The article deals with the problem of computing efficiently the nonlinear response of a rod involving a fractional constitutive model, and exposed to random excitation. The constitutive model is a three-parameter model comprising an instant elasticity modulus, a prolonged elasticity modulus, and a relaxation parameter. The nonlinear term is a linear-plus-cubic force of the Winkler kind. The resulting nonlinear fractional partial differential equation governing the rod displacement has no known exact solution. Thus, the article proposes an approximate analytical solution by relying on the statistical linearization technique. Further, it develops a Boundary Element Method (BEM)-based approach to estimate numerically the rod response statistics. The statistical linearization solution is obtained by representing the rod displacement as the superposition of linear modes of vibration having time-dependent coefficients. In this context, it is shown that the equation governing the time variation of the mode coefficients is a nonlinear fractional ordinary differential equation, whose solution is computed by a surrogate linear system identified by minimizing the response error between the linear system and the nonlinear one in a mean square sense. Relevant Monte Carlo studies pertaining to rods with fixed-fixed, and fixed-free ends show that the proposed analytical solution is in a good agreement with data obtained by the numerical (BEM) approach.

本文讨论了有效计算杆的非线性响应的问题，该杆涉及分数本构模型并暴露于随机激励。本构模型是包括瞬时弹性模量、延长弹性模量和松弛参数的三参数模型。非线性项是温克勒类型的线性加三次力。由此产生的控制杆位移的非线性分数偏微分方程没有已知的精确解。因此，本文依靠统计线性化技术提出了近似解析解。此外，它还开发了一种基于边界元方法 (BEM) 的方法来数值估计杆响应统计数据。统计线性化解是通过将杆位移表示为具有时间相关系数的线性振动模式的叠加来获得的。在这种情况下，表明控制模态系数的时间变化的方程是一个非线性分数常微分方程，它的解是由一个代理线性系统计算的，通过最小化线性系统和非线性系统之间的响应误差来识别均方意义。有关具有固定固定和固定自由端的杆的相关蒙特卡罗研究表明，所提出的解析解与通过数值 (BEM) 方法获得的数据非常吻合。结果表明，控制模态系数随时间变化的方程是一个非线性分数阶常微分方程，其解是由一个代理线性系统计算的，该线性系统通过最小化线性系统和非线性系统之间的响应误差在均方意义下确定. 有关具有固定固定和固定自由端的杆的相关蒙特卡罗研究表明，所提出的解析解与通过数值 (BEM) 方法获得的数据非常吻合。结果表明，控制模态系数随时间变化的方程是一个非线性分数阶常微分方程，其解是由一个代理线性系统计算的，该线性系统通过最小化线性系统和非线性系统之间的响应误差在均方意义下确定. 有关具有固定固定和固定自由端的杆的相关蒙特卡罗研究表明，所提出的解析解与通过数值 (BEM) 方法获得的数据非常吻合。