2 Cumulative Effect of Tol

Design for Manufacture and Assembly Tolerance Analysis

2 CUMULATIVE EFFECT OF TOLERANCES All piece parts are eventually used for assembly. The probable lengths of the resulting combinations are the one to be concentrated upon. In large volume production the problem is complicated by the fact the lengths of the assembly members are unknown to the designer. They may only know that the lengths of individual parts lie somewhere within the specified tolerances. Variations in a dimension occur despite of keeping production conditions as constant as possible. The manner in which these variations are distributed within the tolerance therefore becomes important in determining the length of the resulting assembly. In addition, the drawings are done at one location, but the fabrication and assembly take place at far-distant places under conditions unknown to the designer. Nevertheless, it becomes necessary that the designer makes the best estimate possible so as to find out the probability of assembly variations be. The contacting surfaces should be flat and free of burrs and other imperfections. Thermal effects must also be taken into account. Types of Tolerance Specification: 1. Unilateral tolerance 2. Bilateral tolerance • Equal • Unequal Unspecified tolerances for the present consideration could be employed as ±0.3. The open tolerance for various dimensions are illustrated in the Table 2.2 For ease of calculation, bilateral tolerancing should be used. Unilateral tolerance can be changed temporarily changed to the bilateral form for the purpose of making calculations. Thus a dimension 4.000,+0.000,-0.008 mm would become 3.996±0.004mm. Tolerance will be represented as u and will be used to represent the permissible variation each way from the mean. The other nomenclatures foregoing u are as follows: A Area under distribution curve k Class width f(x) Function of x N Number of pieces u Engineering tolerance un Natural tolerance or 3σ ua Natural tolerance for assembly x Dimension Weighted average of dimension x x z Number of standard deviations between specified dimension x and mean x of normal curve σ Standard deviation or root-mean-square error σa Standard deviation for assembly σt standard deviation for truncated normal curve 2 σ Variance, Square of standard deviation ∑

Summation x

103

Design for Manufacture and Assembly ∞ x

a

Tolerance Analysis

Infinity Mean value

2.1 Arithmetic or Sure-Fit Law The length of a simple part is shown in Fig 1. A mean dimension x is given with the tolerance ´u´ expressed bilaterally. The mean value x a for an assembly is the sum of means of separate part as given by the Eq 2.1. x a = x 1 + x 2 + ….......= ∑ x ……………………………………(2.1) The simplest rule for estimating the tolerance ua for the assembly is merely to add the tolerances ´u´ of the individual parts. The results are given in equation 2.2. ua = u1 + u2 + …………..= ∑u …………………………………….(2.2)

Fig 1 Length of simple part expressed with bilateral tolerance Example1: The lengths for the three parts that are placed in end-to-end assembly are shown in Figure 2 determine the probable tolerance ua for the assembly by the arithmetic rule. Fig 53 Piece parts

Solution: By Eq. (2.2), ua= u=0.050+0.075+0.150=0.275 mm Equation (2.2) usually gives a value too large for ua. In fact, all the parts could simultaneously be at their high limit or all simultaneously at their low limit. Although physically possible, such an event would be of very rare occurrence. In general, some parts will be longer than their mean and sum will be shorter, so that after assembly the result for ua will be less than that indicated by Eq. (2.2).

104

Design for Manufacture and Assembly Tolerance Analysis

The inverse problem sometime occurs. Mere assembly tolerance ua would be specified with piece part tolerances u1, u2 …to be determined. The arithmetic rule may impose a severe restriction of small tolerances on the individual parts.

2.1.1 Assembly When Some Parts Double Back on Others Assemblies are sometimes arranged where some of the parts double back on the others. It can be easily shown that the mean of the resulting assembly is equal to the algebraic sum of the means of the component parts and the tolerance

Fig 54 Assembly tolerance is unchanged if some of the parts double back on the others. Assembly dimension is 35.56±0.011 . on the assembly is merely the sum of the tolerances on the components all taken positively when the arithmetic rule is used. Example 2: Suppose that the three parts of Fig. 54(a) are arranged as in Fig.54(b) and (c). Find the dimensions of the resulting assembly. Solution: In Fig. 54 (b) the parts have lengths that give the largest assembly dimension and in Fig.54 (c) the lengths give the smallest assembly dimension. It is seen that the assembly dimension, 35.56 mm, is equal to the algebraic sum of the means of the parts. The tolerance ua of 0.011 mm. is the sum of the tolerances on the three parts when all are considered as positive.

105

Design for Manufacture and Assembly Tolerance Analysis Example 3: Arrange the three parts of Fig.55 similarly to Fig. 54 Find the value of tolerance u2 if the tolerance of the resulting opening by the arithmetic rule is to be +0.4572 mm. Make drawings to demonstrate the validity of the solution. Solution: The calculations are carried out in the figure. The maximum opening occurs when parts 1 and 2 have their maximum lengths and part 3 has its minimum length. The figure indicates that u2 is equal to 0.010.

u

u

u

Fig. 55 Determination of component tolerance.

