1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Mass-spring-damper system solve for x(t) using power series

  1. Mar 24, 2012 #1
    1. The problem statement, all variables and given/known data
    A mass of 10kg is suspended from vertical spring of stiffens 100N/m and is provided with dashpot damper having damping coefficient of 1000Ns/m.
    The mass is pulled down the distance of 4cm from its equilibrium position and than released.
    Establish the differential equation of motion and solve using power series for variation in displacement with time.


    2. Relevant equations



    3. The attempt at a solution
    Started with
    mx''(t)=-kx(t)-cx'(t)
    and initial conditions
    t0=0,x(t0)=4cm=0.04m, x'(t0)=0,x''(t0)=0 here assume that velocity and acceleration=0
    at t=0
    after solving using power series got:
    x(t)=a0+a1x+a2x2+a3x3+..... a0=0.04 but all the rest of coefficients =0

    That can be right ,please help
     
  2. jcsd
  3. Mar 24, 2012 #2
    its not 0 for the rest for coefficient , u just have to treat the equation as some function and take its derivative, and since u know x' and x u know x'' by the equation , taking the derivative will get u the equation consist of x''' , x'' and x' and x'', x' have already been known
     
  4. Mar 24, 2012 #3
    mx^''=-kx-cx'
    x^''+c/m x^'+k/m x=0
    Putting known values into equation:
    x^''+100/10 x^'+1000/10 x=0
    x^''+10x^'+100x=0
    Now solving using power series:
    Let assume that:
    x(t)=a_0+a_1 t+a_2 t^2+a_3 t^3+..
    t_0=0,x_0=0.04 so a_0=0.04 From initial conditions

    x^' (t)=a_1+〖2a〗_2 t+〖3a〗_3 t^2+〖4a〗_4 t^3+..
    t_0=0,〖x'〗_0=0 so a_1=0 From initial conditions

    x^'' (t)=〖2a〗_2+〖6a〗_3 t+〖12a〗_4 t^2+20a_5 t^3+..
    t_0=0,〖x''〗_0=0 so a_2=0 From initial conditions
    Putting these into differential equation
    x^''+10x^'+100x=0
    〖(2a〗_2+〖6a〗_3 t+〖12a〗_4 t^2+20a_5 t^3+..)+10(a_1+〖2a〗_2 t+〖3a〗_3 t^2+〖4a〗_4 t^3+..)+
    +100(a_0+a_1 t+a_2 t^2+a_3 t^3+..)=0

    Rearranging
    100a_0+10a_1+2a_2+(100a_1+20a_2+6a_3 )t+(100a_2+30a_3+12a_4 ) t^2+
    +(100a_3+40a_4+20a_5 ) t^3=0
    And a_0=0.04 ,a_1=0,a_2=0 -From initial conditions
    0.04+(100a_1+20a_2+6a_3 )t+(100a_2+30a_3+12a_4 ) t^2+(100a_3+40a_4+20a_5 ) t^3=0

    this is where I m now
    but dont konw how to get rest of coefficents
     
  5. Mar 24, 2012 #4

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Are you trying to solve this by power series for some reason? Although that should work, it is certainly the hard way to do a constant coefficient DE.
     
  6. Mar 24, 2012 #5
    hi ,yes its one of task for my math assignment and its specified to solve it using power series
     
  7. Mar 24, 2012 #6
    actually x''(0) is not zero since the object is at the amplitude it should have maximum accelaration toward equilibrium point
     
  8. Mar 24, 2012 #7

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Show your work. Without some indication of where you went wrong, it would be impossible to assist you.

    RGV
     
  9. Mar 25, 2012 #8
    Hi ,have uploaded a pdf file showing what I have done.
    So what initial conditions would you use?
     

    Attached Files:

    • q5.pdf
      q5.pdf
      File size:
      136.3 KB
      Views:
      94
  10. Mar 25, 2012 #9

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    People keep telling you that x''(0) is nonzero, but you don't listen. The DE x'' = -10 x' - 100x gives x''(0) = -100*4/100 = -4.

    That will give you an equation of the form C1*t + C2*t^2 + C3*t^3 + ... = 0, where the C1, C2, C3,... are linear combinations of your a1, a2, a3, ... . The equation is supposed to be an identity in t, so that means that all the coefficients must vanish; that is, we must have C1 = 0, C2 = 0, C3 = 0, ... . Solving those will give you the coefficients a1, a2, a3, ... in the expansion of x(t).

    RGV
     
  11. Mar 25, 2012 #10
    have changed x''(0)
    is that correct now?
     

    Attached Files:

    • q5.pdf
      q5.pdf
      File size:
      182.8 KB
      Views:
      80
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Mass-spring-damper system solve for x(t) using power series
Loading...