Consider the initial value problem my" + cy' + ky=F(t). (0)-0, y'(0)=0 modeling the motion of a spring-mass-dashpot system initially at rest and subjected to an applied force F(t), where the unit of force is the Newton (N). Assume that m=2 kilograms, e 8 kilograms per second, k=80 Newtons per meter, and the applied force in Newtons is if0 ≤t≤ x/2, ift > x/2. a. Solve the initial value problem, using that the displacement y(t) and velocity y'() remain continuous when the applied forse is discontinuous (32)-3 cos (6)-sin(60)) help (formulas) For 0 ≤t≤/2, y(t)- (se(-21) 24 (50 F(t)= {o Fort >=/2, y(t) help (formulas) b. Determine the long-term behavior of the system. is lim y(t)=0? if it is, enter zero. If not, enter a function that approximates y(t) for very large positive values of t For very large positive values of f, y(t)help (formulas)
Consider the initial value problem my" + cy' + ky=F(t). (0)-0, y'(0)=0 modeling the motion of a spring-mass-dashpot system initially at rest and subjected to an applied force F(t), where the unit of force is the Newton (N). Assume that m=2 kilograms, e 8 kilograms per second, k=80 Newtons per meter, and the applied force in Newtons is if0 ≤t≤ x/2, ift > x/2. a. Solve the initial value problem, using that the displacement y(t) and velocity y'() remain continuous when the applied forse is discontinuous (32)-3 cos (6)-sin(60)) help (formulas) For 0 ≤t≤/2, y(t)- (se(-21) 24 (50 F(t)= {o Fort >=/2, y(t) help (formulas) b. Determine the long-term behavior of the system. is lim y(t)=0? if it is, enter zero. If not, enter a function that approximates y(t) for very large positive values of t For very large positive values of f, y(t)help (formulas)
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:Consider the initial value problem
my" + cy' + ky = F(t). (0) 0, y'(0) = 0
modeling the motion of a spring-mass-dashpot system initially at rest and subjected to an applied force F(t), where the unit of force is the Newton (N). Assume that m=2
kilograms, e 8 kilograms per second, k80 Newtons per meter, and the applied force in Newtons is
if 0 ≤ t ≤ x/2,
(50
{o
ift > */2.
a. Solve the initial value problem, using that the displacement y(t) and velocity y'(t) remain continuous when the applied force is discontinuous
For 0 str/2, y(t)
Se(-21)
24
F(t)=
3e (2)-3 cos (6r)-sin(6r)) help (formulas)
Fort> /2, y(t)-help (formulas)
b. Determine the long-term behavior of the system. Is lim y(t) 0? if it is, enter zero, If not, enter a function that approximates y(t) for very large positive values of t
1-400
For very large positive values of t, y(t) 0 help (formulas)
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