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Obtaining mathematical model for the kinetic system

  1. May 20, 2012 #1
    1. The problem statement, all variables and given/known data

    See attachment.

    f(t) is the input force and b1 and b2 are kinetic friction constants. There is no static friction.

    2. Relevant equations

    [tex]ƩF = m \ddot{x}[/tex]
    [tex]F_s=kx[/tex]
    [tex]F_f=b\dot{x}[/tex]

    Ff is the force from friction and Fs is the force from spring.

    3. The attempt at a solution

    [tex]m_1\ddot{x_1}=-b_1\dot{x_1}-k_1x_1-k_1x_2[/tex]
    [tex]m_2\ddot{x_2}=-b_2\dot{x_2}-k_1x_1-k_1x_2-k_2x_2[/tex]
    [tex]m_3\ddot{x_3}=f[/tex]

    I have two questions:

    1) Should I include the friction between m1 and m3 and m2 and m3 in the equation for x3 and why?

    2) When I imagine this system, I think that f(t) should definitely affect x1 and x2, but in my equations it doesn't. Where am I doing wrong?

    Thanks in advance.
     

    Attached Files:

    Last edited: May 20, 2012
  2. jcsd
  3. May 20, 2012 #2
    Let me try, hope somebody will correct me.
    Taking springs as massless.

    m1a=k1x1+b1m1g
    m2a=k2x2+b2m2g-k1x1
    (m1+m2+m3)a=f(t)
     
  4. May 21, 2012 #3
    b1 and b2 are kinetic friction constants, and k1 spring is squeezed from both x1 and x2 so I can change those parts, but your work gave me new ideas, thank you.
     
  5. May 21, 2012 #4
    I have modified it as:

    [tex]m_1\ddot{x_1}=-b_1\dot{x_1}-k_1x_1-k_1x_2[/tex]
    [tex]m_2\ddot{x_2}=-b_2\dot{x_2}-k_1x_1-k_1x_2-k_2x_2[/tex]
    [tex](m_1+m_2+m_3)\ddot{x_3}=f[/tex]

    But still, x1 and x2 don't depend on f and that bothers me.
     
  6. May 21, 2012 #5
    I should have written the effect of k1 in the equation 1 as k1(x1+x2), not k1x1.

    Also, the k2 spring will pull m3 with k2x2 (to balance the forces on the k2 spring). So should I write the equation for x3 as:
    [tex]m_3\ddot{x_3}=f-k_2x_2[/tex]
    or should I write an overall system with (m1+m2+m3) and include the unbalanced forces from b1 and b2?

    I am very confused.
     
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