1. Tolerance Design

2. Design Specifications and Tolerance
Develop from quest for production quality and efficiency
Early tolerances support design’s basic function
Mass production brought interchangeability
Integrate design and mfg tolerances
Tolerances are critical to the successful manufacture and performance of a product over its intended life cycle. Tolerancing techniques have evolved over the years through an ever increasing quest for quality and efficiency. The demands in quality and efficiency have stimulated marked improvements in the manufacturing and servicing processes.
Early tolerances in manufacturing supported the basic functionality of the design. However, with the advent of the industrial revolution, the role of tolerances became more prominent; mass production spurned the need for improved tolerances in order to allow for universal interchangeability of components and assemblies.
In today’s world, about every product has a tolerance and every process is controlled within some specified limit. Effective tolerances in manufacturing has proven to be a very complex component of engineering which can strongly affect quality, cost and time. The engineer’s challenge today is not only to match the appropriate tolerances with the manufacturing processes but also to integrate the design and manufacturing functions to promote cost effective production.

3. Definition
“The total amount by which a given dimension may vary, or the difference between the limits”
- ANSI Y14.5M-1982(R1988) Standard [R1.4]
So what is tolerance? One definition by the ANSI standard, states that tolerance is, “The total amount by which a given dimension may vary, or the difference between the limits”. Tolerances place boundaries on the variation of a target design parameter or set point. They specify the allowable size and shape variations permitted for components and assemblies. Excessively tight tolerances can significantly increase cost; whereas, insufficient or loose tolerances can degrade performance and quality. Tolerances should derive from the customer’s requirements, the engineer’s knowledge and the analysis of the functional capabilities and limitations of the respective designs and their requisite manufacturing processes.
Source: Tolerance Design, p 10

4. Affected Areas Engineering Tolerance Product Design Quality Control
There are three main groups who are constantly concerned with the problems of determining the magnitude, assignment, and build up of tolerances. The areas include:
Product Design
Manufacturing
Quality Control
Product Design has the responsibility of designing with consideration for the maximum possible working tolerances compatible with the functional requirements of the design. Manufacturing has the responsibility of bringing the design into a physical entity with consideration for process methodologies, capabilities and economics. Quality control has the responsibility of measuring and evaluating the final product to ensure the integrity of the product tolerance.
Manufacturing

5. Questions “Can customer tolerances be accommodated by product?”
“Can product tolerances be accommodated by the process?”
When considering tolerance design, two main questions resonate. They are: (read slide)

6. Tolerance vs. Manufacturing Process
Nominal tolerances for steel
Tighter tolerances => increase cost $
Most machine operations have nominal tolerances. This chart show machining operations and nominal tolerances for steel. As the tolerance values get tighter, the cost increases significantly.

7. Geometric Dimensions Accurately communicates the function of part
Provides uniform clarity in drawing delineation and interpretation
Provides maximum production tolerance
Geometrical dimensions and tolerances are used by the designer to specify part and component requirements on a design. Usually, this information is communicated through controlled design or drawing documents such as blueprints. Standards in industry help the engineer establish universal symbols and methodologies for specific functions, values, features, etc. Uses of these standard symbols or combinations of symbols offers the following advantages:
accurately communicates the function of the part,
provides uniform clarity in drawing delineation and interpretation
provides maximum production tolerance.

8. Tolerance Types Size Form Location Orientation
There are various types of tolerances. The main considerations for dimensional tolerances are size, form, location and orientation.
Size tolerances are simply the acceptable dimensional variances that define the part features . Form tolerances considers features about the element and surface. Location tolerances specify the position of the part. Finally, orientation tolerances apply to individual features of the part relative a datum or another feature.

9. Size Tolerances
Size tolerances are simply the acceptable dimensional variances that define the part features . For example, size tolerances might specify the upper and lower limits of a part length or height or diameter.

10. Form Tolerances
Form tolerances considers features about the element and surface such roundness, straightness and flatness.

11. Location Tolerances
Location tolerances specify the position of the part relative to a datum; examples of locational tolerances are for concentricity, symmetry or true position .

12. Orientation Tolerances
Finally, orientation tolerances deal with individual features of the part which are relative a datum or another feature. Examples would include the perpendicularity of two intersecting walls. Others include parallelism and angularity.