The minimum opening occurs when parts 1 and 2 have their minimum lengths and part 3 has its maximum length. The figure shows that u2 is equal to 0.254 mm.

2.32.3 Length of Assembly When Errors are Normally Distributed The Equation of the normal curve is given by ,

106

Design for Manufacture and Assembly Tolerance Analysis 2

1  x −µ  −   2 σ 

1 e 2π σ The variations in the length of the assembly dimension form a distribution that has a variance σ 12 whose value is determined by Eq. (2.3) f(x) =

(

∑ x1 − x1 N

)

2

f 1 ( x1 )

The distribution for another lot to be used in the assembly will have a variance σ 2 The third lot will have the variance σ 32, and so on. Let a part be selected at random from each lot and placed in end-to-end assembly. Assembly lengths will vary depending on the lengths of the parts of which it is composed. A sufficient number of assemblies will form a distribution for the assembly lengths and this distribution will have a variance σ a2. 2.

σ

2 a

=σ

2 1

+σ

2 2

+…=Σ σ 2……………………………………………………….(2.3)

In this equation σ 12 , σ 22 refer to the variances of the piece part distributions regardless of their shape. Variance σ a2 can also be calculated from the assembly histogram in the usual way. However, variance σ a2 as calculated by Eq. (2.3) tells nothing about the shape of the distribution curve for the assemblies. It can be demonstrated, however, that the shape of the distribution curve rapidly approaches the normal form as the number of parts in the assembly increases. This statement is valid regardless of the shape of the distribution curves for the errors of the individual parts. It can be shown, however, that if the piece-part error distribution does not depart too much from the normal form that only a few parts are needed to give the normal form to the assembly curve. Thus only the two parts are needed for truncated normal curves such as those in Fig.2.8(b), (c) and (d). However, for distribution curves such as Fig.2.8 (e) and (f), at least three parts are needed.The natural tolerance un for the normal curve could be taken as three standard deviations σ of the curve, or un=3σ ………………………………..……………………….(2.4) Assume that the assembly has a sufficient number of parts that its distribution curve will have the normal form. Its natural tolerance ua will then have the length ua=3σ

a

………………………………………………………..(2.5)

Example 4: Let the tolerance in Fig 53 natural tolerances with normally distributed errors centered on the mean of the dimension. Find the tolerance ua for the assembly. Solution:

107

Design for Manufacture and Assembly Tolerance Analysis 0.002 For part 1, σ1 = 3 0.003 = 0.001 For part 2, σ 2 = 3 0.006 = 0.002 For part 3, σ 3 = 3 By Eq 2.3 2  0.002  2 2 σ a2 = σ 12 + σ 22 + σ 32 =   + 0.001 + 0.002  3  0.000049 9 0.007 σa = 3 =

By Eq 2.5 ua = 3σa = 0.007 It is noted that 0.007 is a marked reduction from 0.011 as found by arithmetic rule in Example 1. In fact, unless there is full assurance that the errors are normally distributed and centered on the mean dimension with 3σ of the errors equal to the engineering tolerances u , there is a grave possibility that a solution will give too small a value for ua.

2.32.4 Assembly Length Determined By Tolerances Let the error distributions for each of the parts of an assembly have the normal form centered on the mean of the dimension. Let Eq. (2.4) and (2.5) be substituted into Eq(2.3).

ua2 u12 u22 = + + ..... 9 9 9 u

2 a

or = u1

2

+ u22 + …. = ∑ u2 ……………………………………….. (2.6)

It must be carefully noted that this equation is valid only when all piece part errors are distributed normally and centered on the mean of the dimension with 3σ equal to u. Example 5: Let the errors for the parts of Fig. 53 be normally distributed and centered on the mean of the dimension. Find the natural tolerance ua for the assembly by Eq (2.6).

Solution : By Eq.(2.6), ua2 = u1 2 + u22 + u32 = 0.050 2 + 0.075 2 + 0.150 2 = 0.031 ua = 0.176 mm This result is seen to agree with that found in Example 4.

108

Design for Manufacture and Assembly Tolerance Analysis

Care must be exercised in distinguishing between the engineering tolerance u as placed on the drawing by the designer and the natural tolerance un as determined by the accuracy of the production equipment.

2.32.5 Assemblies When Piece-Part Errors Overrun the Tolerances Sometimes the piece-part errors overrun the given tolerances as shown in Fig.2.5. In large-volume production the parts may not be individually checked, If the entire production is assembled , the variation in the length of the assembly will thereby be increased. For the normal distribution the proportion of parts that overrun the tolerances can be represented by the shaded areas in the tails of the curve. These areas can be found by the use of the Table 1.1, as previously shown. Coordinate z is required , which represents the number of standard deviations σ in the space between some chosen value of error x and mean x of the normal curve. Coordinate z is given by the following Equation: z=

x −x

……………………………………………………….(2.7) σ When x is taken at the edge of the engineering tolerance, x- x becomes equal to u and Eq.(2.7) becomes u z = …………………………………………………………..(2.8)

