Advances in Mathematical Physics

Volume 2016, Article ID 8734360, 15 pages

http://dx.doi.org/10.1155/2016/8734360

## On the Motion of Harmonically Excited Spring Pendulum in Elliptic Path Near Resonances

^{1}Department of Mathematics, Faculty of Science, Tanta University, Tanta 31527, Egypt^{2}Department of Physics and Engineering Mathematics, Faculty of Engineering, Tanta University, Tanta 31734, Egypt

Received 30 June 2016; Revised 26 September 2016; Accepted 4 October 2016

Academic Editor: Ciprian G. Gal

Copyright © 2016 T. S. Amer et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The response of a nonlinear multidegrees of freedom (M-DOF) for a nature dynamical system represented by a spring pendulum which moves in an elliptic path is investigated. Lagrange’s equations are used in order to derive the governing equations of motion. One of the important perturbation techniques MS (multiple scales) is utilized to achieve the approximate analytical solutions of these equations and to identify the resonances of the system. Besides, the amplitude and the phase variables are renowned to study the steady-state solutions and to recognize their stability conditions. The time history for the attained solutions and the projections of the phase plane are presented to interpret the behavior of the dynamical system. The mentioned model is considered one of the important scientific applications like in instrumentation, addressing the oscillations occurring in sawing buildings and the most of various applications of pendulum dampers.

#### 1. Introduction

Dynamical system is considered a collection of particles in motion with finite numbers of DOF and can be determined through some processes during a period of time. Chaos theory studies the behavior of dynamical systems that are oversensitive to initial conditions and is considered one of the most important subjects in applied mathematics, physics, and engineering fields. It has several applications ranging from weather forecasts, technology, and physical and life sciences. The kinematical nonlinear systems are of great interest for many outstanding researchers during the last three decades. In [1, 2], the dynamical behavior of such systems is investigated as good models in applied mechanics. Tousi and Bajaj in [3] studied period doubling bifurcations and modulated motion in 2-DOF of a nonlinear system. Moreover, Maewal in [4] examined chaos of the forced response of an excited elastic beam through the numerical solutions of the governing system of his suggested model. In [5], the author used the MS method to construct the expansion of the parametric excitation of two internally resonant oscillators up to the first order. The author determined the solutions of the steady-state case and checked the corresponding stability. In [6], the authors transformed the governing system of the excited buckled beam to approximate one and examined the stability of the equilibrium solutions to obtain Hopf bifurcations and a sequence of period doubling one, leading to chaotic motion. The response of an excited weakly vibrating system with 2-DOF close to resonance is investigated in [7]. The response of the considered system is checked using the averaging method and the numerical integration. In [8], Lee and Park investigated the excitation of spring pendulum using the MS technique and showed that the obtained approximate autonomous system has bifurcations. The effect of the higher-order expansions of this problem with internal resonance was examined in [9]. In [10], the authors investigated the fourth-order approximate solution for the governing system and the stability of such system of a similar problem when the stiffness of the spring becomes nonlinear. In [11], the authors investigated the same problem besides any equilibrium state using MS method and they also examined the general manner of the system through utilizing the basin boundaries of attractors. They observed that these boundaries may be fractal and the damping coefficients have a great effect during the chaotic motion. The problem of the nonlinear behavior of a 2-DOF oscillating system coupled with nonlinear damper and nonlinear spring is studied in [12]. The authors showed that, according to certain values for damping and stiffness coefficients, the oscillating amplitude for the examined model is reduced to give better design for the nonlinear absorber. In [13], Amer and Bek studied the response of the excited elastic pendulum in which the motion of the suspended point is considered in a circular path. The governing nonautonomous system is transformed to autonomous ones up to third order using the MS method. They found that the resulted system has bifurcations leading to chaos. Another pendulum model is examined in [14] when the suspended point moves in a prescribed path. The governing equations of motion have been obtained and solved analytically using the multiple scales method. The authors concentrated on predicting the resonance conditions and they studied these cases. This problem was generalized in [15] to study the vibrations of a rigid body as a pendulum model. Many interested examples for the motion of nonlinear systems can be found in [16–20] and the references herein. However, in [17] the authors investigated the motion of a spring pendulum which is moving in circular path. In [18, 19] they generalized their previous work by considering the effect of a damper to the motion.

In the current work, we extend the previous work in [14] for the nonlinear behavior of kinematically excited spring pendulum in which its suspension point moves in an elliptic path. Our main aim is to reveal the resonances conditions and to outline on a case of simultaneously resonances. Among the perturbation methods, the MS method is used to obtain the modulation equations determining all possible steady-state solutions. The graphical representations for suggested physical parameters of the obtained steady-state solutions are presented. Some examples are given as limit cases from the motion of considered model in order to simulate the dynamical behavior of this model.

#### 2. Description of the Problem

Let us consider a dynamical system which consists of a mass , suspended from one end of a massless linear spring having stiffness and statically stretched length . The other end is attached to the point that moves in an elliptical path in which and represent the lengths of the minor and the major axes of the ellipse, respectively; see Figure 1. Take into account that the corresponding point of (located on the auxiliary circle ) is the point (located on the ellipse) and this point moves with angular velocity . Let us denote the horizontal axis by and the downward one by , in which both of them have the same origin of the ellipse. Let be a moment acts about the point in a anticlockwise direction, indicates the acting force on the mass in the elongation’s direction of the spring, and refers to the damping force acts on along the pendulum length.