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# 1 A, Weakly coupled carts (20 points) m m2 Figure 1: A system of two masses and three springs. A symmetric two degree of

## 1 A, Weakly coupled carts (20 points) m m2 Figure 1: A system of two masses and three springs. A symmetric two degree of freedom system consists of two identical rigid masses m = m = m pictured in Fig. 1. The two outside springs k are identical; the interior spring has a tiny stiffness ek, with < 1. The walls are fixed. a Find the system of coupled equations for the motion of the coordinates 1 and 2 which describe the absolute displacements away from system equilibrium of the two masses. Take m = 1 and k = 1 (how can you justify this choice?). As a check, your K and M matrices should be symmetric. (7 pts) b Find the natural frequencies of vibration and describe the corresponding motion associated with each of these frequencies (find their associated eigenvectors). (8 pts) For = 0.05, plot the motion of the first mass, x1(t), subject to the initial conditions x(0) = 0, x2(0) = 0 and 1(0) = 0, 2(0) = 1. Show the motion over a sufficiently long period to see the full behavior. You might need to use a computer or calculator for this. (5 pts)

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