# Has the asymptotic theory of eigenvalues of infinite matrixes already been applied to vibrations analysis?

My question is reffering to the masses/springs model of a material, like the one presented in this article http://www.laserpablo.com/baseball/Kagan/UnderstandingCOR-v2.pdf. If one treates a long uniform rod as a chain of springs with constant k and N masses with mass m, then predicting the COR (coefficient of restitution) of a compound body amounts to the solution of N coupled differential equations. By taking the limit where N tends to infinity, one can model a real material in this way - K and m correspond to certain expressions of the Young's modulus and te density of the material. The problem can be described in matrix notation this way: let $$(x_1, x_2,..., x_N)$$ be the displacement vector of the N masses (displacement means the contraction of the Nth spring). Then by newton's second law $$m/k \cdot d^2(x_1, x_2,..., x_N)/dt^2$$ is equal to the product of the state vector $$(x_1, x_2,..., x_N)$$ with a certain N on N matrix. Viewed this way, the system has a certain number of natural frequencies $$\omega_1,\omega_2... \omega_N$$ which are the normal modes of the system. I'm not sure how exactly all this is connected to the eigenvalues of this matrix, but asymptotic analysis of the eigenvalues of infinite matrixes must be somehow connected to the solution this model.

Now i want to make clear my question:

• Is there a theoretical upper bound smaller then 1 for the coefficient of restitution of a material body which impacts a perfect surface? it's obvious that the upper bound is 1, but i ask if there is a lower upper bound. This question isn't merely a physical one; this upper bound is a direct result of the asymptotic behaviour of eigenfrequencies of the rod.
• if the theoretical upper bound is not smaller then 1, what is the rate of convergence to 1 as the number N of masses and springs tends to infinity? i ask because maybe the rate of convergence is so slow so even for a system with billions of springs (like a material body) one may get "only" 0.98 (for example, i just dropped a number). So this question is about the "practical limit" of the COR.

I hope i made the point of my question clear.