Harmonic oscillator damped by sliding friction.

Abstract: The motion of a block acted upon by a simple harmonic oscillator (linear spring) and damped by sliding friction is solved exactly. The resulting analysis shows that there are two characteristic parameters that completely determine the motion of the block. This motion is divided into four classifications of oscillation and each is graphically demonstrated. Additional insight is provided through an analysis of the time dependence of the block's mechanical energy. Finally, we discuss the significant differences between the harmonic oscillator damped by friction and the oscillator damped through linear viscous damping.

Introduction

The simple harmonic oscillator damped by sliding friction, as compared to linear viscous friction, provides an important example of a nonlinear system that can be solved exactly. Although this system has been the subject of several articles (1,2,3), we provide some additional insights concerning the analytic solution and its graphical representations.

The system consists of a block with mass m resting on a horizontal surface (See Fig. 1). The block is attached to an immovable wall by a spring with spring constant k. The coefficient of kinetic friction between the block and the horizontal surface is assumed to be constant and will be denoted by [mu]. Clearly no motion is possible if the initial displacement from equilibrium ([x.sub.0]) is such that [kx.sub.0] [less than or equal to] [[mu].sub.s]mg, where [[mu].sub.s] is the coefficient of static friction. For the sake of the derivation, we will assume that [kx.sub.0] > [[mu].sub.s]mg such that we are in the dynamic regime. We also set [[mu].sub.s] = [mu] for simplicity.

Equation of Motion and Its Solutions

The equation of motion for this system is obtained from the horizontal forces acting on the block: the spring restoring force (-kx) and the kinetic frictional force (-[mu]mg x/[absolute value of x]). Inserting these forces into Newton's Second Law and solving for the acceleration, yields the equation of motion for the block,

x(t) = - k/m x(t) - [mu]mg x(t)/\x(t)\. (1)

Unlike the differential equation that governs the harmonic oscillator with linear viscous damping, this equation is nonlinear. Never the less, the frictional force is only dependent upon the direction of the velocity. Since the direction is constant between turning points (namely, points where x(t)=0), Eq. (1) can be solved explicitly between these points. Matching the solutions from each segment produces the general solution.

Introducing the usual definition of angular frequency, [omega] = [square root of (k/m)], and dimensionless parameters of [beta] = [mu]mg/[kx.sub.0] and x(t) = x(t)/[x.sub.0],where [x.sub.0] is a length scale to be identified as the location of the first turning point of x(t) motion, the equation of motion between each turning point can be written as

X(t) = -[[omega].sup.2](X(t) [+ or -] [beta]), (2)

where the top sign of the [+ or -] is for segments with positive velocity and the bottom sign is for segments with negative velocity. Equation (2) is solved readily to yield (1)

X(t) = ([X.sub.0] [+ or -] [beta])Cos([omega]t)+ [V.sub.0]/[omega]Sin([omega]t)[+ or -] [beta] , (3)

where [X.sub.0] is the initial location of the block and [V.sub.0] is its initial velocity in the scaled variable X. Equation (3) tells us that during each time segment the motion of the block is harmonic with constant amplitude whose center of oscillation has been either shifted to the left or to the right of the equilibrium position by an amount [beta]. Equation (3) also shows that the duration of each segment is constant ([tau] [pi]/[omega]) and is independent of the frictional force (i.e, the frictional damping does not affect the oscillation frequency).

If we start with initial velocity [V.sub.0] equal to zero, then the initial turning point is the initial position of the block, [x.sub.0]=[x.sub.0], and can be used as the length scaling factor. For the remainder of the paper we will consider the initial conditions (2) such that [v.sub.0] = 0, and use in place of [x.sub.0]. Matching Eq. (3) at the boundary between consecutive segments yields an equation for the relative position of the block over multiple segments

X(t) = [1- (2n + 1)/[beta]]]Cos([omega]t)+[(-1).sup.n][beta] (4)

where n is the number of previously completed cycles, or the integer part of t/[tau].

