# Damped oscillation equation

1. Apr 5, 2014

### Maxo

1. The problem statement, all variables and given/known data
The equation for a damped oscillation is $$y(t)=Ae^{-\frac{b}{2m}t}cos(\omega't + \phi)$$
We know that y(0)=0.5 and y'(0)=0.

Find the values of A and ø and then plot the oscillation in MATLAB.

2. Relevant equations
See above

3. The attempt at a solution
When derivating y(t) we get

$$y'(t)=Ae^{-\frac{b}{2m}t}(-\frac{b}{2m}cos(\omega't+\phi)-\omega'sin(\omega't + \phi))$$
This, together with the initial values gives that $$A=\frac{1}{2cos(\phi)}$$ and $$\phi=arctan(-\frac{b}{2m\omega'})$$

Is this correct? If not, then what is wrong?

Last edited: Apr 5, 2014
2. Apr 5, 2014

### TSny

This looks correct to me. As a check, you should see that your plot of y(t) satisfies the initial conditions.

3. Apr 5, 2014

### Maxo

Ok, good. Then in other words, we get the following equation: $$y(t)=\frac{1}{2cos(arctan(-\frac{b}{2m\omega'}))}e^{-\frac{b}{2m}t}cos(\omega't + arctan(-\frac{b}{2m\omega'}))$$
Right?

Now I wonder, how can we plot this in MATLAB? Can it be written just like it's written here? Because when I try I don't get a damped oscillation, but an undamped one. What is wrong then?

Last edited: Apr 5, 2014
4. Apr 5, 2014

### TSny

Yes.

I would need to see your code. I don't have MATLAB, but when I plot the function using other software, it looks OK.

5. Apr 5, 2014

### Maxo

Could you please post your code for this? I will try some more and if I still don't manage I will post my code afterwards.