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The author designed a family of nonlinear static electric-springs. The nonlinear oscillations of a massively charged particle under the influence of one such spring are studied. The equation of motion of the spring-mass system is highly nonlinear. Utilizing Mathematica [1] the equation of motion is solved numerically. The kinematics of the particle namely, its position, velocity and acceleration as a function of time, are displayed in three separate phase diagrams. Energy of the oscillator is analyzed. The nonlinear motion of the charged particle is set into an actual three-dimensional setting and animated for a comprehensive understanding.

Analysis of the kinematics and dynamics of a mechanical spring-mass system is a classic physics problem [

Motivated with identifying the missing practical design of mechanical oscillations of an ideal/perfect mass-less oscillator, the author designed a spring made of a static electric field, and called it an electric-spring. The point is a spring made of an electric field is a mass-less spring. In other words, there is no need to make assumptions justifying the smallness of the spring’s mass. This makes the spring ideal. Furthermore, knowing the fact that the strength and the orientation of a static electric field is a function of the distribution of the static electric charge makes the number of the designs limitless. For the sake of transparency and simplicity, in this article we consider a circular charged ring with a uniform positive charge distribution. The field along the axis through the center of the ring and perpendicular to the plane of the ring sustains its direction; it orients itself along the symmetry axis heading outward from the center. The strength of the field however, varies as a function of distance from the center; interestingly, its variation is not linear. Placing a negatively charged particle along a horizontal frictionless axis of the ring exerts a force on the particle accelerating it toward the center of the ring. The particle sling shots through the center of the ring and slides to the opposite side of it. On the other side of the ring because the field is reversed and reoriented, the particle decelerates and slides to a momentarily halt at a symmetrical opposite end. Since there is no loss of energy, continuous repetition of the movement is warranted and results in steady oscillations. The electric-spring in addition to being massless is nonlinear as well. Therefore, without idealizing the setup we are proposing a design resulting in perfect nonlinear oscillations. As mentioned earlier, in general, the shape of the nonlinear electric field is a function of shape of the configuration of the charge distribution. Hence, by replacing the ring with different geometrical shapes such as a square, a rectangle, an ellipse, an n-tagon, and etc. one may fabricate limitless practical nonlinear electric-springs.

This article is composed of seven sections. Section 1 addresses the introduction and motivations, Section 2 outlines the objectives. Section 3 deals with the analysis. Section 4 addresses the energy issues of the oscillations, Section 5 embodies the 3D animation and the accompanied Mathematica code, Section 6 describes the extended design features and Section 7, the last section, shares the conclusions and closing remarks.

We begin with a positively charged ring of radius R. Assuming the ring is made of a conductor; its charge distribution is uniform. Along the symmetry axis of the ring, the electric field, intuitively and justifiably is oriented along the axis heading outward from the center extending on both sides of the ring. The field along the axis due to the symmetry of the ring is distance dependent only. For any off-axis point the field depends on other coordinates as well. Apply the superposition principal to the electric field for a point along the symmetry axis trivially calculates [

We have assumed a right handed Cartesian coordinate system, and the ring is placed in the yz plane with the x axis stretching from left to right. The x is the distance from the center of the ring, Q is the charge on the ring and the value of k=1/(4pe_{0}) º 9.0×10^{9} in MKS units. A plot of E/(kQ) for a 10 cm ring size is shown in

EfieldRing[x_]=

plotEfieldRing=Plot[EfieldRing[x]/.R®10×10^{-2},{x,-1,1}, PlotRange®All,PlotStyle®{Green,Thick},Gri dLines®Automatic, AxesLabel®{“x(m)”,”E/(k Q)(1/m^{2})”}]

Equation (1) is an asymmetric/odd function with respect to x. Its plot displayed in

x<0 is negative. This means the vector field on both sides of the ring orients itself away from the center of the ring. The value of the electric field and its variation vs. distance indeed may be viewed as the characteristic of the electric force that would act on a charged particle if it were placed on the symmetry axis. Now we envision placing a negatively charged particle on the axis. Under the influence of such a nonlinear force the particle attractively accelerates toward the center of the ring. According to

dEfieldRing=D[EfieldRing[x],x]//Simplify;

Solve[dEfieldRing = = 0,x];{{x®-()},{x®}}

Therefore, one objectively may select a ring size, R, to control the position along the symmetry axis where the field is extremum and its impact is the maximum. We now utilize the field; we place a particle with a chosen mass and charge {m,q} along the symmetry axis of the ring. The opposite “sign” of the charged particle, in our case study, a negative charge, experiences an attractive force toward the center of the ring. The electric force accelerates the particle. While sliding, it experiences a position-dependent and a variable nonlinear force. Assuming a frictionless slide, the particle makes it to the center of the ring. With its velocity at the center it slings shots to the opposite side. While sliding on the opposite side of the ring and because of the reoriented field the particle experiences a retarding position-dependent force, decelerating it to a momentarily halt. The lack of friction preserves the energy of the particle, causing the movement to repeat itself endlessly. The particle therefore oscillates; however, because of the non-linearity of the imposed electric force the oscillation is nonlinear. The detailed analysis of the nonlinear oscillations along with related topics of interest follows.

