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  • Sine Bar Question

    At some point in time, someone on this board stated that a sine bar is only accurate on the smaller angles. Something to the effect "that the closer to 45 degrees the larger the error". Do I remember correctly and why is that so?

    Thanks in advance.
    Pete

  • #2
    "only accurate on the smaller angles" is hardly a mathematical statement.

    The equation for a sine bar is:

    sin(theta) = S/L

    where:

    theta = the angle to which the bar is to be set
    S = stack height needed to obtain this angle
    L = bar length

    We can obtain the error equation by taking the derivative of both sides wrt to S. Then:

    dtheta = ds/(L*cos(theta))

    where:

    dtheta = angle error due to
    ds = stack error

    From this we can see that a given stack error will cause a greater angle error as the angle increases because 1/cos(theta) increases as theta increases. That's not the same as "only accurate on the smaller angles".
    Regards, Marv

    Home Shop Freeware - Tools for People Who Build Things
    http://www.myvirtualnetwork.com/mklotz

    Location: LA, CA, USA

    Comment


    • #3
      By example, a stacking error of +0.0001 at 30؛ will produce an angular error of +0.00132؛. That same stacking error at 60؛ will produce an angular error of +0.00229؛.

      There is no error at all if you know precisely what the stacking height is but instrumentation being what it is there will be some amount of error present. If you trust your well-wrung gage blocks you can easily calculate your error bars based on the vendor's tolerance.

      Here's a slide show that demonstrates the 1/cos(x) problem Marv presented.
      http://www.wisc-online.com/objects/V...spx?ID=MSR2202

      BTW, Marv didn't let on in this thread that among his downloadable math tools is a sinebar calculator. It can be found here:

      http://www.myvirtualnetwork.com/mklotz/#math
      Last edited by dp; 02-04-2012, 03:30 PM.

      Comment


      • #4
        Hello Pete. I think I can answer your question from a somewhat practical point of view, although MKlotz did give the correct mathmatical function.
        Take an angle, say 20*, and derive the gage block stack needed for a 5" sine bar. Then figure the same using a 30* angle. All stack values are rounded to 7 decimal places.

        20*=1.7101007"
        30*=2.5000000"

        The difference of the two across a 10* span is .7898993"

        Now figure the same, but this time figure the values for a 35* and a 45* angle.

        35*=2.8678822"
        45*=3.5355339"

        The difference between these two, although still 10* between the two angles, is now .6676517".

        That being said, I cannot give you the mathmatical reason why there are "diminishing returns" between these two examples as the total angle increases. In shop practice, when an angle goes beyond 45*, I (if possible) try to figure the angle from the side 90* opposite this in order to minimize errors. In shop reality, if you know the true geometry of your sine bar and have an accurate set of gage blocks (and the manner of surface to lay them out), the errors would probably fall within the tolerance level that you are holding (alot of "if's" here).
        Hope this helps.

        P.S. Someone please check my math here...

        P.P.S. 1st post on the forum
        Last edited by toolmaker35; 02-04-2012, 03:08 PM.

        Comment


        • #5
          Imperfect knowledge of the length of the sine bar (L) is another source of error. It takes little effort to show that the relevant equation is:

          dtheta = -S*dL/(L^2*cos(theta))

          where dL is the error in the bar length.

          This error has the secant(theta) dependence as well.

          If estimates of both errors are known, the total angle error can be estimated by root-sum-squaring (RSS) them.
          Regards, Marv

          Home Shop Freeware - Tools for People Who Build Things
          http://www.myvirtualnetwork.com/mklotz

          Location: LA, CA, USA

          Comment


          • #6
            TM35, Welcome to the forum.Good post for the first one. Never thought about the stacking error on larger angles, but seldom use a sine bar in that range. My vise is not deep enough to allow it. Keep it up, this is a great place to learn. Bob.

            Comment


            • #7
              Thanks, Bob. I've been browsing for awhile. I like the laid-back atmosphere around here. I hope the OP's question was answered.

              Comment


              • #8
                Okay I think I am halfway there in understanding the issue. What I need is a definition/explanation of what is the "stacking error". This would help me to ask the next question and to explain why this topic is important to me.

                Thanks for the excellent and quick responses.

                Pete

                Comment


                • #9
                  From my point of view, "stacking errors" would actually be the sum of all the errors included in the setup, not just the gage block stack. These in turn can multiply a small error into a larger one. The errors can come from the geometry of the bar itself (distance between rolls, accuracy of the rolls to each other, and how parallel the working surface of the bar is to the centerline of the rolls), the accuracy of the datum plane that is being used to gage from (lower surface, i.e. surface plate), and the accuracy (grade) and condition (burrs, corrosion) of the gage blocks being used to set the stack. Also, keep in mind that when using gage blocks, there will be some rounding to the nearest number when calculating the height of the stack. This too will add to the inaccuracy of the setup. All of these can be worked around (some more than others) if you are trying to gage to very tight tolerances. I realize that all of this is a handfull, but it can be worked around with some forethought. One of the hardest things to do (for me, anyways) is to be realistic on the tolerances that I'm wanting to hold on the actual part, which in turn allows me to pay attention to what matters and what I can ignore. A 32.5* angle +/- .5* is a long ways from a 32.5* angle +/- 7 seconds.

