Analogy to Damped , Driven Oscillator.

[qspeechc]

Hi everyone. I have a project where I need to find a situation this is, or is similar to, a damped oscillator. That is, the Differential Equation (DE) for the system must follow:

x'' + ax' + bx = 0

And, further, it must have some situation corresponding to being 'driven' or 'forced', that is:

x'' + ax' + bx = F

Where F is some term (analogous to a driving force in a mechanical system, which could be a constant, or F=F(x) or F=F(t) or F=F(x,t).

For instance, for a mass on a spring, if x is the displacement, ax' would be the damping term (or a(x')^2), and bx the restoring force. F could be some sinusoidal driving force F(t).

But, I do not want to do the spring on a mass. And not an oscillating electrical circuit either. I though of doing a planet in orbit, moving through some resistive medium, but I cannot think of a driving force. Perhaps another planet coming close to it in its orbit? Would this work? I also thought of a spherical pendulum with two degrees of freedom, but again couldn't think how a driving force comes into it.
Maybe a small ball oscillating (rolling) in a hemi-sphere? If you spin the hemi-sphere does that act as a driving force or not? I think that will only contribute to the damping term, and not correspond to a driving force.

I also thought of doing something from biology. Say, your body's response to a meal. Your blood-sugar levels must decay exponentially right? Or an infection. 'x' could represent the number of bacteria or viruses, then the 'damping' term could be your immune response, and the 'driving force' a drug your taking, but what would the restoring force be (the term proportional to x, bx)?

I am looking for anything that can be modeled by the first DE, and has something corresponding to being driven, that is, the non-homogeneous form,with the "F" on the RHS. It can be from physics, biology, economics, whatever. However, I have only studied physics, and not those other subjects, so it should be something me as an outsider can grasp.

Any help is very much appreciated. Thank you.

 

[trambolin]

Take (linearized) spinning satellite dynamics. They are easy to find. But the system order is 3 if that matters.

 

[HallsofIvy]

An electrical circuit with capacitor of strength C, resistance R, and Inductance L, with external power source proving power at f(t) satisfies the differential equation
$$L\frac{d^2I}{dt^2}+ R\frac{dI}{dt}+ \frac{1}{C}I= f(t)$$
where I(t) is the current in the circuit. Radio waves are produced as a result of resonance in such a circuit.

 

[qspeechc]

Thanks for the suggestion trambolin. If it is in some way analogous to an oscillating system with a damping and driving force, then it will be ok.

HallsofIvy- I could do an electrical circuit, but this is a classical example of the type of system I'm taking about, and has been done to death. I was thinking of doing something more off-the-beaten-track.

 

[trambolin]

well, oscillation part, you are right! It is not oscillating but put some friction in the gyros then you have at least half of it ! Hmm, Not much stuff in nature is oscillating in a linear fashion but. Then a pendulum with joint friction would be also OK! Also there was this liquid mixture, oscillating by colors that would be fancy but I don't remember the name of the reaction.

 

[trambolin]

OK we have Briggs-Rauscher, Belousov-Zhabotinsky, Four-Color Oscillating Reaction etc. These are the names that I could find. But better ask chemistry people they would give you the equations of dynamics... Good luck

 

[qspeechc]

Thanks a lot trambolin. I'm not sure I understand the stuff about satellites. Would it fall under the subject of Celestial Mechanics?

Thanks for showing me the world of oscillating chemical reactions. Wow. Amazing stuff. There are even some clips on Youtube, pretty cool stuff. In my preliminary research (google), it doesn't seem like they are modeled by a DE. But as you say, I should probably ask a chemistry boff.

Thanks again trambolin.

 

[trambolin]

No problem, regarding the DE of the reaction, it should be obviously DE but maybe they are nonlinear. But still, you can pick up an operating point (probably origin) and linearize the equations so your system will be locally realistic (initial conditions from a neighborhood of origin) as it is for small angles in a pendulum case.