Engineering and math

[Le Chaud Lapin]

Analysis using density is often child's play. But look how long it
took to discover as a useful principle. Natural philosophers were
debating the ideas of sharp and blunt as a principle for floating and
sinking, even in Galileo's day, for gosh sake. If someone told me
"density is obvious," and imagine that they are as creative,
imaginative, and smart as those who actually had to discover the idea,
I'd just smile and walk away shaking my head.

Not that I would have ever figured out calculus, but when I was 4, I
remember looking at the edge of a razor blade and trying to tell my
brother that it was not "so thin that it did not have width." We
argued about it for a few minutes until finally I said, "What if you
made some green Jello(R), and you took the razor blade and kept slicing
it over and over, a 500 times? All the little slices would have to add
up. The edges of the Jello would flop and spread out, right?" I
figured it would spread out eventually, but I wasn't sure.

He got frustrated and told my mother that I was talking about slicing
Jello(R) with razor blades and I got a spanking for playing with my
Dad's tools.

-Le Chaud Lapin-

 

[Jonathan Kirwan]

Not that I would have ever figured out calculus, but when I was 4, I
remember looking at the edge of a razor blade and trying to tell my
brother that it was not "so thin that it did not have width." We
argued about it for a few minutes until finally I said, "What if you
made some green Jello(R), and you took the razor blade and kept slicing
it over and over, a 500 times? All the little slices would have to add
up. The edges of the Jello would flop and spread out, right?" I
figured it would spread out eventually, but I wasn't sure.

He got frustrated and told my mother that I was talking about slicing
Jello(R) with razor blades and I got a spanking for playing with my
Dad's tools.

Good imagination, eh? That's important.

Jon

 

[Jens Tingleff]

It is not. A plot of gain and phase vs frequency does not involve any
of the math that is associated with the Fourier transform.

You're not confusing the procedure of plotting a curve with the math which
underlies the analysis which gives the validity of the result, are you?

The math which gives (1 - exp(-t/tau)) as the step response for the system which
has the frequency response 1 / (1 + s tau) *is* the Fourier transform. (Well, I
would call it the Laplace transform, because that is the general method, but the
Fourier transform is the name given to the analysis which gives the response to
inputs which are (composed of) constant-amplitude sinewaves, i.e. described in
the s (or Laplace) domain as being on the imaginary axis - the frequency axis
above.)

Quite complicated answers ensue if you keep asking "why" past the point of - say
- "Kirchoff's current and voltage laws describe a circuit and all you have to do
is solve them."

Is "Frequency on the x-axis" the definition of "Fourier analysis"?

Hah!

of course not, try some thing like
http://en.wikipedia.org/wiki/Laplace_transform if you're curious about this.

(For linear time invariant systems I'd be curious to see a method other than
Laplace transform used to derive the frequency behaviour of the system in
question.)

It's not necessary to have *any* knowledge if it!

(I'll agree with that, actually.)

Well. Are you then content to switch integration methods in Spice when you
simulate an ideal oscillator without knowing why? The answer might very well be
"yes," for you, but I find it pretty comforting to know why some analysis
methods will not give the true result for some problems. (I'm alluding to the
"method=gear" and "method=trap" options in classic Spice, not the newfangled
ones which "just work" )

How do you design stable nonlinear control systems?

Dunno, I'm more of a linear system kind of person myself. How would you
demonstrate the stability of such a design?

John

Best Regards

Jens

 

[Nico Coesel]

I'm an EE and taking classes for my Master's. And I am again realizing
my distaste for math. I don't hate it, I just really struggle with it.
Is this weird?

I have always really enjoyed designing and building electronic things,
and even daydream about it. I analyze everything beyond what I believe
is normal, and am always trying to figure out better/easier ways to do
things, and I'm employed as an engineer. Yet anything beyond
high-school level math drives me nuts. Is it an oxymoron to like
engineering but look at a complex math problem like it's written in
Chinese?

Buy 'Modern Engineering Mathematics' and you'll have a practical guide
to solve mathematical problems. It goes from really simple things like
Pythagoras to Fourier and Laplace. 'Advanced Modern Engineering Math.'
is still on my 'must buy asap' list because the first book doesn't
cover all I need.

 

[Nico Coesel]

Those cathedral builders knew a thing or two actually.

