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By combining a pair of linear springs we devise a nonlinear vibrator. For a one dimensional scenario the nonlinear force is composed of a polynomial of odd powers of position-dependent variable greater than or equal three. For a chosen initial condition without compromising the generality of the problem we analyze the problem considering only the leading cubic term. We solve the equation of motion analytically leading to The Jacobi Elliptic Function. To avoid the complexity of the latter, we propose a practical, intuitive-based and easy to use alternative semi-analytic method producing the same result. We demonstrate that our method is intuitive and practical vs. the plug-in Jacobi function. According to the proposed procedure, higher order terms such as quintic and beyond easily may be included in the analysis. We also extend the application of our method considering a system of a three-linear spring. Mathematica [1] is being used throughout the investigation and proven to be an indispensable computational tool.

Literature articles and text books are flooded with sections describing the characteristics of linear oscillators [

To address the first issue we consider combining two linear springs; the device is shown in

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

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

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

According to the scenario shown in

companied force, Equation (2), the equation of motion is,

First we consider a case assuming x < L, i.e. the initial displacement of the marble from equilibrium is less than the length of the spring; this drops the last term. Sustaining the cubic term only, the equation of motion is an analytically solvable DE with closed form solutions given by Jacobi Elliptic Functions [

.

The 4 × 2 graphic-matrix shown in

It is instructive to compare the oscillations emanating by a cubic force vs. the linear one. This can be done ei-

ther by a direct comparison of the amplitudes or more elegantly by comparing their respective phase diagrams shown in

For the sake of clarity the horizontal scale of