To form the equation describing the motion of the particle we begin with Newton’s second law,. The particle is characterized with {m,q}. At any given time the electric force is the only effective force acting on the particle, , where E(x) is subject to Equation (1). Newton’s law yields,

The over-dot is defined as the derivative of the variable with respect to time, t. Equation (2) describes the motion of the particle. It is quite different from the classic equation describing the linear oscillations of a linear springmass system. It is a highly nonlinear differential equation. As we pointed out in the Introduction, historically, the Duffing equation is used as a pivotal equation describing the non-linear oscillations. A careful analysis of the force at hand reveals its masked correlation with the Duffing force. To unveil the correlation we expand the second term of Equation (2) about the center of the ring. Aside from the constant coefficient this gives:

Series[,{x,0,8}]//PowerExpand x/R^{3}-(3 x^{3})/(2 R^{5})+(15 x^{5})/(8 R^{7})-(35 x^{7})/(16 R^{9})+O[x]^{9}

This shows our proposed mass-less electric-spring is an upgraded description of the Duffing force. The first two terms of the expanded function is known to describe a “soft” spring [

Now we go back to Equation (2). At the outset, it is known that the Duffing equation given in the introduction has no analytic solution. For our superstructure case too, we were not surprised failing to solve Equation (2) analytically. We deploy Mathematica, but it also fails to solve symbolically. Lastly, we supply a set of practical initial conditions and by utilizing Mathematica we seek for a numeric solution. The code leading the numeric solution, along with the relevant computed kinematic quantities such as position, velocity and acceleration i.e. {x(t), v(t),a(t)}, respectively are given. The output is displayed. For the ring and the particle we assumed {R,Q}= {10.×10^{-2} m,4.0 nC}, and {m,q}={10. mg, 3.0 nC}, respectivelyvalues={k® 9. 10^{9},R®10. 10^{-2},Q®4. 10^{-9},m®10. 10^{-6}, q®3. 10^{-9}};

eqn=x”[t]+((k Q q )/m)/.values;

soleqn=NDSolve[{eqn = = 0,x[

{positionx,velocityx,accx}={x[t]/.soleqn,D[x[t]/.soleqn, {t,1}],D[x[t]/.soleqn,{t,2}]};

plotx=Plot[positionx,{t,0,50},AxesLabel®{“t,s”,”x,m”},PlotStyle®Thick, GridLines->Automatic]; plotv=Plot[velocityx,{t,0,50},AxesLabel®{“t,s”,”v,m/s”},PlotStyle® Thick, GridLines->Automatic]; plota=Plot[accx,{t,0,50},AxesLabel®{“t,s”,“a,m/s^{2}”},PlotStyle®Thick, GridLines->Automatic]; Show[GraphicsArray[{plotx,plotv,plota}],ImageSize® 500]

The far left graph of

The particle oscillates about the origin, i.e. the center of the ring. A trained and experienced eye clearly would be able to distinguish the difference between the characteristic of this nonlinear oscillations vs. linear classic simple harmonic oscillations. From this plot alone one may deduce useful information such as period, which for the chosen utilized parameters is 32.0 s. The initial distance of the particle from the center of the ring is 8 R =0.8 m. It takes 8.0 s for the particle to make it to the center. At that

instance, according to the center graph of

To display the differences between the characteristics of the classic simple harmonic motion vs. our designed nonlinear oscillations, we display three phase diagrams. Utilizing Mathematica ParametricPlot we fold the time parameter and plot a set of three kinematic pairs: {x(t),v(t)}, {x(t),a(t)} and {v(t),a(t)}.

plotxv=ParametricPlot[Flatten[{positionx,velocityx}],{t,0,50},AxesLabel®{“x(m)”,“v(m/s)”},PlotStyle®Thick,GridLines®Automatic,PlotRange®All];

plotxa=ParametricPlot[Flatten[{positionx,accx}],{t,0,50},AxesLabel®{“x(m)”,“a(m/s^{2})”},PlotStyle®Thick,GridLines®Automatic,PlotRange®All];

plotva=ParametricPlot[Flatten[{velocityx,accx}],{t,0,50},AxesLabel®{“v(m/s)”,“a(m/s^{2})”},PlotStyle®Thick,GridLines®Automatic,PlotRange®All];

Show[GraphicsArray[{plotxv,plotxa,plotva}],ImageSize®500]

As one may speculate, the shapes displayed in

In the analysis of the classic linear harmonic oscillations one displays the interplay of the kinetic and potential energies of the oscillator vs. time, t. For a conservative setting i.e. when the friction forces are ignored, the sum of these two energies over the span of the oscillations remains constant. Utilizing the same assumption, a nonlinear oscillator preserves its energy as well; however, the time dependent variation of its kinetic and the potential energies are quite different from the linear counter case. We utilize the solution of Equation (2). The potential and kinetic energies of the oscillators are: PE= and KE=1/2 mv(t)^{2}, respectively.

