Are harmonics real?

Hi there,

I am involved in the machinery vibration analysis field.  We use FFTs to
study vibration from rotating machinery.

If you look at the FFT “rules”, an impulse will give a spectrum with
energy at all frequencies, whereas a pulse train will generate harmonics.

If you take that to the mechanical world, I know that a single impact is
like striking a bell - vibration is 'injected' at all frequencies so the
bell rings at all its natural frequencies.

An FFT of the impact will have energy at 'all' frequencies.  However i the
time record has multiple impacts we will see harmonics.

It is the presence of harmonics that I have questions about.  Are they
real.  Is the machine experiencing vibration at those frequencies?  Is the
machine experiencing vibration between those frequencies?  

I think there are three possible explanations:

1.	The harmonics are a remnant of the FFT process.  In actual fact there
is ‘vibration energy’ at the harmonic frequencies and in-between the
harmonics – just as there is with a single impact.  Just because the time
record capture more than one impact does not mean that you are not
injecting energy at ‘all’ frequencies.

2.	The harmonics are real, but the vibration energy is only present at the
harmonic frequencies.  

3.	The harmonics are not real; there is no energy at the harmonics.  Some
believe the peaks are present as a ‘characteristic’ of the FFT process
only.

My experiments suggest that option 2 is correct (therefore the spectrum is
a true representation of the vibration at every discrete frequency),
however my gut told me that option 1 would be true.

My question about harmonics goes further…

A pure sine wave will have not harmonics.  However if I distort the sine
wave then I will see harmonics.  Again the question is whether they are
real or not…

Sorry for such a long post.  I am hoping that someone will know the answer
off the top of their heads.  (I have discussed this question with many of
my colleagues over the years, often resulting in heated discussion, but no
one has come up with a definitive answer/explanation.)

Thanks for your time.

Jason

JasonTranter <jason@ilearninteractive.com> wrote:
 
< I am involved in the machinery vibration analysis field.  We use FFTs to
< study vibration from rotating machinery.
 
< If you look at the FFT ???rules???, an impulse will give a spectrum with
< energy at all frequencies, whereas a pulse train will generate harmonics.
 
< If you take that to the mechanical world, I know that a single impact is
< like striking a bell - vibration is 'injected' at all frequencies so the
< bell rings at all its natural frequencies.

Yes.  And for most bell designs, not harmonically related.
 
< An FFT of the impact will have energy at 'all' frequencies.  However i the
< time record has multiple impacts we will see harmonics.
 
< It is the presence of harmonics that I have questions about.  Are they
< real.  Is the machine experiencing vibration at those frequencies?  Is the
< machine experiencing vibration between those frequencies?  

Many systems have vibrational modes that are harmonically related,
or are close (end effects).  Much of the design of musical instruments
is to make the modes close enough to harmonically related such
that they sound good.

A thin string stretched taught with the ends fixed (violin) has
its modes pretty close to harmonically related.  For air columns
(woodwinds and pipe organs) there is an end effect related to the
diameter of the tube that affects the modes.
 
< I think there are three possible explanations:
 
< 1.      The harmonics are a remnant of the FFT process.  In actual fact there
< is ???vibration energy??? at the harmonic frequencies and in-between the
< harmonics ??? just as there is with a single impact.  Just because the time
< record capture more than one impact does not mean that you are not
< injecting energy at ???all??? frequencies.

They might be remnants.  The FFT has periodic boundary conditions.
If the data is not periodic with the same period, you will get extra
non-harmonic terms.  In the infinite length limit (limit as the
period goes to infinity) those go away.  Sometimes you can get a
good result by doing the transform over a longer interval.
 
< 2.      The harmonics are real, but the vibration energy is only 
< present at the harmonic frequencies.  

You should be able to approximate the system such that you know
if it does or does not have (nearly) harmonic modes.  For solid bars,
for example, the bar stiffness affects the modes.
 
< 3.      The harmonics are not real; there is no energy at the 
< harmonics.  Some believe the peaks are present as a 
< ???characteristic??? of the FFT process only.

