0
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

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.

FIGURES IN THIS ARTICLE
<>
Copyright © 2014 by ASME
Your Session has timed out. Please sign back in to continue.

References

Figures

Grahic Jump Location
Fig. 1

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

Grahic Jump Location
Fig. 2

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

Grahic Jump Location
Fig. 3

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

Grahic Jump Location
Fig. 4

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

Grahic Jump Location
Fig. 5

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

Grahic Jump Location
Fig. 6

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

Grahic Jump Location
Fig. 7

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

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In