Abstract
In this paper, the Homotopy Analysis Method (HAM) with two auxiliary parameters and Differential Transform Method (DTM) are employed to solve the geometric nonlinear vibration of Euler-Bernoulli beams subjected to axial loads. A second auxiliary parameter is applied to the HAM to improve convergence in nonlinear systems with large deformations. The results from HAM and DTM are compared with another popular numerical method, the shooting method, to validate these two analytical methods. HAM and DTM show excellent agreement with numerical results (the maximum errors in our calculations are about 0.002%), and they additionally provide a simple way to conduct a parametric analysis with different physical parameters in Euler-Bernoulli beams. To show the benefits of this method, the effect of different physical parameters on the amplitude is discussed for a cantilever beam with a cyclically varying axial load.
Highlights
The governing equations of beam vibration are generally non-linear making it difficult to solve the nonlinear problem via analytical methods
As it is obvious form these tables, good agreement can be seen between the results of the Homotopy Analysis Method (HAM) with two auxiliary parameters and the numerical method results
Nonlinear vibration of an Euler-Bernoulli beam subjected to a cyclically varying axial load is investigated by analytical methods
Summary
The governing equations of beam vibration are generally non-linear making it difficult to solve the nonlinear problem via analytical methods. Moeenfard et al (2011) developed the homotopy perturbation method (HPM) to analyze the nonlinear free vibration of Timoshenko beams They converted the nonlinear partial differential governing equation to a non-linear ordinary differential equation using Galerkin’s projection method. Ahmadi et al (2014) devoted to the new classes of analytical techniques called the Iteration Perturbation Method (IPM)and Hamiltonian Approach (HA) for solving the equation of motion governing the nonlinear vibration of Euler-Bernoulli beams. DTM applied to solve linear and nonlinear differential equations This method can be used for solution of ordinary and partial differential equations. The results of HAM and DTM are compared with numerical method results and good agreement is achieved
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
