GB/Z 24636.5-2010Geometrical product specifications(GPS) - Statistical tolerance - Part 5: Statistical quality indices of an assembly lot (hole/shaft fits) (English PDF)

1 Scope

This part of GB/Z 24636 specifies the statistical quality indices of an assembly lot (hole/shaft fit) and gives the fit quality indices, the related methods of analysis and the method of statistical tolerance design that guarantees fit quality.

This part applies to hole and shaft fit sizes that have two-sided specification limits and to which statistical process control is applied.

2 Normative references

The provisions of the following documents become provisions of this part of GB/Z 24636 through reference in it. For dated references, subsequent amendments (excluding corrections) or revisions do not apply to this part; however, parties to agreements based on this part are encouraged to investigate the possibility of applying the most recent editions. For undated references, the latest edition applies.

GB/T 1800.1-2009, Geometrical product specifications (GPS) - Limits and fits - Part 1: Bases of tolerances, deviations and fits (ISO 286-1:1988, MOD)

GB/Z 20308, Geometrical product specifications (GPS) - Masterplan (GB/Z 20308-2006, ISO/TR 14638:1995, MOD)

GB/Z 24636.1, Geometrical product specifications (GPS) - Statistical tolerancing - Part 1: Terms, definitions and basic concepts

GB/Z 24636.3-2009, Geometrical product specifications (GPS) - Statistical tolerancing - Part 3: Statistical quality indices of a part lot (process)

GB/Z 24636.4-2009, Geometrical product specifications (GPS) - Statistical tolerancing - Part 4: Statistical tolerance design based on a given confidence level

3 Terms and definitions

The terms and definitions given in GB/Z 24636.1 and the following apply to this part of GB/Z 24636.

3.1.1 fit capability - the measure of the distribution of the clearance or interference after fitting relative to the fit tolerance; the fit capability is six times the standard deviation of the clearance or interference after fitting, denoted 6 sigma-f. Note: sigma-f is the standard deviation of the clearance or interference after fitting, sigma-f being the square root of the sum of the squares of sigma-H and sigma-S, which are the standard deviations of the hole and shaft fit sizes respectively.

3.1.2 fit capability index (FCp) - the ratio of the fit tolerance Tf to the fit capability 6 sigma-f, expressed as FCp = Tf / (6 sigma-f) = (USLf - LSLf) / (6 sigma-f), where Tf is the hole/shaft fit tolerance, TH and TS being the hole tolerance and the shaft tolerance and Tf = TH + TS; USLf is the upper specification limit of the clearance or interference of the hole/shaft fit; and LSLf is the lower specification limit. [GB/T 1800.1-2009, definition 3.14.4] Note: FCp does not reflect the effect on assembly quality of the shift of the distributions of the hole and shaft fit sizes relative to the centres of their respective tolerance zones; it reflects only the potential capability of the final operation forming the hole and shaft fit sizes to meet the fit accuracy requirement, whereas the most important factor affecting the fit size is the distribution characteristic of the hole and shaft sizes.

3.1.3 fit shift parameter (kf) - the ratio of the shift of the mean value mu-f of the clearance or interference after fitting relative to the fit target value Mf to the half value of the fit tolerance, expressed as kf = (mu-f - Mf) / ((USLf - LSLf)/2) = delta-f / (Tf/2) = 2(delta-H - delta-S) / Tf, where mu-f is the mean clearance or interference after fitting; Mf is the fit target value of the assembly size; delta-f is the shift of the mean clearance or interference from the fit target value; delta-H is the shift of the mean of the hole from its target value; and delta-S is the shift of the mean of the shaft from its target value.

3.1.4 fit capability index (FCpk) - the smaller of the two ratios of the distance from the mean of the fit characteristic to each of the two-sided specification limits to the half value of the fit capability, expressed as FCpk = min{(USLf - mu-f)/(3 sigma-f), (mu-f - LSLf)/(3 sigma-f)}; the equivalent formula is FCpk = FCp (1 - |kf|), with 0 <= |kf| <= 1.

3.1.5 fit capability index (FCpm) - the ratio of the fit tolerance to the particular statistic 6 tau-f, expressed as FCpm = Tf / (6 tau-f) = (USLf - LSLf) / (6 times the square root of (sigma-f squared plus (mu-f - Mf) squared)) = FCp / the square root of (1 + delta-f squared), where tau-f is the positive square root of the mathematical expectation of the square of the deviation of the fit characteristic (clearance or interference) from its target value, and delta-f is the standardized shift of the distribution of the assembly size, delta-f = (mu-f - Mf) / sigma-f. Note: the FCpm index reflects in a combined way the effect on assembly quality of both the fluctuation of the hole and shaft operations and the shift from the fit target value, and may be used as an index measuring the centring of the assembly size.