13. Tolerance Buildup
During the assembly of parts, especially with complex parts, small variations in the part dimensions can multiply until the final assembled result is an unacceptable departure from the original design. When one looks at each part dimensions, the variations may seem small by themselves; however, with each added part, the errors can compound to create an defected final part.
Size tolerances is fairly straightforward. However, when added with form, location and/or orientation variances, then it becomes more difficult to predict dimensional fit. Likewise, from this example, the tolerances are linear; however, many design assemblies will not be this straightforward. Most design assemblies will have complex shapes and interweaving parts which will make the design and tolerance analysis much more complicated and nonlinear.

14. Statistical Principles
Measurement of central tendency
Mean
Median
mode
Measurement of variations
Range
Variance
Standard deviation
USL
LSL
tolerance 3s X
Tolerance design utilizes many different methods to understand the machine and process capabilities for a design. Statistical analysis is at the heart of understanding the design and process capabilities and is therefore necessary for effective tolerance design. Statistics is science’s method to organize, characterize and summarize data so that one can use the information to draw conclusions and/or predictions. In statistics, numerical values are used to summarize and give insight about eh data. Key statistical principles include measurements of central tendencies and measurements of variations.
Measurements of central tendency are basically values which tend to cluster around the “middle” of a value set. Three central tendency calculations are: the mean, the median and the mode.
Measurements of variation express significances to the deviation or distance of the results to the mean value. Common measurement of variations are range, variance and standard deviation.

15. Probability Probability Capability Likelihood of occurrence
Relate the mean and variability of the process or machine to the permissible range of dimensions allowed by the specification or tolerance.
Probability theory plays an important role in statistics. In everyday terms, probability can be thought of as the chances or likelihood that a particular event will occur.
Capability studies allow engineers and managers to gain insight on the process, the measurement and the abilities of the entire process. Capabilities define the distribution which describes the output of the process and relate the mean and variability of the process to the permissible range of the dimensions.
We will not discuss any more of the details and mechanics of statistics in this presentation. Please refer to the presentation module titled, “ Statistics Analysis” if you need to understand the principles for this presentation.

16. Tolerance SPC Charting
Many business processes will chart the test results of a product or process throughout the manufacturing processes. These charts give the necessary data which are used to understand the process and capabilities in order to verify improvement, process and capabilities. Here is an example of what a chart may look like during a production run. In this chart, one can quickly see the variations, tendencies and trends of the results.
Figure Source: Tolerance Design, p 125

17. Tolerance Analysis Methods
Worst-Case analysis
Root Sum of Squares
Taguchi tolerance design
There are many different approaches that are utilized in industry for tolerance analysis. The more tradition methods include:
Worst-Case analysis
Root Sum of Squares
Taguchi tolerance method
All these methods will be discussed in further details in the following slides.

18. Initial Tolerance Design
Initial stages of the tolerance design involves research for the recommended or standard tolerances that would typically be assigned to a part according to factors such as function, material, machine and process. Ideally, experienced quality and design engineers who are knowledgeable of the process capabilities will be involved early in the design stage.
Also, one can find standard tolerance information from numerous published literature; the accrued knowledge of engineers from industry and academia have produced many tables and charts about standard tolerances. Finally, one can always find available data from the vendor or manufacturer.
Once a baseline tolerance band has been established, the team can then make further refinements to the tolerances during the design process.
Figure Source: Tolerance Design, p 93

19. References Standard tolerances for mfg processes
Handbook of Product Design for Manufacturing: A Practical Guide to Low-Cost Production, James C. Bralla, Ed. in Chief; McGraw-Hill, 1986
Manufacturing Processes Reference Guide, R.H. Todd, D.K. Allen & L. Alting; Industrial Press Inc., 1994
Standard tolerances for mfg processes
Machinery’s Handbook; Industrial Press
Standard Handbook of Machine Design; McGraw-Hill
Standard Handbook of Mechanical Engineers; McGraw-Hill
Design of Machine Elements; Spotts, Prentic Hall
According to the book, tolerance Tolerance Design, the following sources are highly recommended for aiding in initial tolerance slection: The are: Handbook of Product Design for Manufacturing: A Practical Guide to Low-Cost Production,
And Manufacturing Processes Reference Guide. Additionally, references for tolerances for many different manufacturing processes are:
Machinery’s Handbook
Standard Handbook of Machine Design
Standard Handbook of Mechanical Engineers
Design of Machine Elements
For some products, especially unique products, there may not much documented references available. Therefore, the engineering team will need to work more heavily with manufacturing, quality and the vendor to help establish an appropriate tolerance baseline through discussions, analysis and experimentation.
Figure Source: Tolerance Design, p 92-93

