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Stepside
02-04-2012, 11:44 AM
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?

Pete

mklotz
02-04-2012, 12:08 PM
"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".

dp
02-04-2012, 02:39 PM
By example, a stacking error of +0.0001 at 30º will produce an angular error of +0.00132º. That same stacking error at 60º will produce an angular error of +0.00229º.

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/ViewObject.aspx?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

toolmaker35
02-04-2012, 02:57 PM
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...:o

P.P.S. 1st post on the forum :D

mklotz
02-04-2012, 03:35 PM
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.

Bob Fisher
02-04-2012, 04:03 PM
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.

toolmaker35
02-04-2012, 06:11 PM
Thanks, Bob. I've been browsing for awhile. I like the laid-back atmosphere around here. I hope the OP's question was answered.

Stepside
02-04-2012, 07:06 PM
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

toolmaker35
02-04-2012, 08:00 PM
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. ;)

dp
02-04-2012, 08:46 PM
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

Stepside
02-04-2012, 09:23 PM
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.

Thanks for the input
Pete

dp
02-04-2012, 09:50 PM
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.

Jaakko Fagerlund
02-05-2012, 12:10 AM
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!

Paul Alciatore
02-05-2012, 12:39 AM
"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.

tdmidget
02-05-2012, 01:49 AM
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?:rolleyes::rolleyes:

dp
02-05-2012, 01:58 AM
Sure you don't want to rethink that?:rolleyes::rolleyes:

Do you need a picture?

02-05-2012, 04:32 AM
Sine bars and accurate angle settings are mutually compatible. Sine bars are no different from any other inspection room apparatus: their performance depends on their origianal accuracy, the care taken of them since new, and the care used i their employment.

It's not uncomon for a sine bar to suffer wear on the bottoms of their rolls from being stored in the open and slid around surface plates and machine tables. The flats throw error into small angle setting and the battering and neglect may mean the rolls are disturbed from their seats.

OTH an old sine bar in good condition, calibrated regularly, and kept cased and preserved when not in used will hold its accuracy for many years.

There are limitations to the limits of angle a sine bar can be reliably set. Then there is the accuracy required of the setting. Most angle settings in the open shop are expressed interms of several minutes of error. In a 10" sine bar set at 45 degrees one minute is represented but a stack height change of 0.0004" thereabouts.

If you have precise 6 significant digit knowledge of the sinebar's roll sizes and the distance between them the ange settings of an imperfect sie bar can be compensated mathematically and set to arc seconds should the need arise and the proper metrologoacl precautions taken.

It is true angles exceeding 45 degrees suffer more from a given stack height error and that error if need be must be tracked and compensated for. I've set sine bars to 75 degrees in the past using care and trepidation and the backing of an accurate toolmaker's knee. Beyond 60 degrees approaches the Realm Where Wilde Beastes Dwell.

There is the sine square - like a sine bar but it includes a square feature.

As a follow-on for them that's interested, the Moore Jig Borer people cooked up a double barreled sine bar setting arrangement for their tilt rotary table. 45 degrees and below it's set like a plain old sine bar. Above 45 degrees there is a second set of rolls and the stack height is determined by h = 2 * (cd * Arcsin 1/2a)

I looked for a link but none could I find.

toolmaker35
02-05-2012, 06:43 AM
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!

Thanks, Jaakko. I was reading your response, but couldn't quite get my head wrapped around it, so I wound up on Wikipedia & they have an excellent definition and illustration of just what you are describing under "The Unit Circle" header. You are absolutely correct about it being a function of a circle & not a linear function. I've wondered about the differences of these units before, but never took the time to look up the cause. Thanks, again. :)

Stepside
02-05-2012, 09:14 AM
So here is where I think the discussion has gone.

First-- Stacking error is a function of the quality of the equipment and the care taken by the operator.

Second-- The mathematical values are a derived from the sine bar sweeping an arc from 0 degrees to 90 degrees.

There are a group of students, as well a a lot of grown-ups, who learn much better when they are a presented with a use for a concept. Many of us learn "through our hands" as we manipulate objects and tools. From the exercise of building their own tools and testing them, they will have a better concept of the two ideas stated above.

I welcome any and all comments as well as any additional help.

Thanks for all the input.

Pete

tdmidget
02-05-2012, 11:05 AM
Do you need a picture?
A picture? Why don't you go ahead and make a fool of yourself with a video? Angle blocks are not intended to " set the height by sliding them back and forth until you have the height you need."

You don't "set height" with angle blocks. You stack them to achieve the angle you need and then you don't need a sine bar.

dp
02-05-2012, 11:43 AM
A picture? Why don't you go ahead and make a fool of yourself with a video? Angle blocks are not intended to " set the height by sliding them back and forth until you have the height you need."

You don't "set height" with angle blocks. You stack them to achieve the angle you need and then you don't need a sine bar.

I see your problem. In fact you can and people do. If you don't have adjustable parallels then angle blocks will work just fine. And you can use machinist's jacks, too. The only objective is to precisely establish an angle with your sine bar. It is worth mentioning, to prevent another tangent, that sine bars are setup tools, not work holding tools.

tdmidget
02-05-2012, 11:53 AM
Yeah and you could use gage blocks for a door stop, I suppose. But that's not their intended or proper use. Why would you abuse tools in such a manner?

dp
02-05-2012, 12:17 PM
Yeah and you could use gage blocks for a door stop, I suppose. But that's not their intended or proper use. Why would you abuse tools in such a manner?

