Study on High Performance Averaging Method for Strongly Nonlinear Oscillator.

为了改善平均方法的性能，提出了使用雅可比椭圆函数的平均方法。(1)只使用一项雅可比椭圆函数的椭圆平均方法：为了计算由周期外力、非线性阻尼力和非线性恢复力组成的摄动项Duffing振子的周期解，发展了椭圆平均方法。在这些方法中，只包含雅可比椭圆余弦、正弦或增量函数的一项作为生成解。适当使用CN、SN或DN型椭圆平均法，得到了具有硬化、软化和跳跃弹簧的Duffing振子系统的高精度近似解。(2)二次振子系统的椭圆平均法：为了计算基于二次振子系统的高精度近似解，…(3)利用两个雅可比椭圆函数的组合函数的椭圆平均方法：为了提高椭圆平均方法的精度和扩大椭圆平均方法的适用范围，提出了以雅可比椭圆余弦和正弦函数的组合函数作为生成解的n+CN型椭圆平均方法。该方法比Cn或Sn型椭圆平均法能计算出更高精度的近似解。(4)稳定性判据：提出了最适合椭圆平均法的稳定性判据。(5)椭圆平均法有效性的验证：通过与打靶法计算结果的比较，证明了所提方法的有效性。结果表明，所提出的椭圆平均方法可以较高精度地计算强非线性振子的周期解，具有广泛的适用性。较少

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