Chapter 2. Tolerance analysis
2.2. Tolerance Analysis Process
1
( k
Cp
Cpk = −
( 2-16 )2 ) ( USL LSL
T k y
−
= −
( 2-17 )where
T is the target specification.
k is the number of standard deviations with the process mean departs from target T.
The Cpk index will be thought of a long-term process capability. It can be used to account for the time-based variability of manufacturing. In Six Sigma philosophy the process mean can shift 1.5 standard deviations even when the process is monitored by modern statistical process control over an extended period of time. In that case, Cpk = 1.5.
Figure 2-6 Process capability with shifted mean
2.2. Tolerance Analysis Process
The tolerance analysis process can be summarized and illustrated in Figure 2-7 [16].
Figure 2-7 Tolerance analysis process
Establishing the CTQs
The first step in the process is to identify the critical to quality (CTQ) characteristics.
The CTQs are the requirements that determine the performance of the system or the key dimensions of an assembly. The variation of the key dimensions of an assembly will make great impact on the quality of the system. These CTQs will deploy the requirements of mechanical subassemblies and detail part. The CTQs also determine the factors needed to be analyzed. Figure 2-8 illustrates an one dimensional assembly with five components. In this example the CTQ is that the “gap” must always be great than zero.
Figure 2-8 One dimensional assembly
Draw a Loop Diagram
The second step in the process is to draw a loop diagram. The loop diagram is a graphical representation of each CTQ characteristics. It is also a mathematical model of each CTQ analysis. Each CTQ requires a separate loop diagram. Simple loop diagram are usually horizontal or vertical. For one-dimensional analyses, horizontal loop diagrams will graphically represent the dimensional contributors for horizontal “gap” or target dimension.
The method of drawing a horizontal loop diagram is described below. Vertical loop diagram will be drawn by the same way. Figure 2-9 illustrates the horizontal loop diagram of the example shown in Figure 2-8.
Figure 2-9 Horizontal loop diagram
The steps for drawing the loop diagram are described below.
1. Start from the surface on the left of the gap then followed by a series of features that contribute the dimensional variation of the gap. Stop at the surface on the right of the target dimension.
2. Represent the loop diagram by vector chains. Typically, the displacements of the loop diagram to the left are negative and the displacements of the loop diagram to the right are positive. The displacements are vectors which denote the contributing feature dimensions.
A series of displacements is called a vector chain. When all vectors in the chain are summed, a net positive value indicates clearance, and a net negative value indicates interference.
3. Assign a variable name to each dimension in the loop.
4. Record sensitivities for each dimension. The magnitude of the sensitivity is the value that the target dimension changes when the contributing dimension changes 1 unit. The sign of the sensitivity has been incorporated with the displacement vector. For the one dimensional loop diagram, all of the sensitivities are usually equal to ±1. In the case that a radius is the contributing factor for a diameter, the sensitivity equals to ±0.5.
5. Classify the dimensions as “fixed” or “designed.” A fixed dimension is the one in which we can not control the tolerance, such as a vendor part dimension. A designed dimension is the one in which we can modify the tolerance to change or to tune-up the result of tolerance stack. The designed dimensions are what we are going to design for the tolerance during tolerance allocation.
Converting Dimensions to Equal Bilateral Tolerance
The third step in the process is converting dimensions to equal bilateral tolerance.
Because most of the manufacturing processes are normally distributed, manufacturers will obtain maximum yield of each dimension if the manufacturer aims for nominal dimension.
This helps them to maximize the number of good parts and to minimize the manufacturing cost. The steps for converting to an equal bilateral tolerance are described below:
1. Convert the dimension with tolerances to an upper limit and a lower limit.
2. Substrate the lower limit from the upper limit to obtain the tolerance band.
3. Divide the tolerance band by two to get an equal bilateral tolerance.
4. Add the equal bilateral tolerance to the lower limit or substrate the equal bilateral tolerance from the upper limit to get the mean dimension.
For example, Table 2-1 converts the dimensions and tolerances in Figure 2-8 to the mean dimensions with equal bilateral tolerances.
Table 2-1 Converting dimensions to equal bilateral tolerance Part name Original Dimension
/Tolerances
Mean Dimension with Equal Bilateral Tolerances
A 10.1 +0 / -0.2 10 ± 0.1
B 20.2 +0.1 / -0.5 20 ± 0.3
C 13.8 +0.4 / -0 14 ± 0.2
D 16.0 ± 0.25 16 ± 0.25
E 60.5 +1.0 / -0 61 ± 0.5
Calculating the Mean Value for the Requirement
The third step in the process is calculating the mean value for the requirement. This is also the first step in calculating the variation at the gap. Using the mean value, we can check the validity of the mathematical model of the loop diagram easily. The mean value at the gap
is:
∑
==
ni i i
g
a d
d
1
( 2-18 ) where
d
g is the mean value at the gap.n is the number of independent dimensions in the stack-up.
a
iis the sensitivity factor.
d
iis the mean value of the i
th dimension in the loop diagram.If dg is positive, the mean “gap” has a clearance, and if dg is negative, the mean “gap” has an interference. The sensitivity factor defines the direction and magnitude for the ith dimension.
In the one-dimensional stack-up, this value is usually +1 or -1. For the example the mean value of gap in Figure 2-8, except for part E with positive sensitivity of 1 all the other components have negative sensitivity of -1. The mean gag is:
Gap = (-1)16 + (-1)14 + (-1)20 + (-1)10 + (1)61 = 1
Determine the Method of Analysis
The fourth step in the process is determining the method of analysis. Different method will result in different variation of the gap or target dimension. The two most commonly used traditional tolerance analysis methods are the “Worst Case” model and the “Statistical” model.
The Worst Case model calculates the arithmetic sum of individual tolerance, and it will be described in Section 2.3. There are two traditional statistical methods; the Root Sum of the Squares (RSS) model, and the Modified Root Sum of the Square (MRSS) model. Statistical models will be described in detail in Section 2.4.
Calculating the Variation for CTQs
The last step in the process is calculating the variation for the requirement. During the design process, the design engineer has to make tradeoffs using one of the three traditional models. If the worst case tolerance meets the required CTQs, the tolerance design will stop there. On the other hand, if this model does not meet the requirements, the designer would try to use RSS or MRSS models. It has the risk of certain percentage of products beyond performance requirements.