{PE,KE,totalEnergy}={,1/2 m velocityx^{2},(1/2 m velocityx^{2})} //.values;

Plot[{10^{7} PE,10^{7} KE,10^{7} totalEnergy},{t,0,20},PlotStyle ®{{Thick,GrayLevel[0.5]},{Thick,Black},{Thick,Dashing[{0.02}],GrayLevel[0.2]}},PlotRange®All,AxesLabel®{“t,s”, “Energy,J”},GridLines Automatic]

For the sake of displaying the energies with a meaningful scale the energies are magnified by a factor of 10^{7}. According to

A sinusoidal time dependent function describes the oscillations of a classic linear oscillator. As an example we may envision the swinging movement of a simple pendulum under gravity’s pulls. How does a nonlinear oscillator swing? Applying the solution of Equation (2) and utilizing Mathematica’s animation feature we compose a code to bring the nonlinear oscillations to life. The animation depicts a positively charged ring in red, a negatively charged particle in blue, and the electric field/force in green. The released particle from its initial positive position on the x-axis moves toward the ring. For the chosen ring size, the maximum value of the field/force occurs in the vicinity of its center; at. The animation clearly shows the impact of the nonlinear field. i.e., when the particle makes it to the maximum field, the field jerks the particle and radically changes its movement.

plot3d1=ParametricPlot3D[{0,x, },{x,-1,1},PlotStyle®{Red,Thick},PlotRange®{-1,1},AxesLabel® {“x”,“y”,“z”}];

plot3d2=ParametricPlot3D[{0,x,- },{x,-1,1},PlotStyle®{Red,Thick},PlotRange®{{-1,1},{-1,1},{-1,1}},AxesLabel®{“x”,“y”,“z”}];

Show[{plot3d1,plot3d2}];

plotlinex=ParametricPlot3D[{t,0,0},{t,-1,1},PlotRange

®{{-1,1},{-1,1},{-1,1}},PlotStyle®Thick];

plotliney=ParametricPlot3D[{0,t,0},{t,-1,1},PlotRange ®{{-1,1},{-1,1},{-1,1}},PlotStyle®{Dashing[0.02],Thick}];

plotlinez=ParametricPlot3D[{0,0,t},{t,-1,1},PlotRange ®{{-1,1},{-1,1},{-1,1}},PlotStyle®{Dashing[0.02],Thick}];

Show[{plot3d1,plot3d2,plotlinex,plotliney,plotlinez}];

EfieldRing3D=ParametricPlot3D[{x,0,1/40Evaluate[EfieldRing[x]/.values]},{x,-1,1},PlotStyle®{Green(*GrayLevel[0.5]*),Thick}];

Manipulate[Show[{plot3d1,plot3d2,plotlinex,plotliney, plotlinez,Graphics3D[{Blue,Sphere[{positionx[[

DynamicModule[{=0},Show[{plot3d1,plot3d2,plotlinex,plotliney,plotlinez,Graphics3D[{Blue,Sphere[{positionx[[

Utilizing the concepts described in the previous section, one may readily fabricate limitless electric-springs. Without deviating too far from what we have already described, we introduce a charged double-ring electric-spring.

For instance, by adjusting the values of the charge densi-

ties of the rings, e.g. a positive density for the outer ring and a negative density for the inner ring, we were able to produce a double hump electric field. In

Applying the solution of the equation of motion we plot three phase diagrams similar to the ones displayed in

Comparing

In this article the author introduces a perfect mass-less static electric-spring design resulting nonlinear mechanical oscillations. The core idea of the design is based on the configuration of the electric field of a suitable static charge distribution. Utilizing the symmetry of a circular ring, it is shown that the field along the axis of the ring has the desired characteristic. The magnitude of the field along the axis varies non-linearly with distance making the force that acts on the particle nonlinear as well. It is also shown how the design is modified. For instance the details of a coplanar charged double-ring is discussed and exhibits an interesting non-linearity behavior. A thorough literature search reveals no such practical designs have been proposed earlier. Furthermore, based on the aforementioned detailed discussions one realizes that the number of blue print electric-spring designs are limitless. Therefore, one may design springs to meet any practical needs. The available literature on the mechanical nonlinear physics phenomenon mostly is discussed from a purely theoretical view. In this article we have shown practical and applicable applications. We also would like to point out that in our study because of the slowness of the motion of the charged particle justifiably we ignored the effects of electromagnetic radiation of the oscillator.