Abstract This article develops a Laplace-frequency method to derive the exact closed-form solution for the response evolutionary power spectral density (EPSD) of a simple oscillator subjected to an evolutionary stochastic process characterized by a particular kind of time–frequency ( t , ω ) non-separable EPSD: S F ( t , ω ) = t 2 λ e − 2 γ ( ω ) t Φ ( ω ) . The proposed Laplace-frequency method also gains insightful physics for the response EPSD. Although the transient mean square response can be easily evaluated by taking the integral of the response EPSD with respect to the frequency, a closed-form solution for the mean square response is not always possible. The correctness of the computed mean square response based on numerical integration has been verified by Monte Carlo simulations.

中文翻译：

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评估简单振荡器对特定类型的时频不可分离进化随机过程的响应

摘要 本文开发了一种拉普拉斯频率方法来推导简单振荡器的响应演化功率谱密度 (EPSD) 的精确闭式解，该过程受到以特定类型的时间频率 (t , ω ) 不可分离的 EPSD：SF ( t , ω ) = t 2 λ e − 2 γ ( ω ) t Φ ( ω ) 。所提出的拉普拉斯频率方法也为响应 EPSD 获得了深刻的物理学。尽管通过对响应 EPSD 相对于频率进行积分可以很容易地评估瞬态均方响应，但均方响应的封闭形式解并不总是可能的。Monte Carlo 模拟验证了基于数值积分计算的均方响应的正确性。