Abstract
A vast amount of published work can be found in the field of beam vibrations dealing with analytical and numerical techniques. This paper deals with analysis of the nonlinear free vibrations of beams. The problem considered represents the governing equation of the nonlinear, large amplitude free vibrations of tapered beams. A new implementation of the ancient Chinese method called the Max-Min Approach (MMA) and Homotopy Perturbation Method (HPM) are presented to obtain natural frequency and corresponding displacement of tapered beams. The effect of vibration amplitude on the non-linear frequency is discussed. In the end to illustrate the effectiveness and convenience of the MMA and HPM, the obtained results are compared with the exact ones and shown in graphs and in tables. Those approaches are very effective and simple and with only one iteration leads to high accuracy of the solutions. It is predicted that those methods can be found wide application in engineering problems, as indicated in this paper.
Highlights
Analyzing the nonlinear vibration of beams is one of the important issues in structural engineering
Goorman [7] is given the governing differential equation corresponding to fundamental vibration mode of a tapered beam
To demonstrate the accuracy of the Max-Min Approach (MMA) and Homotopy Perturbation Method (HPM), the procedures explained in previous sections are applied to obtain natural frequency and corresponding displacement of tapered beams
Summary
Analyzing the nonlinear vibration of beams is one of the important issues in structural engineering. Applications such as high-rise buildings, long-span bridges, aerospace vehicles have necessitated the study of their dynamic behavior at large amplitudes. Many researchers have studied tapered beams, which are very important for the design of many engineering structures. The non-linear vibration of beams is governed by a non-linear partial-differential equation in space and time. Goorman [7] is given the governing differential equation corresponding to fundamental vibration mode of a tapered beam. Evensen [6] studied on the nonlinear vibrations of beams with various boundary conditions by using the perturbation method. Pillai & Rao [24] considered the different types of solutions to the nonlinear equation of motion such as
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