In this paper, we consider the dynamical description of a pendulum model consists of a heavy solid connection to a nonelastic string which suspended on an elliptic path in a vertical plane. We suppose that the dimensions of the solid are large enough to the length of the suspended string, in contrast to previous works which considered that the dimensions of the body are sufficiently small to the length of the string. According to this new assumption, we define a large parameter and apply Lagrange’s equation to construct the equations of motion for this case in terms of this large parameter. These equations give a quasi-linear system of second order with two degrees of freedom. The obtained system will be solved in terms of the generalized coordinates and using the large parameter procedure. This procedure has an advantage over the other methods because it solves the problem in a new domain when fails all other methods for solving the problem in such a domain under these conditions. It is one of the most important applications, when we study the slow spin motion of a rigid body in a Newtonian field of force under an external moment or the rotational motion of a heavy solid in a uniform gravity field or the gyroscopic motions with a sufficiently small angular velocity component about the major or the minor axis of the ellipsoid of inertia. There are many applications of this technique in aerospace science, satellites, navigations, antennas, and solar collectors. This technique is also useful in all perturbed problems in physics and mechanics, for example, the perturbed pendulum motions and the perturbed mechanical systems. The results of this paper also are useful in moving bridges and the swings. For satisfying the validation of the obtained solutions, we consider numerical considerations by one of the numerical methods and compare the obtained analytical and numerical solutions.

中文翻译：

---

用大参数程序处理固体摆运动

在本文中，我们考虑了一个摆模型的动力学描述，该模型由与非弹性弦的重固体连接组成，该非弹性弦悬挂在垂直平面的椭圆路径上。我们假设固体的尺寸对于悬挂的弦的长度足够大，这与先前的工作相反，之前的工作认为主体的尺寸对于弦的长度足够小。根据这个新的假设，我们定义了一个大参数，并根据此大参数应用拉格朗日方程来构造这种情况下的运动方程。这些方程式给出了具有两个自由度的二阶拟线性系统。所获得的系统将根据广义坐标和使用大参数过程。此过程相对于其他方法具有优势，因为在这些条件下，当在该域中解决该问题的所有其他方法失败时，它可以解决新域中的问题。当我们研究外力作用下牛顿力场中刚体的慢自旋运动或均匀重力场中重固体的旋转运动或具有足够重力的陀螺仪运动时，这是最重要的应用之一。绕惯性椭圆长轴或短轴的小角速度分量。此技术在航空航天科学，卫星，导航，天线和太阳能收集器中有许多应用。这项技术还可以解决所有物理和力学问题，例如，扰动的摆运动和扰动的机械系统。本文的结果对于移动桥梁和秋千也很有用。为了满足所获得解决方案的有效性，我们通过一种数值方法考虑了数值考虑，并比较了所获得的解析解和数值解决方案。