Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

46, 4, pp. 897-908, Warsaw 2008

PHENOMENON OF FORCE IMPULSE RESTITUTION IN

COLLISION MODELLING

Jerzy Michalczyk

Technical University of Mining and Metallurgy, Cracow, Poland

e-mail: michalcz@agh.edu.pl

The rightness of the Newtonian hypothesis concerning a constat value
of the coefficient of restitution R has been confirmed in the paper with
reference to collisions in which the loss of energy occurs in conseqence
of material damping. For collisions of different nature, when R depends
on the density of the energy flux Φ, the paper points to the possibility
of extending this notion to the case of eccentric collision.The possibility
of describing R as a random function of Φ has been shown as well.

Key words: collision, coefficient of restitution

1. Introduction

Despite its universality in nature and technology, the phenomenon of collision
of bodies has not been fully and satisfactorily described. The starting point
in the analysis of collision is usually Newton’s hypothesis, according to which
the correlation between an impulse Π1 of a contact force F at the stage of
increase in deformation of the bodies and an impulse Π2 of the same force
during the relieving process is given by the following equation

R=
Π2
Π1

(1.1)

where

Π1 =

t0
∫

t1

F dt Π2 =

t2
∫

t0

F dt (1.2)

and R∈ [0,1] is a constant, called the coefficient of restitution.
The fact that impact forces are generally stronger than any forces acting

on the system as well as short duration of the collision make it possible to



898 J. Michalczyk

Fig. 1.

omit non-impact forces and relocations of bodies, and regard the system as a
free body system (it does not refer to any geometrical constraints which are
imposed on motion of the bodies).
For such a system, from equations of conservation of momentum and an-

gular momentum, it is possible to determine kinematic parameters for both
bodies at the end of increase of deformation, when the normal components of
velocities at the collision point are equalised.
On the base of Newton’s hypothesis, which describes the degree of resti-

tution of impulses of collision forces in the second stage of the collision, it is
possible to obtain final values of linear and angular velocities for both bodies
after collision.
Newton’s ”time-free” model of the collision enables us to utilize a lot of

experimental data concerning impulse restitution and give an ”integral look”
on the collision, but makes it impossible to determine the process of force
growth and local deformations. It is also inconvenient in modelling of dyna-
mical phenomena for computer simulation.
From the theoretical point of view, the basic meaning also has an answer

to the question about physical sense of Newton’s idea of a constant value of
the ratio of impulses in both stages of the collision, i.e. to indicate a physical
processwhich leads to thismodel or to prove that this hypothesis has no sense.

2. R=const hypothesis and physical phenomena responsible for
incomplete restitution

An incomplete restitution of instantaneous forces during the collision ismainly
associated with:

1. loss of energy which is associated with a higher or lower degree of stable
deformations (e.g. plastic or brittle)



Phenomenon of force impulse restitution... 899

2. flow of energy from the area of collision to further parts of both bodies
that takes place by excitation of waves and vibrational phenomena

3. loss of energy which is associated with pressing of contamination, e.g.
oxides, dust, etc.

4. loss of energy which results from friction (slip friction – generally asso-
ciated with different velocities of local deformation of both bodies in the
plane of collision) and material damping – in the volumes undergoing
deformation of both bodies.

As for factor No.1, it is possible to asses theoretically the loss of energy
in this case, but it does not lead to a constant value of R. On the contrary
– R forms a decreasing function of relative velocity (Gryboś, 1969, 1971). The
actual value of R is burdened with a margin of error which results from its
susceptibility to factors which are difficult to be described in a deterministic
way due to incomplete repeatibility of the place and angle of contact of both
bodies, deviations in microstructure of surface, etc.

The second factor is only essential with regard to bodies of a flabby struc-
ture. As for bodies of a compact structure, i.e. when the period of natural
vibrations is shorter than the period of contact during collision, this pheno-
menon does not have any significant contrbution to the balance of loss during
the collision (Rayleigh, 1906). In the case of flabby structures, there is an ef-
fect of R on velocity (Gryboś, 1969; Harris and Piersol, 2002), so R is not a
constant value.

As for the third factor, it can only be described within statistical classes,
although there is a strong dependency of statistical characteristics of R on
deterministic factors, e.g. area of collision contact (Michalczyk, 1984).

As for factor No.4, it should be pointed out that the loss of energy, which
results fromslip friction, shows strongdependence on the intensity of collision,
randomproperties ofmicrostructure and cleanness of the surface, so it can not
be considered as a constant value.

