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A high-order, purely frequency based harmonic balance formulation for continuation of periodic solutions: The case of non-polynomial nonlinearities

机译：一种高阶，纯频率谐波平衡配方   周期解的延续：非多项式非线性的情形

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In this paper, we extend the method proposed by Cochelin and Vergez [A highorder purely frequency-based harmonic balance formulation for continuation ofperiodic solutions, Journal of Sound and Vibration, 324 (2009) 243-262] to thecase of non-polynomial nonlinearities. This extension allows for thecomputation of branches of periodic solutions of a broader class of nonlineardynamical systems. The principle remains to transform the original ODE systeminto an extended polynomial quadratic system for an easy application of theharmonic balance method (HBM). The transformation of non-polynomial terms isbased on the differentiation of state variables with respect to the timevariable, shifting the nonlinear non-polynomial nonlinearity to atime-independent initial condition equation, not concerned with the HBM. Thecontinuation of the resulting algebraic system is here performed by theasymptotic numerical method (high order Taylor series representation of thesolution branch) using a further differentiation of the non-polynomialalgebraic equation with respect to the path parameter. A one dof vibro-impactsystem is used to illustrate how an exponential nonlinearity is handled,showing that the method works at very high order, 1000 in that case. Variouskinds of nonlinear functions are also treated, and finally the nonlinear freependulum is addressed, showing that very accurate periodic solutions can becomputed with the proposed method.
机译：在本文中，我们将Cochelin和Vergez提出的方法[用于连续周期解的高阶纯基于频率的谐波平衡公式，声音和振动学报，324（2009）243-262]扩展到非多项式非线性的情况。该扩展允许对更广泛的非线性动力学系统的周期解的分支进行计算。原理仍然是将原始ODE系统转换为扩展的多项式二次系统，以便于应用谐波平衡法（HBM）。非多项式项的变换基于状态变量相对于时间变量的微分，将非线性非多项式非线性转换为与时间无关的初始条件方程，而无需考虑HBM。在此，使用非多项式代数方程相对于路径参数的进一步微分，通过渐近数值方法（求解分支的高阶泰勒级数表示）执行所得代数系统的延续。使用一个自由度振动冲击系统来说明如何处理指数非线性，这表明该方法以很高的阶数工作，在这种情况下为1000。还处理了各种非线性函数，最后解决了非线性自由摆，表明该方法可以计算出非常精确的周期解。

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