Just as in the preceding section the force acting on

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Unformatted text preview: cos ω1 t + b1 sin ω1 t √ x= = a2 cos ω2 t + b2 sin ω2 t x2 1/ 2 1 / 2 , or equivalently, x = b1 (a1 cos ω1 t + b1 sin ω1 t) + b2 (a2 cos ω2 t + b2 sin ω2 t). The motion of the carts can be described as a general superposition of two modes of oscillation, of frequencies ω1 2π and ω2 . 2π Exercises: 2.4.1.a. Consider the mass-spring system with two carts illustrated in Figure 2.5 in the case where k1 = 4 and m = k2 = k3 = 1. Write down a system of secondorder differential equations which describes the motion of this system. b. Find the general solution to this system. c. What are the frequencies of vibration of this mass-spring system? 2.4.2.a. Consider the mass-spring system with three carts illustrated in Figure 2.5 in the case where m = k1 = k2 = k3 = k4 = 1. Show that the motion of this system is described by the matrix differential equation −2 1 0 d2 x = Ax, where A = 1 −2 1 . dt2 0 1 −2 52 Figure 2.6: Three carts connected by springs and moving along a friction-free track. b. Find...
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