Free 173 Best Homemade Tools eBook:  
Become a founding member: 500+ tool plans, full site access, and more.

User Tag List

Page 1 of 4 1 2 3 4 LastLast
Results 1 to 10 of 38

Thread: Extending the range of calipers

  1. #1
    Supporting Member mklotz's Avatar
    Join Date
    Aug 2015
    Location
    LA, CA, USA
    Posts
    2,692
    Thanks
    282
    Thanked 5,121 Times in 1,709 Posts

    mklotz's Tools

    Extending the range of calipers

    Tony Foale has described a radius gauge...

    http://www.homemadetools.net/forum/r...ineering-70673

    based on the principle of measuring a chord and its related sagitta and then using these values to compute the radius of the measured curve.

    The same principle can be applied to measure the diameter of circular objects whose diameter exceeds the capacity of your calipers. (Capacity here can refer to: calipers whose jaws do not open far enough to encompass the diameter to be measured or calipers whose jaws are too short to reach the diameter of the object.)

    The first photo shows a too small caliper attempting to measure the diameter of an object. The caliper's jaws are too short to reach the diameter so it ends up measuring the length of some chord of the circle formed by the object. Note how the object is touching the beam of the caliper; it's pushed as far into the gape of the caliper as it will go.



    This diagram will help to understand the mathematical notation...



    The tips of the caliper jaws are at locations B and D. The line BD is the chord and that is what the calipers are measuring. The sagitta, EC (marked S in the diagram) is the distance from the chord to the circle's circumference. It should be obvious that that is the length of the caliper's jaws. Just to be clear, the following picture shows what you need to measure to find the length of your caliper's jaws...



    (but use calipers not a scale; the picture is meant to show what to measure, not how to measure it.)


    Now, if you know the length of the chord and the sagitta, you can compute the radius of the curve from a simple formula...

    r = (c^2 + 4 * s^2) / (8 * s)

    d = 2 * r = (c^2 + 4 * s^2) / (4 * s)

    c = chord
    s = sagitta
    r = radius
    d = diameter

    How well does it work ? Well, of course, that depends on your instruments and how carefully you measure.

    For the bench block pictured, I made the following measurements...
    c = 2.859
    s = 1.159

    and computed a diameter of 2.922. I then used my height gauge to measure the diameter (the jaws of my larger calipers are too short) and got a diameter of 2.924. YMMV.

    173 Best Homemade Tools eBook
    Last edited by mklotz; Apr 13, 2019 at 10:55 AM.
    ---
    Regards, Marv


    Home Shop Freeware
    http://www.myvirtualnetwork.com/mklotz

  2. The Following 16 Users Say Thank You to mklotz For This Useful Post:

    CookieCrunch (May 28, 2019), DIYer (Apr 11, 2019), DIYSwede (May 20, 2019), homey_g (Jun 16, 2021), Inner (Apr 11, 2019), janders1957 (Apr 13, 2019), Jon (Apr 11, 2019), KustomsbyKent (Apr 11, 2019), old_toolmaker (Jun 11, 2021), Paul Jones (Apr 20, 2019), RetiredFAE (Jun 11, 2021), rgsparber (Apr 11, 2019), Scotsman Hosie (Apr 12, 2019), Seedtick (Apr 11, 2019), Tonyg (Apr 11, 2019), wallaci (Apr 11, 2019)

  3. #2
    Supporting Member rgsparber's Avatar
    Join Date
    Nov 2012
    Location
    Phoenix, AZ
    Posts
    1,049
    Thanks
    567
    Thanked 2,236 Times in 548 Posts

    rgsparber's Tools
    Marv,

    Do you have any error analysis?

    Rick
    Rick

  4. #3
    Supporting Member Toolmaker51's Avatar
    Join Date
    Feb 2016
    Location
    Midwest USA
    Posts
    3,587
    Thanks
    5,540
    Thanked 2,287 Times in 1,422 Posts

    Toolmaker51's Tools
    Quote Originally Posted by rgsparber View Post
    Marv,

    Do you have any error analysis?

