Consider the initial value problemтx" + сx' + kxF(t), x(0) = 0, x'(0) = 0modeling the motion of a damped mass-spring system initially at rest andsubjected to an applied force F(t), where the unit of force is the Newton (N).8 kilograms per second, k = 80 Newtons2 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))/5æ(t) =help (formulas)Determine the long-term behavior of the system (steady periodic solution). Islim x(t) = 0? If it is, enter zero. If not, enter a function that approximates x(t)forvery large positive values of t.For very large positive values of t,x(t) 2 xsp(t) =(-3cos(8t))/10+(2sin(8t))/5help (formulas)

The given initial value problem is mx''+cx'+kx=F(t), x(0)=0, x'(0)=0 The given values are m=2 kg,…

Answered: e motion of a damped mass-spring system…

Math

Advanced Math

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

Section: Chapter Questions

Problem 1RQ

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

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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