They sure do. In France I've been is small churches build around the
year 1000. But these building are quite simple. I also visited to
Borobudur and Pram Banan temples in Indonesia. The rocks of these
temples are not simply stacked, but they are cut like pieces of a 3
dimensional puzzle. Some extensive engineering and planning must have
taken place in order to build something big like a temple using such a
method.

 

[Al]

You're not confusing the procedure of plotting a curve with the math which
underlies the analysis which gives the validity of the result, are you?

The math which gives (1 - exp(-t/tau)) as the step response for the system
which
has the frequency response 1 / (1 + s tau) *is* the Fourier transform. (Well,
I
would call it the Laplace transform, because that is the general method, but
the
Fourier transform is the name given to the analysis which gives the response
to
inputs which are (composed of) constant-amplitude sinewaves, i.e. described
in
the s (or Laplace) domain as being on the imaginary axis - the frequency axis
above.)

Quite complicated answers ensue if you keep asking "why" past the point of -
say
- "Kirchoff's current and voltage laws describe a circuit and all you have to
do
is solve them."


Hah!

of course not, try some thing like
http://en.wikipedia.org/wiki/Laplace_transform if you're curious about this.

(For linear time invariant systems I'd be curious to see a method other than
Laplace transform used to derive the frequency behaviour of the system in
question.)


(I'll agree with that, actually.)

Well. Are you then content to switch integration methods in Spice when you
simulate an ideal oscillator without knowing why? The answer might very well
be
"yes," for you, but I find it pretty comforting to know why some analysis
methods will not give the true result for some problems. (I'm alluding to the
"method=gear" and "method=trap" options in classic Spice, not the newfangled
ones which "just work" )



Dunno, I'm more of a linear system kind of person myself. How would you
demonstrate the stability of such a design?



Best Regards

Jens

I was using hypergeometric functions in grad school. Havn't seen one
since.

Al

 

[John Larkin]

You're not confusing the procedure of plotting a curve with the math which
underlies the analysis which gives the validity of the result, are you?

The math which gives (1 - exp(-t/tau)) as the step response for the system which
has the frequency response 1 / (1 + s tau) *is* the Fourier transform. (Well, I
would call it the Laplace transform, because that is the general method, but the
Fourier transform is the name given to the analysis which gives the response to
inputs which are (composed of) constant-amplitude sinewaves, i.e. described in
the s (or Laplace) domain as being on the imaginary axis - the frequency axis
above.)

When I do a Bode plot anaysis of a closed-loop system, I have nothing
to do with time-domain waveforms, step responses, or any of that. I
plot gain as a function of frequency, and stay purely in the frequency
domain. No "transform" is performed. Plotting amplitude versus
frequency was no doubt done long before Fourier was born; probably any
good piano tuner had the concept already.

Fourier invented an algorithm to map a (basically) time-domain
function into the frequency domain. He didn't invent graphing, and his
transform is not executed when I do a Bode plot. And the Fourier
transform doesn't "validate" graphing gain versus frequency; the
physics of resistors and capacitors does.

John

 

[Jens Tingleff]

On 13 Oct 2006 07:01:40 -0700, Jens Tingleff <[email protected]>
wrote:
[....]

The math which gives (1 - exp(-t/tau)) as the step response for the system which
has the frequency response 1 / (1 + s tau) *is* the Fourier transform. (Well, I
would call it the Laplace transform, because that is the general method, but the
Fourier transform is the name given to the analysis which gives the response to
inputs which are (composed of) constant-amplitude sinewaves, i.e. described in
the s (or Laplace) domain as being on the imaginary axis - the frequency axis
above.)

When I do a Bode plot anaysis of a closed-loop system, I have nothing
to do with time-domain waveforms, step responses, or any of that. I
plot gain as a function of frequency, and stay purely in the frequency
domain. No "transform" is performed. Plotting amplitude versus
frequency was no doubt done long before Fourier was born; probably any
good piano tuner had the concept already.

If you say so. I thought that Fourier was the first to show the output of a
linear system to an input consisting of a combination of periodic signals.

http://en.wikipedia.org/wiki/Fourier_series#Historical_development

Maybe I'm entirely wrong...