    Rick
    Our mathmetician's mathematician, Marv don't need no stinkin' error analysis, lol.
    But he provides next best thing "YMMV".
    Sincerely,
    Toolmaker51
    ...we'll learn more by wandering than searching...

  5. The Following User Says Thank You to Toolmaker51 For This Useful Post:

    that_other_guy (Apr 24, 2019)

  6. #4
    Content Editor
    Supporting Member
    DIYer's Avatar
    Join Date
    Aug 2013
    Posts
    3,079
    Thanks
    649
    Thanked 1,587 Times in 1,423 Posts


    Thanks mklotz! We've added your Circular Object Radius Computation to our Measuring and Marking category,
    as well as to your builder page: mklotz's Homemade Tools. Your receipt:




  7. #5
    Supporting Member tonyfoale's Avatar
    Join Date
    Nov 2016
    Location
    Spain
    Posts
    1,123
    Thanks
    466
    Thanked 2,001 Times in 540 Posts

    tonyfoale's Tools
    Quote Originally Posted by rgsparber View Post
    Marv,

    Do you have any error analysis?

    Rick
    In the PDF from my post, that Marv mentioned, I wrote about that aspect. The same considerations apply in general to Marv's clever variation on that theme. The PDF is here RadiusGauge.pdf

  8. The Following User Says Thank You to tonyfoale For This Useful Post:

    Scotsman Hosie (Apr 12, 2019)

  9. #6

    Join Date
    Nov 2017
    Location
    Southern England
    Posts
    4
    Thanks
    1
    Thanked 1 Time in 1 Post

    joneb's Tools
    Quote Originally Posted by mklotz View Post
    Tony Foale has described a radius gauge...

    http://www.homemadetools.net/forum/r...ineering-70673

    based on the principle of measuring a chord and its related sagitta and then using these values to compute the radius of the measured curve.

    The same principle can be applied to measure the diameter of circular objects whose diameter exceeds the capacity* of your calipers. (Capacity here can refer to: calipers whose jaws do not open far enough to encompass the diameter to be measured or calipers whose jaws are too short to reach the diameter of the object.)

    The first photo shows a too small caliper attempting to measure the diameter of an object. The caliper's jaws are too short to reach the diameter so it ends up measuring the length of some chord of the circle formed by the object. Note how the object is touching the beam of the caliper; it's pushed as far into the gape of the caliper as it will go.



    This diagram will help to understand the mathematical notation...



    The tips of the caliper jaws are at locations B and D. The line BD is the chord and that is what the calipers are measuring. The sagitta, ED (marked S in the diagram) is the distance from the chord to the circle's circumference. It should be obvious that that is the length of the caliper's jaws. Just to be clear, the following picture shows what you need to measure to find the length of your caliper's jaws...



    (but use calipers not a scale; the picture is meant to show what to measure, not how to measure it.)


    Now, if you know the length of the chord and the sagitta, you can compute the radius of the curve from a simple formula...

    r = (c^2 + 4 * s^2)(8 * s)

    d = 2 * r = (c^2 + 4 * s^2)(4 * s)

    c = chord
    s = sagitta
    r = radius
    d = diameter

    How well does it work ? Well, of course, that depends on your instruments and how carefully you measure.

    For the bench block pictured, I made the following measurements...
    c = 2.859
    s = 1.159

    and computed a diameter of 2.922. I then used my height gauge to measure the diameter (the jaws of my larger calipers are too short) and got a diameter of 2.924. YMMV.
    Sorry for being a bit thick but what does the ^ mean I've not seen this symbol used before

  10. #7
    Supporting Member mklotz's Avatar
    Join Date
    Aug 2015
    Location
    LA, CA, USA
    Posts
    2,692
    Thanks
    282
    Thanked 5,121 Times in 1,709 Posts

    mklotz's Tools
    Quote Originally Posted by joneb View Post
    Sorry for being a bit thick but what does the ^ mean I've not seen this symbol used before
    It stands for "raised to the power". As examples...

