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^{-[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,he^{i[tex]\omega[/tex]'t}

where [tex]\hat{x}[/tex]_o,h = |[tex]\hat{x}[/tex]_o,h| e^{i[tex]\phi[/tex]}.

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


A = x_o

B = \frac{\frac{x_o\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
e^{i[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...

 

[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

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

 

[rwisz]

Hello,

Thank you for your question about the damped oscillator. This is a common topic in oscillation and can be a bit challenging to understand at first. Let me try to provide some guidance and explanation for the equations you have listed.

Firstly, the given solution for the damped oscillator is correct. It represents the position of the oscillator as a function of time, where A and B are constants and \hat{x}o,h is the amplitude of the oscillation. The exponential term represents the damping effect, with \beta being the damping coefficient.

Now, to solve for A and B, we can use the initial conditions given in the problem. At t = 0, the oscillator is at position xo and has a velocity of \varthetao. This means that x(0) = xo and x'(0) = \varthetao. If we substitute these values into the given solution for x, we get:

x(0) = e^0(Acos(0) + Bsin(0)) = A = xo

x'(0) = e^0(-\beta/2)(Acos(0) + Bsin(0)) + e^0(A(-\omega')sin(0) + B(\omega')cos(0)) = -\beta/2(A) + \omega'B = \varthetao

Substituting A = xo, we get:

-\beta/2(xo) + \omega'B = \varthetao

Solving for B, we get:

B = (\frac{\beta}{2}xo + \varthetao)/\omega'

Now, to find the amplitude \hat{x}o,h, we can use the Pythagorean theorem:

|\hat{x}o,h| = \sqrt{A^2 + B^2}

Substituting A = xo and B from above, we get:

|\hat{x}o,h| = \sqrt{xo^2 + (\frac{\beta}{2}xo + \varthetao)^2}/\omega'

Finally, to find the phase angle \phi, we can use the tangent function:

tan\phi = \frac{B}{A} = \frac{\frac{\beta}{2}xo + \varthetao}{xo}

I hope this helps in solving the problem. Don't worry if it takes some time to fully understand the damped oscillator, it can be a complex