Using general tolerances that apply to locating dimensions have four inherent problems. These problems are:
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Undesirable tolerance accumulation
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Lack of clear measurement origins
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Tolerancing points in space that cannot be measured
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Wedge shaped tolerance zones
The next drawing illustrates these four problems. Imagine trying to draw a shadowgraph to inspect the R30 on an optical comparator. The general tolerances do not relate to the datums shown on the part so you would be guessing at where to start. Also, since the 50 and the 75 dimensions would use the +0.2 general tolerance, it is not clear where the center of the R30 is. Anyone who has ever tried to inspect a part such as this will tell you that once the part is produced, it is virtually impossible to reproducibly measure the R30 value and its location. Again, once you have the part in your hand, try to determine what origin to use when measuring the 45° angled surface.
These problems may be overcome by using general geometric tolerances to locate features. By using a general profile tolerance, the tolerances apply to the surface of the part, something you can touch. General + tolerances are often to imaginary points in space.
ISO 2768 invokes a set of general tolerances based on a study of common machine shop practices. If the part does not meet these general tolerances, the part might still be accepted if it seems to work in its' function. This approach may work in Europe, but with our present requirements for Cp and Cpk, it would never fly. Avoid referencing this standard. Coplanarity:ISO uses flatness and the term "common zone" to control coplanarity.
Locating plane surfaces
Y14.5 uses profile of a surface to locate planes. Although ISO does not disallow using profile of a surface to locate planes, traditionally position is used. In Y14.5, position may only be used to locate features of size and bounded features such as hex, square and 'D' shaped holes. As mentioned earlier, the ISO definition of a profile of a surface tolerance zone creates a rounded corner condition while Y14.5 creates a sharp corner condition. Concentricity and symmetry
Y14.5 requires that all derived median points must be within the specified tolerance for concentricity and symmetry. This makes inspection very time consuming and should be avoided. ISO's definitions of concentricity and symmetry are identical to the Y14.5 definition of position for features shown coaxial or symmetric. Also, ISO permits the use of maximum material and least material for concentricity and symmetry whereas Y14.5 does not. Additional differences in ISO
ISO as yet does not have symbols for all around, between, controlled radius, counterbore, countersink, depth, statistical tolerance and tangent plane. ISO does not include axis or center plane straightness, composite profile and a mathematically defined datum feature. Datum referencing with position is optional. Ambiguous order of datums is permitted in datum referencing if no vertical lines are present in the feature control frame where datums are referenced. Target lines use different symbology. Angular tolerances do not include form control. A leader when specifying a datum feature or geometric tolerance may indicate a centerline. Numbers are separated from decimal fractions by a comma. The application of geometric tolerances to threads is not defined in ISO. First angle projection is the default in ISO whereas the ASME standards default to third angle projection. The definition of composite does not agree in the two standards.
(Day, 1997 – 2008)
Tolerance Stack
It is of interest to examine how the ASME Y14.5M-1994 standard and its companion ASME Y14.5.1M-1994 treat this subject. The former contains a very short Section 2.16, pp 38-39, which briefly mentions the basic forms of arithmetic and statistical tolerancing in connection with a new drawing symbol indicating a statistical tolerance, namely · ST . This symbol is intro-
duced there for the first time and it is to be expected that future editions of this standard will move toward taking advantage of statistical tolerance stacking. At this point the above symbol indicates that tolerances set with this symbol are to be monitored by statistical process control methods. How that is done is still left up to the user. Other symbols with similar intent are already in use in various companies.
Typically any exposition on tolerancing will include the two cornerstones, arithmetic and statistical tolerancing. We will make no exception, since these two methods provide conservative and optimistic benchmarks, respectively.
Under arithmetic tolerancing it is assumed that the detail part dimensions can have any value within the tolerance range and the arithmetically stacked tolerances describe the range of all possible variations for the assembly criterion of interest.
In the basic statistical tolerancing scheme it is assumed that detail part dimensions vary randomly according to a normal distribution, centered at the midpoint of the tolerance range and with its ±3s spread covering the tolerance interval. For given part dimension tolerances this kind of statistical analysis typically leads to much tighter assembly tolerances, or for given assembly tolerance it requires considerably less stringent tolerances for detail part dimensions, resulting in significant savings in cost or even making the difference between feasibility or infeasibility of a proposed design.
