Background

Since combinations of the Newton’s method and the harmonic balance (HB) method require, at each iteration step, calculating the first or the first- and second-order derivatives of the restoring force function, and expanding the function, its first- and second-order derivatives into Fourier series, the procedural costs are high and sometimes difficult to achieve algebraically. It is thus preferable to avoid expensive re-linearization or computation of the second-order derivative.

Purpose

A new approach is proposed to construct accurate analytical approximation solutions to strongly nonlinear conservative oscillators with odd nonlinearities.

Methods

The approach is based on a combination of a modified Newton method and the HB method. For the modified Newton method, two simplified Newton steps are taken between each Newton step where only one linearization of the restoring force function is required. The resulting equations are solved by applying the HB method appropriately.

Results

Using only one modified Newton iteration step may achieve highly accurate analytical approximation solutions to the strongly nonlinear oscillators. Three examples with physical implications are used to illustrate the proposed method.

Conclusion

Through the modified Newton iteration step, the multiple cumbersome linearizations of the restoring force function are replaced by only one linearization, and the corresponding governing equations can be properly solved by the HB method. The current work is expected to extend to the study of other nonlinear oscillations.

中文翻译：

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强奇数非线性振荡器的修正牛顿-谐波平衡方法

背景

由于牛顿法和谐波平衡（HB）方法的组合要求在每个迭代步骤中计算恢复力函数的一阶或一阶和二阶导数，并扩展该函数，其一阶和二阶为了将导数分解为傅立叶级数，过程成本很高，有时很难通过代数实现。因此，最好避免昂贵的重新线性化或二阶导数的计算。

目的

提出了一种新的方法来构造具有奇数非线性的强非线性保守振荡器的精确解析近似解。

方法

该方法基于改进的牛顿法和HB法的组合。对于改进的牛顿法，在每个牛顿步骤之间需要两个简化的牛顿步骤，其中仅需要一个线性恢复力函数。通过适当地应用HB方法可以求解所得方程。

结果

仅使用一个修改的牛顿迭代步骤就可以实现强非线性振荡器的高精度解析近似解。使用三个具有物理含义的示例来说明该方法。

结论

通过改进的牛顿迭代步骤，恢复力函数的多个繁琐的线性化仅被一个线性化所替代，并且相应的控制方程可以通过HB方法正确求解。预期当前的工作将扩展到其他非线性振荡的研究。