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arXiv:condmat/0210374v1 [condmat.statmech] 17 Oct 2002
Collision of OneDimensional Nonlinear Chains
Shinichiro Nagahiro and Yoshinori Hayakawa
Department of Physics, Tohoku University, Sendai, Japan
(Dated: February 1, 2008)
We investigate onedimensional collisions of unharmonic chains and a rigid wall. We find that the
coefficient of restitution (COR) is strongly dependent on the velocity of colliding chains and has a
minimum value at a certain velocity. The relationship between COR and collision velocity is derived
for lowvelocity collisions using perturbation methods. We found that the velocity dependence is
characterized by the exponent of the lowest unharmonic term of interparticle potential energy.
PACS numbers: 45.50.Tn, 05.45.a, 46.40.f
I. INTRODUCTION
In collisions between two bodies which have internal
degrees of freedom, some of the initial translational en
ergy is transformed into internal energy of the two bod
ies. This is the major cause of energy dissipation. To
characterize the macroscopic features, a phenomenologi
cal parameter, the coefficient of restitution (COR),
η =Kr
Ki, (1)
where Kiand Krare translational kinetic energy before
and after the collision, respectively, is commonly used.
Recent studies of collisions are mainly focused on the
determination of η from microscopic mechanisms.
Hertz developed the theory of collision between fric
tionless elastic bodies [1, 2] based on his static theory of
elastic contact [3]. In the theory, it is assumed that in
lowvelocitycollisions, the deformation of colliding bodies
is given by the static theory and the production of vibra
tion is totally ignored. Hence, the theory gives no infor
mation on energy dissipation. Plastic deformation is one
of the possible means of kinetic energy dissipation during
collisions. Taking this into account, the energy dissipa
tion rate 1−η is found to increase with collision velocity
with a power law with the exponent 1/2 [4]. Considering
viscoelastic properties, the dissipation rate increases with
the exponent 2/5 [5, 6, 7]. These results were confirmed
experimentally [8, 9, 10].
The results presented above are based on quasistatic
approximation, hence it is expected that they are re
stricted to lowvelocity collisions. We believe that more
general results will be obtained through microscopic sim
ulations [11, 12, 13].
Sugiyama and Sasaki [14] and Basile and Dumont
[15] considered collisions between simple onedimensional
chains and a rigid wall. A chain is composed of n identical
point particles which interact with nearestneighbor par
ticles. If we choose linear force as the interaction between
the particles, the COR is independent of collision velocity
and approaches unity in the thermodynamic limit.
For collision between metallic bodies, plastic deforma
tion usually occurs even if collision velocity is low. The
deformation exceeds the elastic regime in which Hooke’s
law remains valid [4, 16]. Hence nonlinear effects are im
portant in realistic collisions. In this study, we choose
nonlinear force as the interparticle interaction and per
form the onedimensional simulations of collision between
a “nonlinear chain” and a rigid wall. We find that the
COR of this collision has a minimum value at a certain
velocity and derive the velocity dependence of the COR
for lowvelocity collisions using perturbation methods.
This paper is organized as follows. In the next section,
we briefly review the studies of Sugiyama and Sasaki [14]
and Basile and Dumont [15]. In section 3, we present our
results for collision between nonlinear chains and a wall.
Finally, we summarize our results.
II. COLLISIONS OF ONEDIMENSIONAL
HARMONIC CHAINS
First we briefly discuss the collision of onedimensional
harmonic chains (model A) with a rigid wall as discussed
by Sugiyama and Sasaki [14]. Consider a chain composed
of n identical point particles labeled j = 1,2,···n. Each
particle in the chain is linked to nearestneighbor parti
cles with a Hooke’s spring, as illustrated in Fig.1. The
Hamiltonian of this system is written as
H =m
2
n
?
j=1
˙ x2
j+1
2mω2
n−1
?
j=1
(xi+1− xi− ℓ)2+ Vw, (2)
FIG. 1: Schematic of a onedimensional chain and a rigid wall.
