Abstract
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 ε 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 θ and φ 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.
Highlights
The pendulum models have attracted scientists and researchers with many descriptions of motions and their analysis as important examples in physics and theoretical and applied dynamics
The most important pendulum motions come from the moving of a heavy particle suspended a light rod which is jointed pivotally at a point on the X-axis which rotates by an angular velocity ω about the horizontal fixed axis
In [1], the authors presented the elastic pendulum problem. They derived the equations of motion and gave real-life examples for elastic pendulum motions
Summary
The pendulum models have attracted scientists and researchers with many descriptions of motions and their analysis as important examples in physics and theoretical and applied dynamics. In [9], the frequencies of the elastic pendulum oscillations are in the ratio 2 : 1 which is named the pulsation The dynamics of this problem and the modulation equations for the resonant motion with small amplitudes are obtained as 3 wave equations. The authors in [12] studied the problem of spring pendulum dynamics in the presence of pendulum absorber using the theory of nonlinear normal patterns and asymptotic numerical procedures. They investigated the dynamics of the pendulum for both low and high vibration amplitudes. The accuracy of these solutions is investigated through a numerical technique and computerized programs
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