3.2.1 superior degree percentage of hole/shaft fit (PC(f)) (percentage of clearances or interferences in the defined middle zone of the fit tolerance) - the ratio of the number of clearances or interferences of the assembly lot that lie within the middle zone (superior zone) of the fit to the number in that assembly lot, expressed as a percentage. The superior degree percentage PC(f) is calculated in the following two cases: a) the middle zone (superior zone) of the fit is the middle third of the fit tolerance zone: PC(f)(Mf plus or minus Tf/6) = {Phi[FCp(1 - 3kf)] - Phi[-FCp(1 + 3kf)]} x 100 %; b) the middle zone (superior zone) of the fit is the middle half of the fit tolerance zone: PC(f)(Mf plus or minus Tf/4) = {Phi[1.5 FCp(1 - 2kf)] - Phi[-1.5 FCp(1 + 2kf)]} x 100 %. Note: to simplify application and to promote the raising of fit quality, the index PC(f)(Mf plus or minus Tf/6) is recommended in preference.

3.2.2 fraction of average fit quality loss (Pql(f)) - the average quality loss caused to society and to the user by the deviation of the clearance or interference of the assembly lot from the target clearance or target interference, expressed as a percentage: Pql(f) = [1/(3 FCp) squared + kf squared] x 100 %.

4 Relation between the statistical quality indices of an assembly lot (hole/shaft fit) and the statistical parameters of the two (hole and shaft) fit sizes

4.1 General. The relation between the statistical quality indices of the assembly lot (hole/shaft fit) and the statistical parameters of the two (hole and shaft) fit sizes is the basis of statistical tolerance design oriented to the fit quality indices, and is also the main basis for predicting the statistical quality indices of the assembly lot (hole/shaft fit) from the statistical parameters of the two fit sizes. The relation between the statistical quality indices and the statistical tolerances of the two fit sizes is the embodiment of the relation between the statistical parameters under the particular condition that the statistical parameters of the hole and shaft fit sizes all reach the limiting values of the statistical tolerances.

4.2 Relation between the fit capability index FCp and the process capability indices of the two fit sizes. FCp = Tf / (6 sigma-f) = (TH + TS) / (6 sigma-f) = 1 / the square root of (1/Cp(H) squared + (1/(r Cp(S))) squared) plus 1 / the square root of (1/Cp(S) squared + (r/Cp(H)) squared), where r is the ratio of the hole tolerance to the shaft tolerance, r = TH/TS.

4.3 Relation between the fit shift parameter kf and the shift parameters kH and kS of the two fit sizes. kf = delta-f / (Tf/2) = 2(delta-H - delta-S) / (TH + TS) = (r kH - kS) / (1 + r). Note: when the hole/shaft tolerance ratio r = 1, kf = 0.5(kH - kS); when r = 1.6, kf is approximately 0.615 kH - 0.385 kS.

4.4 Relation between the superior degree percentage PC(f) and the process capability indices and shift parameters of the two fit sizes. The superior degree percentage PC(f) can be predicted from the process capability indices and the shift parameters of the two fit sizes by substituting the expressions of 4.2 and 4.3 into the formula of 3.2.1.

4.5 Relation between the fraction of average fit quality loss Pql(f) and the process capability indices and shift parameters of the two fit sizes. The fraction of average fit quality loss Pql(f) can likewise be predicted from the process capability indices and the shift parameters of the two fit sizes: Pql(f) = {((r kH - kS)/(1 + r)) squared + [3 (1 / the square root of (1/Cp(H) squared + (1/(r Cp(S))) squared) plus 1 / the square root of (1/Cp(S) squared + (r/Cp(H)) squared))] to the power minus two} x 100 %.

4.6 Relation between the fit capability index FCpk and the process capability indices and shift parameters of the two fit sizes. FCpk = FCp (1 - |kf|) = [1 / the square root of (1/Cp(H) squared + (1/(r Cp(S))) squared) plus 1 / the square root of (1/Cp(S) squared + (r/Cp(H)) squared)] x (1 - |(r kH - kS)/(1 + r)|).

5 Statistical tolerance values of the fit quality indices

5.1 Grading of the superior degree percentage of fit. The statistical tolerance value of the superior degree percentage PC(f) is expressed as a percentage, generally taken to one or two decimal places. Reference may be made to the standardized grading values of the process superior degree percentage in Table 4 of GB/Z 24636.3-2009. The commonly used range of statistical tolerance values is 60 % to 95 %; see Table B.9 of Annex B.

5.2 Grading of the fraction of average fit quality loss. The statistical tolerance value of the fraction of average fit quality loss Pql(f) is expressed as a percentage, generally taken to one or two decimal places. Reference may be made to the standardized grading values of the process fraction of average quality loss Pql in Table 5 of GB/Z 24636.3-2009. The commonly used range of statistical tolerance values is 3 % to 12 %; see Table B.10 of Annex B.

5.3 Grading of the fit capability index. The commonly used range of statistical tolerance values for the grading of the fit capability index is 1.33 to 3.0.

