# Coefficient of restitution of a linear dashpot on a rigid surface

Sean P. Bartz<EMAIL_ADDRESS>_Dept. of Chemistry and Physics, Indiana
State University_
_Terre Haute, IN 47809_

###### Abstract

The linear dashpot model is applied to a single ball bouncing on a rigid
surface. It is shown that when gravity is included the coefficient of
restitution depends on impact velocity, in contrast to previous work that
ignored the effects of gravity. This velocity dependence is most pronounced at
low impact velocities and high damping. Previous work has considered the ball
to be in contact with the floor when the compression is nonzero, while other
analysis terminates the collision earlier, to prevent an attractive force. We
compare these models and propose a hybrid between the two. The hybrid model is
successful in reproducing experimental results for a cart bouncing repeatedly
on a spring.

##  1 Introduction

A spring with a linear dashpot is a convenient model for inelastic collisions,
such as those involving viscoelastic spheres. For the duration of the contact
between the spheres, the equations of motion are the well-known damped
harmonic oscillator. In the absence of gravity, this model yields a
coefficient of restitution that is independent of impact velocity [1, 2],
which simplifies the analysis of collisions. In contrast to a damped harmonic
oscillator, a viscoelastic sphere always exhibits a repulsive force. Ensuring
the absence of an attractive force in the linear dashpot model affects the
calculated collision duration and coefficient of restitution [2, 3].

Gravity can be neglected when two spheres collide in midair, as gravity
affects each sphere equally. However, for a ball bouncing off of a rigid
surface, gravity cannot be neglected, and its inclusion results in a
coefficient of restitution that depends on impact velocity [4].

There are several experimental methods for determining the elastic and damping
properties of a ball, including measuring successive bounce heights [5],
measuring velocities before and after collisions, or measuring the forces
directly [6]. Of particular interest to our analysis is the method of
measuring the time of flight between successive impacts [7, 8, 9], the
duration of an individual impact [10] or the total time and number of bounces
until the ball comes to rest [1]. These experimental techniques typically
assume a constant coefficient of restitution, but we show that the final
bounces are notably affected by the velocity dependence of $\varepsilon$. We
compare our results to previous theoretical models using the linear dashpot,
and experimental data from a cart bouncing on a spring.

##  2 Linear dashpot force models

The spring with a linear dashpot includes a repulsive elastic term that is
proportional to the compression and a dissipative term that is proportional to
the velocity. The compression is defined $\xi=\min[0,R-z],$ where $R$ is the
radius of the ball and $z$ is its vertical position. The linear dashpot force
is defined

$F=-k\xi-\gamma\dot{\xi}.$ (1)

When this force is non-zero, the equation of motion is

$\ddot{\xi}=-g-\omega_{0}^{2}\xi-2\beta\dot{\xi},$ (2)

with the substitutions

$\beta=\frac{\gamma}{2m},\quad\omega_{0}^{2}=\frac{k}{m},\quad\omega=\sqrt{\omega_{0}^{2}-\beta^{2}}.$
(3)

Solving for initial conditions $\xi(0)=0,\dot{\xi}(0)=-v_{0}$ in the case of
low damping ($\beta<\omega_{0}$) gives

$\xi=-\frac{v_{0}}{\omega}e^{-\beta t}\sin\omega
t+\frac{mg}{k}\left(-1+e^{-\beta t}\left(\cos\omega
t+\frac{\beta}{\omega}\sin\omega t\right)\right).$ (4)

When neglecting gravity, only the first term remains. In the case of high
damping, $\beta>\omega_{0}$, the solution to (2) is

$\xi=-\frac{v_{0}}{\Omega}e^{-\beta t}\sinh\Omega
t+\frac{mg}{k}\left(-1+e^{-\beta t}\left(\cosh\Omega
t+\frac{\beta}{\Omega}\sinh\Omega t\right)\right),$ (5)

with $\Omega=\sqrt{\beta^{2}-\omega_{0}^{2}}$. The following analysis focuses
on the low damping case, but the high damping results are found analogously.
High damping results are included in the numerical analysis when appropriate.

