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Homework Help: Link with spring-mass-damper system

  1. Nov 15, 2012 #1
    From this mechanical part (attached), I need to derive a differential equation relating mass 'x' position to hinge 'y' position.

    [itex]\ddot{x}m + b\dot{x} + cx = F[/itex]

    The link l would be rotating. I am confused about considering the displacement of x with reference to y. Would this change the differential equation by replacing x with (x-y) ?

    I derived an equation to relate 'y' with [itex]\theta[/itex] ;

    [itex]y = lsin(\theta)[/itex]

    Finally I also need to derive a differential equation from the previous two to relate the loading torque on the shaft for some rotation theta.

    [itex]\tau = (\ddot{x}m + b\dot{x} + cx)lcos(\theta)[/itex]

    I would really appreciate if you can give me some hints on my work. thanks

    Attached Files:

    Last edited: Nov 15, 2012
  2. jcsd
  3. Nov 15, 2012 #2

    Simon Bridge

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    Damped-driven harmonic motion with feedback?
    Is the servo "active" (it is being driven by a motor and the second mass is just a counter-balance?) ... the it is just providing the f(t) in the standard equations.
  4. Nov 16, 2012 #3
    I need to investigate some tight control on an open loop modelling of a servo motor linked with a spring mass damper system.
    The linkage with spring mass damper is stable with a counterbalance on the other side. The aim is to shift the mass (by the servo) of the s-m-d system and eliminate oscillations as much as possible. The servo is modeled as having a feedback loop with theta. I will also be implementing a PID loop in the servo controller but the spring-mass-damper is open loop.

    I already derived the transfer functions and state space for the servo which is simply a DC motor with a 10:1 gearbox.

    My main confusion right now are the equations in my previous post. I need these in order to simulate everything together in simulink. attached

    Attached Files:

  5. Nov 16, 2012 #4

    Simon Bridge

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    The F in the above relation is being applied at y - y moving up and down is what is generating it.
    If ##\theta## is expected to be small enough that you can discount lateral movement then ##\sin(\theta)\approx \theta##
    considering the idea is to minimize oscillations, that seems reasonable.

    measure x from some equilibrium/rest position and y likewise ... then you'll see better how they interact.
  6. Nov 16, 2012 #5
    I cannot assume this. The system has to remain non linear that is why it will be simulated via simulink. Dont worry, I will not be working any calc with a non linear system.

    Do you mean [itex]y-y_{o}[/itex] ? Then how would the differential equation be?
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