Resonance in a damped triangular potential well

In summary: A. The period is then just the time it takes for the particle to go from x=-B/A to x=0. Since the particle is bouncing off the ground, I'm assuming that the surface reaction acts like an elastic damper, damping the accelerating and decelerating motion of the particle. If you want to investigate this more, you might want to look at the differential equation for a bouncing ball.The equivalent quality factor would be the quality of the driving force. It is not clear to me what you are asking.
  • #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?
 
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  • #2
I don't understand your equation. The periodicity you're searching for arises from the regular impulse the ball receives during its elastic collision with the ground, but I can't find a corresponding term in your equation of motion. I also don't believe the solution to an equation modelling a particle moving due to its own weight and air resistance is parabolic.
 
  • #3
The equation is valid only for x>0. I don't include the delta-like force of the surface reaction, i just assume that whenever the particle arrives at x=0, its speed flips sign. My equation of motion is
[tex]
x(t) = At^2 + Bt
[/tex]
so the particle starts out at x=0 with speed B, moves up, comes down to x=0 at time T=-B/A with speed -B, and then flips back to the initial condition
 

1. What is resonance in a damped triangular potential well?

Resonance in a damped triangular potential well refers to the phenomenon where a system oscillates with a larger amplitude at a specific frequency when driven by an external force. In this case, the system is a particle moving in a triangular potential well that is damped by some external force or friction.

2. How is resonance in a damped triangular potential well different from other types of resonance?

Resonance in a damped triangular potential well is unique because it occurs in a non-linear system. This means that the amplitude of the oscillations is not directly proportional to the frequency of the external force, unlike in linear systems. Additionally, the triangular potential well adds an asymmetry to the system, which can lead to interesting effects.

3. What factors affect the resonance frequency in a damped triangular potential well?

The resonance frequency in a damped triangular potential well is affected by several factors, including the strength of the damping force, the amplitude of the external force, and the shape and depth of the potential well. These factors can all influence the amplitude and frequency of the system's oscillations.

4. How is resonance in a damped triangular potential well used in practical applications?

Resonance in a damped triangular potential well has various applications, such as in mechanical systems, electrical circuits, and even musical instruments. For example, in a guitar, the strings vibrate in a triangular-shaped potential well created by the frets, which produces the desired sound. In mechanical systems, resonance can be both beneficial (e.g. in tuning forks) and detrimental (e.g. in bridge construction).

5. Can resonance in a damped triangular potential well be harmful?

Yes, resonance in a damped triangular potential well can be harmful in some cases. This is because the system can become unstable and exhibit very large amplitudes of oscillations, potentially causing damage or failure. This is why it is important to understand and control resonance in various systems to prevent any negative consequences.

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