Environmental Engineering Reference
In-Depth Information
- radius of the contact surface:
1/ 3
⎛
⎞
3
4
π
rr
(
12
12
12
)
a
=
P k
+
k
[4.38]
⎜
⎟
rr
+
⎝
⎠
2
2
1
−
v
1
−
v
1
2
with:
k
=
k
=
1
2
π
E
π
E
1
2
- relative displacement of the center of the two spheres:
3
4
π
P
(
)
α=+
kk
[4.39]
12
a
It was demonstrated by Lord Rayleigh [TIM 70] that for two spheres in motion,
the duration of the shock is very long in comparison with the vibrations due to the
eigenfrequency of the spheres. It could therefore be considered that the static
equations are applicable during the shock for masses
m
1
and
m
2
and an impact
velocity
v
:
- maximum displacement:
2/5
2
⎡
⎤
⎛
⎞
5
4
v
mm
12
1
α
=
⎢
[4.40]
⎥
⎜
⎟
κ
mm
+
⎢
⎝
⎠
⎥
⎣
⎦
2
1/ 2
⎛
⎞
16
rr
rr
12
⎜
⎟
with:
κ
=
⎜
2
2
+
⎟
(
)
9
π
kk
+
12
⎝
⎠
12
- maximum force:
3/5
2
⎡
⎤
⎛
⎞
5
4
v
mm
3/2
12
1
P
=
κα
=
⎢
κ
[4.41]
⎥
⎜
⎟
κ
mm
+
⎢
⎝
⎠
⎥
⎣
⎦
2
When these formulae are applied to boulder falls [DES 96], the following
conditions are modified:
- boulder:
mmrrE E
=
=
>>
from which
k
=
0;
1
1
1
2
1
2
1
−
v
- terrain:
mr
,
→∞
k
= =
π
k
.
22
2
E