Change the length (time) of the FFT and see what the harmonics do.
If they move, then they aren't real.

(snip)
 
< My question about harmonics goes further???
 
< A pure sine wave will have not harmonics.  However if I distort the sine
< wave then I will see harmonics.  Again the question is whether they are
< real or not???

One reason that harmonics are interesting is that many natural
systems generate harmomically related sounds.  Our aural system
is designed to process such sounds.  

Someone actually did a study once on this, generating sounds 
where the 'octave' was a factor of 2.1 instead of the usual 2.  
That is, stretched scale music.  As expected, it sounds horrible.  

I suppose if you wanted to do the experiment right, you would
take babies and raise them in an environment where only such
stretched (2.1) octave sounds exist and test that their aural
system adapts to the environment or not.  

-- glen
 

JasonTranter wrote:
> Hi there,
> 
> I am involved in the machinery vibration analysis field.  We use FFTs to
> study vibration from rotating machinery.
> 
> If you look at the FFT “rules”, an impulse will give a spectrum with
> energy at all frequencies, whereas a pulse train will generate harmonics.
> 
> If you take that to the mechanical world, I know that a single impact is
> like striking a bell - vibration is 'injected' at all frequencies so the
> bell rings at all its natural frequencies.
> 
> An FFT of the impact will have energy at 'all' frequencies.  However i the
> time record has multiple impacts we will see harmonics.

Only if the train of impulses is strictly periodic. Then you see 
harmonics of the impact frequency.

> It is the presence of harmonics that I have questions about.  Are they
> real.

Yes.

>  Is the machine experiencing vibration at those frequencies? 

The impact causes the machine to vibrate.

> Is the machine experiencing vibration between those frequencies? 

Probably.

> I think there are three possible explanations:
> 
> 1.	The harmonics are a remnant of the FFT process.  In actual fact there
> is ‘vibration energy’ at the harmonic frequencies and in-between the
> harmonics – just as there is with a single impact.  Just because the time
> record capture more than one impact does not mean that you are not
> injecting energy at ‘all’ frequencies.

If the impacts are periodic, the energy between harmonics undergoes 
destructive cancellation.

> 2.	The harmonics are real, but the vibration energy is only present at the
> harmonic frequencies.  

Ultimately, yes. The periodic excitation has to have been applied for 
long enough so that the transients have decayed.

> 3.	The harmonics are not real; there is no energy at the harmonics.  Some
> believe the peaks are present as a ‘characteristic’ of the FFT process
> only.

Some believe that the earth is flat. Believing doesn't make it so.

> My experiments suggest that option 2 is correct (therefore the spectrum is
> a true representation of the vibration at every discrete frequency),
> however my gut told me that option 1 would be true.
> 
> My question about harmonics goes further…
> 
> A pure sine wave will have not harmonics.  However if I distort the sine
> wave then I will see harmonics.  Again the question is whether they are
> real or not…

Of course they are real. You know what a pure sine looks like. You have 
to add the distortion products to it to get whatever weird shape the 
distortion looks like.

> Sorry for such a long post.  I am hoping that someone will know the answer
> off the top of their heads.  (I have discussed this question with many of
> my colleagues over the years, often resulting in heated discussion, but no
> one has come up with a definitive answer/explanation.)

It appears that none of your colleagues understands the math. Arm waving 
is a poor way to analyze physics.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

On Tue, 12 May 2009 12:58:34 -0500, "JasonTranter"
<jason@ilearninteractive.com> wrote:

>Hi there,
>
>I am involved in the machinery vibration analysis field.  We use FFTs to
>study vibration from rotating machinery.
>
>If you look at the FFT &ldquo;rules&#4294967295;?, an impulse will give a spectrum with
>energy at all frequencies, whereas a pulse train will generate harmonics.
>
>If you take that to the mechanical world, I know that a single impact is
>like striking a bell - vibration is 'injected' at all frequencies so the
>bell rings at all its natural frequencies.
>
>An FFT of the impact will have energy at 'all' frequencies.  However i the
>time record has multiple impacts we will see harmonics.
>
>It is the presence of harmonics that I have questions about.  Are they
>real.  Is the machine experiencing vibration at those frequencies?  Is the
>machine experiencing vibration between those frequencies?  