σ

The σ in these equations refers to the σ of the full normal curve. When applied to the normal curve for an assembly, Eq.(2.7) becomes

z=

x a − xa …………………………………………………….(2.9) σa

where σa is the standard deviation of the assembly curve. Example 6: In Fig. 56 suppose that the parts have normally distributed errors, but 4% of them are σ σ = 0.0434 and 4% are larger. If all of smaller than the specified tolerance them are assembled , find the natural tolerance ua for the resulting assembly. Solution: Since the entire production is assembled, the standard deviation to be used are those of the full normal curve. By Table1.1, for A=0.04 and z= -1.75, using Eq.(2-7) for the part on left, x − x1 u1 − 0.0762 σ1 = 1 = = = 0.044 z z − 1.75 Eq (2.7) for the part on right, x − x 2 u21 − 0.1219 σ2 = 2 = = = 0.070 σ z z − 1σ.75 σ σa 2 = ∑ σ2 = 5 * 0.0442 + 2 * 0.0702 = 0.019

109

Design for Manufacture and Assembly σa = 0.140 mm ua = 3 σa = 3 * 0.140 = 0.420 mm.

Tolerance Analysis

Fig 56 Assembly curve for piece parts that overrun their tolerance

2.32.6 Truncated Normal Curve A symmetrically truncated normal curve results when equal amounts of the curve are cut off at each and. Sometimes the distribution curve for a set of errors resembles a truncated curve. For making calculations, its standard deviation σ t will be required. It is obvious that σ t for the truncated portion is smaller than σ for the full normal curve before truncation.

σ σ

σ σ

Fig 57 Truncated Normal Curve. For symmetrical truncation:

σt = −0.12 z 2 + 0.69 z − 0.02 σ z is a positive value

110

Design for Manufacture and Assembly Tolerance Analysis

Values of the ratio

σt can be found from the graphs of Fig.59. such values depend z σ

µ .The empirical equations shown on the figure can be σ used if desired. The curves of Fig 57 do not apply to nonsymmetrical truncation. The lower curve in Fig 57 is convenient , as it enables one to pass directly from z or u/σ to u/σ t. or u/σ as determined by Eq z =

Example 7: Consider the curve of Fig 59, where u is 0.102 and σ is 0.064. Find the standard deviation σ t for the retained central portion of the curve.

∞ ∞

σ

σ

σ

Fig 58. Areas under the Normal curve when spread of production exceeds the engineering tolerance. Solution: Let x be taken at the edge of the tolerance zone. Then , z= By Fig 59,

µ = 0.102/0.064=1.59 σ

σt = 0.776 σ

σ t = 0.776 σ = 0.776*0.064=0.050 mm.

111

Design for Manufacture and Assembly Tolerance Analysis σ

σ

s =

σ

σ

σ

σ

σ

u

σ

σ

u

σ

σ

σ σ

σ

σ

σ

σ

u

u

σ

σ

σ

σ

σ

u

σ

σ σ

σ

σ

σ

=

σ

σ

σ

σ

σ σ

= 0.880

σ

σ

σ

σ

σ

σ σ σ

σ

σ

u

0.5σ

σ

σ

u

1.5 σ σ σ

1.5 σ

σ

σ σ

= 0.744

σ

σ

σ

σ

σ σ σ

σ

σ

u

σ

σ

u

1.5 σ σ σ

0.5σ

1.5 σ

σ

σ σ

= 0.744

112

Design for Manufacture and Assembly Tolerance Analysis

Fig 59 Standard deviations for various distributions of errors. Ratio Obtained.

σt σ

Example 8: A normal curve is given a symmetrical truncation equal to σ of the full normal curve. Find the value of σ t for the truncated curve. Solution: For σ truncation, u=2σ or σ =u/2. µ 2σ z= = =2 By Fig.2.8

σ

σ

σt = 0.88 σ σ t = 0.88 σ = 0.88 *

u 2

= 0.44 u

This result can also be obtained from the lower curve in Fig.4. For z = 2, u u =2.27 σ t= = 0.44u σt 2.27 Normal curves with truncations of 0.5σ , σ ,1.5σ and 2σ are shown in Fig.59. Thus if σ t is equal to 0.44 u, the normal curve has been given a symmetrical truncation equal to σ . Such a rule of thumb applies strictly only to the normal curve but may be approximately true when the frequency is highest near the mean and becomes less on either side.

2.32.7 Assemblies Made from Truncated Normal Curve

µ

113

σ

µ

Design for Manufacture and Assembly Tolerance Analysis

Fig 60 Individual parts and their assemblies Let the piece-part as shown in Fig.60 errors be approximated by a truncated normal curve. Such distributions can occur when normal curve distributions overrun the specified tolerance and the parts are sorted by go/no go gauges and only the acceptable parts are assembled. In working examples it will be necessary to find σt, the standard deviation of the acceptable central portion of the error distribution curves for the piece parts. These are then substituted into Eq.(2.3).