It is interesting to note that as the block oscillates, friction decreases its amplitude in steps. This decrease appears in Eq. (4) at the turning points, as the center of oscillation is shifted toward the block by an amount equal to twice [beta]. When, at a turning point, the block is at a distance less than 2[beta] from the previous center of oscillation, the block stops oscillating and remains stationary at that location. Physically, at this point the static friction has become greater than the restoring spring force and thus prevents the block from moving. In Eq. (4) this occurs at the turning point for which the oscillation amplitude would become negative. Hence, the maximum value of n, or [n.sub.max], is the largest integer which satisfies the equation

N < 1/2[beta] 1 - [beta]). (5)

Using [n.sub.max] we can write the final position of the block (i.e., where it will come to rest) as,

[x.sub.final]/[x.sub.0] = [(-1).sup.[n.sub.max]][2[beta]([n.sub.max] + 1) -1]. (6)

Equations (4) through (6) show that the quantities [beta] and [omega] are the characteristic parameters which completely determine the motion of the system. The parameter w sets the time scale of the oscillation, while the parameter [beta] determines the number of oscillations and final position of the block. Substituting all possible values of [beta] into Eq. (5) we find that there are four significant regions of interest.

First consider the case [beta] = 0, or frictionless oscillation. Figure 2-(a) shows the first seven oscillations3 of this case. As expected for an undamped oscillation, and confirmed by Eq. (4) with [beta] = 0, the oscillations have constant amplitude and are centered at the origin. In the limit that [beta] [right arrow] 0, Eq. (5) shows that the oscillations repeat indefinitely as [n.sub.max] [right arrow] [infinity].

The second region of interest corresponds to non-zero frictional forces that are much smaller than the initial spring restoring force. This is the region associated with 0 < [beta] < 1/3. In this region the block oscillates over a finite number of periods before coming to rest. Figure 2-(b) shows the oscillation of a block with a relative frictional force of [beta] = 0.08. For this frictional force, Eq. (5) gives [n.sub.max] = 5, which accounts for the observed six half cycles (n=0 to 5) before the block comes to rest. Equation (6) gives the final position of the block as [x.sub.final] = 0.04[x.sub.0], which is slightly above the axis as shown in the figure.

When 1/3 [less than or equal to] [beta] < 1, Eq. (5) gives [n.sub.max] = 0. In this region the block comes to rest after one-half period. Using Eq. (6) we see that when [beta] = 1/3, the block comes to rest at the lowest point below the axis, or [x.sub.final] = -1/3[x.sub.0]. When [beta] = 1/2 the block comes to rest at the origin. For larger values of [beta] the block comes to rest at points closer to its starting point. As an example of this, Fig. 2-(c) shows the case for [beta] = 0.65, with [x.sub.final] = 0.30[x.sub.0].

Finally, when [beta] [greater than or equal to] 1, [n.sub.max] = -1, such that not even a half-oscillation is possible. In this case, the static frictional force is greater than the spring's restoring force and the block remains at its initial position. Fig. 2-(d) shows the case for [beta] = 1.10. Since [n.sub.max] = -1, Eq. (6) gives the final position as [x.sub.final] = [x.sub.0].

Energy Consideration

Consideration of the block's mechanical energy as a function of time provides additional insights into the physics of the system. When the spring is initially stretched by a distance [x.sub.0], potential energy is introduced to the system. This gives the system an initial mechanical energy E(0) = 1/2 [kx.sup.2.sub.0]. (7)

Since the frictional force is a non-conservative force, it turns mechanical energy into heat. The amount of converted energy is obtained by integrating the frictional force over the distance the block has traveled,

[DELTA]E(t) = -[mu]mg [[integral].sub.s]ds=-[mu]mg [[integral].sup.t.sub.0]\x(t')\dt'. (8)

The mechanical energy at any given time will be the sum of the spring's potential energy and the block's kinetic energy. It will also be equal to the initial mechanical energy less the energy dissipated, or

1/2 [kx.sup.2] (t) + 1/2 [mx.sup.2] = 1 / 2 [kx.sup.2.sub.0] - [mu]mg [[integral].sup.t.sub.0]\x(t')\dt'. (9)

Note that differentiating Eq. (9) with respect to time yields the equation of motion, Eq. (1). An interesting way to observe the energy evolution of an oscillating block is to examine its phase-space trajectory. If we define the horizontal axis of our phase space as the position of the block x, and the vertical axis as the ratio of the velocity to angular frequency, x[square root of (m/k)], Eq. (9) shows that in a frictionless case the trajectory would be a circle with radius proportional to the square root of the mechanical energy. When friction is included the trajectory becomes an inward spiral. The radius of that spiral is proportional to the square root of the mechanical energy at any given time.

