# 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)?

--

http://usenet.sinaapp.com/
```
```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

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

Cheers

Phil Hobbs

--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC
Optics, Electro-optics, Photonics, Analog Electronics

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?

--

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...
--

http://usenet.sinaapp.com/
```
```On Fri, 11 Dec 2015 12:43:14 -0500 (EST), bitrex

>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/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

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
```