6 Statistical tolerance design of hole and shaft guaranteeing the fit quality indices

6.1 Setting the statistical tolerance using the relation between the fit capability index FCp and the process capability indices of the two fit sizes. 6.1.1 Equal setting of the statistical tolerances of the hole and shaft fit sizes. The following relation holds between the statistical tolerance FCp* of the fit capability index and the statistical tolerances of the process capability indices of the two fit sizes: FCp* = 1 / the square root of (1/Cp(H)* squared + (1/(r Cp(S)*)) squared) plus 1 / the square root of (1/Cp(S)* squared + (r/Cp(H)*) squared) = Cp* (r + 1) / the square root of (1 + r squared). Note 1: this applies where both the hole and the shaft part lots are still to be machined. Note 2: to simplify the problem, the statistical tolerances of the process capability indices of hole and shaft are set equal, that is Cp(H)* = Cp(S)* = Cp*. Note 3: when the hole/shaft tolerance ratio r = 1, FCp* = Cp* (1 + 1)/the square root of 2, approximately 1.414 Cp*; when r = 1.6, FCp* is approximately 1.378 Cp*.

6.1.2 Setting the statistical tolerance of the process capability index of the shaft or hole to be machined from the fit quality requirement and the statistical parameters of the finished part lot. This applies where the hole or the shaft part lot of the hole/shaft fit size has been machined and its statistical parameters in the controlled state obtained, so that a statistical tolerance is set only for the process capability index of the shaft or hole still to be machined. For example, given Cp(H) = 1.20 and r = 1.6, if FCp* is set at 1.70, substitution in formula (14) gives the statistical tolerance of the process capability index of the shaft fit size as Cp(S)* = 1.32. For convenience of application, Table B.1 and Table B.2 of Annex B give the corresponding tables of values of Cp(H), Cp(S) and FCp for r = 1.0 and r = 1.6 respectively.

6.2 Setting the statistical tolerance of the hole/shaft fit using the relation between the fit shift parameter and the shift parameters of the hole and shaft fit sizes. In statistical tolerance design, the statistical tolerances kH* and kS* of the shift parameters of hole and shaft are made equal, that is kH* = kS* = k*. When the absolute values of the hole and shaft shift parameters both reach the statistical tolerance k*, there are two cases: 1) kH and kS are in opposite directions and both reach the statistical tolerance k* in absolute value: when the hole/shaft tolerance ratio r = 1, kf* = 0.5(kH* - kS*) = 0.5(k* - (-k*)) = k*; when r = 1.6, kf* is approximately 0.615 k* - 0.385 (-k*) = k*. 2) kH and kS are in the same direction and both reach the statistical tolerance k*: when r = 1, kf* = 0.5(k* - k*) = 0; when r = 1.6, kf* is approximately 0.615 k* - 0.385 k* = 0.23 k*.

When the statistical tolerances of the shift parameters of the hole and shaft fit sizes are specified, the directions of kH and kS are not known, so it is not possible to give the statistical tolerances of the shift parameters of the hole and shaft fit sizes separately for the two cases; the relation between the statistical tolerances of the hole and shaft shift parameters and of the fit shift parameter can only be established according to the most unfavourable case, case 1). The statistical tolerance of the shift parameters of hole and shaft should therefore be equal to the statistical tolerance of the fit shift parameter, that is, kH* = kS* = kf* holds whether or not the hole and shaft tolerances are equal. Only in this way can it be guaranteed that in any case, as long as the hole and shaft shift parameters meet the statistical tolerance requirement, the fit shift parameter will not exceed its expected range. The relations between the statistical tolerances of the fit capability index and the fit shift parameter and those of the process capability indices and shift parameters of the hole and shaft fit sizes are summarized in Table 1: for a hole/shaft tolerance ratio r of 1, FCp* = 1.414 Cp* and kf* = k*; for r = 1.6, FCp* = 1.378 Cp* and kf* = k*.

Annex A (informative) Statistical tolerance design oriented to the fit quality target and examples of application

Annex A gives the procedure of statistical tolerance design oriented to the fit quality target, together with worked examples of application, showing how the statistical tolerances of the process capability indices and shift parameters of the hole and shaft are set from a required superior degree percentage or fraction of average fit quality loss.

Annex B (informative) Tables of values commonly used in statistical tolerance design oriented to the fit quality target

Annex B gives the tables of values commonly used in statistical tolerance design oriented to the fit quality target, including the corresponding values of Cp(H), Cp(S) and FCp for hole/shaft tolerance ratios of 1.0 and 1.6 (Tables B.1 and B.2), and the standardized grading values of the superior degree percentage (Table B.9) and of the fraction of average fit quality loss (Table B.10).

Annex C (informative) Position in the GPS matrix model

Annex C gives the position of this part in the GPS matrix model of GB/Z 20308, showing the chain of standards to which it belongs and the related general and complementary GPS standards.