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 massspring 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 massspring system?
2.4.2.a. Consider the massspring 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 frictionfree
track.
b. Find...
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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