law within the geometric and loading conditions. In this work, the lumped-parameter theory is generalized for Hollomon’s power-law materials and the lumped-parameters for the corresponding nonlinear restoring force in the spring-like model for the standard geometric and loading conditions of the power-law Euler beams are provided. For each case in the given lumped-parameter model the corresponding effective mass is also calculated. Then, the resulting spring-mass system is solved to validate the solutions as approximations to the corresponding beam system. Numerical validations of the proposed lumped models for the cantilever beam with circular and rectangular cross-sections are presented.