Damped oscillator question

  • Thread starter burock
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  • #1
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Hi,
I have a question about damped oscillator. Actually, although I have read courses about oscillator, I couldn't solve this. I think this is very easy question :(

1. Homework Statement

Consider the solution for the damped ( but not driven ) oscillator,

x = e-[tex]\beta[/tex]t/2(Acos[tex]\omega[/tex]'t + Bsin[tex]\omega[/tex]'t)

= Re e-[tex]\beta[/tex]t/2[tex]\hat{x}[/tex]o,hei[tex]\omega[/tex]'t

where [tex]\hat{x}[/tex]o,h = |[tex]\hat{x}[/tex]o,h| ei[tex]\phi[/tex].

If the oscillator is at the position xo with velocity [tex]\vartheta[/tex]o at time t = 0, show that


A = xo

B = \frac{\frac{xo\beta}{2} + [tex]\vartheta[/tex]o}{[tex]\omega[/tex]'}

|[tex]\hat{x}[/tex]o,h| = [tex]\sqrt{A2 + B2}[/tex]

tan [tex]\phi[/tex] = -[tex]\frac{B}{A}[/tex]

2. Homework Equations

I know that
ei[tex]\phi[/tex] = cos [tex]\phi[/tex] + isin[tex]\phi[/tex]
[tex]\omega[/tex]'2 = [tex]\omega[/tex]o2 - [tex]\beta[/tex]2/4

3. The Attempt at a Solution

I tried to show the third equation. So I put A2 and B2 to the square root. And I changed [tex]\omega[/tex]'2 to [tex]\omega[/tex]o2 - [tex]\beta[/tex]2/4. But I couldn't reach the solution. Also I couldn't find A or B.


This is the first time I am using Latex. I hope I did no mistake.

Thanks for helping...
 

Answers and Replies

  • #2
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I couldn't write B exactly. Rigth B is that :


B = [(beta times Xo / 2) + Vo] / w'

I hope it is clear
 
  • #3
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Is there anyone who can help me?
 
  • #4
vela
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Start by finding R and ϕ such that

[tex]A\cos \omega t + B \sin\omega t = R\cos(\omega t-\phi)[/tex]
 

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