The response to a single strike is analogous to the general solution
to the system differential equation (the system's natural modes).  The
response to multiple strikes is analogous to the particular solution
(the system's response to a particular set of initial conditions), or
more to the point, to the system's response to a specific input.  If
that input has spectral content at certain frequencies, then why would
you not expect the system response to also have content at those same
frequencies?

On Tue, 12 May 2009 12:58:34 -0500, JasonTranter wrote:

> Hi there,
> 
> I am involved in the machinery vibration analysis field.  We use FFTs to
> study vibration from rotating machinery.
> 
> If you look at the FFT “rules”, an impulse will give a spectrum with
> energy at all frequencies, whereas a pulse train will generate
> harmonics.
> 
> If you take that to the mechanical world, I know that a single impact is
> like striking a bell - vibration is 'injected' at all frequencies so the
> bell rings at all its natural frequencies.
> 
> An FFT of the impact will have energy at 'all' frequencies.  However i
> the time record has multiple impacts we will see harmonics.
> 
> It is the presence of harmonics that I have questions about.  Are they
> real.  Is the machine experiencing vibration at those frequencies?  Is
> the machine experiencing vibration between those frequencies?
> 
> I think there are three possible explanations:
> 
> 1.	The harmonics are a remnant of the FFT process.  In actual fact 
there
> is ‘vibration energy’ at the harmonic frequencies and in-between the
> harmonics – just as there is with a single impact.  Just because the
> time record capture more than one impact does not mean that you are not
> injecting energy at ‘all’ frequencies.
> 
> 2.	The harmonics are real, but the vibration energy is only present 
at
> the harmonic frequencies.
> 
> 3.	The harmonics are not real; there is no energy at the harmonics. 
> Some believe the peaks are present as a ‘characteristic’ of the FFT
> process only.
> 
> My experiments suggest that option 2 is correct (therefore the spectrum
> is a true representation of the vibration at every discrete frequency),
> however my gut told me that option 1 would be true.
> 
> My question about harmonics goes further…
> 
> A pure sine wave will have not harmonics.  However if I distort the sine
> wave then I will see harmonics.  Again the question is whether they are
> real or not…
> 
> Sorry for such a long post.  I am hoping that someone will know the
> answer off the top of their heads.  (I have discussed this question with
> many of my colleagues over the years, often resulting in heated
> discussion, but no one has come up with a definitive
> answer/explanation.)

Do you understand the mathematics behind the Fourier transform and the 
FFT?  Harmonics in a signal are as real as those mathematics, and in the 
case of the Fourier transform that's pretty darn real -- you can sum up 
all the energy in a signal in your choice of real time or the Fourier 
frequency domain, and you get exactly the same number (see Parseval's 
Theorem).  If the energy balance works, then its hard to argue with the 
physics.

The FFT case is a bit problematical, because the FFT is only exact if you 
happen to be dealing with a periodic and sampled signal.  What makes the 
FFT fail to be exact isn't because the transform itself isn't exact, it's 
because you're "telling" the math that you're giving it one cycle of a 
sampled periodic wave, and chances are that's not really what you have.

So you can manufacture "ghost" signals, both by aliasing and by spectral 
leakage, and the FFT itself cannot distinguish those signals from the 
real thing.

But the harmonics themselves are real things -- any time you have a 
periodic waveform that's not a perfect sine wave you'll have them, and in 
any physical system there's real energy behind them, so they're as real 
as real can be.

