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TMR4225 Marine Operations, 2009.01.20 Part 2

• Lecture content:– Linear submarine/AUV motion equations

– AUV hydrodynamics – Hugin operational experience

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Motion equations

• How would you write the motion equations for a submarine?

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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

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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

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Surge equation:

sin)('''''2

''2

)''()''(2

'')'''()'''()'''(2

)''''(2

)''(2

22

222

3

224

43

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Sway equation:

sincos)(''''2

))(()(2

)()()()()(2

'2

')''()''(2

''''''||')'''(2

)'''()'''(2

)''(2

22

2/122

2

3

22||

4

43

1

2

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Heave equation:

coscos)(''''2

))(()(2

)()()()()(2

''2

)''()''(2

)''''(''''2

'')'''(2

)''(2

22

2/122

20

2

3

224

43

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2

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Roll equation:

sincos)(coscos)(

''''2

))(()('2

'2

'''')'''()'''('2

||''''')'''(2

)'''(''2

)''(')''(2

22

12

3

4

||225

45

1

2

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Pitch equation:

sin)(coscos)(

''''2

))(()(2

)()()()()(2

'2

'''''')'''(2

'''')'''(2

)'''(''2

'')''(2

22

2

122

3

4

225

45

1

2

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Yaw equation:

sin)(sincos)(

''''2

))(()(2

)()()()()(2

'2

''''')'''(''2

'''')''(2

)'''(2

')''()''(2

22

2

122

3

4

225

45

1

2

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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.

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Effect of bridge fin on hull aft of it and on control planes:

1

2

))(()(2

x

x

FWL dxxtvxwCl

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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

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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

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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)

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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