σ a2 = σ t21 + σ t22

σ t21 = 0.88 (for 1σ truncation) σ1 σ t1=0.44u1 σ t2=0.44u2 σa = σt21 +σt22

= 0.44

u12 + u 22 or

u a =1.32

u12 +u 22

Example 9: Suppose that only the parts falling within the engineering tolerances in Fig 56 are assembled. Find the tolerance ua for the assembly.

Solution : By Table 1.1 , for A= 0.04 and z = 1.75 , by Fig 55,

σt σ

From Example 6, Then By Eq.(2.3),

= 0.82

σt = 0.82 σ σ1 = 0.043, σt1

σ2 = 0.070

for full normal curves.

= 0.82 * 0.043 , σt2

= 0.82 * 0.070

σa 2 = ∑ σt2 = 5(0.82* 0.043)2+ 2 (0.82* 0.070)2

114

Design for Manufacture and Assembly Tolerance Analysis

= 0.013 mm σa = 0.113 mm The assembly distribution can be assumed to be normal , so that ua = 3 σa = 3 * 0.113 = 0.339 mm.

Example 10: Let the error distribution in Fig 54 be truncated normal curves of the form shown in Fig 59 c. Find the natural tolerance ua of the resulting assembly distribution. Solution. In Fig 59(c) , σt = 0.44 u. For part 1, σt1 = 0.44 * 0.051 =0.022 For part 2, σt2 = 0.44 * 0.076 =0.034 For part 3, σt3 = 0.44 * 0.152 =0.067 By Eq.(2.3), σa 2 = ∑ σt2 = 0.0222+0.0342+0.0672 σa = 0.078 ua = 3 σa = 3 * 0.078 = 0.235 mm. This result is intermediate between that of Example 1 using the arithmetic law and Example 4 when the errors were assumed to be normally distributed with 3σ equal to u. Example 11: Let the errors in part1,Fig 54 have a distribution like that of Fig.59 (c) .For part 2 the distribution is that of Fig.59(d), and for part 3, the errors have the distribution of Fig. 59(e).Find the natural tolerance ua for end- to -end assembly. Solution: By Fig. 59, For part 1, For part 2,

σt1 σt2

= 0.440u = 0.440* 0.051 =0.022mm = 0.496u =0.496* 0.076 =0.038mm

For part 3,

σt3

= 0.540u =0.540* 0.152 =0.082mm

By Eq(2.3) σ a2 = ( 0.0224 ) 2 + ( 0.0378 ) 2 + ( 0.0823 ) 2 = 0.009 σa = 0.093 mm The assembly curve can be assumed to have the normal form. By Eq(2.5) ua=3σa = 3*0.093=0.288mm. This example indicates that when the piece part distributions resemble normal curves with the greater amount of truncation, the excepted tolerance on the assembly is increased.

115

Design for Manufacture and Assembly Tolerance Analysis

The difference between the expected tolerance for an assembly when calculated by the arithmetic method and then by statistical method becomes greater as the number of parts in the assembly is increased. Example 12: Same as Example 11 except there are three pieces for each of the parts of Fig 54. Find the excepted tolerance for the end-end assembly. Solution: From Example 11, σ t1 = 0.022 σ t2 = 0.038 σ t3 = 0.082 By Eq(2.3), σ

a

σa ua

= Σ σ t 2 = 3*0.0222 + 3*0.0382 +3*0.0822 = 0.026 = 0.162 mm = 0.485 mm

2

By the arithmetic rule, ua = 3(0.051 + 0.076 + 0.152) = 0.838 mm Hence the variation between the statistical method and the arithmetic rule becomes greater when the number of parts is increased and the arithmetic rule may indicate too large a tolerance for the assembly. The inverse problem that of finding the piece-part tolerances for a specified assembly tolerance, can be handled by the foregoing equations. The following example will aid in visualizing the large difference between arithmetic tolerances and those determined by probability methods. Example 13: It is desired that an end-to-end assembly of four equal parts have a tolerance ua equal to 0.203mm . Find the piece-part tolerances that should be specified by the arithmetic rule, the normal law with a truncation of one standard deviation. Solution: For the arithmetic rule, by Eq.(2.1) ua = Σ u = 4u = 0.203 mm u = 1/4*0.203 = 0.051 mm For the normal distribution , by Eq.(2.7) ua 2 = Σ un 2 = 4un 2 =0.0232 = 0.041 un 2 =1/4*0.041= 0.010 un = 0.102 mm For the truncated normal distribution, σ t =0.44u by Fig.59 (c)

