Math 333 Logarithms Worksheet With Answers Page 5
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13B1. (B & D, 3.9, problem 17) Consider a vibrating system described by the initial value
problem
1
u +
u + 2u = 2 cos ωt,
u(0) = 0,
u (0) = 2.
4
(a) Determine the steady-state part of the solution of this problem.
Solution: Following the notation on p. 203 of B&D, we have m = 1, k = 2, so
ω
=
2, and γ = 1/4. From this we get
0
∆ =
(2
ω
2
)
2
+ ω
2
/16
With F
= 2, we can write the steady-state solution as
0
2
U (t) =
cos(ωt
δ)
(2
ω
2
)
2
+ ω
2
/16
where
2
2
ω
ω
cos δ =
sin δ =
(2
ω
2
)
2
+ ω
2
/16
4 (2
ω
2
)
2
+ ω
2
/16
(b) Find the amplitude A of the steady-state solution in terms of ω.
2
Solution: We have done this already; it’s
.
(2
ω
2
)
2
+ ω
2
/16
(c) Plot A as a function of ω.
Solution: Here is the picture:
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