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2
where xj and ˙ xj are the position and velocity of the j
th particle, respectively, and ℓ is the natural length of
the springs. We assume the chain is homogeneous and
all springs have the same spring constant k(= mω2). Vw
represents the hardcore potential of a rigid wall located
at x = 0 where collision takes place. During the collision
process, only the particle j = 1, which is at the end of the
chain, interacts with the wall since the positional order
of the particles is always kept as xn< xn−1< ··· < x2<
x1. In the following discussion, we assume that the wall
is so rigid that the particle x1simply reverses its velocity
as −˙ x1→ ˙ x1.
Although there is no dissipation term in the Hamilto
nian, vibration energy which remains after the collision
is regarded as the “dissipated” portion of energy. The
COR η is therefore evaluated as
η = 1 −Evib.
E
,(3)
where E is the total energy and Evib. is the vibration
energy which remains after the collision.
In the absence of Vw, the equation of motion is written
as
¨ x = −ω2(Ax + b),(4)
where x = (x1,x2,··· ,xn), b = (−ℓ,0,··· ,0,ℓ) and A
is an n × n matrix of the form
0···
Taking a principalaxis coordinate system in which A is
diagonal, internal vibration of the chain can be repre
sented using n noninteracting fundamental modes. In
the presence of Vw, collision between the chain and the
wall is realized as follows.
j = 1 collides with the wall f(n) times at t1,t2,··· ,tf(n).
The fundamental modes describe equienergy elliptical
orbits in phase space. The orbits are discontinuous at
t = t1,t2,··· ,tf(n).
For the following numerical simulations, we choose ini
tial conditions
A =
1−1
2
−1
0··· 0
··· 0−1
0
...
−1
2
0
−1 0 ··· 0
...
0 −1 1
...
. (5)
Assume that the particle
?xj = ℓ j,
˙ xj = −v0 (j = 1,2,··· ,n).
Before collision, the chain has no internal vibration, i.e.,
zero temperature. Figure 2 shows the collision between
the wall and the chain with n = 19. Each line is a trace of
the trajectory of a particle. In the plot, units on the time
axis are taken as τ = (n−1)ℓ/cl. τ indicates the duration
in which the longitudinal sound wave propagates from
one end of the chain to the other. Let us call the time
tcduring which the collision takes place ”contact time”.
Here we note that contact time is almost equal to 2τ
(6)
?
?
?
?
?
??
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????? ? ???????
?
?
?
?
?
?
?
?
? ? ?? ? ??? ?
?
???????
FIG. 2: Trajectories of the particles in the n = 20 colliding
chain. (Reduced units where cl= τ = 1.)
for large n. COR and contact time are independent of
the collision velocity of the chain because time intervals
△tq = tq+1− tq are independent of initial velocity for
every q and the orbits retain similar forms even if initial
velocity of the chain varies. Therefore, Ej/E, the ratio
of the jth fundamental mode energy to total energy,
does not depend on initial velocity. The COR, which is a
ratio of energy, also does not depend on initial velocity.
Hereafter, we use the reduced unit cl= τ = 1, setting ℓ,
m and k as unity.
In model A, the coefficient of restitution η depends
only on n. An approximate expression of η was derived
by Basile and Dumont[15]. To simplify the problem, we
assume that △tq and the velocity before the qth colli
sion vq are constant for every q. Numerical simulations
show that these are good approximations. Under this
assumption, the vibration energy of the jth mode after
the collision process is given as
Ej=4m
nv2cos2
?πj
2n
?sin2{ωjf(n)△t/2}
sin2{ωj△t/2}
,(7)
where ωj is the frequency of the jth mode, which is
ωj = sin(πj/n). △t, f(n) and v are determined from
numerical simulations [15]. However we can also estimate
these values by solving the collision of the chain with
n = 2:
△t =
π
√2ω
2√2
π
π
2√2v0= 1.11v0
= 2.22/ω(2.31/ω),(8)
f(n) ≃
v =
n = 0.90n(0.867n),(9)
(1.15v0),(10)
where the values in brackets were obtained from a nu
merical simulation of an n = 500 chain. We can estimate
an approximate value of η by using Eqs. (8), (9) and
(10).
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3
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FIG. 3: A loglog plot of dissipation rate versus number of
particles.The solid line has slope −2/3.
where cl= τ = 1.)
(Reduced units
To obtain the asymptotic behavior of 1−η for large n,
we expand the dispersion relation.
ωj=
?
k/mπj
n
?