To examine universal behavior, we define a characteristic velocity
$v^{*}=g\sqrt{k/m}$, and plot the dimensionless quantity $\tilde{v}=v/v^{*}$.
For the sake of comparison, the characteristic velocities for a ping pong ball
and a basketball are 0.01 m/s and 0.03 m/s, respectively. For the spring-
loaded cart experiment [4], $v^{*}=0.129$ m/s.

##  3 Coefficient of restitution

The coefficient of restitution is defined

$\varepsilon=\left|\frac{v_{f}}{v_{0}}\right|,$ (6)

where $v_{0}$ is the impact velocity and $v_{f}$ is the velocity at the end of
the collision between the ball and the floor.

Models differ in the definition of the contact duration, and thus have
different values of $\varepsilon$. One class of models terminates the
collision based on position, when $\xi=0$, which we call models of type I.
Alternatively, the end of the collision can be defined when the spring-dashpot
force (1) becomes zero, which we term model II. Additionally, gravity may be
included or neglected, which we label with subscripts $a$ and $b$,
respectively. In all cases, we solve for the contact duration $\tau$, and
calculate the coefficient of restitution

$\varepsilon=\left|\frac{\dot{\xi}(\tau)}{v_{0}}\right|.$ (7)

The models are compared below.

###  3.1 Contact defined by position

Analyses such as [1, 11, 12] consider the ball to be in contact with the floor
as long as the compression is nonzero. In the zero-gravity case, which we will
designate model Ia, the contact duration is calculated [1]

$t_{Ia}=\frac{\pi}{\omega}$ (8)

and the coefficient of restitution is

$\varepsilon=e^{-\beta t_{Ia}},$ (9)

independent of impact velocity.

Including gravity yields an impact velocity-dependent coefficient of
restitution. We numerically solve (4) with $\xi=0$ for the contact time
$t_{Ib}$, with sample results shown in Figure 1. Figures 2, 3, and 4 show the
resulting $\varepsilon$ for a range of $\beta/\omega_{0}$ for representative
impact velocities.

###  3.2 Contact defined by force

It has recently been shown that the models in Section 3.1 result in an
unphysical attractive spring-dashpot force near the end of the collision [2].
This is avoided by defining the end of the collision when the linear dashpot
force is equal to zero, which occurs before the compression returns to zero.

We examine this model in the absence of gravity. This calculation is relevant
for two colliding spheres that are both moving under the influence of gravity,
and has also been used as an approximation for bouncing on a rigid surface [2,
3].

$t_{IIa}=\frac{1}{\omega}\arccos\left(\frac{2\beta^{2}}{\omega_{0}^{2}}-1\right),$
(10)

and the coefficient of restitution is

$\varepsilon=e^{-\beta t_{IIa}},$ (11)

independent of impact velocity.

Including gravity in the analysis, we solve (4) for the case
$\ddot{\xi}(t_{IIb})=-g$, and the ball’s motion reverts to free fall. This
produces a transcendental equation for $t_{IIb}$

$-g=v_{0}e^{-\beta t_{IIb}}\left(2\beta\cos\omega
t_{IIb}+\frac{\omega^{2}-\beta^{2}}{\omega}\sin\omega
t_{IIb}\right)+ge^{-\beta t_{IIb}}\left(\frac{\beta}{\omega}\sin\omega
t_{IIb}-\cos\omega t_{IIb}\right).$ (12)

This equation is solved numerically for $t_{IIb}$, with sample results shown
in Figure 1. This time is used to calculate the velocity when the force goes
to zero, and the coefficient of restitution is calculated based on this
rebound velocity.

Figure 1: The contact time (dimensionless units) is plotted against damping
ratios for $\tilde{v}=23$. For models that terminate the collision when
compression $\xi=0$ and for the hybrid model, the contact time represents the
duration of nonzero compression. For the force based models, the contact time
represents the duration of nonzero force. The results of model Ib are not
shown, as they are visually indistinguishable from the hybrid model.

