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The Duffing equation describes the oscillations of a damped nonlinear oscillator [1]. Its non-linearity is confined to a one coordinate-dependent cubic term. Its applications describing a mechanical system is limited e.g. oscillations of a theoretical weightless-spring. We propose generalizing the mathematical features of the Duffing equation by including in addition to the cubic term unlimited number of odd powers of coordinate-dependent terms. The proposed generalization describes a true mass-less magneto static-spring capable of performing highly non-linear oscillations. The equation describing the motion is a super non-linear ODE. Utilizing Mathematica [2] we solve the equation numerically displaying its time series. We investigate the impact of the proposed generalization on a handful of kinematic quantities. For a comprehensive understanding utilizing Mathematica animation we bring to life the non-linear oscillations.

In our previous work, for a true, real-life setting we investigate the motion of a point-like charged particle that under the influence of a specially designed electrostatic field exhibits non-linear one-dimensional oscillations [

We consider a variety of electromagnet fields capable of exerting forces on a loose permanent magnetic dipole. This distinguishes the difference between our current investigation vs. what we have already reported [

One of the objectives of our analysis is to run a descriptive parallel describing the interaction of a point-like charged particle with a static electric-spring [

We assume the electric current is the source of the magnetic field. Parallel to our previous work [

here i is the conduction current running through the loop, is the vector position from the current element to the point of interest and, with

being the permeability of a vacuum. Assuming a right handed Cartesian coordinate system for a loop placed in the xy-plane with a looping counter-clock-wise current and with the z-axis being the symmetry axis through the center of the loop, Equation (1) trivially evaluates,

where z = 2pR^{2}ni, with n being the number of turns. For the chosen counter-clock-wise current the direction of the field remains the same on both sides of the loop; it stays oriented along the +z-axis. This is not the case for a uniform charged ring. In other words, the directions of the electric and magnetic fields are the same on one side of the loop and are opposite on the other side. Consequently, one needs to consider the corresponding electric and magnetic forces. Asides from the scaling factors for a loop of size 10.0 cm in

Placing a negative point-like charge along the +z-axis will experience an attractive force towards the center. When it slides to the other side of the loop, according to

For a comprehensive understanding we plot the electric and magnetic forces. Forces are scaled and the vertical

axis is proportional to newtons.

Along the positive z-axis the electric and magnetic forces both are negative; i.e. they act on the mobile pieces (either a point-like charge or a permanent magnet) attractively, pulling them towards the center of the loop. On the contrary, on the other side of the loop, i.e., along the negative z-axis, forces are positive, meaning they act towards the center causing retardation.

It is also noteworthy to mention the functional form of the magnetic and the electric forces are distinctly different. For distances further from the center of the loop the magnetic field behaves as 1/distance^{4} while the electric field falls off as 1/distance^{2}. Moreover, as shown in

This shows the expanded functions fall off either as 1/distance^{2} and/or 1/distance^{4}, respectively. To further

our analysis we utilize the unexpanded format of the magnetic force. Its current format according to Equation (3) is a polynomial; it contains odd powers of distance. Comparing this super structure force vs. the Duffing oscillator force that comes about from an additive perturbative z^{3} term to the Simple Harmonic Oscillator, i.e., az+bz^{3}, with {a, b} being constants, one realizes the unexpanded magnetic force stretches the issues of the Duffing oscillator beyond its classic limits.

With this insight about the magnetic force we envision placing a light permanent cylindrical magnet along the horizontal symmetry axis through the center of a circular looping current. Applying Newton’s law of motion we study its motion. This equation along the horizontal z-axis yields, where F_{B} and are the magnetic and the viscous forces, respectively. For a practical setting we envision positioning a horizontal glass tube through the center of the loop and placing the magnet in it. The air is pumped through fine in-punctured holes from underneath the tube, levitating the magnet. While the magnet under the influence of the electromagnet force slides horizontally in the tube it rubs itself against the air experiencing the viscous force. From our previous work [

To further the analysis of the case at hand the specification of the components in the SI units are tabulated in the values list. The R is the radius of the loop, n is the number of the turns, m is the mass of the magnet, m is its dipole moment, g = G/m is the viscosity per mass, and g is the gravity acceleration.