1 −π2j2
24n2+ ···
?
(11)
Substituting only the leading order of Eq.(11) into
Eq.(7), we have η = 1. Taking the second order into
account, we obtain
1 − η ∼ 0.652 n−2
3,(12)
in the limit n → ∞. This relationship agrees very well
with the numerical result. In Fig. 3, we plot the dissipa
tion rate 1 − η versus n.
III.UNHARMONIC CHAINS
In the case of harmonic chains, COR and contact time
were independent of initial velocity v0. Here, we intro
duce nonlinearity to the springs in the chain (model B).
In order to maintain a universal viewpoint, we first
consider a velocity scale characterized by the nonlinear
ity of the spring. Let U(x) be the potential energy of the
spring, which is chosen to be U(1) = 0 assuming the nat
ural length of the spring to be unity. We can define the
amplitude x∗at which the harmonic term and the sum of
the remaining terms are equal in the Taylor expansion,
i.e., x∗is given by the solution of
1
2
d2U(x)
dx2
????
x=1
(x∗− 1)2= U(x) −1
2
d2U(x)
dx2
????
x=1
(x∗− 1)2.
The velocity which corresponds to x∗is
v∗=
?
2U(x∗).
Hereafter, we discuss the velocity dependence of the col
lisions on the scale of v∗.
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FIG. 4: Coefficient of restitution for the collisions of log,
Toda and LennardJonestype chains as a function of collision
velocity. The chains consist of 100 particles.
Let us choose the following three types of potential and
compare the results of simulation.
(a) LennardJones potential
ULJ(x) =
1
72
??1
x
?12
− 2
?1
x
?6?
(b) Toda potential
Utoda(x) =
a
be−b(x−1)+ a(x − 1)
(c) Logtype potential
Ulog(x) = x − log(x)
For the Toda potential, we set ab = 1 and b = 10. Each
potential has one minimum at x = 1 and the function
forms are similar. However, increasing behaviors of the
repulsive forces derived from the three potentials are, in
a very short distance, different from one another. In Fig.
4, we plot the COR versus initial velocity for each case,
as determined by numerical simulations. In the limit of
small velocity, it is clear that COR approaches the value
obtained for the harmonic chain. The COR decreases
with increasing initial velocity and has a minimum near
v = 1. We note that the COR lies on almost the same
curve for all three types of potential.
Figure 5 shows ndependence of η for different values of
initial velocity. In very lowvelocity collision, as expected
from the result for model A, dissipation rate 1−η will ap
proach to zero in the thermodynamic limit. Near v0= 1,
the COR does not approach unity but remains a con
stant less than unity even in the thermodynamic limit.
We can hence conclude that nonlinearity of the potential
U(x) causes the dissipation, which does not appear in
model A.
Using a technique based on the perturbative theory, we
consider the collision of model B for small initial incident
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4
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FIG. 5: The relationship between dissipation rate and number
of particles in the collision of LennardJonestype chain. Each
plot corresponds to a different initial velocity: v0 = 0.780 (?),
0.390 (♦), 0.160 (?), 0.078 (?) and 0.008 (•).
velocity v0≪ 1. During collision processes, the particle
j = 1 transmits vibration force to its neighbor parti
cle. Let us regard this force as external force F(t) which
acts on the chain. The characteristic frequency of this
force is ωext= 2π/△t ≃ 2√2ω. This frequency is higher
than any frequency of fundamental modes of the chain.
Hence, no fundamental mode is excited by the force. In
this situation, the amplitude of each particle’s vibration
is rapidly damped progressively into the chain, i.e., the
particle j = 1 has the largest amplitude during collision.
It is expected that △t will shorten with increasing initial
velocity. As a first approximation, we take into account
only the change of △t against initial velocity v0 in Eq.
(7). To estimate the velocity dependence of △t, let us
consider the collision of a LennardJonestype chain with
n = 2. In this case, particle j = 1 collides with the wall
two times. The Hamiltonian is
H =m
2(˙ x2
1+ ˙ x2
2) + ULJ(x2− x1− 1) + Vw.(13)
Let x = x2− x1− ℓ and xg = (x2+ x1)/2. For initial
conditions, we adopt Eq. (6). Immediately after the first
collision of particle x1, x and xg and their time deriva
tives become
?x =0
˙ x = −2v0 ,
?xg = ℓ/2
˙ xg =0.
