I was hoping that someone could explain why these different equations can be found from different sources please.
The time dependent position, x(t), of an underdamped harmonic oscillator is given by:
[itex]x(t)=e^{-\gamma t}acos(\omega_{1}t-\alpha)[/itex]
where [itex]\gamma[/itex] is the damping coefficient, and [itex]\omega_{1}[/itex] is the frequency of the damped oscillator. This is the equation I am familiar with and can be found in many explanations of harmonic oscillators, e.g. here.
In contrast, Wolfram gives the following equation (note that this is copied from their website, and it uses different notation to the equation above. In particular [itex]\gamma[/itex] is NOT the damping coefficient):
[itex]x(t)=e^{-\frac{\beta t}{2}}[Acos(\gamma t)+Bsin(\gamma t)][/itex]
I believe that changing this equation to use the same notation as the first thing I posted gives this:
[itex]x(t)=e^{-\gamma t}[Acos(\omega_{1}t)+Bsin(\omega_{1}t)][/itex]
So here is what I'd like to understand:
1) Why are these different? I tried to see if there was a case when [itex]A>>B[/itex], which I found happens if the initial velocity is zero, [itex]\dot{x}(0)=0[/itex], and [itex]\zeta<<\frac{1}{\sqrt{2}}[/itex]. Where [itex]\zeta[/itex] is the damping ratio. In other words, when there is very little damping the Wolfram equation reduces to the shorter equation. Is that correct?
2) Whether or not that's correct, I don't understand why there would sometimes be two terms in the equation. What does that physically represent?
3) I actually started looking at this after getting confused about the overdamped case in a similar way. It also has two terms, and I believe that it reduces to a single term when there is a lot of damping ([itex]\zeta>>1[/itex]). Again, what does it physically mean when there are two terms? Although hopefully understanding one case will lead to an understanding of both, so maybe this question is redundant.
Thanks for any help!
The time dependent position, x(t), of an underdamped harmonic oscillator is given by:
[itex]x(t)=e^{-\gamma t}acos(\omega_{1}t-\alpha)[/itex]
where [itex]\gamma[/itex] is the damping coefficient, and [itex]\omega_{1}[/itex] is the frequency of the damped oscillator. This is the equation I am familiar with and can be found in many explanations of harmonic oscillators, e.g. here.
In contrast, Wolfram gives the following equation (note that this is copied from their website, and it uses different notation to the equation above. In particular [itex]\gamma[/itex] is NOT the damping coefficient):
[itex]x(t)=e^{-\frac{\beta t}{2}}[Acos(\gamma t)+Bsin(\gamma t)][/itex]
I believe that changing this equation to use the same notation as the first thing I posted gives this:
[itex]x(t)=e^{-\gamma t}[Acos(\omega_{1}t)+Bsin(\omega_{1}t)][/itex]
So here is what I'd like to understand:
1) Why are these different? I tried to see if there was a case when [itex]A>>B[/itex], which I found happens if the initial velocity is zero, [itex]\dot{x}(0)=0[/itex], and [itex]\zeta<<\frac{1}{\sqrt{2}}[/itex]. Where [itex]\zeta[/itex] is the damping ratio. In other words, when there is very little damping the Wolfram equation reduces to the shorter equation. Is that correct?
2) Whether or not that's correct, I don't understand why there would sometimes be two terms in the equation. What does that physically represent?
3) I actually started looking at this after getting confused about the overdamped case in a similar way. It also has two terms, and I believe that it reduces to a single term when there is a lot of damping ([itex]\zeta>>1[/itex]). Again, what does it physically mean when there are two terms? Although hopefully understanding one case will lead to an understanding of both, so maybe this question is redundant.
Thanks for any help!