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2.0 Introduction

The tolerancing of machine parts is a necessary design function due to the variability of all manufacturing methods. Tolerancing is the assignment of upper and lower limits to basic dimensions to account for this variability.

The basis for the assignment of tolerances should be the design requirements on critical assembly tolerances which assure proper performance. An assembly clearance or gap is an example of a performance parameter. If the clearance is too small, the parts may be difficult to assemble. If the clearance is too large, the parts may experience impact loads or misalign during operation. Such design parameters may be set by ANSI standards, trade group guidelines, government regulations, company design manuals, prototype testing or customer feedback.

In an assembly, the nominal dimensions of the component parts may be summed to determine the basic size of each clearance. However, the component tolerances may also accumulate, causing the clearance of some assemblies to exceed the design limits and thus fail to pass inspection. Hence, the individual component tolerances of the assembly must be chosen such that their sum does not exceed the critical assembly tolerance limits.

2.1 2-D Tolerance Analysis

Tolerance accumulation or stack-up may be estimated from one of the expressions in Table 2.1. All four of these approximations are available in TI/TOL. Which one you use depends upon customer requirements, process data available, and desired accuracy.

Table 2.1. Assembly Tolerance Accumulation Models

Worst Case
Assures 100% assembly acceptance if all parts are within specification. Costly design model. Requires excessively tight component tolerances
Root Sum Square
Assumes Normal distribution and ±3Sigma tolerances. Some fraction of assemblies will not meet specification. May adjust ZASM to obtain desired acceptance fraction. Less costly. Permits looser component tolerances.
Six Sigma Assembly Drift
The same as the Root Sum Square equation with Zp (equal to the number of process standard deviations in each tolerance) replacing Zi.
Six Sigma Component Drift
Most realistic estimates. Accounts for process mean shifts and their long-term affects on assembly distribution.


In Table 2.1, dU is the predicted variation in the resultant assembly dimension; dxi is the variation in a component dimension; partialdiffU/partialdiffxi is the sensitivity that a variation in dxi has on U; and TASM is the assembly tolerance (or design limit) for variations in dU. ZASM and Zi are the number of standard deviations corresponding to the assembly and component tolerance limits. Zp is the number of process standard deviations in the assign component tolerance. Cpi is the process capability ratio for a component dimension whose manufacturing process is assumed to be centered. Cpki is the process capability index for a component dimension which includes a measure of the shift in the process mean.

TI/TOL calculates the sensitivities from the TI/TOL 2-D vector model and predicts the tolerance accumulation of the assembly variables of interest. If specified limits have been set for an assembly variable, the computed distribution of the variable is used to calculate the number of assemblies which will be out of spec. This is shown graphically in Figure 2.1.

Figure 2.1: Determining the number of out-of-spec assemblies.

2.2 Vector Model of an Assembly

Figure 2.2 shows a kinematic vector model overlaid on the cross section of a mechanical assembly. This device is a locking hub for holding a reel of computer tape on a tape drive. In this example, the plunger slides vertically downward while the wedge on its outer diameter pushes the arm outward until it locks against the inside diameter of a computer tape reel. In the extreme position shown, with the plunger resting on the base, we would like to assure an interference fit between the rubber pad and the reel (shown here as a gap or clearance for clarity).

Figure 2.2: Vector loop model of locking computer tape hub assembly.

Two loops describe the variation in this assembly, one closed and one open, as shown in Fig. 2.2. The open loop is one-dimensional and is set up to compute the variable Gap between the Arm and Reel. The closed loop is two-dimensional and is set up to compute RL, the resultant outer radius of the tape hub. The nominal value of the Gap and RL, along with dependent variables u and F , may be determined by an accurate CAD layout. However, the variations in the Gap, RL, u and F are desired. The resultant variation may be computed in terms of the independent variables and their sensitivities using a worst case or statistical model for tolerance accumulation.

Each 2-D vector loop equation may be resolved into three scalar equations, representing the sum of vector components in the X and Y directions and the sum of the relative rotations between adjacent vectors. The resulting system of equations may be expressed:

hx(xi,uj) = 0

hy(xi,uj) = 0

htheta(xi,uj) = 0

where xi are the manufactured component dimensions, uj are the unknown kinematic lengths and angles.

