# Help with equations of motion

1. Jul 7, 2013

### boeing_737

Hi,

I am having a little bit of conceptual trouble with this problem and would appreciate your help. The problem setup is given in the figure. Let's say we have a slender uniform rigid arm(mass m, length l) in space, with a coordinate system $B$ attached to the left end of the arm as shown. C is the center of mass of the arm. We have a moment $M_{z_b}$ acting about the $\hat{z}_{b}$ axis.

Let $(u,v,w)$ and $(p,q,r)$ be the inertial velocity and inertial angular velocity vectors expressed in $B$. I get the scalar equations of motion as (assuming that the angular velocity is only along $\hat{z}_b$):

$m \dot{u} - \frac{ml}{2} r^2 = F_{x_b}$
$m \dot{v} + \frac{ml}{2} \dot{r} = F_{y_b}$
$m \dot{w} = F_{z_b}$

$0 = M_{x_b}$
$-\frac{ml}{2} \dot{w} = M_{y_b}$
$\frac{ml^2}{3} \dot{r} + \frac{ml}{2} \dot{v} = M_{z_b}$

The applied moment is given as : $M_{z_b}(t) = 160 \left(1 - \cos \left(\frac{2 \pi t}{15} \right) \right)$. For $t > 15, M_{z_b} = 0$. See figure below :

Integrating these equations using MATLAB's ode45, I get the following plot :

From the above figure :
1) There is only one component of angular velocity (yaw rate) which is as expected. But is the magnitude correct (ie should it reach 24 rad/s)?

2) I am not able to figure out what's going on with u. Why is it increasing so rapidly?

Any help would be really appreciated.

yogesh

Last edited: Jul 7, 2013
2. Jul 7, 2013