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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October 2014
Research-Article
Models and Numerical Solutions of Generalized Oscillator Equations
Yufeng Xu,
Yufeng Xu
Department of Applied Mathematics,
School of Mathematics and Statistics,
e-mail: xuyufeng@csu.edu.cn
School of Mathematics and Statistics,
Central South University
,Changsha Hunan 410083
, China
e-mail: xuyufeng@csu.edu.cn
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Om P. Agrawal
Om P. Agrawal
1
Department of Mechanical Engineering
and Energy Processes,
e-mail: om@engr.siu.edu
and Energy Processes,
Southern Illinois University Carbondale
,Carbondale, IL 62901
e-mail: om@engr.siu.edu
1Corresponding author.
Search for other works by this author on:
Yufeng Xu
Department of Applied Mathematics,
School of Mathematics and Statistics,
e-mail: xuyufeng@csu.edu.cn
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,
e-mail: om@engr.siu.edu
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. Oct 2014, 136(5): 050903 (7 pages)
Published Online: July 25, 2014
Article history
Received:
May 2, 2013
Revision Received:
February 19, 2014
Citation
Xu, Y., and Agrawal, O. P. (July 25, 2014). "Models and Numerical Solutions of Generalized Oscillator Equations." ASME. J. Vib. Acoust. October 2014; 136(5): 050903. https://doi.org/10.1115/1.4027241
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