63
Chapter 5
Coefficient of Restitution
The results of the eld studies in Chapter 2 and the discussion of impact mechanics in
Chapter 4 illustrate how rock fall behavior can be dened by values for the normal and
tangential coefcients of restitution (e
N
and e
T
) that are applicable to the particular site con-
ditions. The coefcients of restitution quantify velocity changes during impact and help in
understanding how conditions at the impact points inuence rock fall behavior. Figure5.1
shows three impacts and two trajectories and clearly illustrates the reduced velocity and
height of the second trajectory due to the loss of energy at the rst impact, that is, e
N
< 1.
Figure5.1 shows the typical behavior of a rubber ball; rock falls will always have lower
trajectories than these because of the relatively low e
N
of rock.
Changes in the velocity components during impact and the corresponding coefcients of
restitution are also illustrated in Figure5.2 on the (normal impulse, p
N
-relative velocity, v)
diagram. The normal coefcient of restitution denes the change in normal velocity dur-
ing impact and is related to the impact angle as discussed in this chapter and the inelastic
compression characteristics of the slope materials. The tangential coefcient of restitution
denes the reduction in tangential velocity during impact and is related to the friction force
generated between the slope and the body. This chapter discusses how these coefcients are
correlated with impact conditions and are not purely material properties.
This chapter summarizes the results of the e
N
and e
T
values obtained from the ve eld
studies described in Chapter 2, encompassing four different slope materials and 57 impacts.
A sixth location is a laboratory test where blocks of rock were dropped on to a concrete
oor and the rebound heights were measured to give values for e
N
.
5.1 NEWTONS COEFFICIENT OF RESTITUTION
The concept of the coefcient of restitution, which in this case was the normal coefcient,
was rst developed by Isaac Newton (1686) who suspended spheres of the same material on
pendulums and measured how high they rebounded after impact (Figure5.3); the measure-
ments incorporated corrections for velocity loses due to air friction. Values for the coef-
cient of restitution found by Newton included 0.56 for steel and 0.94 for glass. One of the
purposes of the experiments was to prove the third law of motion: every action has an equal
and opposite reaction.
It was assumed at the time of Newton’s experiments that the coefcients of restitution
were material properties. However, it is now understood that impact between rough, rotat-
ing bodies of different materials, such as rock falls, reductions in velocity depend not only
on the material forming the body but also on the impact conditions such as mass and shape
of the impacting body, and the impact angle and velocity.
64 Rock Fall Engineering
E
R
S
T
V
GCD
A
B
FII
Figure 5.3 Isaac Newton’s measurement of normal coefcient of restitution using impact of spheres sus-
pended on pendulums.
Figure 5.1 Impacts between successive trajectories showing typical inelastic behavior and loss of energy
during impact where second trajectory (on right) is lower than rst trajectory; rock falls will
always have lower trajectories than those shown. (From Micheal Maggs, Wikimedia Commons.)
Velocity, v
Normal
Impulse, p
N
v
iT
v
iN
v
fT
compre ssionrestitution
v
i
v
fN
v
f
v
iT
v
iT
v
iN
v
iN
p
cN
p
fN
+
T
+
N
Figure 5.2 Normal impulse-relative velocity plot showing relationships between changes in normal (N) and
tangential (T) velocity components and coefcients of restitution –e
N
= (v
fn
/v
iN
) and e
T
= (v
fT
/v
iT
).

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