116

Design for Manufacture and Assembly Tolerance Analysis σ a2 = Σ σ 2 = 4*σ t 2 σ a = 2σ t = 2* 0.44u = 0.88u ua = 3*σ a = 3*0.88u = 0.203 mm 0.203 u = ---------- = 0.077 mm 3*0.88 The results in Example 13 indicate that the arithmetic rule demands the smallest tolerances for the piece parts. The tolerances required by assuming normal distributions may be too liberal. The truncated normal assumptions, for σ t = 0.44u ,gives results about halfway between. In an assembly, the dimensions that are more expensive to maintain can be given larger tolerances, while the dimensions that are inexpensive can be given smaller tolerances. The following example illustrates how this can be done using estimates by the designer. Example 14: Suppose in an assembly of three parts that ua must be maintained at 0.229 .Let u1 be the least expensive tolerance and let the designer feel that u2 can be 1.5 u1 and that u3 can be 2 u1.If the piece part errors are assumed to have a truncated normal distribution with σ t =0.44u,find suitable values for the tolerances. Solution: σ t =0.44u σ a = ua /3 = 0.229/3 = 0.076

For part1, σ t1 = 0.44u1 For part1, σ t2 = 0.44u2 = 0.44 * 1.5 u1 = 0.66 u1 For part3 σ t3 = 0.44u3= 0.44 * 2 u1 = 0.88 u1 σ a 2 = Σ σ t 2 = u1 2 (0.442+0.662+0.882)=1.404 u1 2 σ a = 1.184 u1 = 1/3* 0.229=0.076 mm u1 = 0.076/1.184 = 0.064 mm u2 = 1.5 u1 = 0.097 mm u3 = 2u1 = 0.129 mm Check: σ σ σ

t1 t2 t3

σ σ σ

= 0.44u1 = 0.028 = 0.44u2 = 0.043 = 0.44u3 = 0.057 σ

a

2

2

= 0.0008 = 0.0018 t2 2 = 0.0032 t3 ---------------= Σ σ t 2 = 0.0058 t1

2

117

Design for Manufacture and Assembly Tolerance Analysis σ

a

= 0.076

ua = 3σ

a

= 0.229

Example 15: Seven similar parts with normal distributions for the errors are placed in end-to-end assembly. The normal curves overrun the engineering tolerance u of 0.043 .Parts are gauged and only good parts are assembled. The Natural tolerance ua of the assembly must be 0.152mm.Find the proportion of oversize and undersize parts that are discarded. Solution: By Eq.(2.5) σ a = ua/3=0.152/3=0.051 For the assembly, by Eq.(2.3) σ a 2 = Σ σ 2 =7σ t 2 OR σ t 2 = σ

a

2

/7

Then σt=σ

a

/ 7 = 0.051/2.646=0.019 For piece parts

With only u and σ t are known ,let the following proportion be formed: u/σ t = 0.043/0.019 = 2.250 By the lower curve, Fig 4, for u/σ t of 2.250 , then Z = 1.97 By Table 2.1, 1.97 = 0.024 or 2.44% A−−∞

Thus 2.44% of the parts will be undersize and 2.44% will be lower size.

2.33 Solved Examples: 1. Find the resulting dimension in both cases. 1.

2. Fig 2.10 Piece parts

X

118

Design for Manufacture and Assembly Tolerance Analysis

Fig 61 Piece parts The resulting dimension x, will have a tolerance of addition of individual tolerances. (i.e.) parts: U1+u1,U2+u2,U3+u3,…..,Un+un Case 1: By Sure Fit Law: Assembly: Ua = U1±U2 = 30+40 = 70 mm ua = u1+u2 = 0.3+0.4 = 0.7 mm X = 70±0.7 mm Case 2: By Sure Fit Law: Assembly: Ua = U1±U2 = 70-30 = 40 mm ua = u1+u2 = 0.3+0.5 = 0.8 mm X = 70±0.8 mm 2. Find the resultant dimension

σ

σ

σ

σ

Fig 62 Normal Distribution-Piece parts By Normal law: µa = µ 1 + µ 2 = 30 + 40 = 70 mm σ a = σ 12 + σ 22 ua =

u12 + u 22

= ( 0.3) 2 + ( 0.4 ) 2 = 0.5 Resulting Dimension is 70±0.5mm 3. Three parts are to be placed in end-to-end assembly. The tolerances are 0.2286 mm, 0.3048 mm and 0.9144 mm. Form an estimate for the variations in the lengths of the assemblies by using the arithmetic and normal laws.

119

Design for Manufacture and Assembly Tolerance Analysis

Solution: Arithmetic or Sure-fit law: ua = 0.229 + 0.305 + 0.914 = 1.448 mm

Normal law: ua = ( 0.229 ) 2 + ( 0.305 ) 2 + ( 0.914 ) 2 = 0.991 mm 4. Find the resultant dimension X in Fig 63

Fig 63 Dimension of Parts Solution: Parts: U1±u1 ; U2±u2 Assembly:Ua±ua Sure Fit Law : ua=u1+u2 Normal Law : ua= u12 + u22 1σ truncated normal law ua= 1.32* u12 + u 22 Mean dimension of 30.2 and 29.9 = 30.05±0.15 Mean dimension of 20.7 and 19.9 = 20.3±0.4 The resulting length X will be : 1.By sure fit law

Ua=U1+U2 = 50.35 X=50.35±0.55 ua=u1+u2 = 0.55

2.By Normal law

Ua=U1+U2 = 50.35 ua= u12 + u22 =

( 0.15 ) 2 + ( 0.4 ) 2

3.By 1σ truncated Normal law Ua=U1+U2 = 50.35 ua=1.32 u12 + u 22

120

= 0.427

X= 50.35±0.427

Design for Manufacture and Assembly Tolerance Analysis =1.32 ( 2 * 0.15 / 3) + ( 2 * 0.4 / 3) =0.376 2

2

X= 50.35±0.376

5.Analyze the interference of the interference fit 20H7/r6 by the above methods. If the

piece part errors are having 1σ truncated normal distribution, find the percentage of assemblies that have more than 20 microns interference.

