Help with equations of motion

  1. 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 [itex]B[/itex] attached to the left end of the arm as shown. C is the center of mass of the arm. We have a moment [itex]M_{z_b}[/itex] acting about the [itex]\hat{z}_{b}[/itex] axis.


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

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

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

    The applied moment is given as : [itex]M_{z_b}(t) = 160 \left(1 - \cos \left(\frac{2 \pi t}{15} \right) \right)[/itex]. For [itex]t > 15, M_{z_b} = 0[/itex]. 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.

    Last edited: Jul 7, 2013
  2. jcsd
  3. Simon Bridge

    Simon Bridge 15,474
    Science Advisor
    Homework Helper
    Gold Member

    1. it's your math.
    The total change in momentum is the area of the force-time graph ... so you can check.
    I don't know why you are not doing this in polar coordinates.

    2. I imagine because the moment is quite high.
    Have you got any reason to expect u to increase less rapidly?
    What sort of value were you expecting and why?

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