Md. Alal Hosen

A New Analytical Technique for Solving Nonlinear Non-smooth Oscillators Based on the Rational Harmonic Balance Method

In the present paper, a new analytical technique based on the rational harmonic balance method (RHBM) has been introduced to determine approximate periodic solutions for the nonlinear non-smooth oscillator. A frequency–amplitude relationship has also been obtained by a novel analytical way. The standard rational harmonic balance method (SRHBM) cannot be used directly; it is possible if we rewrite the nonlinear differential equations (NDEs). To overcome this previously stated issue, we offered a modified rational harmonic balance method (MRHBM). It is noticed that a MRHBM works very well for the whole range of initial amplitudes and the excellent agreement of the approximate frequencies as well as the corresponding periodic solutions with its exact ones. The method is basically illustrated by the nonlinear non-smooth oscillators, but it is additionally useful for other nonlinear oscillatory problems with mixed parity arising in recent development of nonlinear sciences and engineering.

Higher-order analytical solutions for the equation of motion of a particle on a rotating parabola

In the present paper, a novel analytical technique to obtain higher-order approximate solutions for the equation of motion of a particle on a rotating parabola has been introduced, which is based on an energy balance method (EBM). The results are valid for small as well as large oscillation of initial amplitude. It is highly remarkable that using the introduced technique a third-order approximate solution gives an excellent agreement with the exact ones. The introduced technique is applied to the motion of a particle on a rotating parabola having high nonlinearity to illustrate its novelty, reliability and wider applicability.