Abstract This paper presents an innovative analytical approximate method for constructing the primary resonance response of harmonically forced oscillators with strongly general nonlinearity. A linearization process is introduced prior to harmonic balancing (HB) of the nonlinear system and a linear system is derived by which the accurate analytical approximation procedure is easily and innovatively implemented. The main cutting edge of the proposed method is that complicated and coupled nonlinear algebraic equations obtained by the classical HB method is avoided. With only one iteration, very accurate analytical approximate primary resonance response can be determined, even for significantly nonlinear systems. Another advantage is the direct determination of the maximum oscillation amplitude. This is due to the appropriate form chosen for the approximation with no extra processing required. It is concluded that the result of an initial approximate solution from the two-term (constant plus the first harmonic term) harmonic balance is not reliable especially for strongly nonlinear systems and a correction to the initial approximation is necessary. The proposed method can be applied to general oscillators with mixed nonlinearities, such as the Helmholtz-Duffing oscillator. Two examples are presented to illustrate the applicability and effectiveness of the proposed technique.

中文翻译：

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具有强一般非线性的谐波强迫振荡器的初级共振响应的解析近似

摘要 本文提出了一种创新的解析近似方法，用于构造具有强一般非线性的谐波强迫振荡器的初级共振响应。在非线性系统的谐波平衡 (HB) 之前引入了线性化过程，并推导出了一个线性系统，通过它可以轻松创新地实现准确的解析逼近程序。该方法的主要优点是避免了由经典 HB 方法获得的复杂且耦合的非线性代数方程。只需一次迭代，就可以确定非常准确的解析近似初级共振响应，即使对于显着非线性的系统也是如此。另一个优点是直接确定最大振荡幅度。这是由于为近似选择了适当的形式，不需要额外的处理。得出的结论是，从两项（常数加上一次谐波项）谐波平衡的初始近似解的结果是不可靠的，特别是对于强非线性系统，需要对初始近似值进行修正。所提出的方法可以应用于具有混合非线性的一般振荡器，例如 Helmholtz-Duffing 振荡器。提供了两个例子来说明所提出技术的适用性和有效性。得出的结论是，从两项（常数加上一次谐波项）谐波平衡的初始近似解的结果是不可靠的，特别是对于强非线性系统，需要对初始近似值进行修正。所提出的方法可以应用于具有混合非线性的一般振荡器，例如 Helmholtz-Duffing 振荡器。提供了两个例子来说明所提出技术的适用性和有效性。得出的结论是，从两项（常数加上一次谐波项）谐波平衡的初始近似解的结果是不可靠的，特别是对于强非线性系统，需要对初始近似值进行修正。所提出的方法可以应用于具有混合非线性的一般振荡器，例如 Helmholtz-Duffing 振荡器。提供了两个例子来说明所提出技术的适用性和有效性。