Adequate sampling rate for a transient mechanical event?

Hello,

I am trying to capture the motion of a mechanism with a high speed
camera.  The motion is a transient impact that reverses direction
with about 100 g's of acceleration.

I have the displacement curve as acquired from a laser doppler sensor
and I'm
reading it on an oscilloscope with very high sampling rate.  However,
the camera is a much slower rate (4000 fps).  My question:

How do you determine an adequate sampling rate for a transient event
such
as this?  The Tektronix notes say you need a BW of 5 times the highest
frequency content of your signal. They also try to relate bandwidth to
rise time, but I don't know how that applies here.

I will transform my displacement signal into the frequency domain to 
look at the frequency content next.

Kurt

Kurt Sutherland wrote:

> Hello,
> 
> I am trying to capture the motion of a mechanism with a high speed
> camera.  The motion is a transient impact that reverses direction
> with about 100 g's of acceleration.
> 
> I have the displacement curve as acquired from a laser doppler sensor
> and I'm
> reading it on an oscilloscope with very high sampling rate.  However,
> the camera is a much slower rate (4000 fps).  My question:
> 
> How do you determine an adequate sampling rate for a transient event
> such
> as this?  The Tektronix notes say you need a BW of 5 times the highest
> frequency content of your signal. They also try to relate bandwidth to
> rise time, but I don't know how that applies here.
> 
> I will transform my displacement signal into the frequency domain to 
> look at the frequency content next.
> 
> Kurt

You want to sample fast enough so that the largest picture-to-picture 
difference you see is smaller than the smallest error that you can stand.

Transforming the displacement into the frequency domain will give you 
part of the answer, but it won't help if you can't relate it to your 
required error.

As a first-order cut I'd select an error energy that I can call 
acceptable, then find the frequency that has no more than that much 
energy above it (badly stated, sorry).  You need to sample at twice that 
magic frequency.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Kurt Sutherland wrote:


> I am trying to capture the motion of a mechanism with a high speed
> camera.  The motion is a transient impact that reverses direction
> with about 100 g's of acceleration.

> I have the displacement curve as acquired from a laser doppler sensor
> and I'm
> reading it on an oscilloscope with very high sampling rate.  However,
> the camera is a much slower rate (4000 fps).  My question:

> How do you determine an adequate sampling rate for a transient event
> such as this?  

First, if it is an oscillating system the usual twice the
frequency rule applies.  You definitely don't want to be
below that.

 > The Tektronix notes say you need a BW of 5 times the highest
> frequency content of your signal. They also try to relate bandwidth to
> rise time, but I don't know how that applies here.

The rise time relates to the highest frequency component in the
Fourier transform, which you should sample more than twice the
frequency of.    My first approximation to that, given that
you state the acceleration, is the square root of the ratio
of the acceleration to total displacement, and sample
at twice that frequency.

> I will transform my displacement signal into the frequency domain to 
> look at the frequency content next.

That is a good way, too.

-- glen

Kurt Sutherland wrote:

> Hello,
> 
> I am trying to capture the motion of a mechanism with a high speed
> camera.  The motion is a transient impact that reverses direction
> with about 100 g's of acceleration.
> 
> I have the displacement curve as acquired from a laser doppler
> sensor and I'm
> reading it on an oscilloscope with very high sampling rate. 
> However,
> the camera is a much slower rate (4000 fps).  My question:
> 
> How do you determine an adequate sampling rate for a transient
> event such as this?  

Your event is non-repetitive, therefore the usual 2xmaxFrequency 
rule doesn't hold.

> The Tektronix notes say you need a BW of 5 times the
> highest frequency content of your signal. 

This seems a fair compromise.

However, since you know the origin of your frequency content, you 
are probably able to estimate the highest frequency components in 
your signal.

If I take an airbag as an example (where such high accelerations 
appear), and I want to sense the pressure of the airbag against an 
obstacle like a person's face, then I know that in spite of the 
high acceleration the pressure curve will increase softly, not 
abruptly, which means, that I need not bother about very high 
frequencies.
Your camera and/or your scope reading should give you snapshots of 
the curve, the interpolation should be possible out of the 
knowledge about the signal source.

Thus, I'd try to calculate the really interesting frequency content 
from the curve on the oscilloscope.
Derive the scope curve and smoothen it as much as you'd allow 
according to what you know about the signal.

I'd guess that you end up with an acceptable time distance of let's 
say 0.1ms, so that a sampling rate of 10kS/s might be enough.
Probably, your camera will be fast enough.
To prove that, gather a couple of sequences one after the other,
compare the pictures and watch out for such where the displacement 
changes rapidly between consequent pictures.
If nothing astonishing happens so that you think, I need some more 
pictures in between here, then your sampling rate is probably fast 
enough.
Nevertheless, it depends on what you're really watching...

> They also try to relate
> bandwidth to rise time, but I don't know how that applies here.

Take a low pass filter, apply a step signal to it, and watch the 
response. What you see is the rise time which the filter allows.
Vice versa: if you want to see a response quick enough to monitor an 
event with a sharp rise/fall, you need a filter with high enough 
passband edge. The exact relation depends on the sort of filter 
which you have in your system.

> 
> I will transform my displacement signal into the frequency domain
> to look at the frequency content next.

The result will depend on the decision about sampling rate.
You'll not see higher frequencies in the frequency domain even if 
they're present in the signal - they will only spoil your result.

