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 orce in Newtons is 30 F(t) = - {3⁰ 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 ≤/2, y(t) = Fort > л/2, y(t) = b. Determine the long-term behavior of the system. Is lim y(t) = 0? If it is, enter 1-80 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) ≈
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 orce in Newtons is 30 F(t) = - {3⁰ 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 ≤/2, y(t) = Fort > л/2, y(t) = b. Determine the long-term behavior of the system. Is lim y(t) = 0? If it is, enter 1-80 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) ≈
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), 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
F(t) =
{30
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) =
Fort > л/2, y(t) =
b. Determine the long-term behavior of the system. Is lim y(t) = 0? If it is, enter
1→∞0
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) ~
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