- #1
Irid
- 207
- 1
I have a potential well which is an infinite wall for x<0 and a linear slope for x>0. There is damping proportional to velocity. Basically, it's a ball bouncing elastically off the ground and with air friction included. I wonder if there is some periodic driving force which will cause one particular mass to bounce with a high amplitude and only slightly perturbe all the other masses (kind of a pseudo-resonance?).
My first attempt was to try a sawtooth profile force. The equation of motion is thus
[tex]
m\ddot{x} = -k(u+\dot{x}) + f(T/2-t)
[/tex]
where k and u are constants, T is the period of the driving force and fT/2 is the maximum force. The solution is parabolic in time. I figured that the driving force will sync up with bouncing events iff
[tex]
f = \frac{uk^2}{m}
[/tex]
and so, if I could produce a force with such a slope, only a very particular mass m will oscillate with a high amplitude (which itself depends on the period
[tex]
x_0 = \frac{kuT^2}{8m}
[/tex]
).
To sum up, my line of thought is this: I fix the slope f according to the parameters of my system. The period and driving force amplitude are then determined by my choice of oscillation amplitude x_0.
Could you verify these calculations? Also, would the particle readily sync up with the driving force? Would this system be robust enough to work in a realistic experiment with various perturbations etc.? What would be the equivalent of the quality factor?
My first attempt was to try a sawtooth profile force. The equation of motion is thus
[tex]
m\ddot{x} = -k(u+\dot{x}) + f(T/2-t)
[/tex]
where k and u are constants, T is the period of the driving force and fT/2 is the maximum force. The solution is parabolic in time. I figured that the driving force will sync up with bouncing events iff
[tex]
f = \frac{uk^2}{m}
[/tex]
and so, if I could produce a force with such a slope, only a very particular mass m will oscillate with a high amplitude (which itself depends on the period
[tex]
x_0 = \frac{kuT^2}{8m}
[/tex]
).
To sum up, my line of thought is this: I fix the slope f according to the parameters of my system. The period and driving force amplitude are then determined by my choice of oscillation amplitude x_0.
Could you verify these calculations? Also, would the particle readily sync up with the driving force? Would this system be robust enough to work in a realistic experiment with various perturbations etc.? What would be the equivalent of the quality factor?