For the locking tape hub problem, the single closed loop shown in figure 2.2 gives the following set of nonlinear equations:

hx = b cos(90) + a cos(0) + u cos(theta) + r cos(theta - 90) + (e + i ) cos(theta + F - 90) + g cos(theta + F - 180) + RL cos(theta + F - 270) + h cos(theta + F - 180) = 0

hy = b sin(90) + a sin(0) + u sin(theta) + r sin(theta - 90) + (e + i ) sin(theta + F - 90) + g sin(theta + F - 180) + RL sin(theta + F - 270) + h sin(theta + F - 180) = 0

htheta = 90 - 90 + theta - 90 + F + 0 - 90 - 90 + 90 + 90 = 0

where a,b,e,g,h,i,r and theta are the independent manufactured dimensions and RL,u and F are the dependent kinematic dimensions.

The open loop equation is a one-dimensional equation for the resultant gap between the tape reel radius RT and the hub radius in locked position RL.

Gap = RT - RL

2.3 Linearized Solution for the Assembly Tolerances

There are two steps in solving the vector equations. First, all of the manufactured dimensions are set to their nominal values and the system of vector loop equations are solved for the nominal values of the kinematic variables and assembly resultants. Vector equations are generally nonlinear and must be solved by iterative methods. If the nominal values may be determined from a precise CAD layout, this step may be omitted.

Second, the equations are linearized for small variations about the nominal by Taylor's series expansion, retaining the first order derivatives. Each derivative is evaluated using the nominal dimensions of the dependent and independent variables. The component tolerances dxi are substituted into the equations and solve this system of linearized equations is solved for the corresponding variation in the kinematic variables and resultant assembly dimensions.

The linearized closed loop equations may be written as:

dhx(xi,uj,alphak) = Sigma(F(partialdiffhx,partialdiffxi)dxi) + Sigma(F(partialdiffhx,partialdiffuj)duj) + Sigma(F(partialdiffhx,partialdiffalphak)dalphak) = 0

dhy(xi,uj,alphak) = Sigma(F(partialdiffhy,partialdiffxi)dxi) + Sigma(F(partialdiffhy,partialdiffuj)duj) + Sigma(F(partialdiffhy,partialdiffalphak)dalphak) = 0

dhtheta(xi,uj,alphak) = Sigma(F(partialdiffhtheta,partialdiffxi)dxi) + Sigma(F(partialdiffhtheta,partialdiffuj)duj) + Sigma(F(partialdiffhtheta,partialdiffalphak)dalphak) = 0

where dxi are the specified tolerances on the independent dimensions and duj are the resultant variations in the dependent assembly dimensions. The equations have also been expanded to include form and feature tolerances dalphak. The affects of form tolerances will be discussed in a later section. In matrix notation these equations become:

[A] {dx} + [B] {du} + [F] {dalpha} = {0} Closed Loop Equations

where [A] is the matrix of sensitivities to variations in the independent variables xi

{dx} = the vector of specified variations of the independent variables dxi

[B] = the matrix of sensitivities to variations in the independent variables uj

{du} = the vector of unknown variations of the dependent variables duj

[F] = the matrix of sensitivities to variations in the form variables alphak

{dalpha} = the vector of specified variations of the form variables dalphak

This system of equations may be solved by linear algebra for the unknowns duj:

{du} = -[B-1] [A] {dx} - [B-1] [F] {dalpha}

from which expressions for the predicted accumulation of variations in the duj may be generated:

duj = Sigma(|F(partialdiffuj,partialdiffxi)|dxi) + Sigma(|F(partialdiffuj,partialdiffalphak)|dalphak) (Worst Case)

duj = R(Sigma(F(partialdiffuj,partialdiffxi)dxi)2 + Sigma(F(partialdiffuj,partialdiffalphak)dalphak)2 (Statistical)

The linearized open loop equations in matrix form may be expressed:

{dg} = [C] {dx} + [E] {du} + [G] {dalpha} Open Loop Equations

where C, E and G represent sensitivities to variations in dxi, duj and dalphak, respectively. dg is the non-zero vector of specified assembly gaps or rotational variations.

In general, the open loop equations do not require simultaneous solution. They are evaluated after eliminating the duj by substituting expressions from the closed loop equations.

