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Oscillator mathematcal problem

Started by bitrex 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
<bitrex@de.lete.earthlink.net> 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)?
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
<bitrex@de.lete.earthlink.net> wrote:

>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...
K-Y Jelly ?>:-} ...Jim Thompson -- | James E.Thompson | mens | | Analog Innovations | et | | Analog/Mixed-Signal ASIC's and Discrete Systems | manus | | San Tan Valley, AZ 85142 Skype: skypeanalog | | | Voice:(480)460-2350 Fax: Available upon request | Brass Rat | | E-mail Icon at http://www.analog-innovations.com | 1962 | I love to cook with wine. Sometimes I even put it in the food.
On 12/11/2015 12:43 PM, bitrex wrote:
> 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 >> > Also IIRC non-conservative forces such as viscous damping require > something funky to be added to the Lagrangian... >
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 -- 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
bitrex wrote:
> 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... >
Frank Beard? This is some seriously viscous damping right there... -- Les Cargill