I don't know why it would be considered abuse, but it is a very easy way to transfer a dimension from an object or your granite surface to your machine table. Unless you have the ability to generate an infinite number of angles with your angle block set (an impossibility) and you don't own a set of adjustable parallels or gage blocks, and your jacks are too bulky to be used on the setup bench, you sometimes need to use a simple means to create a dimension that can be used by a sine bar. Back to back angle block pairs are perfectly parallel and infinitely adjustable. And made precisely enough to wring. It is probably one of the least abusive ways of using them.

SGW
02-05-2012, 08:03 PM
The point of view I'd take is that stacking error is not the fundamental difficulty. The difficulty arises because the sine function is not linear. From 0 to 1 degrees the since function changes significantly (0.0174524). The amount of change in the sine value from degree to degree progressively decreases as the angle increases, until from 89 to 90 degrees the change in value is only 0.00015230484.

Therefore, for any given stacking error, the error in the angular setting of the sine bar will be roughly 100 times worse at 89 degrees than at 1 degree. Large angles are far more sensitive to stacking error than small angles. As a compromise, 45 degrees is a good upper limit to go by, before the sensitivity to stacking error gets too great to be practical.

dp
02-05-2012, 10:45 PM
The point of view I'd take is that stacking error is not the fundamental difficulty.

Stacking error is a problem because gage blocks have a fixed minimum increment of adjustment, and that increment creates a more significant error as the angle grows. Analog spacers such as machinist's jacks and adjustable parallels get around this, but then you have the tolerance of your measuring tools to deal with.

We're talking about the side opposite here, as the hypotenuse is fixed by the sine bar and once known doesn't change.

If the gage blocks are well made and very clean the space between them after wringing is a matter of molecules and insignificant even if you gang several together. Less than imperfections on your mill table in all likelihood.

You have though reaffirmed the cosine problem inherent in the sine bar.

SGW
02-06-2012, 08:24 AM
I guess I just see the problem differently. It still seems to me that the fundamental cause of the problem is the non-linearity of the sine function (what you call the "cosine problem.") The sensitivity to any given stack height error increases as the angle increases.

Oh well. We're all talking about the same thing, just from different perspectives.

Stepside
02-06-2012, 10:20 AM
So the next part of precision measurement would be having the sine bar, stack material and the part at a uniform temperature. Do the Sine bar and the stack require matching materials? Is there an accepted range of temperature to measure "ultra precision" parts?

What would be helpful to me would be examples from actual operations. ie "when we measure the bearing race for a turbine we leave the part and the measuring tools in a 68 degree room for 24 hours".

I now understand the Sine bar change of accuracy issue that started this thread. For all that contributed it has been a great help. The temperature issue is just "the frosting on the cake" that will make it a more interesting and informative to the students.

Pete

toolmaker35
02-06-2012, 07:37 PM
Hello Pete. Generally, shop gages that are made from hardened steel, such as sine bars and most gage blocks, can use the rule of thumb of the part expanding 0.000006"/inch/*F of temperature change. Most precision parts are calibrated at 68*F or 20*C in order to give a baseline for this. Taking the example of the original thread using a sine bar and steel gage blocks, a change in temperature from the nominal value of 68* doesn't present a problem to gaging as long as all members used in the gaging have normalized to a given temperature. Both the sine bar and the gage blocks will have "grown" the same amount relative to each other and will still produce the same angle that is sought (checking linear values is a whole 'nuther ballgame though :) ). Problems start coming into this when using dissimilar material with different thermal coefficiants. Two that come to mind are the use of ceramic gage blocks and carbide gage blocks. Both of these materials will "grow" at a different rate than the steel sine bar, thus producing errors in the angle. I glanced through my Machinery's Handbook, but didn't find the thermal coefficiant of these two materials offhand, but I'm sure that they're readily available on the web somewhere.

bob ward
02-06-2012, 10:19 PM
FWIW, I don't have a set of blocks to use with my sine bar, and for the few smallish angles I need to duplicate, morse tapers, 8° ER collet holders, NT tapers, my block is a piece of round turned to the required diameter.

If the radius of my home made block is taller than the roll on the sine bar, I mill a flat along the block.

Paul Alciatore
02-07-2012, 11:24 PM
I don't know.... Unless you have the ability to generate an infinite number of angles with your angle block set (an impossibility) ....

Actually, I wrote an article for one of our sponsor's magazines on how to do exactly that. My magazine collection is in storage right now so I can't find the exact magazine or issue, but I am sure it was within the past five years. I explained how to use the sine bar principle to generate extra odd increments between two angle blocks. All that is needed is a set of angle gauge blocks and some small drills or other consistent, round stock. The drills or round stock does not have to be particularly accurate as you can determine the exact diameter with a mike. I have used this technique on several occasions to generate accurate, odd angles for tapers and such.

If you are interested, the article was titled "Setting Up Accurate Angles - Inexpensively". I am sure Village Press will sell reprints or PM me and I will send a copy via e-mail. Caution, some real math is required, but I do explain it step by step so you can follow it somewhat easily.