Oscillator mathematcal problem

Started by December 11, 2015
Suppose we have the equation for a damped harmonic oscillator
driven by a step input, e.g. x'' + Ax' + x = H(t), where A is the
damping parameter.

Now suppose we allow A to be any _continuous_ function of x, i.e.
steps, delta functions, and their permutations are
disallowed.

Is there a way to determine which such function damps the system
out in the minimum time, _without_ allowing the function to
overshoot the final equilibrium point (i.e. no ringing)?

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On Fri, 11 Dec 2015 11:46:16 -0500 (EST), bitrex

> >Suppose we have the equation for a damped harmonic oscillator > driven by a step input, e.g. x'' + Ax' + x = H(t), where A is the > damping parameter. > >Now suppose we allow A to be any _continuous_ function of x, i.e. > steps, delta functions, and their permutations are > disallowed. > >Is there a way to determine which such function damps the system > out in the minimum time, _without_ allowing the function to > overshoot the final equilibrium point (i.e. no ringing)?
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.
On 12/11/2015 11:46 AM, bitrex wrote:
> > Suppose we have the equation for a damped harmonic oscillator > driven by a step input, e.g. x'' + Ax' + x = H(t), where A is the > damping parameter. > > Now suppose we allow A to be any _continuous_ function of x, i.e. > steps, delta functions, and their permutations are > disallowed. > > Is there a way to determine which such function damps the system > out in the minimum time, _without_ allowing the function to > overshoot the final equilibrium point (i.e. no ringing)? >
Sure, that's a fairly straightforward calculus of variations problem. Cheers Phil Hobbs -- Dr Philip C D Hobbs Principal Consultant ElectroOptical Innovations LLC Optics, Electro-optics, Photonics, Analog Electronics 160 North State Road #203 Briarcliff Manor NY 10510 hobbs at electrooptical dot net http://electrooptical.net
On Fri, 11 Dec 2015 12:08:47 -0500, Phil Hobbs
<pcdhSpamMeSenseless@electrooptical.net> wrote:

>On 12/11/2015 11:46 AM, bitrex wrote: >> >> Suppose we have the equation for a damped harmonic oscillator >> driven by a step input, e.g. x'' + Ax' + x = H(t), where A is the >> damping parameter. >> >> Now suppose we allow A to be any _continuous_ function of x, i.e. >> steps, delta functions, and their permutations are >> disallowed. >> >> Is there a way to determine which such function damps the system >> out in the minimum time, _without_ allowing the function to >> overshoot the final equilibrium point (i.e. no ringing)? >> > >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?
On 12/11/2015 12:20 PM, John Larkin wrote:
> On Fri, 11 Dec 2015 12:08:47 -0500, Phil Hobbs > <pcdhSpamMeSenseless@electrooptical.net> wrote: > >> On 12/11/2015 11:46 AM, bitrex wrote: >>> >>> Suppose we have the equation for a damped harmonic oscillator >>> driven by a step input, e.g. x'' + Ax' + x = H(t), where A is the >>> damping parameter. >>> >>> Now suppose we allow A to be any _continuous_ function of x, i.e. >>> steps, delta functions, and their permutations are >>> disallowed. >>> >>> Is there a way to determine which such function damps the system >>> out in the minimum time, _without_ allowing the function to >>> overshoot the final equilibrium point (i.e. no ringing)? >>> >> >> 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 -- Dr Philip C D Hobbs Principal Consultant ElectroOptical Innovations LLC Optics, Electro-optics, Photonics, Analog Electronics 160 North State Road #203 Briarcliff Manor NY 10510 hobbs at electrooptical dot net http://electrooptical.net
Phil Hobbs <pcdhSpamMeSenseless@electrooptical.net> Wrote in message:
> On 12/11/2015 11:46 AM, bitrex wrote: >> >> Suppose we have the equation for a damped harmonic oscillator >> driven by a step input, e.g. x'' + Ax' + x = H(t), where A is the >> damping parameter. >> >> Now suppose we allow A to be any _continuous_ function of x, i.e. >> steps, delta functions, and their permutations are >> disallowed. >> >> Is there a way to determine which such function damps the system >> out in the minimum time, _without_ allowing the function to >> overshoot the final equilibrium point (i.e. no ringing)? >> > > Sure, that's a fairly straightforward calculus of variations problem. > > Cheers > > Phil Hobbs > > -- > Dr Philip C D Hobbs > Principal Consultant > ElectroOptical Innovations LLC > Optics, Electro-optics, Photonics, Analog Electronics > > 160 North State Road #203 > Briarcliff Manor NY 10510 > > hobbs at electrooptical dot net > http://electrooptical.net >
The Lagrangian is straightforward enough, but what are the appropriate constraints? -- ----Android NewsGroup Reader---- http://usenet.sinaapp.com/
Phil Hobbs <pcdhSpamMeSenseless@electrooptical.net> Wrote in message:
> On 12/11/2015 11:46 AM, bitrex wrote: >> >> Suppose we have the equation for a damped harmonic oscillator >> driven by a step input, e.g. x'' + Ax' + x = H(t), where A is the >> damping parameter. >> >> Now suppose we allow A to be any _continuous_ function of x, i.e. >> steps, delta functions, and their permutations are >> disallowed. >> >> Is there a way to determine which such function damps the system >> out in the minimum time, _without_ allowing the function to >> overshoot the final equilibrium point (i.e. no ringing)? >> > > Sure, that's a fairly straightforward calculus of variations problem. > > Cheers > > Phil Hobbs > > -- > Dr Philip C D Hobbs > Principal Consultant > ElectroOptical Innovations LLC > Optics, Electro-optics, Photonics, Analog Electronics > > 160 North State Road #203 > Briarcliff Manor NY 10510 > > hobbs at electrooptical dot net > http://electrooptical.net >
Also IIRC non-conservative forces such as viscous damping require something funky to be added to the Lagrangian... -- ----Android NewsGroup Reader---- http://usenet.sinaapp.com/
On Fri, 11 Dec 2015 12:43:14 -0500 (EST), bitrex