20. Worst-Case Methodology
Extreme or most liberal condition of tolerance buildup
“…tolerances must be assigned to the component parts of the mechanism in such a manner that the probability that a mechanism will not function is zero…”
- Evans (1974)
Worst-case-analysis is not considered a statistical procedure but is used often for tolerance analysis and allocation. This method provides a basis to establish the dimensions and tolerances such that any combination will produce a functioning assembly. This method compares the part tolerances with the entire assembly tolerances to reveals the extreme or most liberal condition of tolerance build-up; hence, the term “worst-case”. Evans describes it as, “(read slide)”

21. Worst-Case Analysis Ne + Te => Maximum assembly envelope
The maximum worst-case condition, WC max, and the minimum worst-case condition, Wcmin, are expressed here mathematically. Npi is the nominal design value of the ith part, Tpi is the tolerance assigned to the ith nominal part design value and m is the total number of parts.
The worst-case maximum and minimum assembly envelopes are expressed as N sub e plus T sub e and N sub e minus T sub e, respectively. In worst case analysis, one can compare the minimum and maximum assembly envelope to the respective worse case conditions to determine whether a conflict exist.
Suppose that we have the condition that implies a negative clearance between the “stacked” parts and the envelope. Then, one would need to reduce one or more of the design specifications, either the dimension or the tolerance, in order to properly fit the design assembly.
Ne + Te => Maximum assembly envelope
Ne - Te => Minimum assembly envelope
Source: “Six sigma mechanical design tolerancing”, p

22. Assembly gaps
Therefore, we can express the Gmax as the worst case assembly gaps that the design can tolerate and Gmin as the minimum worst-case assembly gap for the assembly. We can also define the nominal assembly gap.
Now that we know the extremes and nominal gaps, we can begin to allocate the tolerances so that the assembly components can be developed for the calculated tolerance conditions.

23. Worst Case Scenario Example
In this example, we see a mating hole and pin assembly. The nominal dimensions are given in the second figure.
Source: Tolerance Design, pp

24. Worst Case Scenario Example
Here, we can see the two worst case situations where the pins are in the extreme outer edges or inner edges. The tolerance stack up can be evaluated as seen here. In this example, the analysis begins on the right edge of the right pin. You should always try to pick a logical starting point for stack analysis. Note that the stack up dimensions are summed according to their sign (the arrows are like displacement vectors).
Source: Tolerance Design, pp

25. Worst Case Scenario Example
From the stack up, we can determine the tolerance calculations as seen in this table. Analyzing the results, we find that there is a nominal gap and tolerance buildup for the worst case in the positive direction. This gives us a total worst-case largest gap of It gives us a worst case smallest gap of which is an interference fit. Thus, in this worst-case scenario, the parts will not fit and one needs to reconsider the dimension or the tolerance.
Largest => = 0.143
Smallest => =
Source: Tolerance Design, pp

26. Non-Linear Tolerances
In some cases, the relationship between the components and the assembly will stack up in a linear relationship. When it occurs, one must determine a function that can be used to define the relationship between the dependant assembly variable, y, and the independent component variables, x sub n Then the general worst case tolerance equations for the nominal dimension and tolerance can be used as shown here. The partial derivatives represent the particular sensitivity that each component dimension and tolerance will induce on the assembly.
Wource: “Six sigma mechanical design tolerancing”, p 104

27. Root Sum-of-Square RSS Assumes normal distribution behavior
Although useful for interchangeability, the worst case analysis, however, is also very conservative and does not ensure 100% producibility. The worst case scenario does not take into account the statistical probabilities related to process capabilities. For example, suppose the probability for a defect is 10 % and there are 5 parts in a linear system. The probability or chances of producing a system with all five “out-of-spec” parts is 0.10 raised to the 5th which equals percent. The such a remote chances of defect, one could conceivably loosen the tolerances, if appropriate.
Therefore, the root sum of squares or RSS approach should be considered to account for the more likely chances of having dimensions which do not all occur at the extreme limits simultaneously. The root mean square utilizes basic statistical methods to add the measure of variability. We postulate a probability distribution, as seen here, for each component in the assembly X is a random variable, f(x) is the probability function of x and mu and sigma are constant parameters. This allows us to study the joint probabilities relative to the specified gap constraints and/or worst-case conditions. The RSS method is used to determine if a functional fit will occur between the mating assemblies.
It assumes a normal distribution and uses the standard z-transform or student t-transform to calculate probability of assembly success.
Wource: “Six sigma mechanical design tolerancing”, p 16