20.021 20H7=

20.012± 0.011

3σ =0.011

20.035± 0.007

3σ =0.007

20.000 20.041 20r6= 20.028

Fig.64 Normal Distribution

σ

a

= σ t21 +σ t22

= 0.44

u12 + u 22

= 0.44 (0.007 ) 2 +(0.004 ) 2 = 0.004 24 Area above 20=0.5+ A20 Solution : By sure fit law: Mean (interference) = µ1 − µ2 or u1 − u 2 = 27.5µ Tolerance on mean interference= u1 + u 2 = 0.011+0.010=20.5µ

121

Design for Manufacture and Assembly Tolerance Analysis i.e., Mean interference = 27.5µ Minimum interference = 7µ Maximum interference = 48µ By Normal law: Mean interference= u1 − u 2 =27.5µ Tolerance on mean interference= u12 + u 22 = (0.011 ) 2 +(0.010 ) 2 = 0.015 = 14.5µ Minimum interference=27.5-14.5=13 µ Maximum interference=27.5+14.5=42µ

Fig 65 (a)Normal Distribution

Fig 65 (b) 1σ Truncated Normal Distribution

By truncated Normal law: Refer 65 (b) µa = µ1 − µ2 =27.5 microns. σ

a

= σ t21 + σ t22

= 0.44 u12 + u 22 = 0.44 (0.0070 ) 2 +(0.0043 ) 2 = 0.00361 = 3.61µ u1 = 2σ = 2 / 3 * 0.011 = 0.007 ; µa + 3σ a = 27 .5 + 3 * 3.61 = 38 .33 µ u 2 = 2σ = 2 / 3 * 0.010 = 0.007 ; µa − 3σ a = 27 .5 − 3 * 3.61 = 16 .67 µ Therefore Maximum Interference = 38.33µ Minimum Interference = 16.67µ 7.The clearance permitted between stator and rotor in a motor assembly is 0.120 to 0.280.assuming hole basis system, determine the design sizes for both the stator bore and rotor to meet the above functional requirements. The nominal size of the rotor may be taken as φ 200.Distribute the machining tolerance equally between the stator and rotor. Solution:

122

Design for Manufacture and Assembly Maximum clearance=0.280mm Minimum clearance=0.120mm Mean clearance = (280-120)/2+0.120 200µ 6σ a=0.280-0.120 =160µ σ a=160/6=26.6µ =0.003mm Hole based system is followed. Therefore, Fundamental deviation of hole is zero i.e., Min hole=basic size of hole

Tolerance Analysis

Fig 66 Stator and Rotor in a Motor Assembly

σ a= σ 12 + σ 22 If we distribute the machining tolerances equally between stator and rotor u1 = u 2 σ1 = σ 2 Therefore σ a= 2σ σ =σ a/ 2 =0.027/ 2 =0.019 and 3σ = 0.056 6σ = 0.113mm 8. Two parts have normally distributed errors centered on their tolerances u of 0.076 mm, 0.122mm as produced, 5% of the parts are undersize and 5% are oversize. Find the natural tolerance for an end-to-end assembly of the two parts if the entire production is assembled. Solution:

123

Design for Manufacture and Assembly Tolerance Analysis

A = 0.05 Z = -1.64 Left

σ1 =

(x

1

−x z

)

= -0.076/-1.64 = 0.047 mm

σ2 =

(x

2

−x z

)

= -0.122/-1.64 = 0.074 mm σ

= Σ σ 2 = 2 * (0.047)2 + 2 * (0.074)2 = 0.015 mm σ a = 0.124 mm ua = 3σ a = 3* 0.124 = 0.372 mm 2 a

9.The assembly tolerance ua for the three parts must be held to 0.508 mm. the designer decided that the tolerance on the second part should be twice as great as for the first part, and for the third part should be three times as great as for the first. Distributions are such that σ t = 0.440u . Find the tolerance that should be specified for each part.

