Scraping: Why does a flat bar hinge at 30%?

[Yan Wo]

I've been watching several scraping videos in preparation for a class I'm taking next week.

A common observation in all the presentations is that the finished piece will hinge at about one third of its length.

I'm very curious as to why that is. Can someone educate me?

Thanks!

 

[TGTool]

If the bar is low in the middle, it will quite clearly pivot on just the ends. Similarly, if it's low on the ends the hinge point will clearly be near the middle. There may be a physical mass/momentum explanation for why a flat surface hinges at about the 30% points but I don't know it. It may also relate to the Bessel or Airy points, the two point support points for a beam to provide the least distortion from sag. Those are not identical, but are close to each other.

 

[Yan Wo]

If the bar is low in the middle, it will quite clearly pivot on just the ends. Similarly, if it's low on the ends the hinge point will clearly be near the middle. There may be a physical mass/momentum explanation for why a flat surface hinges at about the 30% points but I don't know it. It may also relate to the Bessel or Airy points, the two point support points for a beam to provide the least distortion from sag. Those are not identical, but are close to each other.

Thanks! I think you're on to something. The Airy points article you linked shows the points at about 58% of the length. Very close to what's observed.

 

[J_R_Thiele]

I find it interesting that the points that make the ends of the bar parallel (Airy), and the points that give the bar its maximum length (Bessel) are named- but the support point which provide the minimal deflection from gravity are not.

If I need to support a straight edge and measure off of it I am measuring off the bottom or the top, not the ends. It is the minimal deflection from gravity that I need---but it has no short name to remember and look up.

That they are all close is very helpful. From Airy to minimal deflection, if measuring in from one side, the difference is .0119 x the length, so it is not something to loose sleep over.

You do want to pay attention to the nodes of free vibration as mentioned in "other support points of interest". Its hard to get a good measurement from something "ringing like a bell" or picking up machine vibrations.

As to the hinging and its location.

I am sure there are others better able to answer than I- but I do have some practical observations.

If you hinge a straight edge on clean granite, and then do it on a very uniform layer of light bluing, the results are very similar. Put the same straight edge so one end is on bluing and the other is not- or it is on uneven bluing, and the hinging will be very different. No suprise, as the friction between the work and the granite is very different due to the bluing and its thickness. The other factors to pay attention to are surface area and weight distribution over that area. Suppose you are scraping all three sides of a cast iron triangle with angles of 30, 60, and 90 degrees. No two of the sides will hinge the same.

 

[CharlyDE]

I find it interesting that the points that make the ends of the bar parallel (Airy), and the points that give the bar its maximum length (Bessel) are named- but the support point which provide the minimal deflection from gravity are not.

What you are asking for is called Bessel-point. If the deflection by gravity is minimal, the length is maximal. So, the Bessel-Point has both properties: maximum length and minium deflection.

First sentence from Wikipedia in german language for the subject "Bessel-Punkt" translated into english:

The Bessel points are the two symmetrically arranged support points of a longitudinal member at which it undergoes the least possible gravity-induced deformation.

 

[J_R_Thiele]

CharlyDE

"What you are asking for is called Bessel-point. If the deflection by gravity is minimal, the length is maximal. So, the Bessel-Point has both properties: maximum length and minium deflection."

I used to think that also. It clearly would be true for a line in a two dimensional space. When it comes to actual physical bars it apparently is not.

You post prompted me to do some further reading.
This article indicates that the Bessel points and those producing minimal sag are not the same.

Airy Points, Bessel Points, Minimum Gravity Sag, and Vibration Nodal Points of Uniform Beams | Mechanics and Machines

The article references a good discussion about the topic in a thesis.

"A good discussion of Airy points, Bessel points, and minimum gravity sag is given in the thesis by Nijsse at TU Delft, Linear motion systems: a modular approach for improved straightness performance. He makes a good point about accuracy of results by comparing the locations and distortions calculated from various beam models including Euler-Bernoulli, Timoshenko, and plane strain elasticity."

The thesis

https://repository.tudelft.nl/islan...c7d-b54c-1cfafac63485/datastream/OBJ/download

is 255 pages long. The portion relevant to us is on page 49 of the PDF pages. The author states that the named support points are based on prismatic bars using simple beam theory- and that using Plane Stress Theory yields more accurate predictions as to displacement and where to place the supports. It is stated that using the external dimensions of the bar and calculating for the top surface vs the neutral plain for the support location can reduce the gravitational deflection by 35% vs simple beam theory calculations.

The thesis is interesting to read- but far far beyond my knowledge base.

For the work I am capable of doing any of the points will work