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Second-order spectra for quadratic nonlinear systems by Volterra functional series: Analytical description and numerical simulation

机译：Volterra函数级数的二次非线性系统的二阶谱：分析描述和数值模拟

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摘要

Higher-order spectral analysis techniques are often used to identify nonlinearities in complex dynamical systems. More specifically, the auto- and cross-bispectrum have proven to be useful tools in testing for the presence of quadratic nonlinearities based on knowledge of a system's input and output. In this paper, analytical expressions for the auto- and cross-bispectrum are developed using a Volterra functional approach under the assumption of a zero-mean, stationary Gaussian input; proper simplifications are presented when the whiteness of the input signal is also imposed. These formulae show the contributions of the bispectrum in terms of the system frequency response function and elementary physical properties of the system. Simulations based on a stochastic numerical integration technique accompany the analytical solutions for a mechanical mass-spring-damper system possessing quadratic damping and stiffness coefficients and subjected to Gaussian white noise excitation. Subsequent estimates of the bispectrum based on the simulated signals show excellent agreement with theory. These results show how modes may interact nonlinearly producing intermodulation components at the sum and/or difference frequency of the fundamental modes of oscillation. The presence and extent of nonlinear interactions between frequency components are identified. Advantages of using higher-order spectra techniques will be revealed and pertinent conclusions will be outlined.
机译：高阶频谱分析技术通常用于识别复杂动力系统中的非线性。更具体地说，基于系统输入和输出的知识，自动双谱和交叉双谱已被证明是用于测试二次非线性的存在的有用工具。在本文中，在零均值固定高斯输入的假设下，使用Volterra函数方法开发了自动双谱图和交叉双谱图的解析表达式。当还施加输入信号的白度时，将给出适当的简化。这些公式显示了双谱在系统频率响应函数和系统基本物理特性方面的贡献。基于随机数值积分技术的仿真伴随着具有二次阻尼和刚度系数并受到高斯白噪声激励的机械质量－弹簧－阻尼器系统的解析解。随后的基于模拟信号的双谱估计值与理论具有很好的一致性。这些结果表明，模式如何在非线性和基本振荡模式的总和和/或差频率处非线性相互作用，产生互调分量。确定了频率分量之间非线性相互作用的存在和程度。将揭示使用高阶光谱技术的优势，并概述相关结论。

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