(14)
Taking into account the second order of ULJand solving
the equation of motion with the initial conditions of Eq.
(14) in the firstorder perturbation theory, one can ob
tain the interval △t between first and second collisions
of particle x1as
△t ≃
π
√2
?
1 − αv0
v∗
?
, (15)
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106
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0.001
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3
2ωjf(n)△t(1 − α˜ v)]
sin2[1
2ωj△t(1 − α˜ v)]
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6
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5
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5
9
8
;
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=
FIG. 6: Increase of dissipation rates. We plot the dissipation
rate of model B minus that of model A for r = 3,4 and 5.
Fitted lines have slopes 1.34,2.09 and 3.14, respectively.
with α = 0.808. Substituting this into Eq. (7), we have
Ej=4
ncos2
?πj
2n
?sin2[1
.(16)
In the limit n → ∞, ωj can be replaced with the linear
form
ωj= ωπj
n.
(17)
When v/cl≪ 1, Eq. (16) can be approximated as
32
nπ2
j
Ej≃
?n
?2
sin2
?
αvπn
?j
n
??
.
Substituting this into Eq. (3), we have the dissipation
rate
1 − η =
n
?
j=1
Ej
E
≃
64
nπ4
?1
?∞
0
dx1
x2sin2(αvnx)
≃
= Cv,
64
π3αv
0
dxsin2(αx)
x2
(18)
where C =
tion rate 1 − η increases with the power law vpwhere
p = 1. This result agrees with numerical simulation for
lowvelocity collision. However the constant C is not in
accord with the above result (our numerical simulation
gives C = 0.66).
The above result directly depends on the exponent of
the lowest unharmonic term of interparticle potential
energy. In the case of LennardJones potential, the ex
ponent r = 3. The exponent is the same for other types
of chains. Let us discuss more general cases. Suppose
that the interparticle potential energy can be expanded
around its equilibrium position as
32
π2α ≃ 2.61. This implies that the dissipa
U(x) =1
2x2+ cxr+ higher order,(19)
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5
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COR
f(n) / n
?? ? ?
? ?
? ?
?
?
?
? ??
?
?
? ? ?
?
?
?
?
?
?
?
??
?
?
?
FIG. 7: Plot of ratio of f(n) to n and COR versus velocity
for LennardJonestype chain. Both increase at ˜ v ≈ 1.
where the constant c is a positive (negative) number when
r is even (odd). In this case, the contact time shortens
as vr−2
0
. The dissipation rate hence increases as
1 − η ∝
?v
v∗
?r−2
.(20)
Taking potential energy Eq.(19) for the interaction of
particles in model B, we plot numerical results of COR
in Fig. 6 for v ≪ 1. The results agree with Eq. (20).
When the particles in the chain can be regarded as
rigid spheres, interparticle interaction is of the δ function
type. Collision between the two spheres is reduced to a
simple exchange of their momenta. We can exactly solve
the entire dynamics of the collision between the chain
and a rigid wall. The sphere j = 1 collides with the
wall n times and no internal vibration remains after the
collision, i.e., the COR is exactly unity in this case.
In the limit of highvelocity collision of model B, par
ticles interact like rigid spheres because ULJ, Utodaand
Ulog all behave like hard core potentials within a very
short distance. In Fig. 7, we show the COR of a Lennard
Jonestype chain and f(n)/n versus collision velocity on
the same plot. The rate f(n)/n approaches unity for
v > 1. This indicates that particles interact like rigid
spheres in highvelocity collision. Consequently COR in
creases in the highvelocity regime.
which only the onedimensional model exhibits and is un
realistic. In real systems, plastic deformation is crucial
in such highvelocity collisions.
This is a feature
IV.SUMMARY
We have presented a simple onedimensional micro
scopic model of colliding bodies to understand the energy
dissipation process. LennardJones, Toda and Logtype
potentials are chosen as interactions between particles.
We found that the COR depends on the initial velocity
and is minimum at v/v∗ ≃ 1. These behaviors are inde
pendent of the potential form. In lowvelocity collisions,
the relationship between the energy dissipation rate and
collision velocity is derived using perturbation methods.
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