Forced Oscillations (proving diff equation by subsitution)

[hjr]

Homework Statement

By substituting the proper equations I showed that the equation is right when time = phi/w.
Now when I make cos = o and sin = 1 and time = (pi/2 - phi)/w I can't solve the equation.

Homework Equations

If you need to see all the equations i can give it to you but I am pretty sure at this stage they are not needed. The original diff equation is:

m*dx^2 + b* dx + kx = F_nat*cos(wt)


with x equal to:

Asin(wt + phi)

A = F-nat/ mh where h = $$\sqrt{(\omega^{2}-\omega_{0}^{2})^{2} + b^{2}\omega^{2}/m^{2}}$$

tan($$\phi$$) = $$\frac{ (\omega^{2}-\omega_{0}^{2})}{\omega(b/m)}$$

The Attempt at a Solution

when time = (pi/2 - phi)/w

i got:

F-nat/m * (k-mw^2) = f-nat * (w-nat^2 - w^2)

in my book in a different example you can solve for k and get mw^2 but then that side will be zero. But then the natural frequency has to equal the external forces frequency to make that side zero. I just need a hint. Sorry if this is all a mess. If you need anything clarify I will try my best to do it.

 

[rock.freak667]

I am not too entirely sure as to what you are looking to find. Could you please clarify a bit?

 

[ehild]

The solution is x=Asin(wt + phi) if A and phi have the appropriate values which depend on the parameters of the diving force, w and F_nat. You can determine these values by substituting x(t) and its derivatives into the differential equation.This equation has to be true for all values of t. Substitute t=-phi/w, (it is minus!) you get a relation between A, phi and w and F_nat. Substitute t=pi/2-phi/w, you get a second equation. You can solve this system of equations for A and phi in terms of w and F_nat.

ehild

 

[hjr]

Yeah i should of been more clearer, A and phi were already given. I have to prove that the equation is valid for any time t. I did it for t = -phi/w which I was able to show/prove that it does work. Now when i do t=(pi/2-phi/w) i get the solution above but I can't figure out how to get rid of k. The only solution I could think of was that it must be zero on both sides. I am going to edit the question to make it clearer.

 

[hjr]

yea so these are the equations i was given. When i make sin = 0 i can make it work but when cos = o i can't figure out how to get rid of k.

 

[ehild]

What do you mean on w-nat? Or is it w<sub>0<sup>?

As for k, you know that w02=k/m. If you mean f-nat the same as F-nat, the two sides of the equation are equivalent .ehild</sup></sub>