Practice has shown that the results are usually not quite as good as advertised. Assemblies often show more variation in the toleranced dimension than predicted by the statistical tolerancing method. The causes for this lie mainly in the violation of various distributional assumptions, but sometimes also in the misapplication of the method by not understanding the assumptions. Not wanting to give up on the intrinsic gains of the statistical tolerancing method one has tried to relax these distributional assumptions in a variety of ways. As a consequence such assumptions are more likely to be met in practice.
One such relaxation is to allow other than normal distributions. Such distributions essentially cover the tolerance interval with a wider spread, but are still centered on the tolerance interval midpoint. This results in somewhat less optimistic gains than those obtained under the normality assumption, but it usually still yields better results than those given by arithmetic tolerancing, especially for tolerance chains involving many detail parts.
Another relaxation of assumptions concerns the centering of the distribution on the tolerance interval midpoint. The realization that it is difficult to center any process exactly where one wants it to be has led to several mean shift models. In these the distribution may be centered anywhere within a certain small neighborhood around the nominal tolerance interval midpoint, but usually it is still assumed that the distribution is normal and its ±3s spread is within the tolerance limits. This means that while we allow some shift in the detail process mean we either require a simultaneous reduction in part variability or we have to widen the tolerance interval. The mean shifts are then stacked in worst case fashion. The variation of the shifted distributions
is stacked statistically. The overall assembly tolerance then becomes (in worst case fashion) a sum of two parts, consisting of an arithmetically stacked mean shift contribution and a term reflecting the statistically stacked part variation distributions. It turns out that our cornerstones of arithmetic and statistical tolerancing are special cases of this more general model, which has been claimed (Greenwood and Chase, 1987) to unify matters.
However, there is another way of dealing with mean shifts which appears to be new, at least in the form presented here. It takes advantage of statistical stacking of mean shifts and stacking that aggregate in worst case fashion with the statistical stacking of the part variation distributions. A precursor to this can be found in Desmond"s discussion of Mansoor"s (1963) paper. However, there it was pointed out that it leads to optimistic results. We discuss the issues involved and present several variations on that theme.
Other fixes augment the statistical tolerancing method with blanket tolerance inflation factors with more or less compelling reasons. It turns out that one of the above mentioned mean shift treatments results in just such an inflation factor, with the size of the factor linked explicitly to the amount of tolerated mean shift.
When dealing with tolerance stacking under mean shifts one has to take special care in assessing the risk of nonassembly. Typically only one tail of the assembly stack distribution is significant when operating at one of the two worst possible assembly mean shifts. One can take advantage of this by reducing the assembly tolerance by some small amount. We indicate briefly how this is done but refer to Scholz (1995) for more details.
(Scholz, 1995)
Chain Dimensioning
Chain Dimensioning is the method of dimensioning the beginning of the next feature from the end of the last. It seems pretty straight forward but some may not realize that this gives the greatest amount of variation. The illustration below shows a simple revolved part dimensioned using this method. The part shown was modeled to have a mean distance of 3.500? between surfaces A and B. Adding together the tolerances of the three intermediate dimensions, you will see that the actual variance is +/- .015?. This would make the maximum distance between A and B to be 3.515?. This may not seem like a lot but it may be enough to affect the overall performance of the design.
Base Line Dimensioning
Base Line Dimensioning gives you a better result then Chain Dimensioning. With Base Line Dimensioning each feature is dimensioned independently from each other, all off the same origin. By using this method you are creating less variance between features since the actual variance is the tolerance of the two features added together. Using the same part as before, this time using base line dimensioning, the actual variance is +/- .010? because you are only going to add the tolerance of the dimension that affects A to the tolerance of B. This will make the maximum distance between A and B to be 3.510". This is better then before, but we can sill do better.
Direct Dimensioning
Direct Dimensioning will give you better control over the finished dimensions of the part since you will be tolerancing the specific features you wish to control. In this part we have been concerned about the true distance between A and B, so why don"t we just control that dimension. By adding adding a dimension between A and B we cut the variance down to +/-.005?, this would give us a maximum distance of 3.505?. I am not saying that you should not use any of the two previous methods. Each method has it"s place and you should consider what variance you can accept when dimensioning your parts.
Dimensional Limits Related to an Origin
As you saw in the previous sections how you dimension a part can seriously effect the final results. Where a dimension originates from can also have an effect on the final shape of the part. The method shown below designates a feature as the dimension origin using a Dimension Origin Symbol instead of an arrow. This is not the same as designating a "datum" as you would in GT&D (we will cover Datums at a later date) instead this method is used to create a tolerance zone that the feature must lie. Look at the 1.000+/-.100 dimension, the shorter side of the part is being designated at the origin. This means that the tolerance applies to the other side of the dimension, the longer side.