-- 
www.wescottdesign.com

On May 12, 4:08&#4294967295;pm, Tim Wescott <t...@seemywebsite.com> wrote:
> On Tue, 12 May 2009 12:58:34 -0500, JasonTranter wrote:
> > Hi there,
>
> > I am involved in the machinery vibration analysis field. &#4294967295;We use FFTs to
> > study vibration from rotating machinery.
>
> > If you look at the FFT &#4294967295;rules&#4294967295;, an impulse will give a spectrum with
> > energy at all frequencies, whereas a pulse train will generate
> > harmonics.
>
> > If you take that to the mechanical world, I know that a single impact is
> > like striking a bell - vibration is 'injected' at all frequencies so the
> > bell rings at all its natural frequencies.
>
> > An FFT of the impact will have energy at 'all' frequencies. &#4294967295;However i
> > the time record has multiple impacts we will see harmonics.
>
> > It is the presence of harmonics that I have questions about. &#4294967295;Are they
> > real. &#4294967295;Is the machine experiencing vibration at those frequencies? &#4294967295;Is
> > the machine experiencing vibration between those frequencies?
>
> > I think there are three possible explanations:
>
> > 1. The harmonics are a remnant of the FFT process. &#4294967295;In actual fact
> there
> > is &#4294967295;vibration energy&#4294967295; at the harmonic frequencies and in-between the
> > harmonics &#4294967295; just as there is with a single impact. &#4294967295;Just because the
> > time record capture more than one impact does not mean that you are not
> > injecting energy at &#4294967295;all&#4294967295; frequencies.
>
> > 2. The harmonics are real, but the vibration energy is only present
> at
> > the harmonic frequencies.
>
> > 3. The harmonics are not real; there is no energy at the harmonics.
> > Some believe the peaks are present as a &#4294967295;characteristic&#4294967295; of the FFT
> > process only.
>
> > My experiments suggest that option 2 is correct (therefore the spectrum
> > is a true representation of the vibration at every discrete frequency),
> > however my gut told me that option 1 would be true.
>
> > My question about harmonics goes further&#4294967295;
>
> > A pure sine wave will have not harmonics. &#4294967295;However if I distort the sine
> > wave then I will see harmonics. &#4294967295;Again the question is whether they are
> > real or not&#4294967295;
>
> > Sorry for such a long post. &#4294967295;I am hoping that someone will know the
> > answer off the top of their heads. &#4294967295;(I have discussed this question with
> > many of my colleagues over the years, often resulting in heated
> > discussion, but no one has come up with a definitive
> > answer/explanation.)
>
> Do you understand the mathematics behind the Fourier transform and the
> FFT? &#4294967295;Harmonics in a signal are as real as those mathematics, and in the
> case of the Fourier transform that's pretty darn real -- you can sum up
> all the energy in a signal in your choice of real time or the Fourier
> frequency domain, and you get exactly the same number (see Parseval's
> Theorem). &#4294967295;If the energy balance works, then its hard to argue with the
> physics.
>
> The FFT case is a bit problematical, because the FFT is only exact if you
> happen to be dealing with a periodic and sampled signal. &#4294967295;What makes the
> FFT fail to be exact isn't because the transform itself isn't exact, it's
> because you're "telling" the math that you're giving it one cycle of a
> sampled periodic wave, and chances are that's not really what you have.
>
> So you can manufacture "ghost" signals, both by aliasing and by spectral
> leakage, and the FFT itself cannot distinguish those signals from the
> real thing.
>
> But the harmonics themselves are real things -- any time you have a
> periodic waveform that's not a perfect sine wave you'll have them, and in
> any physical system there's real energy behind them, so they're as real
> as real can be.
>
> --www.wescottdesign.com- Hide quoted text -
>
> - Show quoted text -

They could be complex or imaginary, as well as real.

Dirk

Tim Wescott <tim@seemywebsite.com> wrote:
(snip)
 
< Do you understand the mathematics behind the Fourier transform and the 
< FFT?  Harmonics in a signal are as real as those mathematics, and in the 
< case of the Fourier transform that's pretty darn real -- you can sum up 
< all the energy in a signal in your choice of real time or the Fourier 
< frequency domain, and you get exactly the same number (see Parseval's 
< Theorem).  If the energy balance works, then its hard to argue with the 
< physics.