{dg} = [C] {dx} - [E] ([B-1] [A] {dx} + [B-1] [F] {dalpha}) + [G] {dalpha}

= ([C] - [E] [B-1] [A] ) {dx} + ([G] - [E] [B-1] [F] ) {dalpha}

from which expressions for the predicted accumulation of variations in the dgj may be generated:

dgj = Sigma(|F(partialdiffgj,partialdiffxi)|dxi) + Sigma(|F(partialdiffgj,partialdiffalphak)|dalphak) (Worst Case)

dgj = R(Sigma(F(partialdiffgj,partialdiffxi)dxi)2 + Sigma(F(partialdiffgj,partialdiffalphak)dalphak)2 (Statistical)

In the statistical expressions above, if the dxi and the dalphak represent three standard deviations of their respective statistical distributions, then the duj and dgj will also correspond to three standard deviations. These equations may be used to estimate the variance of the resultant assembly distributions. The resulting distribution may be used to compute the percent rejects as shown in Fig. 3.1.

Thus, the variational behavior of the whole population of assemblies may be determined by analyzing just one assembly statistically, rather than analyzing a large sample population one at a time, as is done in Monte Carlo Simulation.

2.4 Numerical Example

The preceding tape hub problem will now be solved using the numerical values for the component nominal and tolerances, which are summarized in table 2.2.

Table 2.2: Nominal dimensions and tolerances of the tape hub lock.

Independent Dependent
a = 1.355 ±0.0015 g = 0.493 ±0.004 u = 0.319 ± ?
b = 0.400 ±0.006 h = 0.200 ±0.008 RL = 1.864 ± ?
r = 0.060 ±0.002 theta = ±0.5 [[ring F = ± ? 75[[ring]] ]] 15[[ring]]  
e = 0.318 ±0.003 RT = 1.856 ±0.004 Gap = ± 0.006 -0.010
i = 0.050 ±0.002        

Taking the partial derivatives of the three hub lock equations yields the following sensitivities (evaluated at the nominal dimensions):

The bold symbols above each column indicates with respect to which variable the derivative has been taken.

Solving for du in terms of the independent variables dx (ignoring form variations for now):

The solution for the dependent assembly variations in terms of the component tolerances:

{du dR dF } =

= {db da dr de di dg dh dtheta}

To estimate dRL, multiply the second row of the B-1A matrix times the vector of component tolerances, taking absolute values for worst case or root sum square for statistical.

dRL = |0.268 db| + |1.0 da| + |1.035 dr| + |1.0 (de+di)| +|0.268 (dg+dh)| + |0.3307 dtheta| = 0.01548 (Worst Case)

dRL = (0.2682db2 +1.0da2 +1.0352dr2 +1.0(de2+di2) +0.2682(dg2+dh2) +0.33072dtheta2)1/2 = 0.00578 (Statistical)

Clearly, worst case is a much more conservative estimate of the variation.

By substituting trial values for the tolerances dxi, these expressions may be used for "what-if" studies and tolerance design. On the other hand, by substituting the design limits for the duj, tolerance allocation algorithms may be used to determine acceptable component tolerances for dxi.

To predict the variation in the resulting gap between the tape reel and hub, we must evaluate the open loop equation. Calculating the nominal gap and resulting variation:

Gap = RT - RL = 1.856 - 1.864 = -0.008 (interference)

dGap = dRT - dRL

The corresponding expressions for worst case and statistical:

dGap = |0.004| + |0.01548| = ±0.0195 (Worst Case)

dGap = R(0.0042 + 0.005782 ) = ±0.0070 (Statistical)

Calculated Gap limits: Min = -0.015 Max = -0.001 Mean = -0.008

Specified Gap limits: Min = -0.016 Max = -0.004 Mean = -0.010

% Rejects: Lower Limit: 0.03% Upper Limit: 4.36%

Thus, we see that the component tolerances as specified will yield gap limits which are too wide and the mean value will not be centered in the specified range. An unacceptable number of assemblies will have insufficient grip to lock a tape reel securely. Adjustments will be needed in the nominals to center the mean properly and some component tolerances will have to be tightened.

Examining the sensitivities and percent contribution of each component will help the designer to decide which tolerances to change. Detailed examples will be presented to show how to use the TI/TOL tolerance design options to accomplish this.

2.5 Approximating Form Variations

Form variations only introduce variation into an assembly at the points of contact between mating surfaces. Figure 2.3 shows a cylinder in contact with a plane. In one assembly, the cylinder might rest on a peak of the surface. In the next assembly, the cylinder could be down in a valley. Thus, the surface waviness in the plane results in a translational variation normal to the surface.