The figure below better illustrates what the tolerance zone created by designating the shorter side as the dimension origin. The entire longer surface indicated must lie within the tolerance zone created.
In case your wondering why it makes a differences as to what side the dimension originates, the view below shows how the part could be made of the other side was designated. Big difference, right?
Designating a Origin
Now that you know what is a dimension origin, you probably want to know how to add the symbol to a dimension. I must admit, this stumped me for a while and I must thank Josh Mings at Solidsmack for helping me figure it out. The key is to make sure that your the intended dimension is not set as a Smart Dimension. If you right-click on the dimension and you see Smart Dimension selected, de-select it. Then on the dimension itself, when you select it, you will see nodes on each arrowhead.
When you right click on the node, you will be presented with the available arrowhead types. Click the Dimenension Origin Symbol, the one that looks like an empty circle.
(Geek, 2008)
Notation and Problem Formulation
The tolerance stacking problem arises in the context of assemblies from interchangeable parts because of the inability to produce or join parts exactly according to nominal. Either the relevant part dimension varies around some nominal value from part to part or it is the act of assembly that leads to variation.
For example, as two parts are joined via matching hole pairs there is not only variation in the location of the holes relative to nominal centers on the parts but also the slippage variation of matching holes relative to each other when fastened.
Thus there is the possibility that the assembly of such interacting parts will not function or won"t come together as planned. This can usually be judged by one or more assembly criteria, say G1,G2, . . .. Here we will be concerned with just one such assembly criterion, say G, which can be viewed as a function of the part dimensions L1, . . ., Ln. A simple example is illustrated in Figure 1, where n = 6 and
G = L1 – (L2 + L3 + L4 + L5 + L6)
= L1 – L2 – L3 – L4 – L5 – L6 (1)
is the clearance gap of interest. It determines whether the stack of cogwheels will fit within the case or not. Thus it is desired to have G > 0, but for functional performance reasons one may also want to limit G from above.
A graphical representation of equation (1) is given in Figure 2, where the various dimensions L1, L6, L5, L4, L3, and L2 are represented by vectors chained together, L1 butting into L6, L6 butting into L5 (after changing direction), L5 butting into L4, L4 butting into L3, and L3 butting into L2.
The remaining gap to make L2 butt up to L1 is the assembly tolerance gap of interest, namely G. This type of linkage is called a tolerance path or tolerance chain. Note that the arrows point right for positive contributions and left for negative ones.
As was pointed out before, the actual lengths Li may differ from the nominal lengths ?i by some amount. If there is too much variation in the Li there may well be significant problems in satisfying G > 0. Thus it is prudent to limit these variations through tolerances. Such tolerances, Ti, represent an "upper limit" on the absolute difference between actual and nominal values of the ith detail part dimension, i.e., |Li – ?i| = Ti. It is mainly in the interpretation of this last inequality that the various methods of tolerance stacking differ.
The nominal value ? of G is usually found by replacing in equation (1) the actual Li"s by the corresponding nominal values ?i, i.e.,
? = ?1 – ?2 – ?3 – ?4 – ?5 – ?6.
If the objective is to achieve a gap G that is positive and not too large (for other functional reasons) then one would presumably design the assembly in such a way that the nominal gap ? satisfies this goal, with the hope that the actual gap G be not too different from ?. Thus the quantity G – ? is of considerable interest. It can be expressed as follows in terms of _i = Li – ?i, the detail deviations from nominal,
G – ? = (L1 – ?1) – (L2 – ?2) – (L3 – ?3)
– (L4 – ?4) – (L5 – ?5) – (L6 – ?6)
= _1 – _2 – _3 – _4 – _5 – _6 .
The main question of tolerance stacking is the bounding of the assembly error or assembly misfit G – ? when given tolerance bounds Ti on the detail part errors, i.e. |_i| = |Li – ?i| = Ti. In the following we will present several such bounds and state under what assumptions they are valid. Before doing so we generalize the above example to a generic tolerance chain and in the process widen the scope to smooth sensitivity analysis problems.