This is true, but it doesn't explain why harmonics, that is,
sines and cosines as basis functions, are so useful.  

One reason is that many physical systems generate harmonically
related signals, though not all.  The normal modes of most
bells are not harmonically related, though they are still sinusoids.
The modes of a drum head with uniform tension are bessel functions.

Another reason is that analog filters (RLC, or mass and spring)
can be described in terms of their response to sinusoidal inputs.
 
< The FFT case is a bit problematical, because the FFT is only exact if you 
< happen to be dealing with a periodic and sampled signal.  What makes the 
< FFT fail to be exact isn't because the transform itself isn't exact, it's 
< because you're "telling" the math that you're giving it one cycle of a 
< sampled periodic wave, and chances are that's not really what you have.

Sometimes you can reduce the effect by extending the time, also
known as zero padding.  If the signal decays smoothly to zero,
then the amplitude of the harmonics usually decreases with
frequency and the result isn't so bad.  If the system is continually
driven that likely won't work.  If it is driven periodically, then
do the transform on that period.

-- glen

On May 12, 12:58&#4294967295;pm, "JasonTranter" <ja...@ilearninteractive.com>
wrote:
> A pure sine wave will have not harmonics. &#4294967295;However if I distort the sine
> wave then I will see harmonics. &#4294967295;Again the question is whether they are
> real or not&#4294967295;
>
> Sorry for such a long post. &#4294967295;I am hoping that someone will know the answer
> off the top of their heads. &#4294967295;(I have discussed this question with many of
> my colleagues over the years, often resulting in heated discussion, but no
> one has come up with a definitive answer/explanation.)

My own intuition between what math shows and what physics does came
while thinking about instantaneous frequency dphi/dt while building
[tutorial] FM radio during lab in college.

If the FT shows energy present at some band, that is actually what
system is doing, whatever that means, and if not, then system is not
doing that. But FT is, in general, time-varying. Striking a bell will
generate FT that will, in theory, have impulses, but in practice, will
have wide spectrum of frequencies present, and energy of of entire
ensembly will decay, in time, not proportionally, while you are
attempting to take FT.

This is why I stopped thinking about frequencies and started thinking
about instantaneous rates of change.  To speak of a "frequency" being
present is to imply that FT will have impulse for that frequency in F-
domain that is constant. But instanteous rates of changes makes dphi/
dt explicitly a function of t, and also have much more intuitive
sense, as you cannot take an arbitrary signal, try to change its phase
quickly, without generating corresponding engery in the F-domain.

So if someone gies you pure sine way at 4440Hz and then add a little
fast-moving dancing squiggly to one of the humps, you could look at
the squiggly, discern that the rate of change of phase of signal has
to be high since the squiggly is "fast-moving", and know that there
will be energy in FT at higher frequency.

On a somewhat related note, it used to drive me *CRAZY* when some
books would write...

"and a sine wave is turned on at t == 0 as input into the LTI
filter...what is the output..", without qualfying that the filter has
zero-state at t == 0, or that the answer is *not* the same as turning
on signal at == -INF.

-Le Chaud Lapin-

On May 12, 1:58&#4294967295;pm, "JasonTranter" <ja...@ilearninteractive.com>
wrote:
> Hi there,
>
> I am involved in the machinery vibration analysis field. &#4294967295;We use FFTs to
> study vibration from rotating machinery.
>
[snip]

You ask a very good question!  I would like to caution you that you
have
assumed that the input-output relationship of your system follows the
linear, time-invariant rule.  Are you sure about this assumption?

Just food for thought:  suppose that you have a saturation effect in
your
system (which makes it nonlinear).  Then given a pure sinusoid input,
the output may have harmonics since the output may end up looking like
a square wave, for example.

