186 BELL SYSTEM TECHNICAL JOURNAL 



Cumulative Assembly Tolerances 



Another problem in choice of tolerances is in those cases where a con- 

 siderable number of parts are additively assembled into a unit as in the case 

 of "spring pileups" used on electrical contacting apparatus such as relays 

 and switches. These consist of considerable numbers of sheet metal springs 

 and insulators alternating and clamped by screws. If the overall toler- 

 ance on such an assembly must be taken as the sum of the tolerances of the 

 individual parts various courses of action are presented, the extremes of 

 which are: 



1. Very small tolerances must be maintained on the individual parts or 



2. Adequate space must be provided in the apparatus for extremely 

 large variations in the assembly. 



Small tolerances on the individual parts may be extremely expensive and 

 large space allowances and provisions in associated parts for variations in 

 the assembly may be a serious design handicap. 



However, it is recognized that there is obviously small probability that 

 all minimum or all maximum parts will appear in any one assembly. It 

 has been found satisfactory in certain types of such pileups to assume that 

 the maximum dimensional variation that will actually be encountered in an 

 assembly will not be greater than 70% of the sum of the part tolerances. A 

 similar situation exists in many kinds of assemblies or associations of toler- 

 ances. 



The statistical relationships involved in this problem are indicated by 

 Fig. 3. The curves show the percentage of the cumulative part tolerances 

 within which 99.7% of the assemblies may be expected to be found with 

 different numbers of similar units in the assembly. The solid line is de- 

 duced from theoretical relationships. It assumes that the parts are all of 

 one kind, that the parts going to assembly are controlled, of normal dis- 

 tribution and the limits are rationally set to represent the actual conditions. 

 The dotted curves have been deduced from relationships which have been 

 proposed as representing rectangular and triangular distributions of in- 

 dividual part tolerances. The curves may not be truly representative of 

 specific cases because of inconsistent selection of limits or erratic distribu- 

 tions. However, they indicate that the 70% rule on pileups is probably on 

 the safe side in most cases and that closer design of assembly or less re- 

 strictive tolerances and cheaper manufacture of piece-parts might be readily 

 possible either (1) by better control, (2) by actual mixing of lots of piece 

 parts or (3) even merely by knowledge of the actual statistical distribution 

 of part dimensions. 



The three points indicated in Fig. 3 show the results of a limited experi- 

 ment in which pileups were assembled from 2083 individual insulators of 

 ^" phenol fiber taken from factory stock. The establishment of curves 



