e motion of a damped mass-spring system initially at rest

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Consider the initial value problem
тx" + сx' + 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).
8 kilograms per second, k = 80 Newtons
2 kilograms, c=
per meter, and F(t) = 40 sin(6t) Newtons.
Assume that m =
Solve the initial value problem.
e^(-2t)((3/10)cos(6t)-(13/30)sin(6t))-(3cos(8t))/10+
(2sin(18t))/5
æ(t) =
help (formulas)
Determine the long-term behavior of the system (steady periodic solution). 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.
For very large positive values of t,
x(t) 2 xsp(t) =
(-3cos(8t))/10+(2sin(8t))/5
help (formulas)
Transcribed Image Text:Consider the initial value problem тx" + сx' + 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). 8 kilograms per second, k = 80 Newtons 2 kilograms, c= per meter, and F(t) = 40 sin(6t) Newtons. Assume that m = Solve the initial value problem. e^(-2t)((3/10)cos(6t)-(13/30)sin(6t))-(3cos(8t))/10+ (2sin(18t))/5 æ(t) = help (formulas) Determine the long-term behavior of the system (steady periodic solution). 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. For very large positive values of t, x(t) 2 xsp(t) = (-3cos(8t))/10+(2sin(8t))/5 help (formulas)
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