Thenwhen R is a constant value?Letus evaluate the loss of energy related
to material damping during collision considered as an ”in-time” process –
development of local deformations inbothbodies.Letusalso assumethat their
contact stiffness (dependence of the force F of mutual interactions on total
deformations in the bodies xw) could be described by the Hertz-Stajerman
model

F = kxpw (2.1)

This model describes a wide range of bodies whose surfaces in the vicinity of
the collision point could be regarded as regularly rotational. The exponent p



900 J. Michalczyk

in equation (2.1) depends on the so-called contact tightness of both bodies,
and for a surface of the second order it is p=3/2.

When the loss of energy ∆L occurs according to amodel ofmaterial dam-
ping in which during the vibration period it is proportional to the maximum
potential energy E of the system

∆L=ΨE (2.2)

where Ψ/2π = η is the loss factor, we can represent the dependence of the
force of mutual interactions F between both bodies as shown in Fig.2.

Fig. 2.

To prove the conformity of the accepted form of the force F with the
model of material damping, let us calculate the loss of energy for the range
xw > 0 (this range refers to change in xw during the collision) using the
model of material damping, and compare it with the loss of energy calculated
by integration of the surface area of half of the hysteresis loop for the force
determined as above.

In the first case, the loss of energy during the collision (continuous line in
Fig.2) is given by

∆L=
1

2
ΨE=

1

2
Ψ

Aw
∫

0

F(xw) dxw =
1

2
Ψ

Aw
∫

0

kxpw dxw =
1

2
kΨ
Ap+1w
p+1

(2.3)

where Aw is the maximum value of deformation xw.



Phenomenon of force impulse restitution... 901

In the second case, the loss of energy described by half of the area of the
hysteresis loop is given by

∆L=

Aw
∫

0

[kxpw− (k−∆k)x
p
w] dxw =

Aw
∫

0

∆kxpw dxw =
∆k

p+1
Ap+1w (2.4)

Comparing (2.3) and (2.4), we obtain

∆k=
Ψ

2
k (2.5)

Therefore, the force F can be described by the following equation

F = kxpw

{

1−
Ψ

4
[1− sgn(xw)sgn(ẋw)]

}

(for the collision sgn(xw)= 1)

(2.6)
We will show that it is possible to obtain an equivalent description of the

collision as a ”time-free” process according to Newton’s formulation charac-
terised by the parameter R, to the description of the collision as a dynamic
process which takes place in a limited time and is characterised by rheological
properties k,∆k of bodies in the vicinity of the contact area.
Let us consider a simple central collision of bodies of masses mi and mj,

as schematically shown in Fig.3, where prior to the collision vi >vj.

Fig. 3.

During the collision

ẍi =−
F

mi
ẍj =

F

mj
(2.7)

By introducing of a relative coordinate xw =xi−xj, we obtain

ẍw = ẍi− ẍj =−
mi+mj
mimj

F (2.8)



902 J. Michalczyk

If we put
mimj
mi+mj

=mw (2.9)

and taking into consideration Eq. (2.1) we obtain (Gryboś, 1969)

mwẍw+kx
p
w =0 (2.10)

This common equation we can interpret in another way – as an equation of a
non-linear oscillator of mass mw, non-linear elasticity (2.1) and initial condi-
tions

xw(0)= 0 ẋw(0)= vw (2.11)

Then, comparing themaximum value of kinetic energy (beginning of motion)
with the maximum potential energy (total deflection Aw), it is possible to
relate the parameters vw and Aw

1

2
mwv

2
w =

Aw
∫

0

kxpw dx (2.12)

From this

mwv
2
w =

2k

p+1
Ap+1w (2.13)

The loss of energy during the collision is given by the Carnot theorem

∆E =
1−R2

2
mwv

2
w (2.14)

Taking into consideration (2.13)

∆E=
1−R2

p+1
kAp+1w (2.15)

On the other hand, the lost energy can be calculated from Eq. (2.3) as the
work of damping forces ∆L. Comparing it with ∆E given by Eq. (2.15), we
obtain

Ψ =2(1−R2) or R=

√

1−
Ψ

2
(2.16)

In the limiting cases: R = 1 for Ψ = 0 and R = 0 for Ψ = 2 (this value
results from the fact that during collision the total loss of energy occurs for
half of the hysteresis loop).
Equations (2.16) show that Newton’s hypothesis R = const is true when

the loss of energy arises from material damping mostly.



