Forced oscillations in a discrete system

[danielakkerma]

Hello Everyone!

Homework Statement

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.

Homework Equations

Dispersion relation for a system of coupled oscillators.
General solution for the displacements of coupled oscillators(discrete).

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 A_n. 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Firstly, it is evident to me that the dispersion relation reproduced above is no longer valid.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

 

[anathema]

Dear Daniel,

Thank you for sharing your thoughts and approach to the problem. It seems like you have a good understanding of the first condition and have made progress in finding a solution. However, as you correctly pointed out, the second condition introduces some new challenges.

In this case, we can no longer assume that all masses will oscillate with the same frequency. Instead, we have two different amplitudes and frequencies for the two end masses. This means that the dispersion relation you derived for the first condition will not be valid anymore.

To find the solution for the second condition, we can still use the general solution you mentioned, but we need to modify it to account for the different frequencies. We can write the general solution as:

ψn = An cos(ω0t + φn)

Where An and φn are constants that need to be determined. To find these constants, we can apply the boundary conditions you mentioned (the non-moving mass at n=N and the masses at n=0 and n=N oscillating with the same amplitude and phase). This will give us a system of equations that we can solve for An and φn.

I hope this helps you make progress in solving the problem. If you have any further questions or need clarification, please don't hesitate to ask.