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Research Papers

# Models and Numerical Solutions of Generalized Oscillator Equations

[+] Author and Article Information
Yufeng Xu

Department of Applied Mathematics,
School of Mathematics and Statistics,
Central South University,
Changsha Hunan 410083, China
e-mail: xuyufeng@csu.edu.cn

Om P. Agrawal

Department of Mechanical Engineering
and Energy Processes,
Southern Illinois University Carbondale,
Carbondale, IL 62901
e-mail: om@engr.siu.edu

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received May 2, 2013; final manuscript received February 19, 2014; published online July 25, 2014. Assoc. Editor: Thomas J. Royston.

J. Vib. Acoust 136(5), 050903 (Jul 25, 2014) (7 pages) Paper No: VIB-13-1144; doi: 10.1115/1.4027241 History: Received May 02, 2013; Revised February 19, 2014

## Abstract

In this paper, we use three operators called K-, A-, and B-operators to define the equation of motion of an oscillator. In contrast to fractional integral and derivative operators which use fractional power kernels or their variations in their definitions, the K-, A-, and B-operators allow the kernel to be arbitrary. In the case, when the kernel is a power kernel, these operators reduce to fractional integral and derivative operators. Thus, they are more general than the fractional integral and derivative operators. Because of the general nature of the K-, A-, and B-operators, the harmonic oscillators are called the generalized harmonic oscillators. The equations of motion of a generalized harmonic oscillator are obtained using a generalized Euler–Lagrange equation presented recently. In general, the resulting equations cannot be solved in closed form. A finite difference scheme is presented to solve these equations. To verify the effectiveness of the numerical scheme, a problem is considered for which a closed form solution could be found. Numerical solution for the problem is compared with the analytical solution, which demonstrates that the numerical scheme is convergent.

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## Figures

Fig. 1

The numerical solutions of Eq. (16) with different h

Fig. 2

The numerical solutions of Eq. (17) with different α

Fig. 6

The numerical solutions of Eq. (20) with different μ

Fig. 7

The numerical solutions of Eq. (20) with different λ

Fig. 4

The numerical solutions of Eq. (18) with different μ

Fig. 5

The numerical solutions of Eq. (18) with different λ

Fig. 3

The numerical solutions of Eq. (18) with different α1 and α2

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