22
Design Issues in Mechanical Tolerance Analysis
ADCATS Report No. 87-5
Reprinted from Manufacturing Review, ASME, vol 1, no 1, Mar. 1988, pp. 50-59
K. W. Chase
Mechanical Engineering Department
Brigham Young University
Provo, UT 84602
W. H. Greenwood
Sandia National Laboratories
Albuquerque, NM 87185
October 26, 1987
ABSTRACT
Tolerance analysis is a valuable tool for reducing manufacturing costs by improving producibility. Several useful methods of selecting design tolerances are presented with examples. The common and more advanced tolerance analysis methods are also reviewed and evaluated. A simple new tolerance analysis model suitable for designers is described as an alternative to the advanced methods. It is much more flexible than the common engineering methods. For example, it can mix statistical and worst case components in the same assembly. Also, it includes a critical manufacturing variable that is often overlooked: "nominal shifts" or biased distributions.
1. Manufacturing Considerations in Engineering Design.
The wise specification of dimensional tolerances for manufactured parts is becoming recognized by industry as a key element in their efforts to increase productivity. Modest efforts in this area can yield significant cost savings with little capital investment. It is a prime example of the success that results from including manufacturing considerations early in the design process. Both engineering design and manufacturing personnel are concerned with the magnitude of tolerances specified on engineering drawings, as shown in figure 1.
Fig. 1. Assignment of tolerances concerns both
Engineering and Manufacturing
Engineers know that tolerance stacking or accumulation in assemblies controls the critical clearances and interferences in a design, such as lubrication paths or bearing mounts, and thus affects performance. Production people know that tight tolerances increase the cost of production. Tolerances also greatly influence the selection of production processes by process planners and determine the assemblability of the final product.
Tolerance specification, then, is an important link between engineering and manufacturing. It can become a common ground on which to build an interface between the two, to open a dialog based on common interests and competing requirements.
However, designers often assign tolerances arbitrarily or base their decisions on insufficient data or deficient models. Any resulting problems must be corrected as they arise during manufacturing planning, tooling and production. Clearly, today's high tech products and growing international competition require knowledgeable design decisions based on realistic models which include producibility requirements. Hence, several issues relative to tolerance specification methods are raised:
1) How can we get Engineering and Manufacturing to communicate their needs effectively?
2) Which tolerance analysis models are both realistic and applicable as design tools?
3) What role should advanced statistical and optimization methods play?
4) How can we get sufficient data on process distributions and costs to characterize manufacturing processes for advanced tolerance analysis models?
In the following discussion, several useful tolerance design tools are described with examples, some of which have not appeared in print before. Some of the limitations of the common engineering models for tolerance analysis are pointed out. In response to these limitations, a simple new model suitable for designers is presented, which has greatly increased flexibility and permits a more realistic representation of actual manufactured parts. Finally, advanced tolerance analysis methods are reviewed, with an evaluation of their potential for use in design.
2. Tolerance Analysis vs. Tolerance Allocation.
A central issue in tolerance specification is that engineers are more commonly faced with the problem of tolerance allocation rather than tolerance analysis. The difference between these two problems is illustrated in figure 2. In tolerance analysis the component tolerances are all known or specified and the resulting assembly tolerance is calculated. In tolerance allocation, on the other hand, the assembly tolerance is known from design requirements, while the component tolerances are unknown. The available assembly tolerance must be distributed or allocated among the components in some rational way. Analytical modeling of assemblies provides a quantitative basis for evaluation of design variations. The influence of the assembly model and the allocation rule used by the designer on the resulting tolerance allocation will be demonstrated.
Fig. 2. Tolerance Analysis vs. Tolerance Allocation.
3. Common Engineering Models for Assembly Tolerances.
If the manufacturing process for a part is known, such as turning or stamping, reasonable tolerances may be selected by following tolerance guidelines for the process. Company design manuals and industry standards also provide useful data. Tolerance build-up in assemblies may then be predicted by tolerance analysis. The basis of tolerance analysis in design is an analytical model for the accumulation of tolerances in a mechanical assembly of component parts. The two most common models used in engineering design are briefly defined below. A more complete treatment may be found in Fortini [1].
a. Worst case.
In a worst limits analysis, the assembly tolerance (TASM) is determined by summing the component tolerances (Ti) linearly. Each component dimension is assumed to be at its max. or min. limit, resulting in the worst possible assembly limits.
