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Finding the normal modes for a oscillating system

  1. Apr 21, 2015 #1
    1. The problem statement, all variables and given/known data
    The system is conformed by two blocks with masses m (on the left) and M (on the right), and two springs on the left/right has the spring constant of k. The middle spring has a spring constant of 4k. Friction and air resistance can be ignored. All springs are massless.
    Find the normal modes.
    Diagram:
    |~m~~~~M~|
    2. Relevant equations
    ##L = T-V ##

    ##T = \frac{1}{2}(m\dot{x}_1^2 + M\dot{x}_2^2) ##
    ##V = \frac{1}{2}[(x_1^2 + x_2^2) + 4(x_1-x_2)^2]##
    ##\frac{\partial{L}}{\partial{x_k}} - \frac{d}{dt} \frac{\partial{L}}{\partial{\dot{x_k}}} = 0##
    ##[A_{ij} - \omega^2 m_{ij}]=0##
    3. The attempt at a solution
    I have tried to guess a solution for the normal modes but of the for ##\eta_1 = x_1 - x_2 ## and ##\eta_2= x_1+x_2## but I does not works. I have tried to add some arbitrary coefficient to ##\eta_1## & ## \eta_2## unsuccessfully. Trying to find the eigenvectors is a pain in the neck since the eigenfrequencies are:##\omega^2 = \frac{5k(M+m) \pm k\sqrt{25(M^2+m^2) -14Mm}}{2Mm}##.
    I read in a textbook that one could find the coefficient for the etas by knowing that the ratios ##\frac{M_{11}}{M_{22}}=\frac{A_{11}}{A_{22}}=\alpha^2## but for this case the first ratio is ##\frac{m}{M}## and the second is 1 .Therefore, this method doesn't help me :/.

    I would appreciate any contribution. Thank you.
     
  2. jcsd
  3. Apr 21, 2015 #2
    Do you have to solve this problem using the Lagrangian?
     
  4. Apr 22, 2015 #3
    The Lagragian is not necessary, but is the only method I know. I believe that if I use forces the problem would complicate more.
     
  5. Apr 22, 2015 #4
    I think the force method is easier but the results are the same.

    I get a slightly different answer for the Eigen values but even then I think you could probably simplify it a little bit:
    ω2 = k(1/M+1/m) [5/2± √(25/4+16/(M/m+m/M+2))]

    I am not aware of any other method except plugging the Eigen values into the equations of motion and solving for the mode shapes.
     
  6. Apr 22, 2015 #5
    Duplicate post.
     
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