Above we had an assembly with a stack of six parts that involved one positive and five negative contributions. This can obviously be generalized to n detail parts with various configurations of positive and negative contributory directions in the tolerance chain. Hence in general we have:
G = a1L1 + a2L2 + a3L3 + . . . + anLn,
F (X1,X2, . . . ,X7) = Y Y
where the coefficients a1, . . . , an are either +1 or -1, independently of each other. Our introductory example had n = 6 and a1 = 1, a2 = . . . = a6 = -1.
This then leads to
as the primary object of tolerance stack analysis.
From here it is only a small step to extending these methods to sensitivity analysis in general. Those not interested in this generalization can skip to the beginning of the next section.
Rather than butting parts end to end and forming an arithmetic sum of ± terms with some resultant output G, we can view this relation as a more general input/output relation. To get away from the restrictive notion of lengths we will use X1, . . .,Xn as our inputs (in place of L1, . . ., Ln) and Y (in place of the gap G) as our output. However, here we allow more general rules of composition, namely
Y = f(X1, . . .,Xn) ,
where f is some known, smooth function which converts the inputs X1, . . .,Xn into the output Y . This is graphically depicted in Figure 3. As an example 7 of such a more general relationship consider some electronic device with components (capacitors, resistances, etc.) of various types. There may be several performance measures for such a device and Y may be any one of them. Given the performance ratings X1, . . .,Xn of the various components, physical laws describe the output Y in some functional form, which typically is not linear.
The design of such an electronic device is based on nominal values, ?1, . . . , ?n, for the component ratings. However, the actual characteristics X1, . . .,Xn will typically be slightly different from nominal, resulting in slight deviations for the actual Y = f(X1, . . .,Xn) from the nominal ? = f(?1, . . . , ?n). Since these component deviations are usually small we can reduce this problem to the previous one of mechanically stacked parts by linearizing f, namely use
Note: for this linearization to work we have to assume that f has continuous first partial derivatives at (?1, . . . , ?n).
Aside from the term a0 we have again the same type of arithmetic sum for our "assembly" criterion Y as we had in the mechanical tolerance stack.
However, here the ai are not restricted to the values ±1. The additional term a0 1 does not present a problem as far as variation analysis is concerned, since it is constant and known.
Again we like to understand how far Y may vary from the nominal ? =
f(?1, . . . , ?n). From the above we have
i.e., just as before, the only difference being that the ai are not restricted to ±1. Since all the tolerance stacking formulas to be presented below will be 1it is based on the nominal and known quantities ?1, . . . , ?n given in terms of these ai and since nowhere use was made of ai = ±1, it follows that they are valid for general ai and thus for the sensitivity problem.
There are situations in which a functional relation Y = f(X1, . . .,Xn), although smooth, is not very well approximated by a linear function, at least not over the range of variation envisioned for the Xi. In that case one could use a quadratic approximation to capture any relevant curvature in f. Tolerance stacking methods using this approach are covered in Cox (1986). These methods are fairly complex and still quite restrictive in the assumptions under which they are valid. Of course, it may be possible to extend these methods along the same lines as presented here for linear tolerance stacks.
As noted above, the linearization will work only for smooth functions f.
To illustrate this with a counterexample, where linearization fails completely, consider the function
which can be viewed as the distance of a hole center from the nominal origin (0, 0). This function does not have derivatives at (0, 0), its graph in 3-space looks like an upside cone with its tip at (0, 0, 0). There can be no tangent plane at the tip of that cone and thus no linearization. Another example where such linearization fails is discussed in Altschul and Scholz (1994). It involves hinge mating and the problem arises due to simultaneous and thus minimum gap requirements.
In presenting the tolerance stacking formulas we will return to using Li and ?i for the part dimensions and nominals. Those that wish to apply these concepts to sensitivity analysis should have no problem replacing
Tolerance Stacking Formulas
In this section we will present various formulas for tolerance stacking. By Tolerance stacking we mean a rule that combines the detail tolerances Ti Into an assembly tolerance Tassy. Typically Tassy is a monotone increasing Function of the Ti. Thus, if the resulting Tassy is too large, one can counteract That by reducing all or some of the Ti, which usually makes for costlier Part production. On the other hand, if Tassy is smaller than required for Successful assembly fit, and then one can loosen the detail tolerances Ti, with Some possibility of cost reduction.
Why do we have more than one formula for tolerance stacking and why So many? One reason for this is that these methods have evolved and are Still evolving, partly responding to economic pressures and partly because of The nature of the problem. Namely, it all depends on what assumptions one Is willing to make.
