# How to calculate the moment of inertia for a half Ship

Discussion in 'Hydrodynamics and Aerodynamics' started by Anum, Jan 27, 2017.

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### AnumJunior Member

Dear All,

I want to simulate the flow around a displacement ship hull with trim and sinkage. Only the half symmetric hull will be modeled. 2 DOF will only allow translational motion along the z-axis and rotational motion around the y- axis. The udf for dynamic mesh is given below.

#include "udf.h"
DEFINE_SDOF_PROPERTIES(ship, prop, dt, time, dtime)
{
prop[SDOF_MASS] = ; I know the mass of half of my ship
prop[SDOF_IYY] = ; This is moment of inertia for a half ship, i do not know how to calculate this value
prop[SDOF_IXX] = 1;
prop[SDOF_IZZ] = 1;
prop[SDOF_ZERO_TRANS_X] = TRUE;
prop[SDOF_ZERO_TRANS_Y] = TRUE;
prop[SDOF_ZERO_TRANS_Z] = FALSE;
prop[SDOF_ZERO_ROT_X] = TRUE;
prop[SDOF_ZERO_ROT_Y] = FALSE;
prop[SDOF_ZERO_ROT_Z] = TRUE;

I want to know how can I calculate the moment of inertia for a half ship (IYY) from the offset data?
(I know the formula for the second moment of water-plane area is ∫ B(x, z)x^2dx from –L/2 to L/2, where B(x, z) is the width of the water plane at the position (x, z), L is the length of ship and z is the draft)

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2. Joined: Feb 2011
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### patzefranpatzefran

Strictly mechanically speaking, It is a little tricky to compute exactly : you have to know the density of mass in the volume of the boat (including the keel, the mast, the rigging and all the equipment !!). It is a volume integral on the whole volume of the boat and all the attached mass :
Iyy = sum of ro(x,y,z)*(x^2+z^2)*dx*dy*dz
were ro is the mass density

It is strange the program need this value, unless you compute instationary flow coupled with ship dynamics !!!!
perhaps it is not mechanical inertia but some other definition !

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### FESBNew Member

If you are making a quasi steady computation of ship resistance, the only thing that will be affected with mass moment of inertia will be the angular velocity of your ship, that is how fast your ship will rotate to the final trim angle. You only need mass moment of inertia around Y-axis, but to calculate it you would need to know the exact distribution of every mass component of your ship, longitudinally and verticaly because formula is I=m*r^2, where r is the distance of every element form center of gravity of your ship. Because this is impossible to know at preliminary phase of ship design, I suggest you to use this little trick:
You know the formula for the radius of gyration (or gyradius) is Kyy=(Iyy/m)^(1/2); where Kyy is gyradius around y-axis, the axis that is important to you. Iyy is the moment of inertia that you need to know, and m is the mass of the ship. From experience we know that Kyy ranges between 0,22*L to 0,28*L, for large displacement ships, and you can take it as 0,25*Length of your ship. So you just turn the equation around and here is your simple formula:
Iyy=Kyy^2*m
Iyy=(0,25*L)^2*m
For half of the ship, you calculate it with half of the mass
So the final formula should be:
Iyy=(0,25*L)^2*(m/2)

Anyways, this moment of inertia has really small effect on results of your CFD resistance analysis because sooner or later ship will find itself in equilibrium position with buoyancy balancing hydrodynamical forces of negative pressure that forms near the stern of your ship, and creates trim...
Only problem can appear if value of moment of inertia is way off real one, so the ship could pitch too fast or too slow...

And yes, it doesn't have anything to do with your second moment of water-plane area, that is the value that is known from the form of the ship itself, and it only affects your longitudinal stability, and it's used for calculating longitudinal metacentric height that will give you force (moment) that will balance hydrodynamical pitching moment

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### patzefranpatzefran

Angular velocity is related to mass moment of inertia thru the Angular acceleration around Y axe , equal to My/Iyy, My being the hydrodynamic moment.
So quasi steady flow is computed by iterating time dependent flow equations ?
Which code do you use ?

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Anum

FESB has given you very good gudiance/advice.

However, what you wish to find, does not need this. All you need is a 3D shape. The distribution of masses etc, is not needed, only the LCG of the overall 3D shape, to establish the floating draft and trim.

What you are appearing to ask is how to "simpsonise" offsets to obtain 3D data such as volume. This is basic 1st year NA stuff....just look in any good ref book like Basic Ship Theory or PNA.

6. Joined: Dec 2014
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### FESBNew Member

CFD softwares like Star CCM+ and Numeca usually require Iyy in resistance calculations because when ship is free to rotate along y axis, they couldn't calculate angular acceleration without it, it means no angular velocity and no rotation. Mass is usually automaticaly calculated because those softwares can easily measure ship's volume for given draft, and the density of fluid is known, and user usually only needs to imput vertical center of gravity, but Iyy, eventhough almost not at all important for results of resistance, and final trim has to be specified manually...

To know that you specified value good enough you only need to check if your ship's rotation angle along y axis is constant some time after ship reaches final speed of your computation.

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### RastapopNaval Architect

He will need Iyy for pitch to be calculated dynamically.

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### Nick_DJunior Member

To include free surface effects the CFD code has to solve in the time domain, as surface waves move. So it needs Iyy to solve the equations of motion.
For a first estimate just either use coffin method or set the material density in your solid modelling program so that the displacement is correct. The solid modelling program will then tell you Iyy.
Even just assume a block and use m/12*(b^2+L^2).
Be a little careful with the value, you can cause numerical pitch instability if you get it wildly wrong, where the simulation continues to oscilate long after you expect it to balance.

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### RastapopNaval Architect

Do not do this.
FESB has already given the standard approximation used when the true value is unknown.

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