bölüm 2 6 dynamic force analysis spatial
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1
ME 302 DYNAMICS OF ME 302 DYNAMICS OF MACHINERYMACHINERY
Dynamic Force Analysis VI
Dr. Sadettin KAPUCU
© 2007 Sadettin Kapucu
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2Gaziantep University
Kinetics of a Rigid BodyKinetics of a Rigid BodyEEquation quation oof f MMotion (otion (EOMEOM))
x
z
y
CG
r
dt
LdF
dt
Vmd CG )(
dt
Vdm CG
dt
HdM CG
dt
dmrxxrd
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3Gaziantep University
Kinetics of a Rigid BodyKinetics of a Rigid BodyEEquation quation oof f MMotion (otion (EOMEOM))
x
z
y
CG
r
dt
LdF
dt
Vmd CG )(
dt
Vdm CG
CzyxCG VxkVjViVdt
Vd
zyx
zyx
VVV
kji
iVV yzzy
jVV xzzx
kVV xyyx
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4Gaziantep University
Kinetics of a Rigid BodyKinetics of a Rigid BodyEEquation quation oof f MMotion (otion (EOMEOM))
x
z
y
CG
r
dt
LdF
dt
Vmd CG )(
dt
Vdm CG
CzyxCG VxkVjViVdt
Vd
zyx
zyx
VVV
kji
iVV yzzy
jVV xzzx
kVV xyyx
yzzyxx VVVmF
zxxzyy VVVmF
xyyxzz VVVmF
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5Gaziantep University
Kinetics of a Rigid BodyKinetics of a Rigid BodyEEquation quation oof f MMotion (otion (EOMEOM))
x
z
y
CG
r
dt
HdM CG
dt
dmrxxrd
yzzyxx VVVmF
zxxzyy VVVmF
xyyxzz VVVmF
zyx
zyx
rrr
kji
irr yzzy
jrr xzzx
krr xyyx
rx
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6Gaziantep University
Kinetics of a Rigid BodyKinetics of a Rigid BodyEEquation quation oof f MMotion (otion (EOMEOM))
x
z
y
CG
r
dt
HdM CG
dt
dmrxxrd
yzzyxx VVVmF
zxxzyy VVVmF
xyyxzz VVVmF
xyyxzxxzyzzy
zyx
rrrrrr
rrr
kji
rxxr
irr yzzy
jrr xzzx
krr xyyx
rx
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7Gaziantep University
Kinetics of a Rigid BodyKinetics of a Rigid BodyEEquation quation oof f MMotion (otion (EOMEOM))
x
z
y
CG
r
dt
HdM CG
dt
dmrxxrd
yzzyxx VVVmF
zxxzyy VVVmF
xyyxzz VVVmF
xyyxzxxzyzzy
zyx
rrrrrr
rrr
kji
rxxr
irrrrrr zxxzzxyyxy
jrrrrrr yzzyzxyyxx
krrrrrr yzzyyzxxzx
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8Gaziantep University
Kinetics of a Rigid BodyKinetics of a Rigid BodyEEquation quation oof f MMotion (otion (EOMEOM))
x
z
y
CG
r
dt
HdM CG
dt
dmrxxrd
yzzyxx VVVmF
zxxzyy VVVmF
xyyxzz VVVmF
dmrxxr dmirrrrrr zxxzzxyyxy
dmjrrrrrr yzzyzxyyxx
dmkrrrrrr yzzyyzxxzx
H
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9Gaziantep University
Kinetics of a Rigid BodyKinetics of a Rigid BodyEEquation quation oof f MMotion (otion (EOMEOM))
x
z
y
CG
r
dt
HdM CG
dt
dmrxxrd
yzzyxx VVVmF
zxxzyy VVVmF
xyyxzz VVVmF
dmrrrrrr
dmrrrrrr
dmrrrrrr
H
H
H
H
yzzyyzxxzx
yzzyzxyyxx
zxxzzxyyxy
z
y
x