Similarly, the cylinder is probably not perfectly round, but exhibits waviness or lobing. If, during assembly, it is placed with a lobe at the contact point, the cylinder center will be higher than average. If a low point on the surface of the cylinder is in contact, the center will be lower. Thus, we see that both surfaces produce independent translational variations which are normal to the surface at the point of contact.

Figure 2.3: Assembly variations caused by form variation.

In contrast, a block on a plane produces rotational variation, also shown in Figure 2.3, since one corner of the block may be higher than the other. The magnitude of the variation depends on the length of the block and the amplitude and period of the waviness.

2.6 Calculating the Contribution of Form Variations

Figure 2.4 shows the example assembly with feature tolerances defined at the mating surfaces. Each tolerance is specified by an ANSI Y14.5 feature control symbol which is added to the assembly drawing. The alphai symbols identify the magnitude of feature tolerance as listed in Table 2.3. Corresponding algebraic terms are added to the kinematic assembly model and serve as independent sources of variation.


Figure 2.4. Feature tolerances added at mating surfaces of an assembly.

Table 2.2 Specified Feature Tolerances for Tape Hub Problem

Feature Feature Type Joint Type Variation Type Characteristic Length
alpha1 perpendicularity rigid rotation 1.355.
alpha2 run out cylinder slider translation N/A.
alpha3 profile cylinder slider translation N/A.
alpha4 perpendicularity planar rotation 0.493
alpha5 flatness planar rotation 0.359
alpha6 parallelism planar rotation 0.359

By introducing small translational or rotational variations at each point of contact, the sensitivity matrix for form variations may be determined.

alpha1 alpha2 alpha3 alpha4 alpha5 alpha6

Solving for du:

alpha1 alpha2 alpha3 alpha4 alpha5 alpha6

{du} = -[B-1] [A] {dx} - [B-1] [F] {dalpha}

Substituting for [A], [F] and [B-1], then taking the middle row of sensitivites to form the tolerance accumulation expression with form tolerances:

dRL = 0.01548 + |.694 dalpha1| + |1.035 dalpha2| + |1.035 dalpha3| + |.397 dalpha4| +|..397 dalpha5| + |.397 dalpha6| = 0.02504 (Worst Case)

dRL = (.00578 +0.2682dalpha12 +1.0dalpha22 +1.0352dalpha32 +1.0dalpha42 +0.2682dalpha52 + 0.33072dalpha62)1/2 = 0.00713 (Statistical)

The corresponding expressions for the gap:

dGap = |0.004| + |0.02504| = ±0.02904 (Worst Case)

dGap = R(0.0042 + 0.007132 ) = ±0.0082 (Statistical)

Calculated Gap limits: Min = -0.0162 Max = +0.0002 Mean = -0.008

Specified Gap limits: Min = -0.016 Max = -0.004 Mean = -0.010

% Rejects: Lower Limit: 0.13% Upper Limit: 7.08%

This represents nearly twice as many rejects as was predicted without form tolerances. Thus we see the necessity for including form variations in assembly tolerance analysis.

The foregoing sections have described a comprehensive system for performing tolerance analysis of 2-D mechanical assemblies. It accounts for all three sources of variation: dimensional, form and feature, and kinematic. It offers a systematic procedure for model-building, integrating with commercial CAD systems, and performing tolerance analysis and tolerance design.

Please note that all of the calculations demonstrated here are performed automatically by TI/TOL. Once the engineering model is created, the equations are derived, dependent variables separated, sensitivities calculated, and expressions for tolerance accumulation computed.

2.7 Motorola Six Sigma Analysis--Accounting for Process Mean Shifts

Six Sigma is the quality program instituted corporate-wide at Motorola, for which they received the first annual Baldridge National Quality award in 1988. Now, many of Motorola's suppliers and other major companies have adopted the same system.

The basic premise of the Six Sigma Program is: in order to achieve high quality in a complex product comprised of many components and processes, each component and process must be produced at significantly higher quality levels in order for the composite result to meet final quality standards.

Stated statistically, suppose there were 1000 dimensions or other characteristics of your product, any one of which could lower the quality of the finished product. If each characteristic were produced to ±3 quality (99.73% acceptable parts or 2700 defects per million), the resultant assemblies would be only (.9973)1000=.067 or 6.7% defect free. To have 99.73% defect free assemblies, you would need to produce each component to a quality of (.9973).001=.9999973 or 99.99973%, which is 2.7 defects per million.

