Analytical solutions for nonlinear oscillatory problems based on balance methods

Abstract

Analytical methods are effective and efficient tools to approximate periodic solutions of nonlinear oscillatory problems. In this thesis, three analytical methods, namely, the harmonic balance method (HBM), the rational harmonic balance method (RHBM) and the energy balance method (EBM), respectively, are presented in modified forms to solve nonlinear oscillatory problems. In this study, first we have applied the modified harmonic balance method (MHBM) to the cubic-quintic Duffing oscillator, the nonlinear oscillator having the square of the angular frequency depends quadratically on the velocity, the nonlinear non-smooth oscillator with non-rational restoring force, the Duffing-relativistic oscillator and the Duffing-harmonic oscillator. Applying the MHBM in all problems, the third order approximate solutions yield almost similar as the corresponding exact solutions. Secondly, we have introduced a new analytical technique based on the RHBM to obtain approximate periodic solutions to the free undamped vibration, nonlinear oscillator with singularity, a nonlinear oscillator with non-rational restoring force, the Helmholtz-duffing oscillator and the Duffing-harmonic oscillator. It is noted that the second order approximate solutions are found very close to the third order approximations obtained by standard HBM. Finally, we have examined our modified energy balance method (MEBM) to some benchmark nonlinear oscillatory problems, namely, the Duffing oscillator, the equation of motion of a particle on a rotating parabola, the simple relativistic oscillator, the stretched elastic wire oscillator (with a mass attached to its midpoint) and the Duffing-relativistic oscillator to determine approximate periodic solutions. The correctness of the MEBM is found much better than the existing solutions. The modified analytical techniques eradicate the limitation of the standard HBM, RHBM and EBM. It is highly remarkable that an excellent accuracy of the approximate periodic solutions has been found by applying all modified analytical techniques which are valid for the whole range of large values of oscillation amplitude as compared with the exact ones. A very simple solution procedure with high accuracy is found in the nonlinear oscillatory problems that illustrates the novelty, reliability and wider applicability of the modified analytical techniques. All of these allow us to conclude that the modified analytical techniques are more convenient, efficient and better alternative than the existing methods for solving nonlinear oscillatory problems arising in nonlinear dynamical systems and engineering.