How Do You Solve a Damped Oscillator Problem with Initial Conditions?

[burock]

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^{-$$\beta$$t/2}(Acos$$\omega$$'t + Bsin$$\omega$$'t)

= Re e^{-$$\beta$$t/2}$$\hat{x}$$_o,he^{i$$\omega$$'t}

where $$\hat{x}$$_o,h = |$$\hat{x}$$_o,h| e^i$$\phi$$.

If the oscillator is at the position x_o with velocity $$\vartheta$$_o at time t = 0, show that


A = x_o

B = \frac{\frac{x_o\beta}{2} + $$\vartheta$$_o}{$$\omega$$'}

|$$\hat{x}$$_o,h| = $$\sqrt{A² + B²}$$

tan $$\phi$$ = -$$\frac{B}{A}$$

2. Homework Equations

I know that
e^i$$\phi$$ = cos $$\phi$$ + isin$$\phi$$
$$\omega$$'² = $$\omega$$_o² - $$\beta$$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 $$\omega$$'² to $$\omega$$_o² - $$\beta$$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...

 

[burock]

I couldn't write B exactly. Rigth B is that :B = [(beta times Xo / 2) + Vo] / w'

I hope it is clear

 

[burock]

Is there anyone who can help me?

 

[vela]

Start by finding R and ϕ such that

$$A\cos \omega t + B \sin\omega t = R\cos(\omega t-\phi)$$