Oscillator mathematcal problem

That's past my pay grade, but I have found that you can kill a ringing LC pretty much dead in under one cycle, by slapping a resistor across it. Something like 0.6 times X-sub-L as I recall.

Sure, that's a fairly straightforward calculus of variations problem.

Cheers

Phil Hobbs

Given enough amplitude, can't you kill it dead in an arbitrarily small time?

Sure. That's why it's straighforward. ;)

When you put in other constraints, e.g. maximum amplitude, slew rate, and so on, it becomes a little less straightforward, but calculus of variations is amazingly powerful.

The way it's usually formulated in physics (Lagrangian and Hamiltonian dynamics) is more complicated than these sorts of problems, because in general even for a solid body you have 6 dimensions and 6 momenta to worry about.

Cheers

Phil Hobbs

The Lagrangian is straightforward enough, but what are the appropriate constraints?

Also IIRC non-conservative forces such as viscous damping require something funky to be added to the Lagrangian...

K-Y Jelly ?>:-} ...Jim Thompson

In Lagrangian mechanics, you make a distinction between holonomic constraints (where there's a conservation law that applies throughout the motion) and nonholonomic ones like viscosity.

But for this case, it's easier not to drag out the full machinery, but just define a functional that optimizes what you care about, e.g. P = integral of x'**2 between some t1 and t2, and make a differential equation for A(t) by requiring that small changes in A make only second order changes in P. What you wind up with is a simple differential equation for A. You can add other constraints, e.g. no overshoot, using a Lagrange multiplier.

I always have to get back into the mindset of taking derivatives with respect to momentum and so on, but this is a simpler case, as I say.

Cheers

Phil Hobbs

Frank Beard? This is some seriously viscous damping right there...

Wait -- you're suggesting that rather than a linear system with A as a constant, you want to use some A(x)?

I suspect that it will be easy to find A(x) for a specific step input, but that as soon as you change the size of the step you'll find that your magical A(x) no longer works.

Nice 1

I think you want velocity damping (like you've got written.) Air drag goes as the velocity squared (over some region) and that gives a long tail. Friction damping (proportional to distance) will (in general) leave you with a non-zero amplitude when it stops.. x doesn't go all the way to zero.

For the fastest I think you want it critically damped... more damping than that and you get a longer decay time.

George H.

Well, for example, if you use Wolfram Alpha to solve the critically damped harmonic oscillator equation x'' + 2x' + x = 0 for x(0) = 1, and the following equation with A(y) = sec(y), y'' + 2sec(y)y' + y = 0, y(0) it is clear from the phase portrait of x vs x' and y vs y' that the latter damps out quicker without overshoot.

Essentially, if discontinus functions were allowed, the way you would get it to equilibrium is if you set up the equation right on the boundary of overshooting, and then slammed up an infinite potential "wall" when the oscillator hit the X axis.

But you can't do that when only continuous functions are allowed, because no potential will grow rapidly enough to stop the oscillator from crossing the axis. So you have to have some kind of damping potential that is a function of x of the right variety that "engages" at the right point - too soon and the oscillator will damp much slower than the oscillator with linear damping, too late and it won't damp significantly faster.

Well, you departed from physical reality once, in exciting the oscillator with a Heaviside 'function'. It is easy to imagine non-physical components (like the complex conjugate of the oscillator impedance) that rapidly remove energy, and the x(t) of the connection node for that 'load' would potentially be a good damping solution. The description 'damps the system out in the minimum time' , however, implies some measure of damping and of a time at which the damping is 'finished', neither of which is provided. So, do you minimize integral_0^infty (|x| *t )dt or integral_0^infty (x**2 *t )dt or t_decay such that x(t) .ge. 0 .and. x(t) .le. x(0)/100 whenever t > t_decay ?

What mathematical definition of decay time do you intend?

For example, using a nonlinear capacitor in an R+C damper (like a too-low-voltage X7R), or a regenerative snubber circuit (meaning: a capacitor coupled into a full wave doubler rectifier, so that the energy is regenerated into a supply rail; [quasi- and] resonant snubbers can exhibit similar behavior), tends to result in a linear rather than exponential decay of the ringing envelope. The rationale being, the nonlinear behavior is able to remove a fixed amount of energy or delta V per cycle, rather than proportionally so. You have the gain of saving the energy (well, in the regenerative case anyway), but the loss of having an uglier transient.

Tim

It might be wise to use the definition of "fall time" for overdamped systems, so 90% to 10% of final value?

Ahh, I kinda lost you in there. For me math is used to approximate reality.. you shouldn't get tied to tight to any one model. If you've got feedback and control (and smarts), then you can slam on the brakes as much as they'll go... and then let off easy as you come to your stopping point, like braking your car at a stop sign.

I don't think I'd use a damped oscillator as a model for that. What are you doing, or thinking of doing?

George H.

Time one "hit" anti-phase and proper amplitude,and the oscillations are dead.

Abso-tuve-ally.