                  Comment


                  • #10
                    Stacking error is the cumulative error in what ever you stack under the free end of the sine bar. This would typically be gage blocks that are precision ground and are wringable - the surfaces are so perfect that squeezing out the last bit of air between them will create a strong surface tension between the blocks and hold them together. If you have a good set of gage blocks they are likely the most accurate tool in your kit.

                    Once stacked the actual dimension and the expected dimension are frequently the same, and they tend to stack more accurately than your micrometers can reasonably test. But they may not. These Starrett page sets explains it:

                    http://starrett-webber.com/GB0.html
                    http://starrett-webber.com/GB46.html

                    Comment


                    • #11
                      Dennis

                      From your answer to my question, one could assume that the most accuracy would be when you used only one block. There would be only a few angles that might use a single block from a set. I realize that every block will create an angle, but there is only a small number of those created will be of use.
                      So one could assume that as the angle becomes steeper, the issue of temperature and expansion from temperature is another large problem.

                      To all who have helped so far:
                      I am working on a project using 3D printers and Laser engravers to create both a Sine bar and an adjustable parallel for high school students. There is no doubt that the results will be somewhat crude. If the students work backwards by matching their Sine bar and parallels to an existing angle they will be able to find the roll centers of their bars. This should open the doors to discussions relating to accurate measurement as well as appropriate tolerances for the job at hand.

                      Any and all help you can give me with this task will be appreciated.

                      Thanks for the input
                      Pete

                      Comment


                      • #12
                        Originally posted by Stepside
                        Dennis

                        From your answer to my question, one could assume that the most accuracy would be when you used only one block. There would be only a few angles that might use a single block from a set. I realize that every block will create an angle, but there is only a small number of those created will be of use.
                        So one could assume that as the angle becomes steeper, the issue of temperature and expansion from temperature is another large problem.
                        They're actually designed to be stacked with high accuracy but you pay a dear price for that accuracy. Another analog method is to use angle blocks. You set the height by sliding them back and forth until you have the height you need.

                        http://starrett-webber.com/AG0.html

                        Two identical angle blocks stacked properly form parallel surfaces and of infinitely varying height constrained by their dimensions. They also wring and so will hold their position. The sine bar can be supported by angle blocks set to any height desired. Accuracy is entirely at the mercy of your measuring instruments.

                        Comment


                        • #13
                          Originally posted by toolmaker35
                          That being said, I cannot give you the mathmatical reason why there are "diminishing returns" between these two examples as the total angle increases.
                          Look at an example of a unit circle (that is a circle with a radius of 1) and draw lines for those angles and you see why there is a difference between 20-30 degrees and 35-45 degrees: Sine and cosine are not linear functions, but instead get their values on a point lying on the unit circle, so the output of those functions is not linear between two linear inputs like in your example

                          And welcome to the forum!
                          Amount of experience is in direct proportion to the value of broken equipment.

                          Comment


                          • #14
                            "Stacking Error" is a poor way of looking at it. What you are concerned with is the total error in both the sine bar itself (how well was it made) and in the stack of blocks or whatever you use to prop one end of the sine bar up. Oh, and then there is any error in the supposedly flat surface that it rests on, so that is three sources of error. All else being equal, a sine bar, by it's very nature will be less sensitive to errors in the height of that "stack" at small angles. Actually it will be less sensitive to ALL of these errors at small angles.

                            A practical solution to this is if you need an angle that is greater than 45 degrees, it is theoretically more accurate to use a 90 degree angle block and use the sine bar from it to subtract a smaller angle from the 90 degrees that it produces. That way you can set up an 89 degree angle (90 - 1 = 89) with the same precision as a 1 degree angle with the sine bar alone. Of course, I am not including any error in the angle block itself which would be added to the error of the sine bar.

                            Another reason to do it this way is because most sine bars do not physically allow such large angles to be set up. The bar itself would interfere with the stack of blocks needed. Or perhaps they are designed this way because you shouldn't use them for angles over 45 degrees. Chicken - egg situation here.
                            Paul A.
                            SE Texas

                            Make it fit.
                            You can't win and there IS a penalty for trying!

                            Comment


                            • #15
                              Originally posted by dp
                              They're actually designed to be stacked with high accuracy but you pay a dear price for that accuracy. Another analog method is to use angle blocks. You set the height by sliding them back and forth until you have the height you need.

                              http://starrett-webber.com/AG0.html

                              Two identical angle blocks stacked properly form parallel surfaces and of infinitely varying height constrained by their dimensions. They also wring and so will hold their position. The sine bar can be supported by angle blocks set to any height desired. Accuracy is entirely at the mercy of your measuring instruments.
                              Sure you don't want to rethink that?

                              Comment

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