- #1
PsychonautQQ
- 784
- 10
Hey PF. This isn't a homework question and I'm hoping this is the right place to ask it, sorry if it isn't!
In the case of a weakly damped harmonic oscillator driven by a sinusoidal force of the form Fe^(iwt). The form of the differential force equation of motion is then given by ma + cv +kx = Fe^(iwt).
The solution to this equation is given as x=Ae^i(wt-y) where y is angular displacement (analogous to theta). We then solve for v and a and plug the values into the force equation and proceeding to set the real and imaginary parts of said force equation equal to each other and then diving those equations by one another.
The result, using the supstitution of c = 2um where m is mass and u is a variable that scales proportionally to the drag coefficient (which is c), can take the form of giving y ( angular deplacement) as a function of angular velocity (w). The equation looks like this:
tan(y) = 2uw / (W^2 - w^2) where W is the undamped nondriven angular velocity given by the equation (k/m)^(1/2) and w is the angular velocity term from the driven force and solution to the differential equation.
Okay... I hope I explained that well enough.. Hoping someone out there is smart enough to know what I'm talking about even though I don't know what I'm talking about lol...
Anyhoo, my question is with understanding the equation. why is y (angular displacement) dependent on the angular velocity at all? Wouldn't angular displacement be a stand alone variable? The way we derived the equation of angular displacement as a function of angler velocity was having the angular displacement term originally take the form of the initial displacement. Why is this dependent on w!? How is it not a stand alone variable? Sorry if this was really repetitive trying to be as clear as possible.
In the case of a weakly damped harmonic oscillator driven by a sinusoidal force of the form Fe^(iwt). The form of the differential force equation of motion is then given by ma + cv +kx = Fe^(iwt).
The solution to this equation is given as x=Ae^i(wt-y) where y is angular displacement (analogous to theta). We then solve for v and a and plug the values into the force equation and proceeding to set the real and imaginary parts of said force equation equal to each other and then diving those equations by one another.
The result, using the supstitution of c = 2um where m is mass and u is a variable that scales proportionally to the drag coefficient (which is c), can take the form of giving y ( angular deplacement) as a function of angular velocity (w). The equation looks like this:
tan(y) = 2uw / (W^2 - w^2) where W is the undamped nondriven angular velocity given by the equation (k/m)^(1/2) and w is the angular velocity term from the driven force and solution to the differential equation.
Okay... I hope I explained that well enough.. Hoping someone out there is smart enough to know what I'm talking about even though I don't know what I'm talking about lol...
Anyhoo, my question is with understanding the equation. why is y (angular displacement) dependent on the angular velocity at all? Wouldn't angular displacement be a stand alone variable? The way we derived the equation of angular displacement as a function of angler velocity was having the angular displacement term originally take the form of the initial displacement. Why is this dependent on w!? How is it not a stand alone variable? Sorry if this was really repetitive trying to be as clear as possible.