Micron-level accuracy over a meter?

[whitmore]

You won't need a meter long lead screw, only a short one on the sub stage. It only need be long enough for your precision adjustment.

My shortest micrometer screw has over 20mm travel; if you can get to half a millimeter accuracy before adjustment,
an elastic linkage from the +/- 10mm down to +/- 0.5mm would be appropriate, and is
fairly easily arranged, using flexures like Dan Gelbart explains. That'd be a nice machining project,
IMHO.

 

[RJSakowski]

Yes, exactly. In reading an original description of Kater's work, the distance between knife edges were compared via a microscope against three different physical scales. As expected, the resulting measurements varied, but were similar to their 5th or 6th significant digits. ("A man with two watches never knows what time it is.")

I will try to do an error budget analysis to see how good an estimate of g I could expect with what's available to me. But I'm not feeling very eager to spend $500 or more to construct my own traveling microscope for the sake of a single measurement, so I'd likely still have to find someone with a large and accurate DRO'd lathe / mill or go to a lab.

Project Gutenberg has a great E-book, "Development of Gravity Pendulums in the 19th Century" by Lenzen and Multhauf that goes into a century of developments in the field. Apparently, gravity pendulums were used up to the 1950's for geological studies, as only then were better instruments finally developed. The modern approach drops weights in a vacuum for a direct measurement of acceleration due to g, and then that's used as a reference for more portable machines, e.g. a weight on a spring or even micromachined devices.

Actually, I recently built a device that measured g via a micromachined accelerometer, but that had poor accuracy for this sort of precision work (it was for a totally different purpose).

Now it becomes interesting!

I see now why you wish to accurately measure the distance between the two knife edges. A question that occurred. If the period of oscillation was determined by comparing to a pendulum clock, they would both be subject to gravity so it would seem to be a circular calibration. The definition of a second in the early nineteenth century was in terms the length of time for the Earth to rotate around the Sun . At that time, the most accurate and reproducible clocks were most likely pendulum based and thus dependent of gravity.

Considering that gravity varies by location due to distance to the center of the Earth, centrifugal force, coriolis force, and localized masses, determining it becomes daunting challenge. The question that I would ask is what is your final objective? If you are looking for an absolute value and make a determination, how do you validate that value? Since gravity varies significantly more than the ppm precision you are looking for, what do you compare to?

The only practical way that I can see would be a calibrated precision load cell and an accurate mass. In practice the load cell could be calibrated by going to a location where an accurate value for the gravitational force was known and checking with your known mass, Mass weights for balances are calibrated in metrology labs to NIST traceable standards so that shouldn't be too difficult, although you may have to take buoyancy into account.

I gather then, that you are attempting to determine an absolute value for gravity rather than looking for variations. Hence the need for an accurate measurement of the distance between the knife edges. If you were willing to use a shorter pendulum, A setup like mine would have the capability to measure that distance, subject of course to the accuracy of the calibration of the mill. The Tormach mill is capable of measurements to 14" and my RF clone can measure to 18.5". I would still make an internal length standard and have that calibrated by a metrology lab. The period will be shorter but time can be accurately measured and clocks can be easily calibrated to an accuracy of one part in 1E14. The length standard should achieve a calibration accuracy in the microinch range. The digital microscope that I have can achieve a resolution of better than 2 microns and a glass scale has resolution to 1 or 5 microns, your choice.

 

[RJSakowski]

Here is a calibration lab for load cells Seriously, an interesting exploration into an allied topic.

 

[zondar]

Now it becomes interesting!

I see now why you wish to accurately measure the distance between the two knife edges. A question that occurred. If the period of oscillation was determined by comparing to a pendulum clock, they would both be subject to gravity so it would seem to be a circular calibration.

Clocks used for scientific purposes were calibrated against the motion of stars, an independent and highly accurate standard. I presume that this was done here too. Of course it wouldn't be perfect, but time is relative, after all. ;-)

Kater's pendulum and the clock (basically the clock's pendulum) were closely, but not perfectly, synchronized. When off slightly, you can compute a period measurement of very high accuracy by measuring the beat frequency: the frequency at which the two pendulums (the clock's and the special one) coincide (the "method of coincidences"). This is similar to how a vernier caliper works.

Considering that gravity varies by location due to distance to the center of the Earth, centrifugal force, coriolis force, and localized masses, determining it becomes daunting challenge. The question that I would ask is what is your final objective? If you are looking for an absolute value and make a determination, how do you validate that value? Since gravity varies significantly more than the ppm precision you are looking for, what do you compare to?

My objective (in principle) is to do as good a job as Kater did.

As for validation, I'd first hope to find a value that is within expectations compared to the average value of g on earth, which is 9.80665 m/s^2 (32.1740 ft/s^2), say +/- 0.3% just to check that I'm in-range.

After that, I could probably find data (i.e. maps) that gives a better value for my particular location. Beyond that, I'd have to find a particular spot or lab at which a high resolution measurement was made, and set up there.

But it's also, and mostly, just to have fun with an interesting challenge!