Fourier invented an algorithm to map a (basically) time-domain
function into the frequency domain. He didn't invent graphing, and his
transform is not executed when I do a Bode plot. And the Fourier
transform doesn't "validate" graphing gain versus frequency; the
physics of resistors and capacitors does.

So, what ties the graphing of gain versus frequency to the physics of resistors
and capacitors?

Best Regards

Jens

 

[John Larkin]

On 13 Oct 2006 07:01:40 -0700, Jens Tingleff <[email protected]>
wrote:
[....]

The math which gives (1 - exp(-t/tau)) as the step response for the system which
has the frequency response 1 / (1 + s tau) *is* the Fourier transform. (Well, I
would call it the Laplace transform, because that is the general method, but the
Fourier transform is the name given to the analysis which gives the response to
inputs which are (composed of) constant-amplitude sinewaves, i.e. described in
the s (or Laplace) domain as being on the imaginary axis - the frequency axis
above.)

When I do a Bode plot anaysis of a closed-loop system, I have nothing
to do with time-domain waveforms, step responses, or any of that. I
plot gain as a function of frequency, and stay purely in the frequency
domain. No "transform" is performed. Plotting amplitude versus
frequency was no doubt done long before Fourier was born; probably any
good piano tuner had the concept already.

If you say so. I thought that Fourier was the first to show the output of a
linear system to an input consisting of a combination of periodic signals.

http://en.wikipedia.org/wiki/Fourier_series#Historical_development

Maybe I'm entirely wrong...

Not worth arguing; but I swear I don't perform Fourier transforms to
do Bode plots, and that I don't sum or superimpose sinewaves of
different frequencies.

So, what ties the graphing of gain versus frequency to the physics of resistors
and capacitors?

The simple expressions of the behavior of resistors and capacitors:
e = i*r and i = C * dv/dt. I doubt that Fourier was aware of these,
and his transform doesn't describe either phenom.

John

 

[Jens Tingleff]

-----BEGIN PGP SIGNED MESSAGE-----
Hash: SHA1

On 13 Oct 2006 07:01:40 -0700, Jens Tingleff <[email protected]>
wrote:
[....]
The math which gives (1 - exp(-t/tau)) as the step response for the
system which has the frequency response 1 / (1 + s tau) *is* the Fourier
transform. (Well, I would call it the Laplace transform, because that is
the general method, but the Fourier transform is the name given to the
analysis which gives the response to inputs which are (composed of)
constant-amplitude sinewaves, i.e. described in the s (or Laplace)
domain as being on the imaginary axis - the frequency axis above.)

When I do a Bode plot anaysis of a closed-loop system, I have nothing
to do with time-domain waveforms, step responses, or any of that. I
plot gain as a function of frequency, and stay purely in the frequency
domain. No "transform" is performed. Plotting amplitude versus
frequency was no doubt done long before Fourier was born; probably any
good piano tuner had the concept already.

If you say so. I thought that Fourier was the first to show the output of
a linear system to an input consisting of a combination of periodic
signals.

http://en.wikipedia.org/wiki/Fourier_series#Historical_development

Maybe I'm entirely wrong...

Not worth arguing; but I swear I don't perform Fourier transforms to
do Bode plots, and that I don't sum or superimpose sinewaves of
different frequencies.

I'm not disputing that. I'm disputing your statement that the Fourier
transform has nothing to do with the frequency response graph, i.e. with
what the graph looks like, not with what you do to draw it.

The simple expressions of the behavior of resistors and capacitors:
e = i*r and i = C * dv/dt. I doubt that Fourier was aware of these,
and his transform doesn't describe either phenom.

Sure. I was just wondering how you go to the frequency domain without using
the Fourier transform. (I think, as I wrote earlier, that a special case of
the Laplace transform - the Fourier transform - is the only way to link the
time and frequency behaviour of linear time invariant circuits, but I'd be
fascinated to hear how you do it).

