Added Mass - Massachusetts Institute of Technology

2.016 Hydrodynamics

Reading #6

2.016 Hydrodynamics

Prof. A.H. Techet

Added Mass

For the case of unsteady motion of bodies underwater or unsteady flow around objects, we

must consider the additional effect (force) resulting from the fluid acting on the structure

when formulating the system equation of motion. This added effect is added mass. Most

floating structures can be modeled, for small motions and linear behavior, by a system

equation with the basic form similar to a typical mass-spring-dashpot system described by

the following equation:

mx + bx + kx = f (t )

(6.1)

where m is the system mass, b is the linear damping coefficient, k is the spring coefficient,

f(t) is the force acting on the mass, and x is the displacement of the mass. The natural

frequency ¦Ø of the system is simply

¦Ø=

k

.

m

(6.2)

In a physical sense, this added mass is the weight added to a system due to the fact that an

accelerating or decelerating body (ie. unsteady motion: dU dt ¡Ù 0 ) must move some

volume of surrounding fluid with it as it moves. The added mass force opposes the motion

and can be factored into the system equation as follows:

mx + bx + kx = f (t ) ? ma 

x

(6.3)

where ma is the added mass. Reordering the terms the system equation becomes:

( m + ma ) x + bx + kx =

f (t )

(6.4)

From here we can treat this again as a simple spring-mass-dashpot system with a new mass

m¡ä = m + ma such that the natural frequency of the system is now

¦Ø¡ä =

k

k

=

m¡ä

m + ma

(6.5)

It is important in ocean engineering to consider floating vessels or platforms motions in

more than one direction. Added mass forces can arise in one direction due to motion in a

different direction, and thus we can end up with a 6 x 6 matrix of added mass coefficients.

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2.016 Hydrodynamics

Reading #6

Looking simply at a body in two-dimensions we can have linear motion in two directions

and rotational motion in one direction. (Think of these coordinates as if you were looking

down on a ship.)

Two dimensional motion with axis (x,y) fixed on the body. 1: Surge, 2: Sway, 6: Yaw

The unsteady forces on the body in the three directions are:

? F1 = m11

du

du1

du

+ m12 2 + m16 6

dt

dt

dt

(6.6)

? F2 = m21

du

du1

du

+ m22 2 + m26 6

dt

dt

dt

(6.7)

? F6 = m61

du

du1

du

+ m62 2 + m66 6

dt

dt

dt

(6.8)

Where F1, F2, and F6, are the surge (x-) force, sway (y-) force and yaw moments

respectively. It is common practice in Ocean Engineering and Naval Architecture to write

the moments for roll, pitch, and yaw as F4, F5, and F6 and the angular motions in these

directions as X4, X5, and X6.

This set of equations, (6.6)-(6.8), can be written in matrix form, F = [ M ]u ,

? m11

F = ?? m21

?? m61

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m12

m22

m62

-2-

? du1 ?

?

?

m16 ? ? dt ?

du

m26 ?? ? 2 ?

? dt ?

?

m66 ?? ?

? du6 ?

?

?

? dt ?

(6.9)

?2005 A. Techet

2.016 Hydrodynamics

Reading #6

Considering all six degrees of freedom the Force Matrix is:

? m11

?m

? 21

?m

F = ? 31

? m41

? m51

?

?? m61

m12

m22

m13

m23

m14

m24

m15

m25

m32

m33

m34

m35

m42

m43

m44

m45

m52

m62

m53

m63

m54

m64

m55

m65

m16 ? ? u1 ?

? ?

m26 ?? ? u2 ?

m36 ? ? u3 ?

?? ?

m46 ? ? u4 ?

m56 ? ? u5 ?

?? ?

m66 ?? ?? u6 ??

(6.10)

We will often abbreviate how we write the Force matrix given in (6.10) using tensor

notation.

The force vector is written as

F = Fi , where i = 1N

, 2, 3 , 4N

, 5, 6 ,

Linear

Forces

(6.11)

Moments

the acceleration vector as

ui = [u1 , u2 , u3 , u4 , u5 , u6 ] ,

(6.12)

and the added mass matrix [ma] as

mij where i, j = 1, 2,3, 4,5, 6 .

(6.13)

A good way to think of the added mass components, mij , is to think of each term as mass

associated with a force on the body in the i th direction due to a unit acceleration in the j th

direction.

For symmetric geometries the added mass tensor simplifies significantly. For example,

figure 2 shows added mass values for a circle, ellipse, and square. In the case of the circle

and square, movement in the 1 and 2 directions yields similar geometry and identical added

mass coefficients ( m11 = m22 ).

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2.016 Hydrodynamics

Circle

m11 = m22 = ¦Ñ d 2

m66 = 0

Reading #6

Ellipse

m11 = ¦Ñ¦Ð b 2

Square

m11 = m22 = 1.51¦Ð¦Ñ a 2

m22 = ¦Ñ¦Ð a 2

m66 = 0.234¦Ð¦Ñ a 4

m66 = ¦Ñ ??? a 2 ? b 2 ???

2

Two dimensional added mass coefficients for a circle, ellipse, and square in 1: Surge, 2: Sway, 6: Yaw

Using these coefficients and those tabulated in Newman¡¯s Marine Hydrodynamics on

p.145 we can determine the added mass forces quite simply.

In three-dimensions, for a sphere (by symmetry):

m11 = m22 = m33 =

1

¦Ñ? = mA

2

(6.14)

ALL OTHER mij TERMS ARE ZERO ( i ¡Ù j ).

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2.016 Hydrodynamics

Reading #6

General 6 DOF forces and moments on a Rigid body moving in a fluid:

Velocities:

G

Translation Velocity : U (t ) = (U1 , U 2 , U 3 )

(6.15)

G

Rotational Velocity : ?(t ) = (?1 , ? 2 , ?3 ) ¡Ô (U 4 , U 5 , U 6 )

(6.16)

All rotation is taken with respect to Origin of the coordinate system (often placed at the

center of gravity of the object for simplicity!).

Forces: (force in the jth direction). ( i = 1, 2, 3, 4, 5, 6 and j, k , l = 1, 2,3)

Fj = ?U i mij ? ¦Å jkl U i ? k mli

(6.17)

Moments: ( i = 1, 2, 3, 4, 5, 6 and j, k , l = 1, 2,3)

M j = ?U i m j +3, i ? ¦Å jkl U i ? k ml +3, i ? ¦Å jkl U kU i mli

(6.18)

Einstein¡¯s summation notation applies!

The alternating tensor ¦Å jkl is simply

¦Å jkl

if any j , k , l are equal

? 0;

?

= ? 1;

if j , k , l are in cyclic order

??1; if j , k , l are in anti-cyclic order

?

(6.19)

The full form of the force in the x-direction (F1) is summed over all values of i:

F = ? U 1 m11 ? U 2 m21 ? U 3 m31 ? U 4 m41 ? U 5 m51 ? U 6 m61

N1

N 









j =1

i =1

i =2

i=4

i =3

i =5

i =6

? ¦Å1kl U1 ? k ml1 ? ¦Å1kl U 2 ? k ml 2 ? ¦Å1kl U 3 ? k ml 3 ? ¦Å1kl U 4 ? k ml 4









i =1

i =2

i =3

(6.20)

i=4

? ¦Å1kl U 5 ? k ml 5 ? ¦Å1kl U 6 ? k ml 6





i =5

i =6

for k , l = 1, 2,3 .

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