Figure 3 shows the phase space plot for the oscillation of Fig. 2-(b). In this figure, when the radial vector is measured from the origin, the energy is continually decreasing. However, the figure can also be divided into two regions. The region given by the curves below the x-axis (P1-P2, P3-P4, and P5-P6) form arcs of concentric circles centered about point A, where ([X.sub.A], [Y.sub.A]) = ([beta],0). The region given by the curves below the x-axis (P2-P3, P4-P5, and P6-P7) form arcs of concentric circles centered at point B, where ([X.sub.B], [Y.sub.B]) = (-[beta],0). Points A and B correspond to the centers of oscillation shown in Fig. 2-(b). Matching the energy at the boundaries gives the correct time dependence of the mechanical energy. In this picture the block comes to rest when the block's position at a turning point (on the x-axis) falls for the first time between points A and B. The remaining energy stored in the static system at this point is evident.

Conclusions

We have analyzed the nonlinear dynamics of a harmonic oscillator damped by sliding (or kinetic) friction and have obtained an exact solution. The dynamics of this system differ considerably from a harmonic oscillator with linear viscous damping. For instance, in the case of the sliding friction, the envelope of the oscillation decays linearly [through the term proportional to (2n+1) in Eq. (4)] and not exponentially as is the case with linear viscous damping. The local oscillation period is independent of the damping for the system with sliding friction, but not for the system with viscous damping. Only in the system with contact friction does the final state of equilibrium retain some stored energy. The motion of the mass subject to kinetic friction is characterized by an acceleration that changes discontinuously at the turning points (where the frictional force abruptly changes direction), whereas all the kinematic variables vary continuously in the case with viscous damping.

The relative magnitudes of the oscillation restoring force ([kx.sub.0]) and the damping force ([mu]mg) provide a means by which we can classify the motion in three distinct regimes: over damped, critically damped, and under damped. It can be seen from Eq. (2) that if the static frictional force is greater than the restoring force, there can be no motion. This is the case of over damping [See Fig. 2-(d)]. From Eq. (5), one can see that for a static frictional force 1/3 < [beta] < 1, there is motion but not a complete oscillation. The block completes only half an oscillation [See Fig. 2-(c)] and can be classified as a critically damped oscillator. For [beta] < 1/3, the block completes at least one full oscillation before coming to rest [See Fig. 2-(b)], and can be classified as an under-damped oscillator. Finally, when [beta] = 0, the block oscillates indefinitely, which is an undamped oscillator.

It will be interesting to study resonance phenomena in the present system. The system presented here may also be useful in experimentally studying sliding friction.

[FIGURE 2 OMITTED]

(1.) This solution is analogous to that of a block oscillating on a spring while under the influence of gravity. The differential equation of motion for that block would be

x(t) = -k/m x(t) - g,

with solution

x(t) = ([x.sub.0] + mg/k)Cos([square root of (k/m t)])- mg/k.

where [x.sub.0] is the initial displacement of the block from the unstretched position of the string, and the block is initially stationary. The force of gravity lowers the equilibrium position of the block, but does not alter the oscillation frequency nor amplitude (as measured from the new equilibrium position).

(2.) Requiring zero initial velocity simplifies the mathematics of our problem, but does not change the physical results. To account for a non-zero initial velocity, Eq. (3) can be used to determine the block motion up to the first turning point. From the first turning point on, our derivation describes the block motion.

(3.) The trajectories shown were obtained by two methods: numerical solution of Eq. (1), and from the analytical solution given by Eq. (4).

References

(1.) C. Baratt and G. L. Strobel. American Journal of Physics 49 (5), 500, (1981).

(2.) I. R. Lapidus. American Journal of Physics 52 (11), 1015, (1984).

(3.) I. R. Lapidus. American Journal of Physics 38 (11), 1360, (1970).