28. RSS method Assembly tolerance stack equation
IN statistics, it is arithmetically wrong to simply add the standard deviations linearly. Instead, one must sum by variance by adding the squares of the variances and taking the square root. Similarly, tolerance stacking works in a similar fashion.
Wource: “Six sigma mechanical design tolerancing”, p 128

29. Pool Variance in RSS
Process capability relates the design specification and the standard deviation. For normal distribution with an unknown standard deviation, one would use the adjusted standard deviation calculations which considers the capability value, Cp.
In the RSS approach, each component variance in the assembly is assumed to be independent of every other component variance; thus, each component will possess its own Cp value. This allows us to formulate a more realistic calculation of the variances, called pooling of the variances, which results in the sigma gap formula seen here. From the formula, one can see that the standard deviation of the gap assembly is expressed as the square root of the pooled variances from the envelope the assembly is to fit within and the sum of the component variances.
Wource: “Six sigma mechanical design tolerancing”, p 128

30. Probability
Now that we have a function model of the gap variance, we can use the Z-transform to calculate the actual probability of exceeding the limits of the gap. First, let us consider a the minimum critical condition which is an interference fit. In the first formula seen here, ZQ is determined when Q is subtracted from the nominal gap and compared to the standard deviation of the assembly gap. Q represents the assembly gap between the envelope and the mating assembly. In this case, we assume that Q=0, in order to have an interference fit represented by line-to-line contact. Thus, ZQ is simply the number of standard deviations away from the nominal that relates to a condition of interference. This Z value is then used to look up the corresponding probability in the Z-chart.
Wource: “Six sigma mechanical design tolerancing”, p 128

31. Probability for Limits
In a similar fashion, one can calculate the Z value for the maximum condition in which we exceed the maximum assembly gap. We can replace Z with Gmin and Gmax to determine the Z values at the gap limits.
Wource: “Six sigma mechanical design tolerancing”, p 128

32. Dynamic RSS
Processes, over time rarely remain centered between bilateral limits. Often times, process means will shift due to internal and external factors in the process and machine. To account for the inevitable shifts, many use the dynamic RSS to approach tolerance design. In dynamic RSS, we use the long-term capability index, Cpk, instead of Cp. Calculations for the Z values are the same as the previous except we simply replace Cpk for Cp.
Wource: “Six sigma mechanical design tolerancing”, p 128

33. Nonlinear RSS
Nonlinear Root sum of squares determination is similar to the linear methods only the tolerance is expressed as shown here. In addition, Cpk is used to for the adjusted sigma.
Wource: “Six sigma mechanical design tolerancing”, p 128

34. RSS Example Largest => 0.05 + 0.051 = 0.101
Let us return to the same hole and pin problems from before and consider the RSS method of analysis. The results for the same problems show a nominal gap and a tolerance buildup for the RSS case in the positive direction. For the largest gap, we would have a total gap of When at the smallest gap, the result is a which is still an interference fit. Although, technically, this scenario would not work out, it is still significantly better than the worst case scenario case.
Largest => = 0.101
Smallest => =
Wource: “Six sigma mechanical design tolerancing”, p 128

35. Taguchi Method Input from the voice of the customer and QFD processes
Select proper quality-loss function for the design
Determine customer tolerance values for terms
in Quality Loss Function
Determine cost to business to adjust
The Taguchi approach to tolerance is also another well practiced methodology that looks to tie the manufacturing tolerances based on the customer tolerances. This method recognizes that many products have fallen short of customer expectation because of differences between engineering terms and the customer’s expectation of the product’s features and stability of performance. Therefore, Taguchi’s method is build around the relationship established between the customer tolerances and engineering tolerances though the mathematics of the quality loss function.
The chart illustrates the basic steps of the Taguchi tolerancing process. We will discuss it in further detail in the next few slides.
Calculate Manufacturing Tolerance
Proceed to tolerance design
Wource: “Six sigma mechanical design tolerancing”, p 21

36. Taguchi Voice of customer Quality function deployment
Inputs from parameter design
Optimum control-factor set points
Tolerance estimates
Initial material grades
The first step in the process include getting input from the parameter design. The voice of the customer and Quality function Deployment processes help provide detailed insight into the customer expectations. Other inputs from the parameter design include:
Optimum control-factor set point, tolerance estimates determined from engineering analysis, and initial material grades selected.
Wource: “Six sigma mechanical design tolerancing”, p 22

