As I remember from physics lessons, a system can go from one state to another in 3 ways. 1st way: gradually from one equilibrium to another, like a ball can roll down a mountain, 3rd way: swinging around a new equilibrium, like the thermostat works from the heating. The second way was a kind of muffled vibration, the way the iris of our eye reacts to a sudden increase in light. The iris suddenly gets smaller, but goes a little further, gets bigger again, but a little too big, gets smaller again, etcetera, etcetera, but with smaller and smaller differences until the new balance is maintained. I think this is called a second order system. What formula or rules does this follow?
Answer
A system on which you put let us say a constant input will evolve to a constant regime response via a transition behavior (transient behaviour). This applies to all systems regardless of order. In the case of a 2nd order system, this transitional behavior can take three forms:
– overdamping : in that case the transition behavior is a sum of 2 different exponentials
so something of the form : C1 exp(-at) + C2 exp(-bt)
– critical damping: then the poles of the system coincide, and the transition behavior exists
of two parts of the form: C1 exp(-at) + C2.t.exp(-at)
– damped vibration; the poles are then complex of the form a+jb and aj/b, and the transition behavior is then of the vrom: C1 exp(-at) cos(bt) + C2 exp(-at) sin(bt)
(in all these cases a is itself a positive number, so the exponentials always cancel out
The order of the system is determined by the order of the derivative in the differential equation that breaks the system. For a damped mass-spring system with damping and a force f
m. y” + c. y’ + ky = f
This is 2nd order system
If you hang three such masses together via springs and dampers, each mass is described in itself by a 2nd order diff equation (some of Newton’s), but those three will be coupled. The total order is then 6.
I wrote an article about this on wikipedia (NL-language): LTC system
Answered by
prof.dr. Paul Hellings
Department of Mathematics, Fac. IIW, KU Leuven
Old Market 13 3000 Leuven
https://www.kuleuven.be/
.