Solution: ua = 0.508 mm u2 = 2 u1 u3 = 3 u1 σ t = 0.44u σ

t1

=

ua 3

0.508 3 = 0.169 mm σ t2 = 0.44 u2 = 0.88 u1 σ t3 = 0.44 u3 = 1.32u1

=

σ

= σ t2 =u12 (0.442 +0.882 +1.322) = 2.711 u12 σ a = 0.169 mm 2 a

124

Design for Manufacture and Assembly Tolerance Analysis

u1 = 0.103 mm u2 = 0.206 mm u3 = 0.309 mm

10.Three parts in an end-to-end assembly have a natural tolerance of ua for the assembly of 0.419 mm. The entire production has to be assembled. If piece part errors are normally distributed, find the proportion of production that fulfills the engineering tolerance of u = 0.152 mm. Solution: ua =0.419 mm u = 0.152 mm σ

a

σ

a

σ

2 t

=

σ

t

= 0.081 mm

= ua/3 = 0.419/3 = 0.139 mm

u

σt

2

=Σ σ

t

=3σ

2

2 t

σ a2 3

= 0.152/0.081 = 1.889

From Table 2.1 z = 1.33 For z = 1.33 A−z∞ = 0.092 = 91.8% 11.Find the net dimensions in Fig.67

125

Design for Manufacture and Assembly Tolerance Analysis

Fig.67 Drill Bush

126

Design for Manufacture and Assembly Tolerance Analysis

Solution: Drill bush to datum[A] gives: From bush I.D(G1) to O.D(H1) From H1 to E1 From E1 to E2 From E2 to H2 From bush O.D (H2) to I.D(G2) Net Dimension

= 0.0±0.01(concentricity) = 0.0±0.00(assume perfect fit) = 38.0±0.01 = 0.0±0.00(assume perfect fit) = 0.0±0.01(concentricity) --------------= 38.0±0.03(acceptable)

The desired location of hole F2 as measured from datum [A] is 23.5±0.1 mm and tracing the link from the drill bush(G2) to datum[A] gives: From bush I.D(G2) to O.D(H2) From H2 to E2 From bush at hole E2 to column D1

= 0.0±0.01(concentricity) = 0.0±0.00(assume perfect fit) = -38.0±0.01–21.0±0.01+20.0±0.01 = -39.0 ±0.03 = -10.0±0.00 = -12.5±0.05+85.0±0.05 = 72.5±0.10 ----------------------23.5±0.14(violated)

From column D1 to dowel pin From dowel pin B to datum [A] Net Dimension

Considering tolerance stacks in the vertical orientation(downwards as positive),it can be seen that the desired location of hole F2 as measured from datum [B] is 23.5±0.1mm and tracing the link from the drill bush G to datum [B] gives: From bush I.D(G) to O.D(H) From H to E From bush at hole E to column D From column D1 to dowel pin B From dowel pin B to datum [B] Net Dimension

= 0.0±0.01(concentricity) = 0.0±0.00(assume perfect fit) = -45.0±0.01 = 58.0±0.00 = 10.5±0.2-47.0±0.05 = -36.5±0.25 ----------------------= -23.5±0.27(violated)

Problems:

1. Five similar parts are placed in end-to-end assembly. Parts overrun the engineering tolerance of 0.127 mm, with 2% of the parts being oversize and 2% undersize. Only parts within tolerance are assembled. Find the natural tolerance ua for the assembly. [ua =0.371mm] 2. The engineering tolerances for the two parts in end-to-end assembly are u1=0.076 mm and u2=0.102 mm. Drawings are marked “95% of parts to fulfill given tolerances”. The entire production is to be assembled. Assume that the errors are normally distributed and centered on the tolerances. Find the natural tolerance for

127

Design for Manufacture and Assembly Tolerance Analysis

the resulting assembly. [ua = 0.193 mm] 3. The natural tolerance ua for an assembly is to be 0.305. There are four parts with similar tolerances. Drawings are marked “98% of parts to fulfill given tolerances”. Assume that the part errors are normally distributed and centered on the tolerances. Find the tolerance u that should appear on the drawing. The entire production is to be assembled. [u = 0.117 mm] 4. Seven parts in end-to-end assembly are to have a natural tolerance ua for the assembly of 0.154 mm. Part errors are normally distributed but overrun the tolerances, with 0.62% of them too small and 0.62% too large. Only the good parts are assembled. Find the engineering tolerance u that should appear on the drawing. [u = 0.051 mm]

5. Seven parts in end-to-end assembly have truncated normal curves for the error distributions, with 1% of the parts discarded as too large and 1% discarded as too small. Engineering tolerance u = 0.051 mm. Only the good parts are assembled. Find the natural tolerance ua for the assembly. [ua = 0.163 mm] 6. Six parts in an end-to-end assembly are dimensioned u = 0.048 mm. Errors have truncated normal curve distributions. The assembly tolerance ua = 0.152 mm. Find the percent of over length and under length parts that are discarded. [1.79%] 7.A case study illustrating cumulative effect of tolerances