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10Gaziantep University
Kinetics of a Rigid BodyKinetics of a Rigid BodyEEquation quation oof f MMotion (otion (EOMEOM))
x
z
y
CG
r
dt
HdM CG
dt
dmrxxrd
yzzyxx VVVmF
zxxzyy VVVmF
xyyxzz VVVmF
dmrrrrrr
dmrrrrrr
dmrrrrrr
H
H
H
H
yzzyyzxxzx
yzzyzxyyxx
zxxzzxyyxy
z
y
x
dmrrrrrr
dmrrrrrr
dmrrrrrr
H
H
H
H
yzyzyxzxxz
zyzzyxyxyx
zxzxzyxyyx
z
y
x
22
22
22
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11Gaziantep University
Kinetics of a Rigid BodyKinetics of a Rigid BodyEEquation quation oof f MMotion (otion (EOMEOM))
x
z
y
CG
r
dt
HdM CG
dt
dmrxxrd
yzzyxx VVVmF
zxxzyy VVVmF
xyyxzz VVVmF
dmrrrrdmrr
dmrrdmrrdmrr
dmrrdmrrdmrr
H
H
H
H
yxzyzyxzx
zyzzxyxyx
zxzyxyzyx
z
y
x
22
22
22
dmrrrrrr
dmrrrrrr
dmrrrrrr
H
H
H
H
yzyzyxzxxz
zyzzyxyxyx
zxzxzyxyyx
z
y
x
22
22
22
z
y
x
yxyzxz
zyzxxy
zxyxzy
z
y
x
dmrrdmrrdmrr
dmrrdmrrdmrr
dmrrdmrrdmrr
H
H
H
H
22
22
22
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12Gaziantep University
Kinetics of a Rigid BodyKinetics of a Rigid BodyEEquation quation oof f MMotion (otion (EOMEOM))
x
z
y
CG
r
dt
HdM CG
dt
dmrxxrd
yzzyxx VVVmF
zxxzyy VVVmF
xyyxzz VVVmF
dmrrrrdmrr
dmrrdmrrdmrr
dmrrdmrrdmrr
H
H
H
H
yxzyzyxzx
zyzzxyxyx
zxzyxyzyx
z
y
x
22
22
22
z
y
x
yxyzxz
zyzxxy
zxyxzy
z
y
x
dmrrdmrrdmrr
dmrrdmrrdmrr
dmrrdmrrdmrr
H
H
H
H
22
22
22
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13Gaziantep University
Kinetics of a Rigid BodyKinetics of a Rigid BodyEEquation quation oof f MMotion (otion (EOMEOM))
x
z
y
CG
r
dt
HdM CG
dt
dmrxxrd
yzzyxx VVVmF
zxxzyy VVVmF
xyyxzz VVVmF
z
y
x
yxyzxz
zyzxxy
zxyxzy
z
y
x
dmrrdmrrdmrr
dmrrdmrrdmrr
dmrrdmrrdmrr
H
H
H
H
22
22
22
z
y
x
zzzyzx
yzyyyx
xzxyxx
z
y
x
III
III
III
H
H
H
H
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14Gaziantep University
Kinetics of a Rigid BodyKinetics of a Rigid BodyEEquation quation oof f MMotion (otion (EOMEOM))
x
z
y
CG
r
dt
HdM CG
dt
dmrxxrd
yzzyxx VVVmF
zxxzyy VVVmF
xyyxzz VVVmF
z
y
x
zzzyzx
yzyyyx
xyxyxx
z
y
x
III
III
III
H
H
H
H
HxkHjHiHdt
Hdzyx
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15Gaziantep University
Kinetics of a Rigid BodyKinetics of a Rigid BodyEEquation quation oof f MMotion (otion (EOMEOM))
x
z
y
CG
r
dt
HdM CG
dt
dmrxxrd
z
y
x
zzzyzx
yzyyyx
xyxyxx
z
y
x
III
III
III
H
H
H
H
HxkHjHiHdt
Hdzyx
zyx
zyx
HHH
kji
iHH yzzy
jHH xzzx
kHH xyyx
Hx
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16Gaziantep University
Kinetics of a Rigid BodyKinetics of a Rigid BodyEEquation quation oof f MMotion (otion (EOMEOM))
x
z
y
CG
r
z
y
x
zzzyzx
yzyyyx
xyxyxx
z
y
x
III
III
III
H
H
H
H
HxkHjHiHdt
Hdzyx
kHHjHHiHHHx xyyxxzzxyzzy
22xyyzzyyyzzyxzxzzxyxyxxxx IIIIIIM
