which gives:
E ¼
1
2
_
z
2
þ VðzÞ ) _
E ¼ € z þ
dV
dz
_
z;
ð3:305Þ
or, substituting V from (3.300) and inserting into (3.296):
_
E ¼ Àh 1 _
z
2
À h 2 _
z
3 M z
ð Þ À h 3 _
z
4 for z 6 ¼
p
2
þ jp; j ¼ 0; 1; . . .;
E þ À E À ¼ À 1 À R
2
À
Á
E À À
p
2
8
for z ¼
p
2
þ jp:
ð3:306Þ
Dividing the first equation with _
z and using (3.304), we obtain an equation where
time is eliminated, while instead z takes the role of the independent variable:
dE
dz
¼ À h 1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 E À VðzÞ
ð
Þ
p
À 2 h 2 E À VðzÞ
ð
Þ M z
ð Þ
À h 3 2 E À VðzÞ
ð
Þ
ð
Þ
3=2
for z 6 ¼
p
2
þ jp;
E þ À E À ¼ À 1 À R
2
À
Á
E À À
p
2
8
for z ¼
p
2
þ jp:
ð3:307Þ
Under assumptions (3.299) the system has the general form (3.258), and can thus
be averaged using (3.259), giving:
dE 1
dz
¼ qðE 1 Þ;
ð3:308Þ
where
qðE 1 Þ ¼ À h 1
1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2E 1 À
p 2
4
r
þ
2
p
E 1 arcsin
p
2
ffiffiffiffiffiffiffi ffi
2E 1
p
(
)
À
1 À R
2
p
E 1 À
p
2
8
À h 3
2E 1 À
p
2
4
3=2
þ
3
4
E 1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2E 1 À
p 2
4
r
þ
3
p
E
2
1 arcsin
p
2
ffiffiffiffiffiffiffi ffi
2E 1
p
(
)
:
ð3:309Þ
Stationary solutions to (3.308) determines periodic oscillations of the mass on
the belt with two impacts per oscillation period. The physical meaning can be
illustrated by considering the case h 3 ¼ h 3 ¼ 0 (which implies h 2 ¼ h 2 ¼ 0), i.e.
with friction monotonically decreasing with increased interface velocity. In that
3.9 Vibro-Impact Analysis Using Discontinuous Transformations
195