Consider the initial value problem mx" + cx' + kx = F(t), x(0) = 0, x'(0) = 0 modeling the motion of a damped mass-spring 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 = 10 Newtons per meter, and F(t) = -130 sin(2t) Newtons. Solution. Find the complementary solution, using C1, C2 for arbitrary constants if needed. xc= e-21 (C1 cos(t) + C2 sin(t)) help (formulas) Guess the form for the steady periodic solution, using A, B for arbitrary constants if needed. Find the steady periodic solution. x sp = A cos(2t) + B sin(2t) help (formulas) x sp 8 cos(2t) sin (2t) help (formulas) Express the steady periodic solution in the form xsp = C cos(wt a) with a = [0,2π). x sp √65 • COS 2 t- (1)) help (formulas) Find the general solution. x = help (formulas) Determine the long-term behavior of the system. Is lim x(t) =0? If it is, enter zero. If not, enter a function that approximates x(t) for very large positive values of t. 0047 For very large positive values of t, x(t)≈ help (formulas)
Consider the initial value problem mx" + cx' + kx = F(t), x(0) = 0, x'(0) = 0 modeling the motion of a damped mass-spring 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 = 10 Newtons per meter, and F(t) = -130 sin(2t) Newtons. Solution. Find the complementary solution, using C1, C2 for arbitrary constants if needed. xc= e-21 (C1 cos(t) + C2 sin(t)) help (formulas) Guess the form for the steady periodic solution, using A, B for arbitrary constants if needed. Find the steady periodic solution. x sp = A cos(2t) + B sin(2t) help (formulas) x sp 8 cos(2t) sin (2t) help (formulas) Express the steady periodic solution in the form xsp = C cos(wt a) with a = [0,2π). x sp √65 • COS 2 t- (1)) help (formulas) Find the general solution. x = help (formulas) Determine the long-term behavior of the system. Is lim x(t) =0? If it is, enter zero. If not, enter a function that approximates x(t) for very large positive values of t. 0047 For very large positive values of t, x(t)≈ help (formulas)
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.7: Applications
Problem 18EQ
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Transcribed Image Text:Consider the initial value problem
mx" + cx' + kx = F(t), x(0) = 0, x'(0) = 0
modeling the motion of a damped mass-spring 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 = 10 Newtons per meter, and F(t) = -130 sin(2t) Newtons.
Solution. Find the complementary solution, using C1, C2 for arbitrary constants if needed.
xc=
e-21 (C1 cos(t) + C2 sin(t)) help (formulas)
Guess the form for the steady periodic solution, using A, B for arbitrary constants if needed.
Find the steady periodic solution.
x sp
=
A cos(2t) + B sin(2t) help (formulas)
x sp
8 cos(2t) sin (2t)
help (formulas)
Express the steady periodic solution in the form xsp = C cos(wt a) with a = [0,2π).
x sp
√65
• COS
2 t-
(1))
help (formulas)
Find the general solution.
x =
help (formulas)
Determine the long-term behavior of the system. Is lim x(t) =0? If it is, enter zero. If not, enter a function that approximates x(t) for very large positive values of t.
0047
For very large positive values of t,
x(t)≈
help (formulas)
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