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Multiple mass-spring-damper system

  1. Oct 30, 2004 #1
    I posted this in the general forum but it probably belongs here.
    This is a tricky one!

    I'm unsure about the forces on the ground level.

    The problem is a 2-spring-damper system with three levels. A ground level, level 1 and level 2. Then a force is applied to the ground level simulating an earth quake (eg the force could be sinusoidal). See the attachment for a diagram.

    These are the forces I have resolved:

    -level 2 mass:
    gravitational force, m0*g (down)
    Spring Force2, K1*(x1-x0) (up)
    Damper Force2, b1*(x1'-x2') (up)
    Inertial Force, m*x0'' (down)

    -level 1 mass:
    gravitational force, m*g (down)
    Spring Force1, K1*(x2-x1) (up)
    Damper Force1, b1*(x2'-x1')(up)
    Inertial Force, m*x1'' (down)
    Spring Force2, K1*(x1-x0) (down)
    Damper Force2, b1*(x1'-x2') (down)

    -level ground mass:
    driving force (earthquake) = some function eg sine
    g-force, mg (down)
    Spring Force1, K1*(x2-x1) (down)
    Damper Force1, b1*(x2'-x1')(down)
    Inertial Force = m*x2'' (down)

    I have tried to solve this using simulink, but for the positions of each level they fall towards negative oblivion. I am thinking I either have forgotten some force added too many.

    Have I left anything out? *thinking*

    Attached Files:

    • mass.JPG
      File size:
      12.2 KB
  2. jcsd
  3. Oct 30, 2004 #2
    Hi, this looks a bit like active suspension control...

    i get following equations :

    m0x0" = -k1(x0-x1)-b1(x0'-x1') + U
    m1x1" = k1(x0-x1) + b1(x0'-x1') - U - k1(x1-x2) - b1(x1'-x2')
    m2x2" = k1(x1-x2) + b1(x1'-x2') - U

    Ofcourse you need to add the overall gravitational force which you implemented correctly in my opinion.

    x : position
    x' : velocity
    x" : acceleration

    just for clarity...

  4. Oct 30, 2004 #3
    m0 is level two
    m1 is level 1
    m2 is groundlevel

    Just to make sure, ok ???

  5. Oct 30, 2004 #4
    Thanks Marlon for the reply. What about when I add a driving force on ground level?
    This is assuming that the springs are fully relaxed? I have calculted the displacements due to the weight of each level. What should be done to may this system dynamic?


    edit: oops for some reason I thought U=mg. U= driving force!! :yuck: But still does this consider the initial displacement of the springs?
    Last edited: Oct 30, 2004
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