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Homework Help: 2 DOF oscillator max force reponse

  1. Apr 25, 2012 #1
    (NOTE, this is not a homework problem, but it sure seems like one)


    I am mounting a component onto structure and I need to determine the maximum force input into the component.

    My system can be represented by a base driven two degree of freedom oscillator:


    I need to determine the force applied to m2:

    F = k2*(x2-x1) + c2*(x2-x1)

    This force needs to be a function of (m1,m2,c1,c2,k1,k2,y) and not of (x1,x2).
    Basically, for a given input y, what will the be force response on m2.

    Every time I solve the system of equations, my result is a function of x1 and/or x2.

    Thank you in advance for your help!

    p.s. If it is easier, feel free to remove the dampers from the system. An un-damped system will work for my purposes.
  2. jcsd
  3. Apr 25, 2012 #2
    As for an attempt at the solution:

    my matrix system of equations is...

    [m1 0 ; 0 m2] [x''1 ; x''2] + [k1+k2 -k2 ; -k2 k2] [x1 ; x2] = [0 ; k1] [0 y]

    Interestingly enough, the force I need to recover is naturally in the lower portion of the equation giving:

    F = k1*y - m2*x''2

    I make the reduction to

    F = k1*y + Wn^2*m2*x2

    but still was not able to remove all of the x2's and x1's (after making a number of substitutions).
  4. Apr 26, 2012 #3

    rude man

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    Homework Helper
    Gold Member

    You have two equations and two unknowns (x1 and x2). You can solve for them as functions of m1,m2,c1,c2,k1,k2, and y.

    (I would use the Laplace transform, but there is of course a time-domain way to solve them also. I don't have that info on hand.
  5. Apr 26, 2012 #4


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    Staff: Mentor

    Your mechanical system can be modeled as an electrical circuit (an "analog" system). Use a simulator (Spice) to characterize the maximal force versus various "input signals".

    Assuming that voltage represents velocity and current represents force, the mechanical elements have particular electrical analogs:

    Spring = Inductor
    Damper = Resistor
    Mass = Capacitor
    Force = Current
    Velocity = Voltage

    Force inputs become current sources. Initial velocities (of masses) become initial voltages on capacitors. Velocity inputs are okay -- they become voltage inputs, so if you want to determine how your system responds to a given motion driving it, characterize the motion in terms of its velocity.


    In the above circuit Io represents the force that's applied to the mass M2.

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