Fewer assumptions entail broader applicability but one also will get less Out of a tolerance stack analysis, i.e., one will wind up with fairly wide Assembly tolerance limits or, when trying to counteract that through the Ti, With very tight and thus costly detail tolerance requirements.
With more knowledge about the manufacturing processes one may feel Comfortable with more assumptions, resulting in tighter assembly tolerance Limits or, if those can be relaxed, with looser detail tolerance requirements.
Thus it is very important to be aware of the assumptions behind the various Methods. We will begin the presentation of stacking methods with the Worst case or arithmetic method, which tends to be most conservative. This Is followed by the conventional RSS or statistical tolerance stacking method, Which tends to be on the optimistic side. This results from imposing some
Rather stringent assumptions. If the arithmetic stacking method gives satisfactory Assembly tolerance results, then there is little motivation to try any Of the other methods, except possibly to relax detail tolerances to achieve Cost reduction. If the RSS method does not give satisfactory assembly tolerance Results, then any of the other methods will not make matters any Better. Then the only recourse is to tighten detail tolerances or, if that is Not feasible, change the design.
After discussing these two basic and well known methods we will discuss Several hybrid tolerance stacking methods which impose assumptions which Are more likely to be met in practice. As a result the assembly tolerances lie Somewhere between those corresponding to the two classical methods.
Arithmetic or Worst Case Tolerance Stacking
The validity hinges solely on the above assumption. Thus, no matter how the detail dimensions Li deviate from their nominal values ?i within the Constraint |Li – ?i| = Ti, the difference |G – ?| is guaranteed to be bounded
By T This guarantee is the main strength of this method. However, one Should not neglect to make sure that the assumptions are met, i.e., detail Parts need to be inspected to see whether |Li – ?i| = Ti or not.
The main weakness of the method is that Tarith
assy grows more or less linearly
With n. This is most easily seen when the detail part tolerance contributions
By inverting this we get
Tdetail = Tarithassy/n
,
Which tells us how to specify detail tolerances from assembly tolerances. As assemblies grow, i.e., as n gets large, these requirements on the detail
Tolerances can get quite severe.
The linear growth of Tarith assy results from assuming a devil"s advocate position
In deriving the formula for Tarith assy , namely by always taking the detail Part variation in such a way as to make the assembly criterion G differ as Much as possible from ?. This is the reason for the method"s alternate name: Worst case tolerancing.
If the detail tolerances are not all the same, it is more complicated to Arrive at appropriate detail tolerances satisfying a given assembly tolerance Requirement. For example, suppose
So that
For i = 2, . . . , n. Thus relaxing or tightening Tarith assy by some factor affects all Detail tolerances Ti by the same factor.
One may also want to treat the detail tolerances Ti in a more differentiated Manner, i.e., leave some as they are and reduce other significantly in order to Achieve the desired assembly tolerance. This easily done in iterative fashion Using the forward formula (2).
The above considerations on how to set detail tolerances based on assembly Tolerance requirements can be carried out for the other types of tolerance Stacking as well and we leave it up to the reader to similarly use the various Tolerance stacking formulas in reverse.
RSS Method or Statistical Tolerancing
Under this method of tolerance stacking a very important new element is Added to the assumptions, namely that the detail variations from nominal Are random and independent from part to part. In some sense this is a Reaction to the worst case paradigm of the previous section which many feel Is overly conservative. It is costly in the sense that it often mandates very Tight detail tolerances.
That all deviations from nominal should arrange themselves in worst case Fashion to yield the most extreme assembly tolerance is a rather unlikely Proposition. However, it had the benefit of guaranteeing the resulting assembly Tolerance. Statistical tolerancing in its classical form operates under two Basic assumptions:
Centered Normal Distribution: Rather than assuming that the Li can Fall anywhere within the tolerance interval [?i – Ti, ?i + Ti], even to The point that someone maliciously and deliberately selects parts for Worst case assemblies, we assume here that the Li are normal random Variables, i.e., vary randomly according to a normal distribution, centered On that same interval and with a ±3s spread equal to the span
Normal Distribution over Tolerance Interval
Of that interval, so that 99.73% of all Li values fall within this interval, See Figure 4. The nature of the normal distribution is such that The Li occur with higher frequency in the middle near ?i and with less Frequency near the interval endpoints. The match of the ±3s spread With the span of the detail tolerance span is supposed to express that Almost all parts will satisfy the detail tolerance limits.
