Abstract A computationally efficient method for determining the response of non-linear stochastic dynamic systems endowed with fractional derivative elements subject to stochastic excitation is presented. The method relies on a spectral representation both for the system excitation and its response. Specifically, first the ordinary non-linear differential equation of motion is transferred into a set of non-linear algebra equations by employing the harmonic balance method. Next, the response Fourier coefficients are determined by solving these non-linear equations. Finally, repeated use of the proposed procedure yields the response power spectral density. Pertinent numerical examples, including a fractional Duffing and a bilinear oscillator, demonstrate the accuracy of the proposed method.

中文翻译：

---

具有分数阶导数且受有色噪声影响的非线性系统的响应谱密度确定

摘要 提出了一种计算效率高的方法，用于确定非线性随机动态系统的响应，该系统具有随机激励下的分数阶导数元素。该方法依赖于系统激励及其响应的光谱表示。具体来说，首先将运动的普通非线性微分方程转化为一组非线性代数方程，采用调和平衡法。接下来，通过求解这些非线性方程来确定响应傅立叶系数。最后，重复使用所提出的程序会产生响应功率谱密度。相关的数值示例，包括分数 Duffing 和双线性振荡器，证明了所提出方法的准确性。