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Unformatted text preview: cos ω1 t + b1 sin ω1 t
a2 cos ω2 t + b2 sin ω2 t
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
2π and ω2
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
A = 1 −2 1 .
52 Figure 2.6: Three carts connected by springs and moving along a friction-free
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- Winter '14