Phenomenon of force impulse restitution... 903

The above relationship allow one to determine the coefficient of restitu-

tion R on the base of rheological properties of bodies and enable finding an
equivalence between Newton’s ”time-free” approach with the description of col-

lision as a force interaction with a limited value.
Substituting (2.16)1 in (2.6), we finally obtain

F = kxpw

{

1−
1−R2

2
[1− sgn(ẋw)]

}

(2.17)

This approach is advantageous in computer simulation of collision.
If the loss of energy arises from other phenomena thanmaterial damping,

relationship (2.17) is still true if we put an adequate value of R.

3. Coefficient of restitution as a function of generalised density of

energy flux during collision

The occurence of plastic deformations, microslides, crushing of contamina-
tions, etc. cause that the coefficient of restitution, in general, exhibits strong
dependece onmass, velocity and shape of bodies in the region of the collision
point.
For overall consideration of the abovementioned factors, it is convenient to

use the idea of density of energy flux Φ during collision, whichwas introduced
by Bagrejev (1964) for central collinear collision

Φ=
mwv

2
w

2r3w
(3.1)

where
mw – reducedmass determined from equation (2.9)
vw – relative velocity prior to collision, vw = vi−vj
rw – reduced radius of curvature in the contact point

rw =
rirj
ri+rj

(3.2)

We will prove that it is possible to generalize the idea of density of energy
flux formulated for the process of central collision about the eccentric collision.
Let us first calculate the acceleration of points Oi, i = 1,2 of contact of

both bodies in the case of an eccentric collision – Fig.4.
This acceleration can be found from Euler’s equation

Ii
dωi
dt
+ωi× Iiωi =Mci+ri×F (3.3)



904 J. Michalczyk

Fig. 4.

where Ii is the tensor of inertia in the central reference frame Cixiyizi, i=1,2

Ii =







Ixi −Ixiyi −Ixizi
−Iyixi Iyi −Iyizi
−Izixi −Iziyi Izi






(3.4)

Hence
dωi
dt
= I−1i (Mci+ri×F −ωi× Iiωi) (3.5)

and the acceleration of point Oi

aoi =
F i+Fci
mi

+[I−1i (Mci+ri×F i−ωi× Iiωi)]×ri+ωi×(ωi×ri) (3.6)

where Fci, Mci are external forces and moments applied to mi. Neglecting
limited value terms in (3.6) we get

a0i =
F i

mi
+[I−1i (ri×F i)]×ri (3.7)

The projection of a0i against the direction of F i is then

ani =a0ini =
{F i

mi
+[I−1i (ri×F i)]×ri

}

ni =Fi
{ni

mi
+[I−1i (ri×ni)]×ri

}

ni

(3.8)
Hence, we can consider the term

mwi =
Fi
ani
=

1
{

ni
mi
+[I−1i (ri×ni)]×ri

}

ni

i=1,2 (3.9)

as the reducedmass of mi in the collision point.



Phenomenon of force impulse restitution... 905

Therefore, the reducedmass mw inBagrejev formula (3.1) canbeexpressed
for a general case of colliding solid bodies without friction as

mw =
mw1mw2
mw1+mw2

=
( 1

mw1
+
1

mw2

)

−1
=

(3.10)

=
( 1

m1
+
1

m2
+{[I−11 (r1×n1)]×r1}n1+{[I

−1
2 (r2×n2)]×r2}n2

)

−1

If xi, yi, zi are central principal axes of inertia of m1,m2, then

I
−1
i =







1/Ixi 0 0
0 1/Iyi 0
0 0 1/Izi






i=1,2 (3.11)

and

mw =
[ 1

m1
+
1

m2
+
1

Ix1
(z1On1j1−y1On1k1)

2+
1

Iy1
(z1On1i1−x1On1k1)

2+

+
1

Iz1
(y1On1i1−x1On1j1)

2+
1

Ix2
(z2On2j2−y2On2k2)

2+ (3.12)

+
1

Iy2
(z2On2i2−x2On2k2)

2+
1

Iz2
(y2On2i2−x2On2j2)

2
]

−1

where ii, ji, ki are the versors of reference frames, and xiO, yiO, ziO are
coordinates of the point O, i=1,2.
Anothermethod of calculation of Φ, more convinient if the impulse Π1 of

the collision force at the first stage of collision is known, may be derived by
transformation of the Bagrejev formula

Φ=
mwv

2
w

2r3w
=
vwmwvw
2r3w

=
vwΠ1
2r3w

(3.13)

where vw is now the normal component of relative velocity of collision points
O1 and O2 of m1 and m2, respectively (Michalczyk, 1984).

