# Resonance in forced oscillations

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1. Dec 1, 2016

### green-beans

1. The problem statement, all variables and given/known data
Consider the differential equation:
mx'' + cx' + kx = F(t)
Assume that F(t) = F_0 cos(ωt).
Find the possible choices of m, c, k, F_0, ω so that resonance is possible.
2. Relevant equations

3. The attempt at a solution
I know how to deal with such problem when there is no damping, i.e. when we ignore cx'. In this case if cx'=0, resonance occurs when ω=k/m. However, now I have damping and as far as I understand resonance will be possible in case of underdamping, so when the complemtary function has complex solutions. My reasoning behind this was that in underdamping we get increasing oscillations (which is kind of the case with resonance). So, I solved the equation with the case of underdamping and resonance should occur when the denominator in my general solution is equal to 0. From what I got if (k-ω2m)2 + ω2c2=0 then resonance should take place. But this gives the relation between the unknowns and it does not take F_0 into consideration. So, I am not sure if I am right to assume ubderdaming and whether the solution is in fact the dependence relation.
P.S. I suspect this is wrong since I did not get a relation for F_0
Thanks a lot!

2. Dec 1, 2016

### RUber

I don't think F_0 should play, since the resonance should only appear when the right hand side is oscillating at the same rate as the homogeneous solution.

3. Dec 1, 2016

### rude man

Obviously F_0 does not come into play, it's just a scaling coefficient.
With damping your denominator will never be zero. The idea is to find the "natural" oscillation frequency with damping [which is < √(k/m)], then make that equal to ω. That give you max. oomph per unit F_0.