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Research Papers

# Nonlinear Free Vibration of a Symmetrically Conservative Two-Mass System With Cubic Nonlinearity

[+] Author and Article Information
T. Pirbodaghi1

School of Mechanical Engineering, Sharif University of Technology, 11365-9567 Tehran, Iranpirbodaghi@mech.sharif.edu

S. Hoseini

School of Mechanical Engineering, Sharif University of Technology, 11365-9567 Tehran, Iran

1

Corresponding author.

J. Comput. Nonlinear Dynam 5(1), 011006 (Nov 18, 2009) (6 pages) doi:10.1115/1.4000315 History: Received September 20, 2008; Revised May 12, 2009; Published November 18, 2009; Online November 18, 2009

## Abstract

In this study, the nonlinear free vibration of conservative two degrees of freedom systems is analyzed using the homotopy analysis method (HAM). The mathematical model of such systems is described by two second-order coupled differential equations with cubic nonlinearities. First, novel approximate analytical solutions for displacements and frequencies are established using HAM. Then, the homotopy Padé technique is applied to accelerate the convergence rate of the solutions. Comparison between the obtained results and those available in the literature shows that the first-order approximation of homotopy Padé technique leads to accurate solutions with a maximum relative error less than 0.068 percent for all the considered cases.

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## Figures

Figure 1

A conservative two degree of freedom system

Figure 2

The effect of auxiliary parameter ℏ on the frequency (m=k=k1=k2=1, a=5, b=1)

Figure 3

Displacement x1(t) for k=0, m=k1=k2=1, a=5, and b=1

Figure 4

Displacement x2(t) for k=0, m=k1=k2=1, a=5, and b=1

Figure 5

Displacement x1(t) for m=k=k1=k2=1, a=3, and b=1

Figure 6

Displacement x2(t) for m=k=k1=k2=1, a=3, and b=1

Figure 7

Displacement x1(t) for m=k=k1=1, k2=5, a=5, and b=1

Figure 8

Displacement x2(t) for m=k=k1=1, k2=5, a=5, and b=1

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