4. Random model of the coefficient of restitution

The strong dependency of R on many difficult to determine and variable
factors like microstructure of the surface, its local hardenings and contami-
nations, changes in the place and angle of collision, etc. results in that the



906 J. Michalczyk

deterministic model of the coefficient of restitution as a function of density of
energy flux R(Φ) does not give satisfactoty description of collision. It is espe-
cially important, e.g., in the investigation of stability problems of vibratory
machines.

On the other hand, the description of R as a random variable, e.g. Egłajs
(1971), leads to loss of functional dependence R on collision conditions.

We suggest to desrcibe R as a random function R(Φ), which enables one
to take into consideration both deterministic and random properties of R.

Laboratory works (Michalczyk, 1984) on properties of this function led to
the following conclusions:

• most often, the probability distribution of the randomvariable R, which
corresponds to Φ, can be considered as a normal cut outside the range
[0,1]

• due to rapid decay of the normalized autocorrelation function, which
corresponds to distant values of Φ, there is a practical possibility, to
approximate the random function R(Φ) as a process with independent
values.

Therefore, wemay describe the random function R(Φ) as a randomvaria-
ble R(Φ) with the probability density

f(R;R0(Φ),σ(Φ))=































0 for R< 0

exp
(

−
[R−R0(Φ)]

2

2σ2(Φ)

)

1
∫

0
exp
(

−
[R−R0(Φ)]2

2σ2(Φ)

)

dR

for R∈ [0,1]

0 for R> 1

(4.1)

where R0(Φ) and σ(Φ) are the expected value and standard deviation of the
non-cut distribution of the random variable R(Φ), respectively.

5. Conclusions

• The carried out analysis showed the rightness of Newton’s R = const
hypothesis about the restitution of impulses of instantaneous forces du-
ring collision when the loss of energy results frommaterial damping.

For such a case, relationships (2.16) between R and Ψ for a broad class
of non-linear characteristics of contact forces were determined.



Phenomenon of force impulse restitution... 907

• When energy dissipation is influenced by other factors, like stable defor-
mations, changes of the coeficient of restitution, R can be desribed as a
function of Bagrejev density of energy flux Φ.

It is shown in the paper that it is possible to generalize the idea of Φ
for eccentric collision, see Eqs. (3.10), (3.13) which allows one to use
experimental data obtained for central collision analysis.

• The rheological model based on Hertzian contact forces, which makes
it possible to obtain the desired value of R in simulation studies, is
proposed, see (2.17).

• It is possible to take into account the strong dependency of R on both
measure of collision intensity Φ and random factors, if the coefficent
of restitution is described as as a random function R(Φ). In practical
considerations, it is possible to present this function as a process of
idependentvalues andone-dimensionalGaussianprobabilitydistribution
cut outside the range [0,1].

References

1. Bagrejev V.V., 1964, Uprugo-plasticheskǐı udar massivnykh tel, Trudy MI-
ZhDT, 193

2. EgłajsV., 1971, Issledovanievibroudarnǒı sistemy so sluchǎınymkoefficentom
vostanovleniia,Voprosy Dinamiki i Prochnosti, Zinatije

3. Gryboś R., 1969, Teoria uderzenia w dyskretnych układach mechanicznych,
PWN,Warszawa

4. Gryboś R., 1971, Zależność maksymalnej siły uderzenia od współczynnika
restytucji,Mechanika Teoretyczna i Stosowana, 9, 1, 263-283

5. Harris C., Piersol A., 2002, Harris’ Shock and Vibration Handbook,
McGraw-Hill, NewYork

6. Michalczyk J., 1984,Wybrane zagadnieniadynamiki nadrezonansowychma-
szyn wibracyjnych, ZN AGH, 5, Kraków

7. Rayleigh J.W., 1906, On the production of vibrations by forces of relatively
long duration with application to the theory of collisions,Phil. Mag., 11



908 J. Michalczyk

Zjawisko restytucji impulsów sił chwilowych w modelowaniu zderzeń

Streszczenie

W pracy wykazano, że Newtonowska idea współczynnika restytucji R = const
jest słuszna w odniesieniu do zderzeń, w których strata energii powstaje głównie na
skutek tłumienia materiałowego.
Dla zderzeń o innym charakterze, dla których R zależy od gęstości strumienia

energii Φ, wskazano na możliwość rozszerzenia tego pojęcia na przypadek zderzenia
mimośrodowego.
W pracy wskazano również na celowość opisania R jako funkcji losowej parame-

tru Φ i podano formułę na siłę kontaktową zapewniającą uzyskanie założonej warto-
ści R podczas symulacji cyfrowej procesu zderzenia.

Manuscript received February 25, 2008; accepted for print May 19, 2008