I gather then, that you are attempting to determine an absolute value for gravity rather than looking for variations. Hence the need for an accurate measurement of the distance between the knife edges. If you were willing to use a shorter pendulum, A setup like mine would have the capability to measure that distance, subject of course to the accuracy of the calibration of the mill. The Tormach mill is capable of measurements to 14" and my RF clone can measure to 18.5". I would still make an internal length standard and have that calibrated by a metrology lab. The period will be shorter but time can be accurately measured and clocks can be easily calibrated to an accuracy of one part in 1E14. The length standard should achieve a calibration accuracy in the microinch range. The digital microscope that I have can achieve a resolution of better than 2 microns and a glass scale has resolution to 1 or 5 microns, your choice.

Thank you for the offer. Just thinking out-loud:

A shorter pendulum is certainly possible, and it would open up more avenues for a length measurement. A meter-long pendulum swings with a period of about a second. A half-second period would be 43.5 cm or about 17 inches.

There would be a penalty somewhere, but my guess is not very much in the resolution of the period measurement given more modern techniques of measuring time.

I expect I'd measure the period with an oscilloscope (which I have) and a light sensor of some sort, which would be cut by a thin blade carried at both ends of the pendulum. I think this could get me down to under 100 ns resolution without too much trouble, and possibly under 10 or even 1 if done right. With this and some averaging, I think I could hope for 7+ digits of time resolution. If that worked out, I think the period measurement component is not the hardest part, and using a smaller pendulum (within reason) would not necessarily become the major limitation (I think L will remain that). Later pendulums were indeed smaller than Kater's, too.

But I kind of like the idea of sticking with the historical meter-long pendulum.

 

[zondar]

My shortest micrometer screw has over 20mm travel; if you can get to half a millimeter accuracy before adjustment,
an elastic linkage from the +/- 10mm down to +/- 0.5mm would be appropriate, and is
fairly easily arranged, using flexures like Dan Gelbart explains. That'd be a nice machining project,
IMHO.

That video was positively fascinating! Thanks. (Plus I'm weirdly jealous of his Einstein-like accent. Lol.)

 

[RJSakowski]

Clocks used for scientific purposes were calibrated against the motion of stars, an independent and highly accurate standard. I presume that this was done here too. Of course it wouldn't be perfect, but time is relative, after all. ;-)

Kater's pendulum and the clock (basically the clock's pendulum) were closely, but not perfectly, synchronized. When off slightly, you can compute a period measurement of very high accuracy by measuring the beat frequency: the frequency at which the two pendulums (the clock's and the special one) coincide (the "method of coincidences"). This is similar to how a vernier caliper works.



My objective (in principle) is to do as good a job as Kater did.

As for validation, I'd first hope to find a value that is within expectations compared to the average value of g on earth, which is 9.80665 m/s^2 (32.1740 ft/s^2), say +/- 0.3% just to check that I'm in-range.

After that, I could probably find data (i.e. maps) that gives a better value for my particular location. Beyond that, I'd have to find a particular spot or lab at which a high resolution measurement was made, and set up there.

But it's also, and mostly, just to have fun with an interesting challenge!




Thank you for the offer. Just thinking out-loud:

A shorter pendulum is certainly possible, and it would open up more avenues for a length measurement. A meter-long pendulum swings with a period of about a second. A half-second period would be 43.5 cm or about 17 inches.

There would be a penalty somewhere, but my guess is not very much in the resolution of the period measurement given more modern techniques of measuring time.

I expect I'd measure the period with an oscilloscope (which I have) and a light sensor of some sort, which would be cut by a thin blade carried at both ends of the pendulum. I think this could get me down to under 100 ns resolution without too much trouble, and possibly under 10 or even 1 if done right. With this and some averaging, I think I could hope for 7+ digits of time resolution. If that worked out, I think the period measurement component is not the hardest part, and using a smaller pendulum (within reason) would not necessarily become the major limitation (I think L will remain that). Later pendulums were indeed smaller than Kater's, too.

But I kind of like the idea of sticking with the historical meter-long pendulum.

I used to build timers for equestrian events. For calibration pf the timer clock, I used an eight decade frequency counter which in turn was calibrated to WWV. WWV carrier frequency is accurate to a part in 1E14. You should have no trouble measuring the pendulum period. I would use a photodetector generating a pulse every rime the pendulum broke the beam. The pulse could trigger a counter to count the pulses and update a timer each time the beam was broken. Count for as many passes as needed to get the precision you want. A pendulum with a half second period and a well setup photodetector should give you ppm accuracy with less than 1,000 passes.

I installed optical homing on my Tormach 770 to improve homing repeatability some years ago. I used an Omron OPB829DZ optointerupter, combined with regulated emitter LED current, comparison to a regulated reference voltage via an LM311 comparator, and well designed optics. This provides consistent positioning to within +/- .0001". While I am dealing in position, you can convert tha precision to time by calculating the velocity of the pendulum as it breaks the beam. For best accuracy, I would place the detector near the center of the swing in order to have the highest velocity.

