The periodic outer force F = sin wt is applied to the mass-spring system from Task 2. The equation becomes y" = -3y – cy' + sin wt, c > 0. (a) Find a periodic solution of this equation in the form y(t) = a cos wt + b sin wt (the answer will depend on c, w). Prove that the amplitude of this solution is ((3 – w²)² + c²w²)-0.5. (b) Fix c (you may assume that c is small) and find w = wmaz > 0 such that the amplitude of this periodic solution reaches its maximum at w. Comment: you will see that for larger c, the amplitude monotonically decreases as w increases, and thus there is no maximum point wmaz > 0. Hint: you may wish to denote v = w when searching for the maximum, but remember that v > 0. (c) For w = 1,c = 2, write out the general solution of this equation (cv. Task 2a) and show that all solutions tend to the periodic solution as t → +00.
The periodic outer force F = sin wt is applied to the mass-spring system from Task 2. The equation becomes y" = -3y – cy' + sin wt, c > 0. (a) Find a periodic solution of this equation in the form y(t) = a cos wt + b sin wt (the answer will depend on c, w). Prove that the amplitude of this solution is ((3 – w²)² + c²w²)-0.5. (b) Fix c (you may assume that c is small) and find w = wmaz > 0 such that the amplitude of this periodic solution reaches its maximum at w. Comment: you will see that for larger c, the amplitude monotonically decreases as w increases, and thus there is no maximum point wmaz > 0. Hint: you may wish to denote v = w when searching for the maximum, but remember that v > 0. (c) For w = 1,c = 2, write out the general solution of this equation (cv. Task 2a) and show that all solutions tend to the periodic solution as t → +00.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:The periodic outer force F = sin wt is applied to the mass-spring system from Task 2. The equation becomes
y" = -3y – cy' + sin wt,
c> 0.
(a) Find a periodic solution of this equation in the form y(t)
this solution is ((3 – w²)² + c²w²)-0.5
= a cos wt +b sin wt (the answer will depend on c, w). Prove that the amplitude of
(b) Fix c (you may assume that c is small) and find w = wmax > 0 such that the amplitude of this periodic solution reaches its maximum at w.
Comment: you will see that for larger c, the amplitude monotonically decreases as w increases, and thus there is no maximum point wmgr > 0.
Hint: you may wish to denotev = w when searching for the maximum, but remember that v > 0.
(c) For w = 1, c= 2, write out the general solution of this equation (cv. Task 2a) and show that all solutions tend to the periodic solution as
t → +0.
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