1 tmr4225 marine operations, 2009.01.20 part 2 lecture content: –linear submarine/auv motion...
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1
TMR4225 Marine Operations, 2009.01.20 Part 2
• Lecture content:– Linear submarine/AUV motion equations
– AUV hydrodynamics – Hugin operational experience
2
Motion equations
• How would you write the motion equations for a submarine?
3
Motion equations
• How would you write the motion equations for a submarine?
– Integro-differensial equations• Unified theory
– Quasi-static non-linear differential equations• 6 degrees of freedom
• Second order non-linear forces
• 3rd order non-linear forces
• Mixed non-linearities
4
Submarine and AUV motion equations
• Motion equations are usually written as
• 6 degrees of freedom equations
• Time domain formulation
• Simplified sets of linear equations can be used for stability investigations – stick-fixed stability
5
A possible calm water simulation model for submarines and
AUVs:
Based on M. Gertler and G.R. Hagen model (published 1967) with modifications as reported by J.P. Feldman 1975
Non-linear damping represented by the cross-flow drag principle
Calculation of effect of vortices from bridge fin on stern control planes (function of lift on bridge fin)
In addition: Separate subroutines for control planes hydrodynamics and propulsion
6
Surge equation:
sin)('''''2
''2
)''()''(2
'')'''()'''()'''(2
)''''(2
)''(2
22
222
3
224
43
BWXXXXXUl
wXvXl
wqmXvrXml
pqymprzmXrXxmqXxml
rymqzmluXml
bpsprudprres
wwvv
wqvr
GGrprrGqqG
GGu
7
Sway equation:
sincos)(''''2
))(()(2
)()()()()(2
'2
')''()''(2
''''''||')'''(2
)'''()'''(2
)''(2
22
2/122
2
3
22||
4
43
1
2
BWYYYYUl
dxxtvxwCl
dxxvxwxvxhxC
uvYl
upYurYmwpYml
qrzmpymrymppYpqxmYl
rYxmpYzmlvYml
bpsprudpr
x
x
FWL
l
d
v
prwp
GGGppGpq
rGpGv
8
Heave equation:
coscos)(''''2
))(()(2
)()()()()(2
''2
)''()''(2
)''''(''''2
'')'''(2
)''(2
22
2/122
20
2
3
224
43
1
2
BWZZZZUl
dxxtvxvCl
dxxvxwxwxbxC
uwZuZl
vpmZuqZml
rpymxmqzmpzml
pymqZxmlwZml
bpsprudpr
x
x
FWL
l
d
ww
vpq
GGGG
GqGw
9
Roll equation:
sincos)(coscos)(
''''2
))(()('2
'2
'''')'''()'''('2
||''''')'''(2
)'''(''2
)''(')''(2
22
12
3
4
||225
45
1
2
BzWzByWy
KKKKUl
dxxtvxwCzl
uvKl
vpymuqymwpzmKurKzmupKl
ppKprIqIrIpqIqrIIKl
vKzmwymlrKIqIpKIl
BGBG
bpsprudpr
x
x
FWL
v
GGGwprGp
ppxyyzyzzxyzqr
vGGrzxxypx
10
Pitch equation:
sin)(coscos)(
''''2
))(()(2
)()()()()(2
'2
'''''')'''(2
'''')'''(2
)'''(''2
'')''(2
22
2
122
3
4
225
45
1
2
BzWzBxWx
MMMMUl
dxxtvxxvCl
dxxvxwxwxxbxC
uwMl
vpxmwpzmvrzmuqxmMl
qpIrIpIqrIrpMIIl
wMxmuzmlrIpIqMIl
BGBG
bpsprudpr
x
x
FWL
l
d
w
GGGGq
yzzxzxxyrpxz
wGGyzxyqy
11
Yaw equation:
sin)(sincos)(
''''2
))(()(2
)()()()()(2
'2
''''')'''(''2
'''')''(2
)'''(2
')''()''(2
22
2
122
3
4
225
45
1
2
ByWyBxWx
NNNNUl
dxxtvxxwCl
dxxvxwxvxxhxC
uvNl
upNwqymvrymurNxmwpxml
rqIpIqIrpIpqNIIl
umyvNxmlqIpNIrNIl
BGBG
bpsprudpr
x
x
FWL
l
d
v
pGGrGG
zxxyxyyzpqyx
GvGyzpzxrz
12
Cross flow drag (sway):
dxxvxwxvxhxCl
d 2/122 )()()()()(
2
Cd(x) Local cross-flow drag coefficient.
h(x) Local height of hull at location x.
v(x) Local velocity in y-direction, equals v+xr.
w(x) Local velocity in z-direction, equals w-xq.
13
Effect of bridge fin on hull aft of it and on control planes:
1
2
))(()(2
x
x
FWL dxxtvxwCl
14
Methods for estimating forces/moments
• Theoretical models– Potential flow, 2D/3D models– Lifting line/lifting surface– Viscous flow, Navier-Stokes equations
• Experiments– Towing tests (resistance, control forces, propulsion)– Oblique towing (lift of body alone, body and rudders)– Submerged Planar Motion Mechanism– Cavitation tunnel tests (resistance, propulsion, lift)– Free swimming
15
Methods for estimating forces/moments
• Empirical models– Regression analysis based on previous experimental results using
AUV geometry as variables
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EUCLID Submarine project
•MARINTEK takes part in a four years multinational R&D programme on testing and simulation of submarines, Euclid
NATO project “Submarine Motions in Confined Waters”. •Study topic:
•Non-linear hydrodynamic effects due to steep waves in
shallow water and interaction with nearby boundaries.
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Force measurement system - submarine
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Testing the EUCLID submarine in waves
• Model fixed to 6 DOF force transducer
• Constant speed
• Regular waves
• Submarine close to the surface
19
Numerical study of bow plane vortex
Streamlines released at bow plane for 10 deg bow plane angle (Illustration: CFDnorway)
Streamlines released at bow plane for -10 deg bow plane angle (Illustration CFDnorway)
20
Initial simulation of turning circles with linear model
Basis for initial simulations:
Mass and moments of inertia from QinetiQ’s report. Hydrodynamic mass and moment of inertia from potential theory panel code
WAMIT. Linear and most important cross coupling coefficient from model tests, CFD
calculations and estimation. Only sway and yaw equations. Only geometric rudder angle applied in subroutine for rudder effect. Propeller net thrust equals resistance – u constant. Simulations made for rudder angles 5, 10 and 15 degrees.
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Linear motion equations
• Linear equations can only be used when– The vehicle is dynamically stable for motions in horisontal and
vertical planes
– The motion is described as small perturbations around a stable motion, either horizontally or vertically
– Small deflections of control planes (rudders)
– For symmetric bodies the 6 DOF equations can be split in two sets of motion equations
• 2 DOF coupled heave and pitch
• 3 DOF sway, yaw and roll
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Dynamic stability
• Characteristic equation for linear coupled heave - pitch motion:– ( A*D**3 + B*D**2 + C*D + E) θ = 0
• Dynamic stability criteria is:– A > 0, B > 0 , BC – AE > 0 and E>0
• Found by using Routh’s method
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Dynamic stability (cont)
• For horisontal motion the equation (2.15) can be used if roll motion is neglected
• The result is a set of two linear differential equations with constant coefficients
• Transform these equations to a second order equation for yaw speed
• Check if the roots of the characteristic equation have negative real parts
• If so, the vehicle is dynamically stable for horisontal motion