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Tracking the position of trebuchet payload

  1. Nov 3, 2012 #1
    1. The problem statement, all variables and given/known data

    For the parts of the Trebuchet shown, develop expressions for the position, velocity,
    and acceleration, x, x', and x'' of the payload as a function of the arm angle θ and its
    derivatives θ', θ''. Use the vector tools in Mathematica to help develop the
    expressions needed. Note that a closed kinematic loop is present during this phase of
    the motion if the rope remains taut. By either using max/min calculus tools or plots of
    your own design, find a set of parameters L, R, and H that will provide the maximum
    velocity x' at the point the payload lifts off the ground. Provide plots showing the
    position, velocity and acceleration of the payload as a function of the arm angle θ.
    Assume the arm angle increases as a quadratic function of time. Bonus points are
    available if animations of the motion are provided. Provide discussion of what is
    observed as your analysis proceeds. Use the Mathematica notebooks as the report
    medium. Report on your work immediately near each plot. Label all axes and title
    each plot.

    Problem diagram is trebuchet.jpg

    2. Relevant equations

    v = dr/dt + ω cross r
    a = dv/dt + ω cross v

    3. The attempt at a solution

    My coordinate system is trebwork.jpg

    For position of the payload I have:

    r = La1 + Rb1

    Where

    A-Frame:

    a1 = sinθn1 - cosθn2
    a2 = cosθn1 + sinθn2

    and

    B-Frame:

    b1 = -cosβa1 - sinβa2
    b2 = -sinβa1 + cosβa2

    and

    ω(Frame:N-B) = θ'a3
    ω(Frame:N-A) = β'b3

    My vector loops is:

    r = La1 + Rb1 - xn2 + Hn1 = 0

    I derive this to find velocity and again for acceleration.

    First, how do I solve for β?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     

    Attached Files:

  2. jcsd
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