One-dimensional assemblies:
T= S T (1)
Multi-dimensional assemblies:
T= S (2)
where Xi are the nominal component dimensions and f (Xi) is the assembly function describing the resulting dimension of the assembly, such as the clearance or interference. The partial derivatives represent the sensitivity of the assembly tolerance to variations in individual component dimensions.
b. Statistical.
Component tolerances add as the root sum squared (RSS). The low probability of the worst case combination occurring is taken into account statistically, assuming a Normal or Gaussian distribution for component variations. Tolerances are commonly assumed to correspond to six standard deviations (6s)
One-dimensional assemblies:
T= [S T]1/2 (3)
Multi-dimensional assemblies:
T= [S (¶f/¶x)T] (4)
More general case: (other than 6s tolerance distributions)
T= C[ S (¶f/¶x)] (5)
where Z is the number of standard deviations desired for the specified assembly tolerance and Zdescribes the expected standard deviations for each component tolerance. Cis a correction factor frequently added to account for any non-ideal conditions. Typical values for Care 1.4 or 1.5.
These common tolerance accumulation models have serious limitations, which will be discussed later.
4. Tolerance Allocation Methods
The rational allocation of component tolerances requires the establishment of some rule upon which to base the allocation. The following are examples of useful rules:
a. Allocation By Proportional Scaling.
The designer begins by assigning reasonable component tolerances based on process or design guidelines. He then sums the component tolerances to see if they meet the specified assembly tolerance. If not, he scales the component tolerances by a constant proportionality factor. In this way the relative magnitudes of the component tolerances are preserved.
Example 1. Worst Case Allocation by Proportional Scaling.
The following example is based on the shaft and housing assembly shown in figure 3. Initial tolerances for parts B, D, E, and F are selected from tolerance guidelines for the turning process, such as figure 4 [2].
Fig. 3. Shaft and housing assembly.
Fig. 4. Tolerance range of machining processes.
Tolerances are chosen from the middle of the range for each part size. The retaining ring (A) and the two bearings (C and G) supporting the shaft are vendor-supplied, hence their tolerances are fixed and must not be altered by the allocation process. The critical clearance is the shaft end-play, which is determined by tolerance accumulation in the assembly. The vector diagram overlaid on the figure is the assembly loop that controls the end-play. The average clearance is the vector sum of the average part dimensions in the loop:
Initial tolerance specifications:
Required Clearance = .020 +/-.015
Average Clearance = -A + B - C + D - E + F - G
= -.0505 + 8.000 - .5093 + .400 - 7.711 + .400 - .5093
= .020
Dimension A B C D E F G
Average .0505 8.000 .5093 .400 7.711 .400 .5093
Tolerances(+/-)
Design .008 .002 .006 .002
Fixed .0015 .0025 .0025
The clearance tolerance is obtained by computing the assembly tolerance sum by worst limits:
T= + T+ T+ T+ T+ T+ T+ T
= + .0015 + .008 + .0025 + .002 + .006 + .002 + .0025
= .0245 (too large)
Solving for the proportionality factor:
T= .015 = .0015 +.0025 +.0025 + P (.008 + .002 + .006 + .002)
P = .47222
Note that the fixed tolerances were subtracted from the assembly tolerance before computing the scale factor. Thus only the four design tolerances are re-allocated:
T= .47222 (.008) = .00378 T= .47222 (.006) = .00283
T= .47222 (.002) = .00094 T= .47222 (.002) = .00094
Each of the design tolerances has been scaled down to meet assembly requirements as shown in figure 5. This procedure could also be followed assuming a statistical sum for the assembly tolerance (equation 3), in which case the tolerances would be scaled up. The results are summarized in Table 1.
b. Allocation By Constant Precision Factor
Parts machined to a similar precision will have equal tolerances only if they are the same size. As part size increases, tolerances generally increase approximately with the cube root of size [1]:
Tolerance T= P (D) (6)
where Dis the basic size of the part and P is the Precision Factor.
Fig. 5. Tolerance allocation by proportional scaling.
Based on this rule of thumb, the tolerances can be distributed according to part size as follows. Compute the Precision Factor:
Worst Limits Statistical
P = P = (7)
Then compute the component tolerances:
T= P D, T= P D, etc.
Example 2. Statistical Allocation by Precision Factor.
Compute the assembly tolerance for the shaft/housing assembly by a statistical sum:
T= T+ T+ T+ T+ T+ T+ T
.015= (.0015+.0025+.0025) + P
.400+ 7.711+.400)
Again, the fixed tolerances are subtracted from the assembly tolerance before computing the precision factor.
