r IÀI 0 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r 2
JC þ 4s 2
M n max
q
¼
P
pR 2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ 16f 2
ð
Þ csc 4 a À csc 2 a
p
ð11:68Þ
It can be seen from the above equation that csc a is a decreasing function and is
always greater than 1. Therefore, when the load P, inclination angle a and f are
determined, the stress at the edge of I À I
0 section decreases with the increase of a.
Point B and I À I
0 section edge points, under specific conditions who is the
dangerous point, should be determined according to the specific axial load value
and inclination a, and its change law can be obtained through the curve analysis.
When estimating the axial load and tangential load, the axial load P comes from
the impulse obtained when the piston impacts the drill bit. For convenience of
calculation, it is assumed that during the collision process, the drill bit is rigid body,
while the piston has elastic deformation and is completely elastic collision. The
piston length is L, the cross-sectional area is A, the mass is m, the elastic modulus is
E, and the initial impact velocity of the piston is v. From the piston just collided to
the velocity decreased to 0, the impact force increases linearly from 0 to Q. When
the velocity is 0, the piston has the maximum deformation DL, according to
Hooke’s law, there is
DL ¼
QL
EA
ð11:69Þ
The piston-to-bit collision is completely elastic, that is, the work done by the bit
reaction force is all converted into the deformation energy of the piston. Regardless
of heat, according to the law of conservation of energy, there is
1
2
Q Á DL ¼
1
2
mv
2
ð11:70Þ
From Eqs. (11.69) and (11.70), it is obtained that
Q ¼
ffiffiffiffiffiffiffiffiffi ffi
mEA
L
r
v
ð11:71Þ
The number of cylindrical teeth on large diameter DTH hammer is about 90, set
n ¼ 90.
According to the previous assumption, each cylindrical tooth is subjected to an
average force, so
P ¼
Q
n
ð11:72Þ
The piston mass m ¼ 248 kg, impact velocity v ¼ 7:1 m=s, cross-sectional area
A ¼ 0:615 m
2 , length L ¼ 0:6 m and E ¼ 2:1 Â 10
11 Pa of a DTH hammer
252
11 Pneumatic Down-the-Hole Hammer
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r 2
JC þ 4s 2
M n max
q
¼
P
pR 2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ 16f 2
ð
Þ csc 4 a À csc 2 a
p
ð11:68Þ
It can be seen from the above equation that csc a is a decreasing function and is
always greater than 1. Therefore, when the load P, inclination angle a and f are
determined, the stress at the edge of I À I
0 section decreases with the increase of a.
Point B and I À I
0 section edge points, under specific conditions who is the
dangerous point, should be determined according to the specific axial load value
and inclination a, and its change law can be obtained through the curve analysis.
When estimating the axial load and tangential load, the axial load P comes from
the impulse obtained when the piston impacts the drill bit. For convenience of
calculation, it is assumed that during the collision process, the drill bit is rigid body,
while the piston has elastic deformation and is completely elastic collision. The
piston length is L, the cross-sectional area is A, the mass is m, the elastic modulus is
E, and the initial impact velocity of the piston is v. From the piston just collided to
the velocity decreased to 0, the impact force increases linearly from 0 to Q. When
the velocity is 0, the piston has the maximum deformation DL, according to
Hooke’s law, there is
DL ¼
QL
EA
ð11:69Þ
The piston-to-bit collision is completely elastic, that is, the work done by the bit
reaction force is all converted into the deformation energy of the piston. Regardless
of heat, according to the law of conservation of energy, there is
1
2
Q Á DL ¼
1
2
mv
2
ð11:70Þ
From Eqs. (11.69) and (11.70), it is obtained that
Q ¼
ffiffiffiffiffiffiffiffiffi ffi
mEA
L
r
v
ð11:71Þ
The number of cylindrical teeth on large diameter DTH hammer is about 90, set
n ¼ 90.
According to the previous assumption, each cylindrical tooth is subjected to an
average force, so
P ¼
Q
n
ð11:72Þ
The piston mass m ¼ 248 kg, impact velocity v ¼ 7:1 m=s, cross-sectional area
A ¼ 0:615 m
2 , length L ¼ 0:6 m and E ¼ 2:1 Â 10
11 Pa of a DTH hammer
252
11 Pneumatic Down-the-Hole Hammer