    x^2 means x-squared
    x^3 means x-cubed
    x^n means x raised to the n power

    In handwritten math the exponent would just be written as superscript. Superscripts and subscripts aren't common on plain keyboards so many programming languages use the circumflex that is on most keyboards to indicate exponentiation.
    Last edited by mklotz; Apr 12, 2019 at 11:00 AM.
    ---
    Regards, Marv


    Home Shop Freeware
    http://www.myvirtualnetwork.com/mklotz

  11. The Following 3 Users Say Thank You to mklotz For This Useful Post:

    elk-a-holic (Apr 12, 2019), joneb (Apr 12, 2019), Toolmaker51 (Apr 12, 2019)

  12. #8
    Supporting Member mklotz's Avatar
    Join Date
    Aug 2015
    Location
    LA, CA, USA
    Posts
    2,692
    Thanks
    282
    Thanked 5,121 Times in 1,709 Posts

    mklotz's Tools
    Quote Originally Posted by rgsparber View Post
    Marv,

    Do you have any error analysis?

    Rick
    Well, since we have the complete relationship of the variables expressed as an equation, constructing an error analysis is straightforward although it does require a bit of differential calculus.

    We have from my previous post:

    d = c^2/(4s) + s

    Taking the derivative of d wrt c we have:

    Dc = dd/dc = c/(2s)

    and the derivative wrt s is:

    Ds = dd/ds = 1 - c^2/(4s^2)

    The error in d due to an error of magnitude Ec in c, which we'll label Ed, is given by:

    Edc = Dc * Ec

    Similarly, the error in d due to an error Es in s is given by:

    Eds= Ds * Es

    The total error in d is then given by the RSS (root-sum-square) of Edc and Eds...

    Ed = sqrt (Edc^2 + Eds^2)

    As my freshman calculus teacher used to say, it's left as an exercise for the student to fill in the numbers and obtain a final numerical value.
    ---
    Regards, Marv


    Home Shop Freeware
    http://www.myvirtualnetwork.com/mklotz

  13. The Following User Says Thank You to mklotz For This Useful Post:

    Paul Jones (Apr 20, 2019)

  14. #9

    Join Date
    Mar 2019
    Posts
    13
    Thanks
    10
    Thanked 2 Times in 2 Posts
    Hi Marv,
    That is a neat way to get the diameter of a cylinder that is larger than the capacity of your calipers. However in the above equation, the first term in brackets needs to be divided by the second term in brackets. The division sign between the two terms in the equation above is missing. Should be as follows:
    d = 2*r = (c^2+4*s^2)/(4*s) or simply d = c^2/4*s +s
    Thanks for the idea.
    MrMetal

  15. #10
    Supporting Member mklotz's Avatar
    Join Date
    Aug 2015
    Location
    LA, CA, USA
    Posts
    2,692
    Thanks
    282
    Thanked 5,121 Times in 1,709 Posts

    mklotz's Tools
    Quote Originally Posted by MrMetal View Post
    Hi Marv,
    That is a neat way to get the diameter of a cylinder that is larger than the capacity of your calipers. However in the above equation, the first term in brackets needs to be divided by the second term in brackets. The division sign between the two terms in the equation above is missing. Should be as follows:
    d = 2*r = (c^2+4*s^2)/(4*s) or simply d = c^2/4*s +s
    Thanks for the idea.
    MrMetal
    Thanks for that. I have no idea how that sign disappeared. I'll fix it immediately.
    ---
    Regards, Marv


    Home Shop Freeware
    http://www.myvirtualnetwork.com/mklotz

  16. The Following User Says Thank You to mklotz For This Useful Post:

    Paul Jones (Apr 20, 2019)

Thread Information

Users Browsing this Thread

There are currently 1 users browsing this thread. (0 members and 1 guests)

Bookmarks

Posting Permissions

  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
  •