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Forced oscillations in a discrete system

  1. Nov 22, 2013 #1
    Hello Everyone!
    1. The problem statement, all variables and given/known data
    There exists a very large (discrete) system of N coupled masses, each of mass "m", where every pair is connected via a spring of constant "K". Assuming all motion is horizontal, find the amplitude of the oscillations of an nth mass in the system, under the following conditions:
    1). The mass at n=0 is made to oscillate with amplitude A0 and ang. frequency ω0, while the mass at n=N is bound and fixed.
    2). Both the masses at n=0 & n=N are made to oscillate with amplitude A0, with the phase between the displacements being 0, but with ang. frequencies ω' ≠ ω0.
    2. Relevant equations
    Dispersion relation for a system of coupled oscillators.
    General solution for the displacements of coupled oscillators(discrete).
    3. The attempt at a solution
    I was initially perturbed by the problem, as unlike in similar questions, the inter-mass distance had not been given here.
    However, in terms of 1)., assuming that when system reaches a steady-state all the masses therein vacillate with the same frequency -- that of the mass at n=0, ω0, I derived the dispersion relation:
    [tex] \omega_0^2 = \frac{2K}{m}(1-\cos(ka)) \Rightarrow k\cdot a = \cos^{-1}(\frac{2K-m\omega_o^2}{2K}) [/tex]
    Where I included the separating distance "a" between the masses, but eliminated it through the dispersion relation.
    Next, I applied the general solution to such a problem.
    If I let: [tex] \psi_n = A_n \cos(\omega_0 t) ~ \vert ~ A_n = A\sin(kna)+B\sin(kna) [/tex]
    I would then insert into these equations my boundary condition(e.g. the mass at n=N is non-moving; the mass at n=0 has an amplitude of A0, and so forth). Most importantly, I could rid all these formulae from either a(and in this case, even k -- the wave number) and arrive at the solution without adding anything onto the initial parameters I was given. Thereafter, I could find A,B(the amplitudes of motion), and lastly find An. Seems straightforward...
    ***
    However, my difficulty arises when solving 2).
    Firstly, it is evident to me that the dispersion relation reproduced above is no longer valid. There are now two different omegas, at different locations. Furthermore, even the general solution won't work here, as the statement that all masses will eventually oscillate with the same frequency no longer applies.
    In short, I am lost here, and would very much appreciate any aid or guidance you can throw my way.
    Thank you for your attention,
    Daniel
     
    Last edited: Nov 23, 2013
  2. jcsd
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