# Question about Lagrangean dynamics(inertia matrix)

1. Jul 9, 2012

### supernova1387

I have a 3 degree of freedom system and my equation of motion is like this:

M(q)q_dd + C(q,q_d)q_d+G(q)=0

M(q): inertia matrix
C(q,q_d): Coriolis-centrifugal matrix
G(q): potential matrix.

where q_d is the first derivative of q etc and q is a vector of my variables.

q=[θ,γ,a]T

"θ" and "γ" are angles in (rad) and "a" is length in (m).

Now my question is this: I have 3 equations with 3 unknowns but 2 variables are in terms of angles and 1 in terms of distance, so the elements of my inertia matrix don't have the same units. Is that wrong? I mean each equation is consistent in units. The first 2 have units of kg(m/s)^2 while the 3rd has units of kg(m/s^2). I can make the whole thing dimensionless but I read somewhere that the inertia matrix should be symmetric(which it is symmetric at the moment). If I make my equation of motion dimensionless then inertia matrix won't be symmetric anymore. What shall I do?

Any suggestions are welcome

2. Jul 9, 2012

### AlephZero

It's perfectly OK to have variables in different units. Just go with the consistent and symmetric matrices.

If you want to see a fully worked out example of this sort of thing, Google for the the formulation for the mass and stiffness of a cantilever beam with displacement and rotation variables at each end. You should be able to find it worked out both by "mechanics of materials" and Lagrangian methods, and of course both will give the same matrices if all the other assumptions about the system are the same.

3. Jul 10, 2012

### supernova1387

Thank you for your reply. I think this will happen whenever one variable is measuring displacement while the other is measuring rotation even for 2 DOF systems.