Treating the Solid Pendulum Motion by the Large Parameter Procedure

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.

Dynamical Stability of a 3-DOF Auto-Parametric Vibrating System

“…For a variety of parametrical locations, the adjusted amplitudes b j have been displayed with the detuning parameter 2 , as seen in Figs. 21,22,23,24,25,26, in accordance with the above used data. Figures 21,22,23,24,25,26 are graphed, respectively, when the variations of the damping coefficients c j (j = 1, 2, 3) and the frequencies j are considered.…”

“…The study of the movement of the pendulum suspension point, whether on a certain curve or even if the point is fixed, has captured the attention of various scientists [21][22][23][24][25][26][27][28]. The nonlinear motion of a vibrating pendulum was investigated in [21] under the impact of exterior harmonic force and moment in the spring path and at the suspension point, respectively.…”

“…A comparison of analytical and numerical outcomes revealed that both approaches yielded satisfactory results for non-resonant oscillation. In [23] and [24], the fourth-order approach of Runge-Kutta is used to explore the numerical solutions of a vibrating pendulum connected to a rigid body near the positions of equilibrium, whereas the approaches of small and large parameters are employed in [25] and [26], to get the analytical solutions of a linked rigid body with a pendulum.…”

On the solutions and stability for an auto-parametric dynamical system

“…The scientific studies on the motion of vibrating dynamical systems have been increased during the last two centuries [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] due to its significance and applications in practical life like vibrating buildings, electric motors, airplanes, locomotive engines, and missiles. Such problems can be solved whether analytically or numerically according to the nature of the governing system of motion.…”

“…The approach of Routh-Hurwitz [21] for the nonlinear stability is utilized to study the stability of the investigated model. The numerical solutions of a vibrating rigid body's pendulum close to the equilibrium locations are examined in [12,13] using the Runge-Kutta algorithm [22], while the methods of small and large parameter [20] are used in [14,15] to obtain the analytic solutions of a connected rigid body with pendlum, respectively. Whereas, the location of equilibrium of a vertical triple pendulum exposed to an asymmetric iterative force is investigated in [16].…”