m
d
2 y r
dt 2 ¼ p HA
p
4
D
2
À d
2
À
Á À p A0
V A0
V A0 À
p
4 D 2 y r
n p
4
D
2
À mg
ð12:11Þ
Simultaneous solving Eqs. (12.10) and (12.11), the dynamic model of the rising
stage is obtained as follows:
Q p t 1 ¼
Z
C d A R x
j s þ 2P HA0 C d xcosR
p HA0 À p c
ð
Þ
ffiffiffiffiffiffiffiffiffiffiffiffi
2p HA0
q
s
dt
þ
p HA0
p HA
1
n À1
"
#
V HA0 þ
p
4
D
2
À d
2
À
Á
y r
m
d
2 y r
dt 2 ¼ p HA
p
4
D
2
À d
2
À
Á À p A0
V A0
V A0 À
p
4 D 2 y r
n p
4
D
2
À mg
ð12:12Þ
12.2.1.3 Dynamics Model of Descending Process
In the descent stage, the hammer core accelerates to descend under the double
action of compressed nitrogen and gravity. The dynamic equation is as follows:
m
d
2 y f
dt 2 ¼ mg þ p A1
p
4
D
2
À Bv À p LA
p
4
D
2
À d
2
À
Á
ð12:13Þ
where
p A1 Pressure of a certain working moment in nitrogen chamber;
p LA Pressure of low-pressure accumulator at a certain working moment;
y f
Descending displacement of hammer core.
The ideal gas equations of the nitrogen chamber are
p A1 V
n
A1 ¼ p A0 V
n
A0
ð12:14Þ
V A1 ¼ V A0 À DV
ð12:15Þ
DV ¼
p
4
D
2 H À y f
À
Á
ð12:16Þ
where H—Hammer stroke.
Substituting Eqs. (12.15) and (12.16) into Eq. (12.14), the pressure of nitrogen
chamber in a certain working moment is
p A1 ¼ p A0
V A0
V A1 À
p
4 D 2 H À y f
À
Á
"
# n
ð12:17Þ
282
12 Pneumatic–Hydraulic Pile Driving Hammer