# Differential Equations

posted by Smith

Consider the system shown below of two masses of mass m, coupled together between two fixed walls via springs with varying spring constants.

Let x(t) and y(t) be the horizontal displacements of the two masses as a function of time.

(a)Write down a system of differential equations for x'' and y''.

(b)For the case when m = 1, k1 = 1 and k2 = 2, calculate the eigenvalues associated with this system, which are associated with the characteristic frequencies of vibration.

(c)Plot the eigenvalues as a function of k2 over the range 0 < k2 < 3. Discuss the physical interpretation of the changes to the eigenvalues as k2 is increased.

1. uu

k2=0

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