Notes_08_05
Multiplanar Balancing
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Y
Y
X
Z
plane
for A
plane
for B
in-plane primary balancing (static balancing)
FIMB _ i  m i ri 2 cos i
X
FBAL _ A  m A rA 2 cos  A
X
FBAL _ i  m B rB 2 cos  B
X
FIMB _ i  m i ri 2 sin i
Y
FBAL _ A  m A rA 2 sin  A
Y
FIMB _ B  m B rBi 2 sin  B
Y
forces in X
 mi ri cos i   m A rA cos A  mB rB cos B  0
Eq. 1
forces in Y
 mi ri sin i   m A rA sin A  m B rB sin B  0
Eq. 2
secondary dynamic balancing to prevent shaking moments
M IMB _ i   FIMB _ i a i   m i ri z i  2 cos  i
Y
X
M IMB _ i  FIMB _ i a i    m i ri z i  2 sin  i
X
Y
moments in Y
 mi ri z i cos i   mA rA z A cos A  0
zB  0
Eq. 3
moments in X
 mi ri z i sin i   m A rA z A sin A  0
zB  0
Eq. 4
Notes_08_05
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Notes_08_05
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m A rA z A sin A   mi ri z i sin i 
m A rA z A cos A   mi ri z i cos i 
tan  A 
 m i ri z i sin  i 
 m i ri z i cos  i 
tan A  tan i
for only one imbalance
m A
rA z A sin  A   m A rA z A cos  A    m i ri z i sin  i    m i ri z i cos  i 
m A
rA z A    m i ri z i sin  i    m i ri z i cos  i 
m A
2
2
2
rA  
2
2
 m i
2
ri z i sin  i    m i ri z i cos  i 
for only one imbalance
2
2
zA
m A rA   m i ri  z i
zA
m B rB sin B   mi ri sin i   m A rA sin A
m B rB cos B   mi ri cos i   mA rA cos A
tan  B 
 m i ri sin i   m A rA sin  A
 m i ri cos i   m A rA cos  A
m B rB    mi ri sin i   m A rA sin A 2   mi ri cos i   m A rA cos A 2
2