The coefficient of restitution is found by computing the velocity of the ball
at time $t_{IIb}$. Typical results are shown in Figures 2, 3, and 4 for a
range of damping ratios $\beta/\omega_{0}$. The coefficients of restitution
are strictly less than the result from [2].

To compare the contact times between these two models, we calculate the
acceleration in the model including gravity, using the contact time from the
gravity-free model

$\ddot{\xi}(t_{IIa})=ge^{-\beta t_{IIa}}.$ (13)

This acceleration is greater than zero, indicating that the ball is still in
contact with the floor. This shows that the contact duration is greater when
gravity is included, as seen in Figure 1.

Figure 2: The coefficient of restitution for the various models is plotted as
a function of ball parameter $\beta/\omega_{0}$ for a dimensionless impact
velocity $\tilde{v}=23$. The curves terminate where elastic capture occurs.

###  3.3 Hybrid model

Instantaneous velocities are difficult to measure, so experiments typically
infer coefficients of restitution from bounce heights or time between impacts.
[19]. Doing so implies that the impact velocity and rebound velocity are both
measured when the compression is zero, and the ball’s position is $z=R$. This
assumption is compatible with models of type I, but causes a discrepancy with
models of type II. In these models, $z<R$ when the force goes to zero, so the
bounce height is lower than predicted, while the flight time is longer.

Figure 3: The coefficient of restitution is plotted as a function of ball
parameter $\beta/\omega_{0}$ for a dimensionless impact velocity
$\tilde{v}=15$. The curves terminate where elastic capture occurs.

We present a hybrid model, which avoids unphysical attractive forces, but
calculates the rebound velocity $v_{1}$ when the compression is zero. We
calculate the ball’s position $z=R-\xi(t_{IIb})$ and velocity
$v_{f}=\dot{\xi}(t_{IIb})$ at the time the force goes to zero, based on (12).
Afterward, the ball is in free-fall, and we use energy conservation to
calculate the velocity when $z=R$

$v_{1}=\sqrt{v_{f}^{2}-2g\xi(t_{IIb})},$ (14)

and the time it takes the ball to traverse this distance, $\Delta t=2v_{1}/g$,
which is needed when calculating the time between successive bounces, as in
Section 4. The coefficients of restitution for this model fall between models
Ib and IIb, as seen in Figures 2-4.

###  3.4 Inelastic capture

Inelastic capture occurs when two objects remains in contact after a
collision. In model Ia, inelastic capture occurs for $\beta/\omega_{0}\geq 1$,
as the contact time (8) diverges. In model IIa, inelastic capture never
occurs, as the contact time (10) is always finite.

In the models that include gravity, there is a critical impact velocity for a
given value of $\beta/\omega_{0}$, below which the ball is inelastically
captured. This impact velocity is found when there is no real solution to the
collision time (4) and (12), respectively. In the hybrid model, the critical
impact velocity is found when the rebound velocity (14) goes to zero. These
critical velocities are shown in Figure 5. Model IIb and the hybrid model
allow for bouncing in the high damping case, but in model Ib, inelastic
capture occurs for all velocities at high damping.

At the critical impact velocity, $\varepsilon\rightarrow 0$ in model Ib and
the hybrid models, as seen in Figures 2-4, but $\varepsilon$ in model IIb
terminates at a finite value. This suggests the hybrid model is a more
sensible method to remove the unphysical attractive force from the linear
dashpot models.

Figure 4: The coefficient of restitution is plotted as a function of ball
parameter $\beta/\omega_{0}$ for a dimensionless impact velocity
$\tilde{v}=5$. The curves terminate where elastic capture occurs.  Figure 5:
For a range of damping ratios $\beta/\omega_{0}$, the cutoff velocity is
plotted, scaled by $g/\omega_{0}$ to make it dimensionlesss. Below the cutoff
velocity, the ball is inelasticically captured.