> 
> Kurt

Bernhard

Kurt Sutherland wrote:
> Hello,
> 
> I am trying to capture the motion of a mechanism with a high speed
> camera.  The motion is a transient impact that reverses direction
> with about 100 g's of acceleration.
> 
> I have the displacement curve as acquired from a laser doppler sensor
> and I'm
> reading it on an oscilloscope with very high sampling rate.  However,
> the camera is a much slower rate (4000 fps).  My question:
> 
> How do you determine an adequate sampling rate for a transient event
> such
> as this?  The Tektronix notes say you need a BW of 5 times the highest
> frequency content of your signal. They also try to relate bandwidth to
> rise time, but I don't know how that applies here.
> 
> I will transform my displacement signal into the frequency domain to 
> look at the frequency content next.
> 
> Kurt


The 5X Nyquist is about right.  There is a AL Nuttal paper somewhere 
that goes into a rigorous analysis on why you need about 4.5X instead of 
2X for the transient case.  I've never actually seen the paper so I 
can't outline his reasoning.  The fact that Tektronix gives a similar 
ball park number would tend to make me accept it.

Fundamentally a transient has infinite bandwidth so the strictly band 
limited requirement in Nyquist sampling doesn't hold.  Actually the 
strictly band limited requirement of Nyquist sampling is never perfectly 
meet in practice.

I recall discovering the first time that I used a Nicolet digital 
O-scope that oversampling was needed to get a nice looking trace. 
Nicolet used a polynomial interpolation scheme to display the actual 
waveform.

"Kurt Sutherland" <kurtsutherland@comcast.net> wrote in message
news:30d41dc9.0405051333.1dccfe19@posting.google.com...
> Hello,
>
> I am trying to capture the motion of a mechanism with a high speed
> camera.  The motion is a transient impact that reverses direction
> with about 100 g's of acceleration.
>
> I have the displacement curve as acquired from a laser doppler sensor
> and I'm
> reading it on an oscilloscope with very high sampling rate.  However,
> the camera is a much slower rate (4000 fps).  My question:
>
> How do you determine an adequate sampling rate for a transient event
> such
> as this?  The Tektronix notes say you need a BW of 5 times the highest
> frequency content of your signal. They also try to relate bandwidth to
> rise time, but I don't know how that applies here.
>
> I will transform my displacement signal into the frequency domain to
> look at the frequency content next.


Kurt,

It looks like you have the right idea.
The displacement measurement, with its high sample rate, should give you
adequate information.

Another way to look at it was already suggested - how much displacement
error can you stand between frames?

Maybe instead of thinking about sampling in time we should be thinking about
sampling in space.  Then you need to sample at a rate that is greater than
2x the spatial frequency or at a spatial interval that is less than 1/2 the
spatial separation of important features.  Or, in your case, that is less
than the allowable position error of the measurement.

If you know the displacement as a function of time (sampled at a very high
rate) then perhaps you can relate the two:

Let's pose the question this way:
A high-speed object at velocity v0, at time t0, starts to decelerate at time
t0.  It reverses direction and just reaches velocity -v0 = v1 at time t1.
What is the (maximum) displacement when the velocity reaches zero between t0
and t1?
How accurately must that displacement be measured?

There are two approaches that I'm sure someone will bring up:
1) You can sample the position with a camera and interpolate the position
from those frames.  If you know a lot about the acceleration, you might do a
very good job with this approach.  I'll not discuss this right now.

2) If you must brute force the measurement by only taking sample frames and
assume there is no interpolation, then consider this:
If you only sample at t0 and t1, then you haven't got a very good
measurement of the maximum displacement.
If you sample at intervals of (t1-t0)/2 then it could be much better but
only if you can register the samples at exactly t1 and t0 and only if the
acceleration is constant.
Assuming you can't do this we might assume that you are perfectly misaligned
with t1 and t0 - so it would be better to sample at intervals of (t1-t0)/4
and capture displacements at t0, t1 and [t0 + (t1-t0)/2] as well as at [t0 +
(t1-t0)/4] and at [t0 + 3*(t1-t0)/4]
But you can't know that you are perfectly misaligned either.  So even the
approach above could yield position uncertainties as great as perhaps the
equivalent displacement to a time interval of (t1-t0)/8 (using linear
interpolation)
..... and so on.

You didn't say how you're going to use the frames.  Given that you have the
high rate oscilloscope displacement data and know what you're going to do
with the frames, I'd say you already have enough information to decide what
to do with the camera.

It does appear that synching the camera with the "event" could help by as
much as a factor of 2 over the worst case arbitrary camera timing.

Also, if the experiment is repeatable, how about taking multiple frame
sequences with a variety of synch points relative to the "event" and
interleaving whole frame sets with appropriate temporal registration?  The
result would likely be somehwat "noisy" but could have some benefit.

The latter could be viewed as follows:
What if the experiment were noiseless / repeatable and was actually
continuous and periodic.  It just keeps happening over and over again with
the same frequency.
Then you could sample the periodic result at an interval that is slightly
shorter than the period - and not harmonically related ... like 3/pi of the
period.  The longer samples are taken, the more "filled" in the period with
samples.

Now, this argues that the sample interval  never has to be more than
slightly shorter than 2*(1/2) the period, no matter the frequency content of
the signal - as long as one can spend enough time sampling.  That seems to
fly in the face of common thinking.  I guess the "trick" is very accurately
knowing what the period is so that succeeding samples can be registered with
samples from preceding periods.  Normally we don't allow ourselves to "know"
that in general DSP processing as compared to computing a Fourier Series.

Fred