Differential equations of forced oscillation and resonance

[MissP.25_5]

How do I derive A? As you can see in the attachment, I tried to substitute x and expand the equation but I got stuck. How do I get rid of the δ and cos and sin to get the result in the end? Please help!

 

[SteamKing]

I would leave (omega*t-delta) alone while calculating the various derivatives. The sines and cosines with this argument do not need expanding until you have done you calculations.

 

[MissP.25_5]

I would leave (omega*t-delta) alone while calculating the various derivatives. The sines and cosines with this argument do not need expanding until you have done you calculations.

OK, what do I do next? I don't expand it, and here's what I got.

 

[SteamKing]

Where did the plain 'ω' come from in the last line of your calculations?

Specifically, the term (ωe^2 - ω^2)?

 

[SteamKing]

Disregard last post. I see now.

 

[SteamKing]

OK, what do I do next? I don't expand it, and here's what I got.

Now you expand the cosine and sine terms and equate same to the RHS of the equation.
In your first attempt, you differentiated δ w.r.t. time. This was incorrect. The phase angle δ is constant w.r.t. time, which was one reason your original derivation got so unwieldy.

 

[MissP.25_5]

Now you expand the cosine and sine terms and equate same to the RHS of the equation.
In your first attempt, you differentiated δ w.r.t. time. This was incorrect. The phase angle δ is constant w.r.t. time, which was one reason your original derivation got so unwieldy.

So, you're saying that I don't have to bother the one with the delta? That would mean doing partial differentiation, right?

 

[SteamKing]

There is no partial differential involved. If you take the derivative of cos(ωt-δ), you will get -ω*sin(ωt-δ). The δ represents a constant phase angle; it is not a function of t.

Now that you have your LHS in terms of sin and cos, now is the time to expand, for instance, cos(ωt-δ) using the angle difference formulas. You then solve for the coefficients of the sine and cosine terms on the LHS which correspond to whatever sine and cosine terms you have on the RHS.

 

[MissP.25_5]

There is no partial differential involved. If you take the derivative of cos(ωt-δ), you will get -ω*sin(ωt-δ). The δ represents a constant phase angle; it is not a function of t.

Now that you have your LHS in terms of sin and cos, now is the time to expand, for instance, cos(ωt-δ) using the angle difference formulas. You then solve for the coefficients of the sine and cosine terms on the LHS which correspond to whatever sine and cosine terms you have on the RHS.

So you mean, equate LHS and RHS and then substitute them back into the equation? But the right hand side only has F/cos(ω_e*t).

 

[SteamKing]

Look at these notes:

http://web.pdx.edu/~larosaa/Ph-223/Lecture-Notes-Ph-213/PH-213_Chapter-15_FORCED_OSCILLATIONS_and-RESONANCE_%28complete-version%29.pdf

By the time you get to p. 6, you should see the method illustrated.

 

[MissP.25_5]

There is no partial differential involved. If you take the derivative of cos(ωt-δ), you will get -ω*sin(ωt-δ). The δ represents a constant phase angle; it is not a function of t.

Now that you have your LHS in terms of sin and cos, now is the time to expand, for instance, cos(ωt-δ) using the angle difference formulas. You then solve for the coefficients of the sine and cosine terms on the LHS which correspond to whatever sine and cosine terms you have on the RHS.

Ok, now what should I do? The right hand side only has cos(w_e*t). And even so, if I did equate the sin and cosine, what should I do with it? I would still have cos(w_e*t) on the right side, don't I?

 

[SteamKing]

If you read the attached notes from post#10, you would see how to handle this.

 

[MissP.25_5]

If you read the attached notes from post#10, you would see how to handle this.

Thanks for the post. That really hepls a ton. But my answer is a little different. How come my denominator and numerator are inverted?

 

[SteamKing]

It's hard to tell from your posted calculations. After expanding the cosine and sine terms which contain the phase angle δ, on the LHS there will be sin δ and cos δ terms mixed in with the cos(ωet) and sin (ωet) terms. You use the phase angle triangle to determine sin δ and cos δ in terms of the other known quantities before solving for A.

 

[MissP.25_5]

It's hard to tell from your posted calculations. After expanding the cosine and sine terms which contain the phase angle δ, on the LHS there will be sin δ and cos δ terms mixed in with the cos(ωet) and sin (ωet) terms. You use the phase angle triangle to determine sin δ and cos δ in terms of the other known quantities before solving for A.

But in my calculation, I already determined the sin δ and cos δ, see I drew the triangle?

 

[MissP.25_5]

It's hard to tell from your posted calculations. After expanding the cosine and sine terms which contain the phase angle δ, on the LHS there will be sin δ and cos δ terms mixed in with the cos(ωet) and sin (ωet) terms. You use the phase angle triangle to determine sin δ and cos δ in terms of the other known quantities before solving for A.

Hey, I got it!Look! Thank you soooo much!