What is the optimal frequency for a forced damped oscillator?

[gpax42]

Homework Statement

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)

Homework 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...

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 =(

 

[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)?

 

[gpax42]

thats my first gray area... I am 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))

 

[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)

 

[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) ?

 

[tiny-tim]

Hi gpax42!

-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.

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 Z_m² = ((45 - ω²)² + 36ω²)/ω²,

which is the denominator of A and B …

so I think it is correct. ​

 

[gpax42]

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