##  4 Modeling experimental data

The effects of the velocity-dependent coefficient of restitution are most
clearly seen in experiments where an object is allowed to bounce multiple
times until inelastic capture. In [4], the authors used a spring-loaded cart
on an inclined track as a practical model of a spring with a linear dashpot.
We use the data from that experiment to test the various models

The mass and elastic coefficient are typically measured directly, but the
damping coefficient $\beta$ must be estimated from the bouncing motion. In
this experiment, $m=0.506$ kg and $k=255$ N/m. The authors use $\varepsilon$at
high velocity and the results of model Ia to estimate $\beta=1.79$ s-1.

We use each model recursively, solving for the successive impact velocities,
and totaling the contact time and flight time between each bounce until
inelastic capture. The damping coefficient $\beta$ is determined by a best fit
to the bounce time data [4], shown in Figure 6. The results from the various
models are visually indistiguishable, so only the hybrid model is plotted. The
values of $\beta$ differ at the 10% level as shown in Table 1.

Model | Best fit value of $\beta$ (s-1)
---|---
A | 1.95
B | 1.78
C | 1.94
D | 1.97
Hybrid | 1.87
Table 1: The damping coefficients for the various models are fit by matching
the time interval between bounces to the experimental data, as shown in Figure
6.

The differences between models is apparent when these values of $\beta$ are
used to calculate $\varepsilon$ for the range of impact velocities in this
experiment. Unsurprisingly, the models that account for gravity reflect the
observed velocity dependence of the coefficient of restitution, as seen in
Figure 7. The hybrid model and model Ib fit the data equally well. The success
of the hybrid model suggests that the rebound velocities were inferred from
successive bounce heights, or measured directly when the cart returns to its
position of initial impact.

Figure 6: The results from the various models are visually indistinguishable
on the plot, so only the hybrid model is shown.  Figure 7: The coefficient of
restitution $\varepsilon$ is calculated in various models and compared to
experimental data from [4]. The velocity dependence is seen in the
experimental data and well matched by the models that include gravity.

##  5 Analysis

In this paper we compare several spring-dashpot models for a ball bouncing on
a rigid surface. We show that the inclusion of gravity produces a velocity-
dependent coefficient of restitution, in contrast to the linear dashpot models
for colliding spheres. We compare different definitions for the termination of
the collision, and their effects on the coefficient of restitution and the
contact duration. The differences in these models are most apparent at low
damping and low impact velocities.

Comparison to experiment shows these effects. For the experiment studied here,
the inclusion of gravity has a greater effect on the predictions than the
choice of termination condition for the bounce. Although model Ib includes an
unphysical attractive force, it gives results just as good as the hybrid
model. Absent an independent measurement of $\beta$, these experimental
results cannot discriminate between the two models. The results in Figures 2,
3, and 4 suggest gravity will also be of increased importance at larger values
of $\beta/\omega_{0}$, and differences between the termination conditions are
likely to be observable. Collecting data for a cart with a spring with greater
damping or for an “unhappy” ball [13] and comparing to the predictions of
various models would make a good project for an undergraduate student.

We show the importance of relating model choice to the method of experimental
measurement. Although model IIb’s avoidance of the attractive force is
theoretically sound, measuring the rebound velocity at the instant $F=0$ is
often impractical. The hybrid model corrects for this difficulty, and is also
more accurate in modeling repeated bounces. However, if measuring contact
duration with a force sensor, model IIb would be the preferred choice.

These results suggest that models of collisions of multiple spheres should
account for impact velocity dependence when the lower ball is in contact with
the floor and gravity is present [3, 14, 15, 16]. These results may also be
relevant for modeling granular materials under gravity or another driving
force, [17] while gas models typically lack such a driving force [18] and can
neglect this effect.

Observations of a cart bouncing on a spring provide a direct comparison to the
linear spring and dashpot model, but bouncing spheres are more accurately
described by a Hertzian force model $F\sim\xi^{3/2}$, with a dissipative force
that may also be nonlinear [11, 19]. These equations of motion cannot be
solved analytically, but a student interested in numerical techniques could
take a similar approach to the various termination conditions to the one
outlined here, and test which produces more accurate predictions, particularly
in cases where the ball’s damping coefficient is high.

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