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Mon 21 Apr, 2008 09:32 am
Hello everyone. Hope your having a fine day.
This is for a project I need to do. My problem is this (please could everyone, biologists too, bear with me!):
We know for a mass-spring system, with damping, the Differential Equation (DE) governing this is:
x'' + ax' + bx = 0
where 'x' is the displacement, ax' the damping term (force due to friction) and bx the resoring force (force due to the spring). The system can also be driven by some arbitrary force F, giving:
x'' + ax' + bx = F.
Now, I need to find some system, it doesn't matter what, that can be written as those two DEs. For instance, an oscillating electrical circuit is governed by the same set of DEs for the driven and un-driven states.
So this system needs a 'restoring force' (bx), and a 'damping term' (ax'), and must have some state corresponding to being driven. I do not want to do the mass-spring system or the electrical circuit- they have both been don to death.
I am looking for any ideas. It can be from any field- physics, biology, economics, whatever! However I am a physics student, so if it isn't physics, it should be something I can grasp.
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)?
Any help is much appreciated, thank you.
Hi Quincy. Lots of apps for differential equations with downloadable simulations can be found here:
http://demonstrations.wolfram.com/NonlinearWaveEquationExplorer/
I don't know whether wave formations are of interest to you; they have a lot of apps in financial functions using Fourier transforms. Look around in that site, you'll find many others.
Probably the most common is "the wave equation." Set a=0 and the second order equation becomes.
x"+bx=C
Used extensively in chemistry, electronics, power generation, communications.
Rap
Thanks for the suggestions guys.
The wave equation looks interesting. There might be two problems with this: its a PDE (partial differential eqn), and this is for an ODE (ordinary diff eqn) course. I should check with my proffessor though, because he is trying to shove some PDE stuff down our throats (which he isn't supposed to be doing). Also, there isn't anything corresponding to a damping term, which is necessary to have for this project.
Quincy - if the wave equations suggested by RapRap and me aren't suitable, my only other idea is from astrophysics (Raprap would probably be able to add to your DE lists) and it seems to meet your specs:
Quote:.......... secondary star is tidally distorted and mass transfer is initiated. Matter is lost through the first Lagrange point and flows towards the more massive compact object. Eventually this mass builds up to form an accretion disk. ....
http://library.wolfram.com/infocenter/Demos/5956/
Not sure if you have to download extra software, or if your professor allows its use - better ask.
You could do a pendulum or quantum harmonic oscillator. Or perhaps a diatomic molecule - maybe in an electric or magnetic field.
/Kayyam
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