Best Regards

Jens

- --
Key ID 0x09723C12, [email protected]
Analogue filtering / 5GHz RLAN / Mandriva Linux / odds and ends
http://www.tingleff.org/jensting/ +44 1223 211 585
"What a plan! Simple, yet insane!" Monsters Inc.
-----BEGIN PGP SIGNATURE-----
Version: GnuPG v1.4.2.2 (GNU/Linux)

iD8DBQFFNHkeimJs3AlyPBIRAgTCAJ9UotFR4i0t7ki9Y/4U+fKNjCl0YwCgxR3C
xh5klFF7J01+hup+xwbfzT4=
=L2cZ
-----END PGP SIGNATURE-----

 

[joseph2k]

Indeed, a lot of practical engineering is more seat-of-the-pants math
in terms of understanding the implications rather than doing
complicated problems as they were done in school. You need to know
what it is going to look like, but you can resort to the computer for
the actual numbers.

That is, assuming someone's already written the software, and that you
have it available. Ironically, I do more "engineering math" in support
of hobbies than I do for "work"... things like machining parts for
musical instruments, or arguing about movements in ballroom dancing -
when there aren't textbook methods to resort to and you are exploring a
new field with methods borrowed from another, then you end up deriving
(or rederiving) a lot of things from basic relationships.

It was similar but different for me, I enjoyed algebra, less so geometry,
and when i got to Calculus it drove me nuts; when i finally wrapped my mind
around differential equations (about halfway through the second semester)
my world changed. Suddenly i could see differential equations everywhere
and in everything.
Probably have a hell of a time trying to do the math now, but the worldview
change persists.

 

[Giorgis]

Yes. There is no engineering without math - just guesswork. We aren't
cathedral builder from the middle ages.

You'd be amazed at th elevel of matrhs and logistics required for those
cathedral or any large buildings from antiquity.

G

 

[John Larkin]

Sure. I was just wondering how you go to the frequency domain without using
the Fourier transform. (I think, as I wrote earlier, that a special case of
the Laplace transform - the Fourier transform - is the only way to link the
time and frequency behaviour of linear time invariant circuits, but I'd be
fascinated to hear how you do it).

By not starting in the time domain at all. No domain crossing, no
transform executed.

John

 

[joseph2k]

You'd be amazed at th elevel of matrhs and logistics required for those
cathedral or any large buildings from antiquity.

G

The Professors analyzing them now use all kinds of fancy math. They could
not keep their jobs if they did not. The architects that created them
barley used anything better than lame algebra, if that. Algebra did not
really exist in the 11th through 15th centuries, yet they still built
cathedrals still standing today.

 

[Jens Tingleff]

By not starting in the time domain at all. No domain crossing, no
transform executed.

Fine. This is also what I would do if I wanted a frequency response, I assure
you

So you have the physics of the elements (as you described, in the part you
snipped ) in the time domain. What are you using in the frequency domain?

Best Regards

Jens

 

[Homer J Simpson]

The Professors analyzing them now use all kinds of fancy math. They could
not keep their jobs if they did not. The architects that created them
barley used anything better than lame algebra, if that. Algebra did not
really exist in the 11th through 15th centuries, yet they still built
cathedrals still standing today.

But they used rules of thumb and experience, not math. And too many times
they fell down. It's a wonder that so many survived.

 

[RST Engineering \(jw\)]

The ark was built by talented and knowledgable amateurs.

The Titanic was built using the best engineering practice and calculations
of the day.


Jim

 

[Jim Thompson]

The ark was built by talented and knowledgable amateurs.

The Titanic was built using the best engineering practice and calculations
of the day.


Jim

[snip]

With bulkheads with no ceilings.

And sunk by a cowboy intent on setting a speed record ;-)

...Jim Thompson

 

[RST Engineering \(jw\)]

Yup. CAD has nothing on a swizzle stick dipped in soy sauce and a clean
napkin.

Jim

The ark was built by talented and knowledgable amateurs.

The Titanic was built using the best engineering practice and calculations
of the day.


Jim

[snip]

With bulkheads with no ceilings.

And sunk by a cowboy intent on setting a speed record ;-)

 

[Jim Thompson]

Yup. CAD has nothing on a swizzle stick dipped in soy sauce and a clean
napkin.

Jim

That's what I like about Macaroni Grill, table "cloth" is a sheet of
white butcher paper... surprises the hell out of them when I roll it
up and take it with me, sauce stains and all ;-)

The ark was built by talented and knowledgable amateurs.

The Titanic was built using the best engineering practice and calculations
of the day.


Jim

[snip]

With bulkheads with no ceilings.

And sunk by a cowboy intent on setting a speed record ;-)

...Jim Thompson