Julius

On Tue, 12 May 2009 13:27:17 -0700, bellda2005 wrote:

> On May 12, 4:08&nbsp;pm, Tim Wescott <t...@seemywebsite.com> wrote:
>> On Tue, 12 May 2009 12:58:34 -0500, JasonTranter wrote:
>> > Hi there,
>>
>> > I am involved in the machinery vibration analysis field. &nbsp;We use FFTs
>> > to study vibration from rotating machinery.
>>
>> > If you look at the FFT &ldquo;rules&rdquo;, an impulse will give a spectrum with
>> > energy at all frequencies, whereas a pulse train will generate
>> > harmonics.
>>
>> > If you take that to the mechanical world, I know that a single impact
>> > is like striking a bell - vibration is 'injected' at all frequencies
>> > so the bell rings at all its natural frequencies.
>>
>> > An FFT of the impact will have energy at 'all' frequencies. &nbsp;However
>> > i the time record has multiple impacts we will see harmonics.
>>
>> > It is the presence of harmonics that I have questions about. &nbsp;Are
>> > they real. &nbsp;Is the machine experiencing vibration at those
>> > frequencies? &nbsp;Is the machine experiencing vibration between those
>> > frequencies?
>>
>> > I think there are three possible explanations:
>>
>> > 1. The harmonics are a remnant of the FFT process. &nbsp;In actual fact
>> there
>> > is &lsquo;vibration energy&rsquo; at the harmonic frequencies and in-between the
>> > harmonics &ndash; just as there is with a single impact. &nbsp;Just because the
>> > time record capture more than one impact does not mean that you are
>> > not injecting energy at &lsquo;all&rsquo; frequencies.
>>
>> > 2. The harmonics are real, but the vibration energy is only present
>> at
>> > the harmonic frequencies.
>>
>> > 3. The harmonics are not real; there is no energy at the harmonics.
>> > Some believe the peaks are present as a &lsquo;characteristic&rsquo; of the FFT
>> > process only.
>>
>> > My experiments suggest that option 2 is correct (therefore the
>> > spectrum is a true representation of the vibration at every discrete
>> > frequency), however my gut told me that option 1 would be true.
>>
>> > My question about harmonics goes further&hellip;
>>
>> > A pure sine wave will have not harmonics. &nbsp;However if I distort the
>> > sine wave then I will see harmonics. &nbsp;Again the question is whether
>> > they are real or not&hellip;
>>
>> > Sorry for such a long post. &nbsp;I am hoping that someone will know the
>> > answer off the top of their heads. &nbsp;(I have discussed this question
>> > with many of my colleagues over the years, often resulting in heated
>> > discussion, but no one has come up with a definitive
>> > answer/explanation.)
>>
>> Do you understand the mathematics behind the Fourier transform and the
>> FFT? &nbsp;Harmonics in a signal are as real as those mathematics, and in
>> the case of the Fourier transform that's pretty darn real -- you can
>> sum up all the energy in a signal in your choice of real time or the
>> Fourier frequency domain, and you get exactly the same number (see
>> Parseval's Theorem). &nbsp;If the energy balance works, then its hard to
>> argue with the physics.
>>
>> The FFT case is a bit problematical, because the FFT is only exact if
>> you happen to be dealing with a periodic and sampled signal. &nbsp;What
>> makes the FFT fail to be exact isn't because the transform itself isn't
>> exact, it's because you're "telling" the math that you're giving it one
>> cycle of a sampled periodic wave, and chances are that's not really
>> what you have.
>>
>> So you can manufacture "ghost" signals, both by aliasing and by
>> spectral leakage, and the FFT itself cannot distinguish those signals
>> from the real thing.
>>
>> But the harmonics themselves are real things -- any time you have a
>> periodic waveform that's not a perfect sine wave you'll have them, and
>> in any physical system there's real energy behind them, so they're as
>> real as real can be.
>>
>> --www.wescottdesign.com- Hide quoted text -
>>
>> - Show quoted text -
> 
> They could be complex or imaginary, as well as real.

Hey!  It's _my_ job to be a smartass!

"Real" as in the "can be traced back to physical reality" sense, not 
"real" as in the "expressed with real numbers".

Jeeze.

-- 
www.wescottdesign.com