Deviations from nominal are not a deliberate act but inadvertent and Due to forces not under our control. If these forces are several and Influence the final deviation from the nominal value in independent Fashion, then there are theoretical reasons (the central limit theorem of Probability theory) supporting a normal distribution for Li. However, It may not always be reasonable to assume that this normal distribution Is exactly centered on the nominal value. This objection is the starting Point for some of the hybrids to be discussed later.
Independent Detail Variation: The independence assumption is probably The most essential cornerstone of statistical tolerancing. It allows For some cancellation of variation from nominal.
Treating the Li as random variables, we also demand that these random Variables are (statistically) independent. This roughly means that the deviation Li – ?i has nothing to do with the deviation Lj – ?j for i _= j. In particular, the deviations will not be mostly positive or mostly
Negative. Under independence we expect to get a mixed bag of negative And positive deviations of various sizes which essentially leads to some Variation cancellation in the adding process. Randomness alone does Not guarantee such cancellation, especially not when all part dimension Show random variation in the same direction. This latter phenomenon Is exactly what the independence assumption intends to exclude.
Typically the independence assumption is reasonable when part dimensions Pertain to different manufacturing/machining processes. However, Situations can arise where this assumption is questionable. For Example, several similar/same parts (coming from the same process) Could be used in the same assembly. If this process is affected by a Mean shift, then this mean shift will accumulate in worst case fashion For all parts coming from that process. Thermal expansion also tends To affect different parts similarly.
Under the above assumptions of centered normality and independence we Can give the following statistical tolerance stacking formula
Where the latter formulation holds when ai = ±1 for all i = 1, . . . , n. The Term RSS for this type of stacking stems from its abbreviation for Root Sum Of Squares.
Typically Tstatassy is significantly smaller than Tarithassy . For n = 3 the magnitude Of this difference is easily visualized and appreciated by a rectangular Box with side lengths T1, T2 and T3. To get from one corner of the box to The diagonally opposite corner, one can traverse the distanceT21 + T22 + T23 Along that diagonal or one can go the long way and follow the three edges With lengths T1, T2, and T3 for a total length Tarith assy = T1 + T2 + T3 as in Figure 5.
This reduction in assembly tolerance comes at a small price. Whereas Tarithassy bounds the assembly deviation |G – ?| with absolute certainty, the
Statistical tolerance stack Tstat assy bounds |G-?| only with some high assurance, Namely with .9973 probability. The crookedness of .9973 results from the fact That the variation of G around ? is again normal2 and that ±Tstat assy represents
a ±3s range for that variation. The 3 in 3s is a nice round number, but The probability content (.9973) associated with it is not. One cannot have it Both ways.
The small price, going from absolute certainty down to 99.73%, is not all. Recall that normal part variation, centered on the tolerance interval with Ti = 3si, and independence of variation from part to part are assumed as Well.
RSS Method with Inflation Factors
Practice has shown that arithmetic tolerancing tends to give overly conservative Results and that the RSS method is too optimistic, i.e., is not living up To the proclaimed 99.73% assembly fit rate. This means that actual assembly Stack variations are wider than indicated by the ? ±Tstat assy range. The reasons
For this have been examined from various angles. We list here
Independence: An important aspect of statistical tolerance stacking is the Independence of variations from nominal for the detail parts participating In an assembly.
3si = T i: Does the ±Ti range really represent most or all of the detail part Variation?
Normality: Is the detail part variation reasonably represented by the normal Distribution?
Centered process: Is the process of part variation centered on the nominal, the midpoint of the tolerance interval?
One reason for a reduction in the efficacy of statistical tolerance stacking could be that the independence assumption is violated. We will not dwell on that issue too much except for some very specific modes of dependence such as random mean shifts or tooling errors. Dependence can take so many forms that it is difficult to cope with it in any systematic way. However, we will return to this later when we discuss mean shifts that are random.
One other possible reason for the optimism of the RSS method is that one basic premise, namely Ti = 3si, is not fulfilled. This can come about when manufacturing process owners, asked for the kind of tolerances they can hold, sometimes will respond with a ±Ti value which corresponds to a ±2si range.