22xzxzzxzzxxyxzyzyyyzyxxyy IIIIIIM
22yxxyyxxxyyzzzzxyyzzyxxzz IIIIIIM
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17Gaziantep University
Kinetics of a Rigid BodyKinetics of a Rigid BodyEEquation quation oof f MMotion (otion (EOMEOM))
x
z
y
CG
r yzzyxx VVVmF
zxxzyy VVVmF
xyyxzz VVVmF
22xyyzzyyyzzyxzxzzxyxyxxxx IIIIIIM
22xzxzzxzzxxyxzyzyyyzyxxyy IIIIIIM
22yxxyyxxxyyzzzzxyyzzyxxzx IIIIIIM
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18Gaziantep University
Kinetics of a Rigid BodyKinetics of a Rigid BodyEEquation quation oof f MMotion (otion (EOMEOM))
x
z
y
CG
r
zyyyzzxxxx IIIM
zxzzxxyyyy IIIM
yxxxyyzzzz IIIM
If x,y,z are chosen such that they are principal axes than product of mass moment of inertia vanishes.
0 yzxzxy III
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19Gaziantep University
Gyroscopic Action in MachinesGyroscopic Action in Machines
zyyyzzxxxx IIIM
zxzzxxyyyy IIIM
yxxxyyzzzz IIIM
x
y
zp
s
Body coordinate frame
y
z
Instantaneous body coordinate frame
ts
2
2
1mrI xx 2
4
1mrII zzyy
ps
sx sinpy
y
zz
y
cospz
Magnitudes of the ws and wp are constant
0x
coscos sppy
sinsin sppz
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20Gaziantep University sin21 2spz mrM
cos21 2spy mrM
0 xM
Gyroscopic Action in MachinesGyroscopic Action in Machines
x
y
zp
s
Body coordinate frame
y
z
Instantaneous body coordinate frame
ts
2
2
1mrI xx 2
4
1mrII zzyy
ps
sx sinpy
y
zz
y
cospz
Magnitudes of the ws and wp are constant
0x
coscos sppy
sinsin sppz
zyyyzzxxxx IIIM
zxzzxxyyyy IIIM
yxxxyyzzzz IIIM
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21Gaziantep University
sin21 2spz mrM
cos21 2spy mrM
0 xM
Gyroscopic Action in MachinesGyroscopic Action in Machines
x y
zp
sy
z
y
zz
y
spmr 221
M
p
sPrecession
Spin
M
o90
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22Gaziantep University
Gyroscopic Action in MachinesGyroscopic Action in Machinesp
s
tHH p *
s
M
H
Angular momentum
smrH 221
p
tpH
pspspot
mrIHt
H 2
2
1*lim
MH
M
Active gyroscopic couple
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23Gaziantep University
Gyroscopic Action in MachinesGyroscopic Action in MachinesExample 1Example 1 p
s
NmIM ps 2.9284.6*7.104*014.0
A uniform disc of 150 mm diameter has a mass of 5 kg. It is mounted centrally in bearings which maintain its axle in a horizontal plane. The disc spins about it axle with a constant speed of 1000 rpm. while the axle precesses uniformly about the vertical at 60 rpm. The directions of rotation are as shown in Figure. If the distance between the bearings is 100 mm, find the resultant reaction at each bearing due to the mass and gyroscopic effects.