Utilizing the magnetic force associated with this field the equation of motion yields

where. This is a highly nonlinear differential equation. We were unable to solve this equation analytically, so did Mathematica. We then deploy Mathematica numeric skim. Assuming a set of practical initial conditions utilizing NDSolve successfully we solve the equation. First we analyze the viscous free case.

values = {k–>1.0 × 10^{–}^{7}, R ® 10.5 × 10^{–}^{2}, n ® 200.0, I ® 2.0, m ® 2.0 10^{–}^{3}, m ® 1.75, g ® 0, g ® 9.8};

Utilizing the numeric solution of the equation at hand, we form the velocity and the acceleration of the magnet. Plots of these quantities are shown in

The left graph of ^{4} while the latter is proportional to z. Consequently the sinusoidal oscillations of a SHM is being replaced with a smoother “sinusoidal” function. Since, velocity and the acceleration respectively are the slopes of the position and the velocity with respect to time, the interpretation of the middle and the last plots of

Taking advantage of graphic capabilities of Mathematica by folding the time axis we display a set of phase diagrams. These are shown in

Putting all this information into perspective we close this section by mentioning that we have an understanding about how a linear oscillator oscillates; e.g. oscillations of a small angled swing, but what about oscillations of a nonlinear oscillator? To make the formulation meaningful, utilizing Mathematica animation we animate the oscillations. A snap shot of the oscillations is shown in

In this subsection we consider a case where in addition to the magnetic force the loose magnet along with the same previous initial conditions experiences also a speed dependent viscous force. From our previous work we utilize the value of g = G/m. Applying NDSolve we solve the associated ODE describing the motion of the magnet. Utilizing this solution we also form its associated velocity and acceleration. These are displayed in

Values g = {k–>1.0 × 10^{–7}, R®10.5 × 10^{–2}, n ® 200.0, I ® 2.0, m ®2.0 × 10^{–3}, m ® 1.75, g ® 2.0, g ® 9.8};

The impact of the viscous force is severe as expected. The viscous force dampens the oscillations bringing the oscillator to its final expected position, the center of the loop. Similar to what we discussed in the previous case, interpretation of the middle and the last graphs are straight forward.

We also plot useful graphs similar to those displayed in

In this subsection we show the analysis of nonlinear oscillations may be extended to more sophisticated and interesting cases. We discuss one such case in detail and leave the design of other cases to the interest of the reader. Here we consider a set of three parallel loops separated from one another with different distances. In general the loops may have different sizes, different number of turns, different currents and they may run in different directions. For a set of parameters tabulated in the values3 list a plot of the magnetic field vs. the distance from the center of the middle loop is shown in

Values 3 = {k ® 1.0 × 10^{–}^{7}, R3 ® 20.0 × 10^{–}^{2}, R1 ® 15.0 × 10^{–}^{2}, R2 ® 10.0 × 10^{–}^{2}, x3 ® 20.0 × 10^{–}^{2}, x2 ® 10.0 × 10^{–}^{2}, i3 ® 20.0, i1 ® 10.0, i2 ® 20.0, n1 ® 200.0, n2 ® 200.0, n3 ® 200.0, m ® 2.0 × 10^{–}^{3}, m ® 1.75, g ® 0.}.

The field along the symmetry axis is distorted. The impact of the distortion on the movement of the magnet is beyond ones intuition. To form an opinion about the movement of the magnet one needs to solve the equation of motion. We report our viscous free case and to keep the length of the manuscript manageable we do not report the cases for the viscous fluid.

As we discussed in the previous cases, with the numeric solution of the position as a function of time we form its velocity and acceleration. These quantities are shown in

At a first glance the time dependent position of the loose magnet, the left plot of the

Here we consider the energy of the nonlinear oscillator. For a chosen scenario such as case 2 i.e. a single loop current and viscous fluid, the evaluation of the energy of the system is somewhat straight forward. Similarly, the energy of the different scenarios discussed in this text may also be evaluated. The energy of the mobile magnet is composed of two pieces: kinetic and potential. The total

energy is E = K.E. + P.E., where and; utilizing z(t) we evaluate and 1/(R^2+z^2(t))^{3/2} . For the case at hand the time series of the energies are shown in

Descriptively speaking, at the beginning the stationary magnet is far from the ring possessing a zero total energy. Under the influence of a weak magnetic field it begins being pulled towards the center of the loop. While in motion it gains kinetic and potential energies. The viscosity of the fluid dissipates the energy, dampening it to final rest position. The gradual deterioration of the energy is noticed with the jagged edges of the red curve.

Utilizing a static magnetic field we propose a design producing oscillations for a magneto static spring-mass system. A magneto-spring is made of magnetic field and inherently is a mass-less spring. This augments the classic mechanical view of the spring-mass system by passing the needed assumptions about the mass-less spring. The price we pay for our invention is the mathematical challenges we encounter with solving the nonlinear ODEs describing the motion. This by itself is a positive adventure. In light of the fast growing Computer Algebra System industry, such as Mathematica the author believes the numeric solutions are as powerful and informative as analytic ones. Had it not been for the former the insights that we gained by addressing the numeric mathematical challenges of the project at hand we would have left the problem unanswered.

The author would like to mention the current research project is a complementary analysis to our previously published work [

The author would like to thank Mrs. Nenette Sarafian Hickey for carefully reading over the manuscript and making useful editorial comments.