Consider the initial value problem 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, c = 8 kilograms per second, k = 80 Newtons per meter, and the applied force in Newtons is my" + cy' + ky = F(t), y(0) = 0, y'(0) = 0 if 0 < t < π/2, [20 F(t) = 10 if t > π/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. help (formulas) For 0 ≤t≤ π/2, y(t) For tπ/2, y(t) = 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 t-∞0 values of t. = For very large positive values of t, y(t) ~ help (formulas) help (formulas)
Consider the initial value problem 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, c = 8 kilograms per second, k = 80 Newtons per meter, and the applied force in Newtons is my" + cy' + ky = F(t), y(0) = 0, y'(0) = 0 if 0 < t < π/2, [20 F(t) = 10 if t > π/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. help (formulas) For 0 ≤t≤ π/2, y(t) For tπ/2, y(t) = 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 t-∞0 values of t. = For very large positive values of t, y(t) ~ help (formulas) 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
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, c = 8 kilograms per second, k = 80 Newtons per meter, and the applied force in Newtons is
if 0 < t < π/2,
if t > π/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 ≤ t ≤ π/2, y(t)
For tπ/2, y(t)
my" + cy' + ky = F(t), y(0) = 0, y'(0) = 0
=
=
F(t) =
20
0
help (formulas)
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.
t→∞0
For very large positive values of t, y(t) ≈
help (formulas)
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