Homework Help: Forced Damped Oscillator

1. Jan 26, 2010

gpax42

1. The problem statement, all variables and given/known data

Find the frequency that gives the maximum amplitude response for the forced damped oscillator d$$^{2}$$x/dt$$^{2}$$ + 6dx/dt + 45x = 50cos($$\omega$$t)

2. Relevant equations

I'm really confused by this problem, but I know that the amplitude can be found by taking the $$\sqrt{c_{1}^2+c_{2}^2}$$ with c$$_{1}$$ and c$$_{2}$$ being parameters of the general solution...

3. The attempt at a solution

I suppose I want to maximize my c$$_{1}$$ and c$$_{2}$$ values. And this can be done by modifying the value of $$\omega$$. So, my only guess as to how I could solve this problem is through manipulation of the Method of Undetermined Coefficients, and see for what values of $$\omega$$ my c$$_{1}$$ and c$$_{2}$$ become largest...

If anyone could offer me any suggestions involving different strategies for solving this problem, i would greatly appreciate it

the superscripts above some of my "c" parameters should be subscripts, i'm not sure why they keep getting turned into superscripts, sorry =(

Last edited: Jan 26, 2010
2. Jan 27, 2010

tiny-tim

Hi gpax42!

(have an omega: ω and a square-root: √ and try using the X2 and X2 tags just above the Reply box )

What do you have as the general solution for the full equation (ie, including ω)?

3. Jan 27, 2010

gpax42

thats my first gray area... im fine with the general solution of the homogenous equation, but I can't use the method of undetermined coefficients to solve the full equation seeing that $$\omega$$ isn't a constant

the general solution for the homogenous part is e$$^{-3t}$$(C_1*cos(6t) + C_2*sin(6t))

4. Jan 27, 2010

tiny-tim

(what happened to that ω i gave you? and try using the X2 and X2 tags just above the Reply box )

Yes, your general solutiuon is correct.

Now look for a particular solution of the form Acosωt + Bsinωt.

(and ω is constant … it's a constant you can choose, but once you choose it it's constant)

5. Jan 27, 2010

gpax42

taking that trial solution and its respective first and second derivatives and plugging those back into the original oscillation equation i get...

-ω$$^{2}$$Acos(ωt)-ω$$^{2}$$Bsin(ωt)-ω6Asin(ωt)+ω6Bcos(ωt)+45Acos(ωt)+45Bsin(ωt) = 50cos(ωt)

when i isolate out the common terms I'm left with the equations...

-ω$$^{2}$$A+6ωB+45A = 50

-ω$$^{2}$$B-6ωA+45B = 0

....

I then tried solving or A and B in terms of ω but got ridiculous solutions.
Is it simply supposed to be A = 50/45 = 10/9 and B = 0

which would give me a particular solution of

$$\frac{10}{9}$$cos(ωt) ?

6. Jan 27, 2010

tiny-tim

Hi gpax42!
Yes, the solution for A and B is pretty horrible …

but if you look at http://en.wikipedia.org/wiki/Damped_harmonic_oscillator#Sinusoidal_driving_force",

you'll find that Zm2 = ((45 - ω2)2 + 36ω2)/ω2,

which is the denominator of A and B …

so I think it is correct.

Last edited by a moderator: Apr 24, 2017
7. Jan 27, 2010

gpax42

ahhh, i understand the problem completely now ... thank you very much for all your help tiny tim!