Literature articles and text books are flooded with sections describing the characteristics of linear oscillators [2]. One quick review of these resources reveals these are confined limited to mechanical systems. In the area of electro and magneto-dynamics, the author thoroughly has investigated scenarios of nonlinear oscillators [3,4]. Laboratory setups conducive to these scenarios have been proposed, and for the magneto-dynamic case the validity of the theoretical model is qualified with actual data [5]. For the sake of completeness, therefore, it is essential to fill in the gap considering a practical scenario conducive to a nonlinear mechanical oscillator. Although the latter is the main motivation of tackling the issue, however, in the course of analyzing the problem we stumbled on a mathematical observation constituting our secondary objective.

To address the first issue we consider combining two linear springs; the device is shown in Figure 1. The springs are identical in shape and have the same stiffness. They are fastened to two separate supports at one end and jointly hold a marble at the other end. In the absence of gravity, the springs exert a nonlinear force on the marble along the horizontal direction when it is pulled away symmetrically from equilibrium. The non-linear force could be a combination of cubic and quintic coordinate-dependent terms; details are discussed in the next section.

Releasing the marble somewhere off from equilibrium results in oscillations along the horizontal direction. The corresponding equation of motion is a nonlinear differential equation. The issues concerning the solution of the equation will be discussed later. Nonetheless, it is worthwhile mentioning not only we solve the equation utiliz-

ing traditional methods, we propose a fresh semi-analytic method resulting in the same output. With these objectives we craft this four-section article. In addition to Motivation and Objectives, in Section 2, along with the physics of the problem we formulate the problem. In Section 3 we present the Analysis along with the associated descriptive graphs. We close the article with conclusions and closing remarks.

As shown in Figure 1, the marble is pulled from its initial equilibrium position to a point such as G, stretching the linear springs with stiffness k beyond their relaxed length L. The differential length elongation of each spring is, yielding the net force

For displacement x < L the quantity in the parentheses in Equation (1) may be replaced with

One realizes this simple mechanical device is capable of exerting nonlinear forces; the forces are a combination of cubic and quintic coordinate-dependent terms. For the sake of “completeness”, in order to include a linear coordinate-dependent force we modify the previous design by adding a third identical spring, shown in Figure 2.

For this three-spring system the net force is,

The criteria x < L suggests that the impact of the third term and terms with powers higher than the fifth are to be insignificant; this is qualified later on. Nonetheless, the equation of motion of the marble under the influence of the force given by Equation (3) along the x-axis yields,

with this equation in hand we analyze the impact of the force terms on the kinematic characteristics of the mobile marble. First, by turning off the linear term, a scenario subject to Figure 1, we focus on the impact of the cubic and quintic terms. Then, we consider a case including the combination of the linear and the cubic terms.