37. Quality Loss Function
Identify customer costs for intolerable performance
Quadratic quality loss function
At the heart of the Taguchi method is the quality loss function. The quality loss function identifies the customers cost for intolerable performance. It accounts for the loss to both the customer and the business as a function of performance variability in the design.
The quadratic loss function is described here. In the formula, L(y) is the loss in dollars due to a deviation away from the target performance a function of the measured response, y, of the product; m is the target value of the product’s response; and k is an economic loss function called the the quality loss coefficient and is calculated as A zero over Delta zero squared. .
The typical quality loss function is also illustrated here. From the figure, one can see that at y=m, the loss is zero; the loss increases as y moves from m. As the curve approaches the customer tolerance limits, Delta zero, the cost for the poor performance increase. Thus, A zero is the cost at failure. Note that the quadratic function is shown here for example purposes and is not indicative of all quality loss function behavior; the quality loss function can and would change depending on the customer’s tolerance and usage environment.
Wource: “Six sigma mechanical design tolerancing”, p 208

38. Cost of Off Target and Sensitivity
Cost to business to adjust off target performance
Sensitivity, b
The next step, one needs to determine the cost to the business to adjust the off-target performance values back onto target during the manufacturing process, and the sensitivity between the customer tolerance and manufacturing tolerances. Once the function and its limits are established, the engineering team need to determine the a safety factor, phi, to prevent off-target performance values. The company also needs to quantify what expense is worth funding to remedy the off-target performance. Therefore, the safety factor can be described as the square root of the average loss in dollars when a product characteristic exceeds customer tolerance limits over the average loss in dollars when a product characteristics exceeds the manufacturing and/or design tolerance limits.
Sensitivity is the change in the high-level customer observable characteristic or a product-level engineering characteristic, y, when a unit change occurs from the target set point. The relationship of the of the sensitivity in the manufacturing loss, A, is equal to the loss function for variability in component x.
Wource: “Six sigma mechanical design tolerancing”, p

39. Manufacturing Tolerance
Accounting for the safety factor and the sensitivity, we can link the customer limits to the development of design element limits. The result is delta, the manufacturing tolerance based on the Taguchi equation.
Now, once this tolerance is calculate, one can proceed forward to the design process where further tolerance design can be optimized.

40. Summary Importance of effective tolerances Tolerance Design Approaches
Worst-Case analysis
Root Sum of Squares
Taguchi tolerance method
Continual process
Involvement of multi-disciplines
The importance of tolerances in engineering has increased significantly since the advent of the industrial revolution. Proper tolerances can significantly affect the product quality, cost, ease of process and time to market. Effective tolerance design also seeks to match the appropriate tolerances with the manufacturing processes as well as promotes cost effective production.
In this presentation, we discussed three methods to establish tolerance. They are:
Worst-Case analysis
Root Sum of Squares
Taguchi tolerance method
Although this methods provide a good method for tolerance design, we should recognize that processes are hardly every static. One should always look to available resources, such as vendors, manufacturing and quality control as well as statistical analysis and experimentation to maintain optimal tolerance design and control.

41. Credits
This module is intended as a supplement to design classes in mechanical engineering. It was developed at The Ohio State University under the NSF sponsored Gateway Coalition (grant EEC ). Contributing members include:
Gary Kinzel…………………………………. Project supervisor
Phuong Pham.……………. ………………... Primary author
Reference:
“Six Sigma Mechanical Design Tolerancing”, Harry, Mikel J. and Reigle Stewart, Motorola Inc. , 1988.
Creveling, C.M., Tolerance Design, Addison-Wesley, Reading, 1997.
Wade, Oliver R., Tolerance Control in Design and Manufacturing, Industrial Press Inc., New York, 1967.

42. Disclaimer
This information is provided “as is” for general educational purposes; it can change over time and should be interpreted with regards to this particular circumstance. While much effort is made to provide complete information, Ohio State University and Gateway do not guarantee the accuracy and reliability of any information contained or displayed in the presentation. We disclaim any warranty, expressed or implied, including the warranties of fitness for a particular purpose. We do not assume any legal liability or responsibility for the accuracy, completeness, reliability, timeliness or usefulness of any information, or processes disclosed. Nor will Ohio State University or Gateway be held liable for any improper or incorrect use of the information described and/or contain herein and assumes no responsibility for anyone’s use of the information. Reference to any specific commercial product, process, or service by trade name, trademark, manufacture, or otherwise does not necessarily constitute or imply its endorsement.