(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the steady-state oscillation of the mass–spring system

modeled by the given ODE. Show the details of your

calculations.

2. Relevant equations

1. y'' + 6y' + 8y = 130 cos 3t

2. 4y’’ + 8y’ + 13y = 8 sin 1.5t

3. The attempt at a solution

1. cos(3t) at the end means the basic angular frequency is 3 radians per second. Hence the steady-state oscillation frequency is 3/2pi Hz = 0.477 Hz.

2. solve for homogeneous differential equation,

4y'' + 8y' + 13y = 0

propose y = e^(ct)

y' = ce^(ct)

y'' = c²e^(ct)

substitute into mass-spring motion equation,

4y'' + 8y' + 13y = 0

4c²e^(ct) + 8ce^(ct) + 13ce^(ct) = 0

e^(ct)(4c² + 8c + 13) = 0

of course for unique solution, it must be e^(ct) ≠ 0

4c² + 8c + 13 = 0

c = -1 ± (3i)/2

homogeneous solution is

y(t) = e^(-t) (A sin (3t/2) + B cos (3t/2))

where A and B is an arbitrary constants which dependent to boundary conditions

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# Homework Help: Find the steady-state oscillation of the mass–spring system modeled by the given ODE.

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