Study on High Performance Averaging Method for Strongly Nonlinear Oscillator.

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Averaging methods using the Jacobian elliptic function are developed in order to improve performances of the averaging method. In this research, the following results were achieved.(1)Elliptic Averaging Method Using Only One Term of the Jacobian Elliptic Function : Elliptic averaging methods are developed in order to calculate periodic solution of Duffing oscillator with perturbed term which is composed of a periodic external force, nonlinear damping and nonlinear restoring force. In these methods, only one term of the Jacobian elliptic cosine, sine or delta function is incorporated as generating solution. Highly accurate approximate solution for the systems based on Duffing oscillators with hardening, softening and snap-through spring are obtained by using elliptic averaging method of cn, sn or dn type properly.(2)Elliptic Averaging Method for Systems Based on Quadratic Oscillator : In order to calculate highly accurate approximate solution for systems based on quadratic oscillator, a … More n elliptic averaging method, which has been improved through the use of squared Jacobian elliptic sine function (sn^2), is developed.(3)Elliptic Averaging Method Using Combined Function of Two Jacobian Elliptic Functions : In order to improve the accuracy and to expand applicability of elliptic averaging method, Elliptic averaging method of sn+cn type, which use the combined function of Jacobian elliptic cosine and sine function as generation solution, are developed. This method can calculate more highly accurate approximation solution than elliptic averaging method of cn or sn type. Furthermore, approximate solution of higher harmonic and sub-harmonic vibration can be calculated by using elliptic averaging method of sn+cn type.(4)Stability Criterion : Most suitable stability criterion for elliptic averaging method is proposed.(5)Confirmation of Validity of Elliptic Averaging Method : The efficiency of the developed methods is demonstrated by comparing the computational results with those obtained by the shooting method. It is confirmed that the developed elliptic averaging methods can calculate the periodic solution with high accuracy of strongly nonlinear oscillator and have a wide range of applicability. Less