(1 point) 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 = 2 kilograms, c = 8 kilograms per second, k = 80 Newtons per meter, and the applied force in Newtons is 30 if 0 ≤t≤/2, F(t) 0 ift > 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 ≤ts/2, y(t) Fortπ/2, y(t) = 0 (-3/8)(e^(-21))cos(6t)-(1/8)(e^(-2t))sin(6t)+3/8 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-∞ For very large positive values of t, y(t) ≈0 help (formulas)
(1 point) 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 = 2 kilograms, c = 8 kilograms per second, k = 80 Newtons per meter, and the applied force in Newtons is 30 if 0 ≤t≤/2, F(t) 0 ift > 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 ≤ts/2, y(t) Fortπ/2, y(t) = 0 (-3/8)(e^(-21))cos(6t)-(1/8)(e^(-2t))sin(6t)+3/8 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-∞ For very large positive values of t, y(t) ≈0 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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Question
Transcribed Image Text:(1 point) 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 = 2 kilograms, c = 8 kilograms per second, k = 80 Newtons per meter, and the applied force in Newtons is
30
if 0 ≤t≤/2,
F(t)
0
ift > 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 ≤ts/2, y(t)
Fortπ/2, y(t) = 0
(-3/8)(e^(-21))cos(6t)-(1/8)(e^(-2t))sin(6t)+3/8
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-∞
For very large positive values of t, y(t) ≈0
help (formulas)
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