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13.42 Design Principles for Ocean Vehicles

Prof. A.H. Techet

Spring 2005

1.Coupled Equation of Motion in Heave and Pitchx3

x5

Once we have set up the simple equation of motion for a vessel in heave it is natural to extend this

discussion to the coupled heave-pitch equations of motion. Considering a ship floating on the free

surface in waves. This ship will naturally heave and pitch due to the incident waves. It is not

guaranteed that these two motions will be independent, however. Thus it becomes necessary to

consider the motions together.

The body boundary condition must be properly specified, thus we need to know both the linear and

angular velocities of the vessel:

3 55 3 5( )B

dx dxk r j z xx x xV

dt dt

= + = ,

G G (1)

It follows that the velocity normal to the vessel hull is

5 3 5 3 5( ) ( )x z z x zB n n z n x n n z n xx x x xV = + = +

G (2)

where the position vector, rG

, crossed with the unit outward normal, n , is

4

5

6

n

n r n

n

= G

(3)

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In general the potential functions do not change significantly from our previous cases. However it

is necessary to account for the pitching motion in the radiation component of the total potential.

( ) ( ) ( ) ( )I D Rz t x z t x z t x z t , , = , , + , , + , , (4)

3 53 5

( ) ( ) ( ) ( )i t i t i t I D

Re a x z x z e x z e x z ex x

= , + , + , + , , (5)

where the motions in the heave (3) and pitch (5) directions are

3 3( ) i tx t Re ex

= (6)

5 5( ) i tx t Re ex

= . (7)

The boundary conditions for the potentials, I ,

D , and

3 are the same as in the pure heave case.

We must now, however, consider also the boundary conditions for the radiation potential due to

pitching motions. Following the same formula from last lecture the equation of motion and

boundary conditions at the free surface, sea floor and on the body are:

1. 2 5 0 = 2. 52

50

zg

+ =

3. 5 0z

= on z H=

4. 5 5n n = ; on the body

In addition to the force in the vertical direction, we also need to formulate the pitching moment

(about the y-axis) acting around the center of gravity. First we can re-write the vertical force

dependent on the two motions as:

3( )T

zS

F t n dst

=

(8)

( ){ }3 53 5i t z I DSRe i e n a dsx x = + + + (9)

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l l l( ) l( ){ }3 3 3 33 5 35i t i t i t

I DRe a e e ex xF F F F

= + + + (10)

l l 2 2

33 33 35 353 3 3 5

i t i t i t

I DRe a e e A i B e A i Bx xF F

= + + + (11)

Next, we can find5

F , the moment on the body due to pitching and heaving motion.

2 5

( )S

F r n dSt

= =

G(12)

where

5 5( )T

SF t n dS

t

=

(13)

l l l( ) l( ){ }5 5 3 53 5 55i t i t i t

I DRe a e e ex xF F F F

= + + + (14)

l l 2 2

53 53 55 555 5 3 5

i t i t i t

I DRe a e e A i B e A i Bx xF F

= + + + (15)

The hydrostatic forces in the two directions are

3 33 3 35 5hF C x C x= (16)

5 53 3 33 5hF C x C x= (17)

where the coefficients,ij

C , are based on the vessel geometry. Taking the pitch motion about the

center of gravity (usually the c.g. is near midships), where 0cg

x = , we can write the equations of

motions in heave and pitch:

2

33 32 h

d xm F F

dt= + (18)

2

55 52 h

d xI F F

dt= + (19)

The system of equations can now be written in matrix form assuming a harmonic input and an LTI

system:

l l

l l

3 333 33 33 335 35 35 3 3

5 553 55 53 55 53 55 5 5 5

I D i t

I D

m A A B B C C xx x F Fa e

A I A B B C C xx x F F

++ + + = ++

(20)

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Simply expressed in matrix notation, the system of equations for the coupled pitch and heave

motions for a freely floating body can be rewritten as

[ ] [ ] [ ] l i tA x B x C x a F e + + + = (21)

where the vector motion is comprised of the linear heave motion,3, and the angular pitch motion,

5x .

3

5

i txx Re ex

=

(22)

Writing the equation in matrix form allows us to better determine the amplitude response function(transfer function) between the forcing and the ship motions:

[ ] [ ] [ ]{ }l

2

ax F

M A i B C =

+ + +(23)