A perspective on conditional spectrum-based determination of response statistics of nonlinear systems基于条件谱确定非线性系统响应统计的观点
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Poramo B, Spanos PD, 2024. A perspective on conditional spectrum-based determination of response statistics of nonlinear systems. Probabilistic Engineering Mechanics, 78: 103704.DOI: 10.1016/j.probengmech.2024.103704
本研究聚焦于多项式非线性系统的频域随机响应特性确定。具体地，给出系统位移和速度的平稳功率谱密度表示。通过回顾条件功率谱概念，假设响应过程各态历经且为虚拟谐波，并以振幅和相位为特征实现这一目标。尝试从理论上阐明现有的条件谱公式。特别地，结合原非线性系统响应振幅的平稳概率密度函数中的时间平均近似来解释这一概念。结果表明，准确定义响应振幅的平稳概率密度，并合理处理振幅在频域中分布的贡献，可修正平稳响应功率谱密度的近似值。该处理需要对符合响应振幅的各值相关系统响应的平稳随机响应替代谱密度进行平均。半解析结果与高成本蒙特卡罗模拟结果相比，在形状和相关频率范围方面均很有优势，即使对强非线性系统也是如此。关键词: 非线性振动, 统计线性化, 随机平均, 平稳响应, 条件功率谱This work focuses on determining the stochastic response properties, in the frequency domain, of a general class of nonlinear systems with polynomial nonlinearities. Specifically, the results are presented in terms of the stationary power spectral densities of the system's displacement and velocity. This is pursued by revisiting the conditional power spectrum concept, with the assumption that the response process is both ergodic and pseudo-harmonic and characterized by an amplitude, and a phase. A theoretical elucidation of an existing formula for the conditional spectrum is attempted. In particular, this concept is interpreted in conjunction with the time averaging approximation made in the definition of the stationary probability density function of a response amplitude quantity, associated with the original nonlinear system. It is shown that a proper definition of the stationary probability density of the response amplitude, along with a reasonable treatment of the distribution over the frequency domain of the amplitude contribution, lead to an improved approximation of the stationary response power spectral density. The treatment involves the averaging of a population of surrogate spectral densities of stationary random responses conforming with the system responses associated with individual values of the amplitudes of the responses. The semi-analytical results have been quite favourably juxtaposed with a large suite of à propos Monte Carlo simulations, both in terms of the shape and of the range of the involved germane frequencies, even for strongly nonlinear systems.
Keywords: Nonlinear vibrations; Statistical linearization; Stochastic averaging; Stationary response; Conditional power spectrum.![]()
图 1: 给定振幅间隔下响应位移计算的响应叠加: (a) 响应位移阶段的选取; (b) 响应振幅的稳态概率密度函数; (c) 原激励; (d) 相应功率谱密度; (e) 选取的激励; (f) 相应功率谱密度Fig. 1. Superposition of responses computed for a given amplitude interval of the response displacement Δa centred around a_i: (a) Selection of segments of the response displacement; (b) Stationary PDF of the response amplitude; (c) Original excitation; (d) Corresponding PSD; (e) Selected excitation; (f) Corresponding PSD
图 2: 白噪声激励下的 Van der Pol 型非线性振子Fig. 2. Oscillator with Van der Pol-like nonlinearity ε = 2/σ_x0^2; m =1; k = 13.05; ζ = 1%, excited by white noise (S_0 = 0.3)
图 3: 白噪声激励下的 Van der Pol 型非线性振子Fig. 3. Oscillator with Van der Pol-like nonlinearity ε = ρ/σ_x0^2 = 0÷8/σ_x0^2; m =1; k = 4; ζ = 2%, excited by white noise (S_0 = 1)
图 4: 白噪声激励下的 Van der Pol 型非线性振子Fig. 4. Oscillator with Van der Pol-like nonlinearity ε = 0.8 ≈ 8/σ_x0^2; m =1; k = 4; ζ = 2%, excited by white noise (S_0 = 1)
图 5: 白噪声激励下的 Duffing 非线性振子Fig. 5. Oscillator with Duffing nonlinearity ϕ = 0.2/σ_x0^2; m =1; k = 13.05; ζ = 1%, excited by white noise (S_0 = 0.3)
图 6: 白噪声激励下的 Duffing 非线性振子Fig. 6. Oscillator with Duffing nonlinearity ϕ = 2/σ_x0^2; m =1; k = 13.05; ζ = 1%, excited by white noise (S_0 = 0.3)
图 7: 白噪声激励下的 Duffing 非线性振子Fig. 7. Oscillator with Duffing nonlinearity ϕ = ρ/σ_x0^2 = 0÷8/σ_x0^2; m =1; k = 4; ζ = 2%, excited by white noise (S_0 = 1)
图 8: 白噪声激励下的 Duffing 非线性振子Fig. 8. Oscillator with Duffing nonlinearity ϕ = 0.1 ≈ 1/σ_x0^2; m =1; k = 4; ζ = 2%, excited by white noise (S_0 = 1)
Fig. 9. Oscillator with damping nonlinearity ε = 0.4 ≈ 4/σ_x0^2; stiffness nonlinearity ϕ = 0.1 ≈ 1/σ_x0^2; m =1; k = 4; ζ = 2%, excited by white noise (S_0 = 1)
作者信息 | Authors
Beatrice Pomaro, 通讯作者 (Corresp.)意大利帕多瓦大学 (University of Padova) 土木环境与建筑工程系Email: beatrice.pomaro@unipd.it
美国工程院院士, 中国科学院外籍院士
美国莱斯大学 (Rice University) 机械与土木工程系
律梦泽 M.Z. Lyu | 编辑 (Ed)
P.D. Spanos | 审校 (Rev)
陈建兵 J.B. Chen | 审校 (Rev)
彭勇波 Y.B. Peng | 审校 (Rev)