 

[zondar]

I used to build timers for equestrian events. For calibration pf the timer clock, I used an eight decade frequency counter which in turn was calibrated to WWV. WWV carrier frequency is accurate to a part in 1E14. You should have no trouble measuring the pendulum period. I would use a photodetector generating a pulse every rime the pendulum broke the beam. The pulse could trigger a counter to count the pulses and update a timer each time the beam was broken. Count for as many passes as needed to get the precision you want. A pendulum with a half second period and a well setup photodetector should give you ppm accuracy with less than 1,000 passes.

Yes, you just reminded me that I have a superbly-accurate counter available that I could borrow. It would do a lot of the work for me, such as averaging periods. Can't average for too much at a time, though, since the period changes slightly with the amplitude of the pendulum. I certainly would have to average, though, since there is constant geological noise all around me from cars and trucks and such.

Only the period is needed here, but yes, I'd measure at the bottom of the swing so the time spent breaking the beam is the shortest.

Another thought about judging the value obtained: The value of g at sea level is 9.80665 m/s^2, and equations to compensate for latitude (the earth is not round) and altitude (I am a few hundred feet above sea level) are easy to find. The equations won't compensate for local geological features, but I could start with that and see how far off I am.

 

[zondar]

I installed optical homing on my Tormach 770 to improve homing repeatability some years ago. I used an Omron OPB829DZ optointerupter, combined with regulated emitter LED current, comparison to a regulated reference voltage via an LM311 comparator, and well designed optics. This provides consistent positioning to within +/- .0001".

I looked at that device. Natively, it shows a peak-to-peak displacement distance of 0.05 inches when the flag is in the middle. I'm not sure that's good enough, but maybe I'm not interpreting it properly.

But also, the aperture is so narrow that it would be trouble with a free-swinging pendulum (hitting it would be disaster). The pendulum would be balanced on a knife edge, not in a bearing that would limit its freedom to only pass through a narrow gap.

Also, I'd be afraid the narrow gap would cause air turbulence as the blade passes it. It would be very important to allow the pendulum to swing without any unnecessary outside interference. I think I'd need to engineer something with a significantly wider gap. But something like this sure would be convenient. I'll have to investigate alternatives.

Edit: It looks to me like that device can be severed at its gap into two independent parts (transmitter and receiver). In that case, the gap could be extended arbitrarily. Used in relative darkness, I think that would be effective. I'll poke around at similar devices, but this one looks like it could be workable after all. The counter would do the discrimination.

 

[RJSakowski]

Yes, you just reminded me that I have a superbly-accurate counter available that I could borrow. It would do a lot of the work for me, such as averaging periods. Can't average for too much at a time, though, since the period changes slightly with the amplitude of the pendulum. I certainly would have to average, though, since there is constant geological noise all around me from cars and trucks and such.

Calculating the velocity would be more difficult than measuring the period, but yes, I'd measure at the bottom of the swing so the time spent breaking the beam is the shortest.

Another thought about judging the value obtained: The value of g at sea level is 9.80665 m/s^2, and equations to compensate for latitude (the earth is not round) and altitude (I am a few hundred feet above sea level) are easy to find. The equations won't fully compensate for local geological features, but I could start with that and see how far off I am.

The velocity wasn't too difficult to calculate. The equations which are use to calculate the period of the pendulum will will give you the velocity.
Θ = Θo cos(2π/T*t) where Θ is the angle from the vertical in radians, Θo is the maximum angle, T is the period in seconds, and t is time in seconds. The angular velocity is dΘ/dt = -2πΘo/T sin(2π/T*t). When t = 0, the cosine =1 and Θ = Θo .when t = T/4, sin(π/4) =1 and dΘ/dt = 2πΘo/T. If T is 1 second and the amplitude is 0.1 radians (5.7º), the angular velocity as the pendulum swings through center is 2π*0.1/1 =.628 radian/sec. At a midway distance of 500 mm, the linear velocity will be 314 mm/sec

If the photodetector can resolve position to +/- .0001" or +/-0.0025 mm, this would be equivalent to a time resolution of 8 µsec. and timing over 10 periods would give ppm accuracy.

 

[zondar]

I found a few more sensors that have larger gaps, which I think are needed to avoid the possibility of collisions and to keep air turbulence to a minimum:

• Optech OPB315WZ has a has a slot width of almost an inch.
• Omron EE-SPWL311 is a split design that can read at up to a meter apart, though the gain drops dramatically over that distance. It's also quite a bit more expensive than the common slotted ones at about $100.

As pointed out, given the timing accuracy of modern electronics, the period measurement probably would not be the limiting factor.

I might start a new thread in the "projects" section. I have very limited machinist's equipment; basically just a tiny Sherline lathe and a few other common tools. (I'm also in the middle of building a small CNC mill, which will be a distraction for a while.)

The pendulum isn't complicated in principle, but I'm an almost total newbie when it comes to machining, so achieving critically-high squareness and parallelism of the knife-edges, etc., might be a bit challenging for me. For practice, I'll probably start with a mock-up using some aluminum I have around before investing in Invar, etc.

It should be a fun project involving many different aspects: historical research, materials, machining, electronics, a bit of math, etc.