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Trying to model the acceleration of a system due to an impulse forcing function

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  1. Nov 2, 2014 #1
    1. The problem statement, all variables and given/known data
    My team and I are working on a design project to design/modify a device that can go on hikes for paraplegic/quadriplegic people.

    Here is the current design (not designed by us):

    http://i.imgur.com/RAOq28k.png
    http://i.imgur.com/zgA3aCg.png

    We are thinking of adding a spring damping system to reduce shock on the passenger. We calculated that we would need a k = 87.3 kN/m and b = 4.137 kNs/m to meet some of the specifications. The mass of the entire device is about 100 kg. Using these we want to find the acceleration felt by the rider after going over a bump. So we decided to model it using an impulse. We calculated that on average the bumps would create an acceleration of 1.36 m/s^2 on the rider without a damping system. Thus, we want to find out what the actual acceleration would be with a spring and damper.

    3. The attempt at a solution
    ln82f3z.jpg

    Plugging this into wolfram gives us:
    x(t) = .064461*e^(-20.685t)*sin21.098t
    (http://www.wolframalpha.com/input/?i=x''+++41.37*x'+++873*x+=+1.36*delta(t),+x(0)+=+0,+x'(0)+=+0)

    Taking the second derivative gives us the acceleration:
    x''(t) e^(-20.685 t) (-56.2631 cos(21.098 t)-1.11236 sin(21.098 t))
    (http://www.wolframalpha.com/input/?i=second+derivative+of+.064461*e^(-20.685t)*sin21.098t)

    This means that the magnitude of the maximum acceleration will be 56.3 m/s^2 which does not make sense since the input was only 1.36 m/s^2. Any idea what is going on here?

    In fact I found the solution to x(t) online and it corresponds with what I got from wolfram:
    jS186MP.png
     
  2. jcsd
  3. Nov 3, 2014 #2
    Anyone know?
     
  4. Nov 3, 2014 #3
    I've spent the last few hours thinking about it. I'll try and ask my mechanics professor tomorrow, if he's around.
     
  5. Nov 10, 2014 #4
    Well, I've talked with my professor about solving this, and tried solving it some different ways. Even still, at the time of the impulse I'm also getting a spike in the acceleration around 50 m/s^2.

    My best guess is that this comes from using the dirac function as your impulse. Otherwise, I really don't know.
     
    Last edited: Nov 10, 2014
  6. Nov 11, 2014 #5

    rude man

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    Homework Helper
    Gold Member

    Careful with your Dirac delta function. It's making your equation inconsistent dimension-wise since the dimensions of δ(t) is T-1. It's not dimensionless.

    To give the impulse term the correct dimensions, which is force, your impulse must be defined as a force-time or momentum input multiplied by δ(t) . So your impulse term must look like (FΔt)δ(t) or Δpδ(t) and you somehow have to come up with an estimate of how much force over how much time is applied to the system, or how much change of momentum.
     
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