In this new study, we investigate the motion of a forced mass-spring-damper in a vertical position. First, the classical Lagrangian as well as the classical Euler–Lagrange equation of motion are constructed. Then the fractional Euler–Lagrange equation is derived by extending the classical Lagrangian in the fractional sense. In this extension, a new form of the fractional derivative is employed including a general function as its kernel. The derived fractional Euler–Lagrange equation is then converted into a system of linear algebraic equation by designing an efficient matrix approximation approach. The numerical findings are reported verifying the theoretical investigations. According to the results, some remarkable thinks are achieved; indeed, the numerical simulations show that different aspects of the system under study can be explored with regard to the flexibility found in selecting the kernel contrary to the traditional fractional models.

在这项新的研究中，我们研究了垂直方向上受力质量弹簧阻尼器的运动。首先，构造了经典的拉格朗日运动以及经典的欧拉-拉格朗日运动方程。然后，通过在分数意义上扩展经典拉格朗日方程，得出分数Euler–Lagrange方程。在此扩展中，采用了一种新形式的分数导数，其中包括通用函数作为其内核。然后，通过设计一种有效的矩阵近似方法，将所得的分数阶Euler-Lagrange方程转换为线性代数方程组。报告的数字结果证实了理论研究。根据结果​​，得出了一些非凡的想法。确实，