Reasons for this could be limited exposure to actual data. Values outside the ±2si range are hardly ever experienced3 and if they do occur they may be rationalized away as an abnormality and then disappear from the conscious record. Thus, if Ti is specified with the misconception Ti = 2si, then Ti is too small by a factor 1.5. To correct for this, Bender (1962) suggests to multiply
the ±Tstat assy value by 1.5and calls this process "benderizing," i.e.,
The assumptions behind this formula are the same as those for (3) except that detail part tolerances correspond to ±2si rather than ±3si normal variation ranges.
This inflation factor 1.5giv es up a fair amount of the gain in Tstat assy. In fact, for n = 2 it is more conservative than arithmetic tolerance stacking, since
Of course, some may say that we should have used a 1.5fac tor on the right side as well, because those tolerances are also misinterpreted. The rationale for the inflation factor is not altogether satisfactory, since it is based on ignorance and suppositions about meanings of Ti. What we have here is mainly a communications breakdown. If we do not have data about the part process capabilities, any tolerance analysis will stand on weak legs. If we have only limited data, then it should still be possible to avoid the mixup of 2si with 3si variation ranges. In fact, upper confidence bounds on 3si, based on limited data, will be quite conservative and thus should lead to conservative values Tstatassy when using such confidence bounds for Ti.
Although the normality assumption is well supported by the central limit theorem4, there are processes producing detail part dimensions which are not normally distributed. Some such processes come about through tool wear, where part dimensions may start out at one end of the tolerance range and, as the tool wears, eventually wind up at the other end. The collection of such parts would then exhibit a more uniform distribution over the tolerance range.
Some people have simply postulated a somewhat wider distribution over the ±Ti tolerance range mainly for the purpose of obtaining an inflation factor to the RSS formula, see Gilson (1951), Mansoor (1963), Fortini (1967), Kirschling (1988), Bjorke (1989), and Henzold (1995). Several such distributions are illustrated in Figure 6 with the corresponding inflation factors c. Of course, one may find that different detail part variations warrant different inflation factors. Using such inflation factors c = (c1, . . . , cn) for the n detail parts leads to the following modified statistical tolerance stacking formula:
The underlying assumptions are that the part variations are independent and are characterized by possibly diverse distributions centered on the part tolerance intervals. These distributions, not necessarily normal, mostly cover the respective part tolerance intervals, either completely or by their ±3si ranges, see Figure 6.
The interpretation of this assembly tolerance stack is as before. Namely, one can expect that 99.73% of all assembly G gap values fall within ? ± Tstatassy(c). Although the individual contributors to the stack may no longer be normally distributed we can still appeal to the central limit theorem to conclude that G is approximately normally distributed. Since the word "limit" in central limit theorem implies that the number of terms being added should be at least moderately large, it is worth noting that in many situations one can get fairly reasonable normal approximations already for n = 2 or n = 3 stacking terms.
One notable problem case among the distributions featured in Figure 6 is the uniform distribution. In that case the sum of two uniformly distributed random variables will in general have a trapezoidal density, which on theface of it cannot qualify as being approximately normal. If the two uniform distributions have the same width then this trapezoidal density becomes triangular. See the left side of Figure 7 where the top panel gives the cumulative distribution and its normal approximation and the bottom panel shows the corresponding densities for the sum of two random variables, uniformly distributed over the interval (0, 1). The right side of Figure 7 shows the analogous comparisons for the sum of three such uniform random variables.
Although the density comparison shows strong discrepancies for the sum of two uniform random terms, there appears to be much less difference for the cumulative distribution, since the undulating errors, visible for the densities, cancel out in the probability accumulation process. Thus the central limit theorem could be appealed to even in that case, if one is content
with somewhat rougher probability approximations
Note also that the normal approximation spreads out further than the approximated distribution. This would result in conservative assembly risk assessment. Rather than .27% of assemblies falling out of tolerance (under the normal approximation) it would be actually less under uniform detail part variation.
Before using inflation factors based on specific distributions one should make sure that such distributions are really more appropriate than the customary normal distribution. Such judgments should be based on data. If one has such validated concerns they may affect just one or two such contributors in (5) and leaving most other c factors equal to one. Note that c factors larger than one increase the assembly tolerance stack.
We view formula (5) mainly as a useful extension to formula (3) for just such situations where normality does not hold for all detail part dimensions.
This way the behavior of one part process alone will not preclude us from performing a valid statistical tolerance analysis.
If one uses such distributions solely for getting some sort of inflation or protection factor without having any other justification, one should drop that distribution pretense and just admit to using an inflation factor for just such protection purpose.
