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首页> 外文期刊>International journal of non-linear mechanics >Closed-form numerical formulae for solutions of strongly nonlinear oscillators
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Closed-form numerical formulae for solutions of strongly nonlinear oscillators

机译：强非线性振荡器解的闭式数值公式

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摘要

Based on Gauss-Kronrod quadrature rule, this paper provides closed-form numerical formulae of the period, periodic solution and Fourier expansion coefficients for a class of strongly nonlinear oscillators. Firstly, the period derived in the form of definite integral is addressed by a new equation constructed according to the fundamental theorem of calculus. Then, an approximate closed-form expression of the period can be established by employing only a low-order Gauss-Kronrod quadrature formula. Changing the lower limit of the integral, the closed-form expression becomes a numerical formula that can give the periodic solution of the system. After this, according to the partial integration rule, the calculation of the Fourier coefficients is derived in a very concise form. In general, the relative error of the approximate period can be reduced to le-6 only by employing a 31-point Kronrod rule. Error magnitude of the period indicates the maximum error level of the periodic solution and Fourier coefficients. In addition, the proposed formulae are stable convergent and the exact solutions being their convergence limits. Three very typical examples are given to illustrate the usefulness and effectiveness of the proposed technique.
机译：本文基于高斯-克朗罗德正交定律，为一类强非线性振荡器提供了周期，周期解和傅立叶展开系数的闭式数值公式。首先，通过根据微积分基本定理构造的新方程来解决以定积分形式导出的周期。然后，可以仅通过使用低阶高斯-克朗罗德（Kaurod-Kronrod）正交公式来建立周期的近似封闭式表达式。改变积分的下限，闭合形式的表达式变为可以给出系统周期解的数值公式。此后，根据局部积分规则，以非常简洁的形式得出傅立叶系数的计算。通常，仅通过采用31点Kronrod规则，就可以将近似周期的相对误差减小到le-6。周期的误差幅度表示周期解和傅里叶系数的最大误差水平。此外，所提出的公式是稳定的收敛性，而精确解是其收敛性极限。给出了三个非常典型的例子来说明所提出技术的有用性和有效性。

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