Solve Small Oscillations in 1 Spring System

[CNX]

A horizontal arrangement with 1 spring in between the two masses, 1 spring connecting each mass to opposite fixed points:

k 3m k 8m k
|----[]----[]----|

I solved the eigenvalue/eigenvector problem for the dynamical matrix D where $$V = 1/2 D_{ij} w_i w_j$$ and the w's are mass-weighed-coords. So I have the frequencies for the normal modes.

$$(D-I\omega)\cdot \vec{b} = 0$$

How do I get the particular solution for when a single mass is given a initial speed 'u'? Without getting the amplitude vectors for each normal mode I don't see how to get the particular solution as there are too many unknowns?

$$x_i (t) = \sum_{j} c_{i}^{(j)} \cos (\omega_j t + \delta_j)$$

i.e 6 unknowns and and 4 IC's x1(0)=0, x2(0)=0, v1(0)=u, v2(0)=0

 

[turin]

I would assume that the default initial condition is implied as zero displacement and zero velocity for both masses. So, if they tell you that one of the masses is given a kick of velocity u, then I would assume that the initial conditions are otherwise zero. Then, use your transformation to interpret this initial condition in terms of the eigenbasis.

 

[CNX]

I'm not sure how to go about doing that...

How is what I did different than solving $$(V-\omega^2 T)\cdot \vec{a}=0$$ for eigenvalues $\omega_1,\omega_2$? i.e. where T is constructed form the KE and V matrix is constructed from the PE, and is similar to my dynamical matrix D above.

 

[turin]

I don't know what your notation means, exactly, but it looks OK. It would help if you would tell me what eigenfrequencies you got and what is your transformation matrix to the eigenbasis so that I can see if you're on the right track (and to double check that I didn't make a mistake either).