Some of the distributions portrayed in Figure 6 require some comments or explanation. The uniform distribution can in some sense be viewed as a most conservative description of variation over a fixed interval. Among all symmetric, unimodal5 distributions over such an interval it has the most spread or the largest standard deviation si.
The trapezoidal density is uniform on the interval [?i – kTi, ?i + kTi], where k is some number in [0, 1], and the density falls off linearly to zero over [?i + kTi, ?i + Ti] and [?i – Ti, ?i – kTi]. The uniform and triangular density are special cases of the trapezoidal one.
The elliptical density6 consists of half an ellipse and is characterized by the requirement that one axis of the ellipse straddles the interval ?i ±Ti and its other half axis has length 2/(pTi).
Aside from the normal distribution the Student t-density is the only one among the illustrated distributions which has an unbounded range. This raises the issue of how to match up the range of such distributions with the finite range [?i-Ti, ?i+Ti]. In the normal case it has been traditional to take Ti = 3si with the normal distribution centered on ?i. With that identification 99.73% of all detail parts of type i will vary within [?i-Ti, ?i+Ti]. In the case of the Student t-distribution we have two options. We can either scale the t-distribution to match the probability content of .9973 over [?i -Ti, ?i +Ti] or we can again let Ti = 3si. In the former approach we will wind up with c-factors that are less than one, because each si would typically be much smaller than Ti/3. The trouble with this approach is that with limited data it is very difficult to establish that [?i – Ti, ?i + Ti] captures 99.73% of all detail part dimensions.
The other approch, namely Ti = 3si, is much easier to implement with limited data and it leads to a c-factor which is one. The ease derives from the fact that standard deviations can be estimated with fairly limited data.
However, the smaller the data set, the less certain we can be about the standard deviation estimate.
One detraction with using Ti = 3si is that we will tend to see more detail parts out of tolerance. In using statistical tolerancing ideas there is no need to guarantee that all detail parts are within tolerance as is required under arithmetic tolerancing. In statistical tolerancing we only need to control the amount of part variation. Occasional detail parts which fall out of tolerance do not need to be sorted out. They actually may average out just fine in the assembly. Note that the two t-distributions illustrated in Figure 6 have different degrees of freedom and thus different detail fall-out rate.
The beta density comprises a rich family of shapes and for its mathematical form we refer to Scholz (1995). Here we only considered symmetric beta densities with paramters a = ÃY and standard deviation si = Ti/v(2a + 1.)
(Scholz, 1995)
Application
Today, most manufacturing companies are abandoning their corporate standards on dimensioning and tolerancing in favor of internationally recognized standards. The two major choices in standards today are the Collection of ISO standards or ASME Y14.5M-1994. There is currently about a 70% overlap in these two standards. Most drawing requirements may be specified by staying inside this overlap
The reasons for this transition include the cost of:
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Maintaining corporate standards
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Educating vendors and employees
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Customizing new technology
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Defending interpretations
These costs are greatly reduced by using a globally recognized standard. The challenge for the corporation is to select a standard that is adequate for their applications and demographics. Many companies are being forced to use ISO standards across the board. There seems to be an assumption that anything ISO is good. Unfortunately, not all ISO standards are mature. This is certainly true in the area of dimensioning and tolerancing. Those being required to adopt the ISO standards for dimensioning and tolerancing should thoroughly understand their current state, direction for the future and current limitations. When a company has only modest design requirements, there is sufficient overlap in the ASME Y14.5 and ISO standards to adequately define undemanding parts. For more complex applications committing to the ASME Y14.5 standard or creating a corporate addendum, which supplements either the ISO or ASME, standards may be required.
Several factors need to be considered when choosing a direction for your company's standard. This matrix illustrates many of these factors.
(Day, 1997 – 2008)
Conclusion
Most manufacturing companies are abandoning their corporate standards on dimensioning and tolerancing in favor of internationally recognized standards. The two major choices in standards today are the Collection of ISO standards or ASME Y14.5M-1994. We learn some of the methods most used and we differentiate the both to choose wish one is appropriate for you company or job.
Sources
http://www.theswgeek.com/2008/09/03/standards-wednesday-tolerance-accumulation/
http://www.tec-ease.com/tce.htm
http://www.tec-ease.com/the-new-gd&t-article.htm
http://www.stat.washington.edu/fritz/Reports/isstech-95-030.pdf
Autor:
Alan Viezcas
28 de abril de 2009
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