# Homework Help: Engineeing Dynamics - Inertia Dyad of Half Cylinder

1. Nov 6, 2012

### Green Lantern

I'm getting desperate. The professor has assigned a project and has not clearly explained how to derive the answers. I'm doing the best I can but his TA's and recommended tutors for the class are always incapable of reproducing the answers either. It's a junior level dynamics class, but he's actually turned the class into a machine dynamics class taught in senior or masters level. The work needs to be done in mathematica, but all I need is help with theory since that's the part we don't learn in lecture.

Anyways, here is the question...

1. The problem statement, all variables and given/known data

Develop the mass center expression and the inertia matrix and inertia dyadic for a half cylinder, then let the inner diameter approach the outer diameter to develop the same for a thin half shell. Use Mathematica for your work.

2. Relevant equations

Ii,i = ∫m(rj2+rk2)dm

3. The attempt at a solution

First is the Inertia-matrix of the half cylinder. I'm not sure how to derive all the terms, but I did my best.

{y2 + z2, -x y, -x z}
{-x y, x2 + z2, -y z}
{-x z, -y z, x2 + y2}

Then I write my position vector:

BrP = x b[1] + y b[2] + z b[3]

Where B is a point in the body frame, P is the endpoint under evaluation, and r is the vector r. b[1,2,3] are unit vectors in the body frame.

Now is where I think I went wrong, if not before. The Volume is:

V = ∫0W0H-L/2L/2rdxdydz

Then mass is:

m = ∫∫∫ρrdxdydz

So center of mass is:

C = (1/m)∫∫∫ρr(x,y,z)rdxdydz

Then I get lost even more...

S.O.S

2. Nov 7, 2012

### SteamKing

Staff Emeritus
If x,y,z are cartesian coordinates, dV = dxdydz, not r dxdydz