To achieve the high quality levels required for world competition in the electronics industry, Motorola has mandated ±6 quality for all processes (.002 defects per million). However, they also recognize that shifts and drifts in the mean of the processes are expected, so they have introduced a modified process model which includes an allowance for accumulated mean shifts. The result is a net quality level of ±4.5 (3.4 defects per million).

Of course, tolerance analysis of assemblies is only one component of the complete Motorola quality management system, but the Six Sigma tolerance analysis model is so versatile and powerful that it can have a major impact on cutting production cost and improving quality.

Assigning tolerances based on statistical tolerance analysis alone is not sufficient to assure that reject rates will remain within quality limits. Naturally occurring shifts in the mean of a process can produce biased distributions which can result in increased assembly problems and a greater number of rejects than anticipated.

The process mean is difficult to control. Tool wear causes the mean to drift, creating a zone of uncertainty, as shown in Figure 2.6. Shifts can also occur due to setup error. Deliberate bias toward the least material condition may be introduced to account for tool wear. Bias toward the maximum material condition may be used to allow material for rework.

In an assembly, mean shifts can also accumulate just as tolerances do. The resulting assembly distribution can be pushed so far off center that significant rejects may occur even though all component parts are produced within tolerance specs.

Even when the mean shifts do not accumulate or bias the component distributions, they can still cause an increase in rejects. For large quantity production, the mean may drift around within the zone of uncertainty, causing the distribution to spread. The long-term distribution of the process will be greater than that determined by typical sampling procedures.

Figure 2.6. The location of the mean is not known precisely.

To account for all these effects, the designer needs to use a more complete model. TI/TOL has incorporated the Motorola model for tolerance accumulation, which includes a mean shift factor, kdyn, to account for dynamic mean shift. Dynamic mean shifts describe the long term increase in the spread of the distribution due to drifting about the center.

The long term tolerance accumulation model may be expressed as:

dU = ZasmR(Sigma(f(Ti,3Cpki))2) <= Tasm

where, xi is the nominal component dimension, dU is the resultant assembly variation, Ti is the component tolerance. Zasm is the desired number of standard deviations of the assembly distribution to be included in the specified assembly tolerance Tasm. Set Zasm = 3.0 for the usual assumption of ±3 assembly tolerance limits. Cpki is the Process Capability Index used by manufacturing to quantify non-centered processes. Set Cpki = 1.0 for the usual assumption of ±3 component tolerance limits. Each component Cpk may be calculated from the expressions:

Cpk = Cp(1-kdyn)

Cp = f(USL - LSL,6)

Cp is the Process Capability Ratio. It is the ratio of the width of the tolerance zone (USL - LSL) to the 6Sigma width of the process. When the tolerance limits are ±3, (USL - LSL) = 6 and Cp = 1.0. When Cp = 1.0 and kdyn = 0, then Cpki = 1.0 and the component tolerances are ±3 and centered. If the process limits are set at ±6 and kdyn = 0.25, then the mean could be shifted as much as one fourth of the tolerance zone, or 1.5, as shown in Figure 2.7.

Figure 2.7. Effect of a mean shift where kstat=0.25 on ±6 tolerance limits.

The Six Sigma model can provide a common ground for interaction between engineering and manufacturing. The model forces the engineer's attention on manufacturing considerations, and he must communicate with manufacturing to get the needed model data. He can convey the critical design parameters to manufacturing in a form that permits freedom to alter tolerances without violating design requirements. Manufacturing can communicate meaningfully to engineering in terms of the two quality assurance parameters most commonly used in statistical process control: mean and variance.

Summary of Six Sigma Advanced Analysis

1. The TI/TOL Motorola Six Sigma model can perform statistical tolerance analysis of assemblies for any specified quality level. Individual quality levels for each component may also be specified in terms of the Process Capability Ratio (Cp) or the standard deviation of the process.

2. The mean shift for each component process may be specified and the resultant effect on the assembly yield analyzed. A dynamic mean shift may be specified. Dynamic mean shift is a random bias due to long-term variations in the process mean, such as tool wear or setup variations over numerous production runs. It causes an increase in the variance or the spread in the process distribution and a decrease in Cpk.

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