In this paper, an improved version of the analog equation method (AEM) is proposed, ideally suitable for non-linear dynamic analysis of time-fractional beams with discontinuities. Various sources of non-linearity will be considered as well as different discontinuities, such as those associated with external point supports (shear force discontinuity) and local flexural flexibility (rotation discontinuity). The main idea of the proposed approach is to reformulate the classical AEM by considering an analog equation with an unknown space-time dependent fictitious load and a spatial generalised operator, which includes the classical 4th-order differential operator and generalised functions modelling the discontinuities along the beam. Consistently, the solution to the analog equation is represented by an integral form involving appropriate Green’s functions of the generalised operator. As in the classical AEM formulation, the integral representation of the solution is then approximated dividing the beam in finite elements and considering a constant value of the fictitious load within every beam element. Substituting the so-built approximate solution in the original equation of motion yields a system of fractional differential equations governing the discrete values of the fictitious load, solved by employing a Newmark integration scheme in conjunction with a G-1 algorithm to account for the fractional-derivative memory effects. Computational efficiency and accuracy will be demonstrated against the classical AEM formulation.

中文翻译：

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具有不连续性的时间分数梁非线性动力分析的改进模拟方程法

在本文中，提出了模拟方程法 (AEM) 的改进版本，非常适用于具有不连续性的时间分数梁的非线性动态分析。将考虑各种非线性源以及不同的不连续性，例如与外部点支撑（剪切力不连续性）和局部弯曲柔性（旋转不连续性）相关的那些。所提出方法的主要思想是通过考虑具有未知时空相关虚拟载荷和空间广义算子的模拟方程来重新制定经典 AEM，其中包括经典 4 阶微分算子和建模沿线不连续性的广义函数光束。一贯地，模拟方程的解由一个积分形式表示，该形式涉及广义算子的适当格林函数。与经典的 AEM 公式一样，解的积分表示然后近似地将梁划分为有限元，并考虑每个梁单元内虚拟载荷的恒定值。代入原始运动方程中如此构建的近似解，产生一个分数阶微分方程组，控制虚拟载荷的离散值，通过采用 Newmark 积分方案和 G-1 算法来解决分数-衍生记忆效应。将针对经典的 AEM 公式证明计算效率和准确性。与经典的 AEM 公式一样，解的积分表示然后近似地将梁划分为有限元，并考虑每个梁单元内虚拟载荷的恒定值。代入原始运动方程中如此构建的近似解，产生一个分数阶微分方程组，控制虚拟载荷的离散值，通过采用 Newmark 积分方案和 G-1 算法来解决分数-衍生记忆效应。将针对经典的 AEM 公式证明计算效率和准确性。与经典的 AEM 公式一样，解的积分表示然后近似地将梁划分为有限元，并考虑每个梁单元内虚拟载荷的恒定值。代入原始运动方程中如此构建的近似解，产生一个分数阶微分方程组，控制虚拟载荷的离散值，通过采用 Newmark 积分方案和 G-1 算法来解决分数-衍生记忆效应。将针对经典的 AEM 公式证明计算效率和准确性。代入原始运动方程中如此构建的近似解，产生一个分数阶微分方程组，控制虚拟载荷的离散值，通过采用 Newmark 积分方案和 G-1 算法来解决分数-衍生记忆效应。将针对经典的 AEM 公式证明计算效率和准确性。代入原始运动方程中如此构建的近似解，产生一个分数阶微分方程组，控制虚拟载荷的离散值，通过采用 Newmark 积分方案和 G-1 算法来解决分数-衍生记忆效应。将针对经典的 AEM 公式证明计算效率和准确性。