x
z
y
Given:
mxCCWsradorrpmN
CCWsradorrpmNmrormmd
pp
s
1.0,/28.660
,/7.1071000,075.0150
We know that mass moment of inertia of the disc, about an axis through its centre of gravity and perpendicular to the plane of disc,
222 014.02/075.0*52/ kgmmrI Gvroscopic couple acting on the disc
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24Gaziantep University
Gyroscopic Action in MachinesGyroscopic Action in MachinesExample 1Example 1 p
s
axiszalongNmM 2.9
A uniform disc of 150 mm diameter has a mass of 5 kg. It is mounted centrally in bearings which maintain its axle in a horizontal plane. The disc spins about it axle with a constant speed of 1000 rpm. while the axle precesses uniformly about the vertical at 60 rpm. The directions of rotation are as shown in Figure. If the distance between the bearings is 100 mm, find the resultant reaction at each bearing due to the mass and gyroscopic effects.
x
z
y
Given:
mxCCWsradorrpmN
CCWsradorrpmNmrormmd
pp
s
1.0,/28.660
,/7.1071000,075.0150
2014.0 kgmI
The direction of the reactive gyroscopic couple is shown in Figure.
NxMF 921.0/2.9/ pLet F be the force at each bearing due to the gyroscopic couple.
s
z
o90
M
The force F will act in opposite directions at the bearings as shown in Figure.
F F
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25Gaziantep University
Gyroscopic Action in MachinesGyroscopic Action in MachinesExample 1Example 1 p
s
axiszalongNmM 2.9
A uniform disc of 150 mm diameter has a mass of 5 kg. It is mounted centrally in bearings which maintain its axle in a horizontal plane. The disc spins about it axle with a constant speed of 1000 rpm. while the axle precesses uniformly about the vertical at 60 rpm. The directions of rotation are as shown in Figure. If the distance between the bearings is 100 mm, find the resultant reaction at each bearing due to the mass and gyroscopic effects.
x
z
y
Given:
mxCCWsradorrpmN
CCWsradorrpmNmrormmd
pp
s
1.0,/28.660
,/7.1071000,075.0150
2014.0 kgmI NF 92
F F
Now let RAw and RBw be the reaction at the bearing A and B respectively due to the weight of the disc. Since the disc is mounted centrally in bearings, therefore.
NkgRR BWAW 5.2481.9*5.25.22/5
W BWRAWR
Total reaction at the bearings, therefore.
)(5.675.2492
)(5.1165.2492
downwardsNRFR
upwardsNRFR
BWB
AWA
Gyroscopic action alters the bearing forces.
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26Gaziantep University
Gyroscopic Action in MachinesGyroscopic Action in MachinesExample 2Example 2
A simple configuration of olive crusher is shown in figure. Radius of the crusher disc is r. Mass of the disc is m. Whole system is rotating with an angular velocity of w. Assume that there is no slip between the ground and the disc. What would be the crushing force?
rR s **
If there is no slip between the ground and disc than
R
r
r
Rs
*
s
p Precessional velocity simply will be
The disc moment of inertia will be2
2
1r
g
WI
p
M
Gyroscopic moment must be act on the system in this direction
The crushing force F is shown in the figure
F
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27Gaziantep University
Gyroscopic Action in MachinesGyroscopic Action in MachinesExample 2Example 2
A simple configuration of olive crusher is shown in figure. Radius of the crusher disc is r. Mass of the disc is m. Whole system is rotating with an angular velocity of w. Assume that there is no slip between the ground and the disc. What would be the crushing force?
R
rs
WRFRrg
Wps 2
2
1
p
M
The moment about O can be written as F
O
psrg
WWF 2
2
1
g
rWF
22
1