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

In summary, the conversation is about a question regarding a damped oscillator and its solution. The solution involves equations for position, velocity, and phase, and the goal is to find the values for A, B, and φ. The conversation includes attempts at solving the problem and requesting help.
  • #1
burock
3
0
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...
 
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  • #2
I couldn't write B exactly. Rigth B is that :B = [(beta times Xo / 2) + Vo] / w'

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

[tex]A\cos \omega t + B \sin\omega t = R\cos(\omega t-\phi)[/tex]
 
  • #5



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
 

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

1. How does damping affect the behavior of an oscillator?

Damping is a force that acts to decrease the amplitude of oscillations in a system. In an oscillator, damping can cause the amplitude of the oscillations to decrease over time, eventually leading to the system reaching equilibrium and coming to rest.

2. What is the equation for a damped oscillator?

The equation for a damped oscillator is: x(t) = A*e^(-bt)*cos(wt + φ), where x(t) is the displacement of the oscillator at time t, A is the initial amplitude, b is the damping coefficient, w is the angular frequency, and φ is the initial phase angle.

3. How does the damping coefficient affect the motion of a damped oscillator?

The damping coefficient, b, determines the rate at which the amplitude of the oscillations decreases. A larger damping coefficient leads to a faster decrease in amplitude and a shorter period of oscillation, while a smaller damping coefficient leads to a slower decrease in amplitude and a longer period of oscillation.

4. Can a damped oscillator ever reach a state of sustained oscillation?

Yes, a damped oscillator can reach a state of sustained oscillation if the damping coefficient is small enough. In this case, the system will continue to oscillate with a constant amplitude and frequency, but it will take longer for the amplitude to decrease to zero compared to a system with a larger damping coefficient.

5. How is damping in an oscillator related to energy dissipation?

Damping in an oscillator is related to energy dissipation because as the amplitude of oscillations decreases, the kinetic energy of the system is converted into other forms of energy, such as heat or sound. This process of energy dissipation is what ultimately causes the oscillator to reach equilibrium and stop oscillating.

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