Solving for the precision factor: P = .004836
Re-allocating:
T= .004836 (8.00)= .00976 T= .004836 (7.711)= .00955
T= .004836 (.400)= .00356 T= .004836 (.400)= .00356
The Precision Factor method is similar to the Proportional Scaling method, except there is no initial allocation required by the designer. Instead, the tolerances are initially allocated according to the nominal size of each component dimension, then scaled to meet the specified assembly tolerance. This procedure could also be followed assuming a worst limits sum for the assembly tolerance (equation 1). The results are summarized in Table 1.
Table 1. Comparison of Allocation Methods
Proportional Precision Factor
Original Worst Stat Worst Stat
Part Tolerance Case 6s Case 6s
A .0015* .0015 .0015 .0015 .0015
B .008 .00378 .01116 .00312 .00967
C .0025* .0025 .0025 .0025 .0025
D .002 .00094 .00279 .00115 .00356
E .006 .00283 .00837 .00308 .00955
F .002 .00094 .00279 .00115 .00356
G .0025* .0025 .0025 .0025 .0025
ASSEMBLY TOL. .0150 .0150 .0150 .0150
PROP. FACTOR .472221.39526 .00156 .004836
* Fixed tolerances
5. Tolerance Allocation Using Optimization Techniques
A promising method of tolerance allocation uses optimization techniques to assign component tolerances such that the cost of production of an assembly is minimized. This is accomplished by defining a cost-vs.-tolerance curve for each component part in the assembly. The optimization algorithm varies the tolerance for each component and searches systematically for the combination of tolerances which minimizes the cost.
Figure 6 illustrates the concept simply for a three component assembly. Three cost-vs.-tolerance curves are shown. Three tolerances (T, T, T) are initially selected. The corresponding cost of production is C+ C+ C. The optimization algorithm tries to increase the tolerances to reduce cost, however, the specified assembly tolerance limits the tolerance size. If tolerance Tis increased, then tolerance Tor Tmust decrease to keep from violating the assembly tolerance constraint. It is difficult to tell by inspection which combination will be optimum. The optimization algorithm is designed to find it with a minimum of iteration. Note that the values of the set of optimum tolerances will be different when the tolerances are summed statistically than when they are summed by worst limits.
a. Cost-vs.-Tolerance Functions.
The key factor in optimum tolerance allocation is the specification of cost-vs.-tolerance functions. Several algebraic functions have been proposed, as summarized in
Fig. 6. Optimal tolerance allocation for minimum cost.
Table 2. The constant coefficient A may include setup cost, tooling, material, prior operations, etc. The B term determines the cost of producing a single component dimension to a specified tolerance.
Table 2. Proposed Cost-of-Tolerance Models
Cost Model Author Ref
Reciprocal Squared A + B/tol Spotts [3]
Reciprocal A + B/tol Chase&Greenwood [4]
Exponential A e Speckhart [5]
Little has been done to verify the form of these curves. Manufacturing cost data are not published since they are so site-dependent. Even companies using the same machines would have different costs for labor, materials, tooling and overhead.
Jamieson [6] reported a government study in which relative costs were determined for actual parts for several metal-removal processes. This seems to be the same data presented as a case study by Trucks [2]. Jamieson correlated the process/cost results with the tolerance-vs.-size chart of figure 4. This data was curve fit by regression analysis using each of the models shown in Table 2. Typical results are shown in figure 7. The reciprocal tolerance curve appears to fit the machining process data the best.
Fig. 7. Comparison of cost-vs.-tolerance models.
b. Tolerance Allocation by Lagrange Multipliers
A closed-form solution for the least-cost component tolerances was developed by Spotts [3]. He used the method of Lagrange Multipliers, assuming a cost function of the form C=A+B/tol2. Chase and Greenwood [4] extended this to cost functions of the form C=A+B/tol as follows:
+ l = 0 (i = 1, . . n)
+ l = 0 (i = 1, . . n)
l =
Eliminating l by expressing it in terms of T:
T= T (8)
Substituting into the assembly tolerance sum:
T= T+ S T
T1= (9)
Substitute this result in equation 8 to obtain the minimum cost tolerances. The numerical results for the example problem are shown in Table 3. The Setup Cost is coefficient A in the cost function. The Reference Cost and Reference Tolerance are used to compute coefficient B.