Consider the initial value problem my??+cy?+ky=F(t), y(0)=0, y?(0)=0 modeling th
ID: 2858171 • Letter: C
Question
Consider the initial value problem
my??+cy?+ky=F(t), y(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=2kilograms, c=8 kilograms per second, k=80 Newtons per meter, and F(t)=40e?t Newtons.
Solve the initial value problem.
y(t)= _________
Determine the long-term behavior of the system. Is limt??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 t, y(t)? ____________
Explanation / Answer
(a)
Given F(t)=40e^(-t)
D.E :2y"+8y'+80y=40e^(-t)
=>y"+4y'+40y=20e^(-t)
take laplace transform over both sides
L.H.S={s²L(y)-sy(0)-y'(0)}+4{sL(y)-y(0...
R.H.S=20/(s+1)
=>L(y){s²+4s+40}=20/(s+1)
=>y=L^-1 [20/{(s+1)(s²+4s+40)}
20/{( s+1).(s²+4s+40)} = {20/(37(s+1)) - 20(s+3)/(37(s²+4s+40)) }
=20/(37(s+1)) -20(s+2)/[37{(s+2)²+36)}] -20/[37{(s+2)²+36}]
Hence y(t)=L^-1 [20/(37(s+1)) -20(s+2)/[37{(s+2)²+36)}] -20/[37{(s+2)²+36}]]
=20(e^(-t))/37 +(10/37 + 5i/111)e^(-2-6i)t + (10/37 - 5i/111)e^(-2-6i)t
(b) when t ->infinity
y(t)=0