Fig. 68 Monoblock Pump Assembly

128

Design for Manufacture and Assembly Tolerance Analysis

Table 2.1 Proportion of total area between - ∞ and z

0.00 Z -0.01 -0.02 -0.03 -0.04 -0.05 -0.06 -0.07 -0.08 -0.09

0.5000 A−z ∞ 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641

-0.40 Z -0.41 -0.42 -0.43 -0.44 -0.45 -0.46 -0.47 -0.48 -0.49

0.3446 A−z ∞ 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121

-0.80 Z -0.81 -0.82 -0.83 -0.84 -0.85 -0.86 -0.87 -0.88 -0.89

129

0.2119 A−z ∞ 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867

-1.20 Z -1.21 -1.22 -1.23 -1.24 -1.25 -1.26 -1.27 -1.28 -1.29

0.1151 A−z∞ 0.1131 0.1112 0.1093 0.1075 0.1057 0.1038 0.1020 0.1003 0.0985

Design for Manufacture and Assembly -0.10 -0.11 -0.12 -0.13 -0.14 -0.15 -0.16 -0.17 -0.18 -0.19 -0.20 -0.21 -0.22 -0.23 -0.24 -0.25 -0.26 -0.27 -0.28 -0.29 -0.30 -0.31 -0.32 -0.33 -0.34 -0.35 -0.36 -0.37 -0.38 -0.39

0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483

-0.50 -0.51 -0.52 -0.53 -0.54 -0.55 -0.56 -0.57 -0.58 -0.59 -0.60 -0.61 -0.62 -0.63 -0.64 -0.65 -0.66 -0.67 -0.68 -0.69 -0.70 -0.71 -0.72 -0.73 -0.74 -0.75 -0.76 -0.77 -0.78 -0.79

0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451 0.2420 0.2389 0.2358 0.2327 0.2297 0.2266 0.2236 0.2207 0.2177 0.2148

-0.90 -0.91 -0.92 -0.93 -0.94 -0.95 -0.96 -0.97 -0.98 -0.99 -1.00 -1.01 -1.02 -1.03 -1.04 -1.05 -1.06 -1.07 -1.08 -1.09 -1.10 -1.11 -1.12 -1.13 -1.14 -1.15 -1.16 -1.17 -1.18 -1.19

0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170

-1.30 -1.31 -1.32 -1.33 -1.34 -1.35 -1.36 -1.37 -1.38 -1.39 -1.40 -1.41 -1.42 -1.43 -1.44 -1.45 -1.46 -1.47 -1.48 -1.49 -1.50 -1.51 -1.52 -1.53 -1.54 -1.55 -1.56 -1.57 -1.58 -1.59

Tolerance Analysis 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559

-2.00 Z -2.01 -2.02 -2.03 -2.04 -2.05 -2.06 -2.07 -2.08 -2.09 -2.10

0.0228 A−z ∞ 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183 0.0179

-2.40 Z -2.41 -2.42 -2.43 -2.44 -2.45 -2.46 -2.47 -2.48 -2.49 -2.50

0.0082 A−z ∞ 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064 0.0062

-2.80 Z -2.81 -2.82 -2.83 -2.84 -2.85 -2.86 -2.87 -2.88 -2.89 -2.90

0.0026 A−z∞ 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019 0.0019

Table 2.1 (continued)

-1.60 Z -1.61 -1.62 -1.63 -1.64 -1.65 -1.66 -1.67 -1.68 -1.69 -1.70

0.0548 A−z ∞ 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455 0.0446

130

Design for Manufacture and Assembly -1.71 -1.72 -1.73 -1.74 -1.75 -1.76 -1.77 -1.78 -1.79 -1.80

0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367 0.0359

-2.11 -2.12 -2.13 -2.14 -2.15 -2.16 -2.17 -2.18 -2.19 -2.20

0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143 0.0139

-2.51 -2.52 -2.53 -2.54 -2.55 -2.56 -2.57 -2.58 -2.59 -2.60

0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048 0.0047

-2.91 -2.92 -2.93 -2.94 -2.95 -2.96 -2.97 -2.98 -2.99 -3.00

-1.81

0.0351

-2.21

0.0136

-2.61

0.0045

-3.01

-1.82

0.0344

-2.22

0.0132

-2.62

0.0044

-3.02

-1.83

0.0336

-2.23

0.0129

-2.63

0.0043

-3.03

-1.84

0.0329

-2.24

0.0125

-2.64

0.0041

-3.04

-1.85

0.0322

-2.25

0.0122

-2.65

0.0040

-3.05

-1.86

0.0314

-2.26

0.0119

-2.66

0.0039

-3.06

-1.87

0.0307

-2.27

0.0116

-2.67

0.0038

-3.07

-1.88

0.0301

-2.28

0.0113

-2.68

0.0037

-3.08

-1.89

0.0294

-2.29

0.0110

-2.69

0.0036

-3.09

-1.90

0.0287

-2.30

0.0107

-2.70

0.0035

-3.10

-1.91

0.0281

-2.31

0.0104

-2.71

0.0034

-3.11

-1.92

0.0274

-2.32

0.0102

-2.72

0.0033

-3.12

-1.93

0.0268

-2.33

0.0099

-2.73

0.0032

-3.13

-1.94

0.0262

-2.34

0.0096

-2.74

0.0031

-3.14

-1.95

0.0256

-2.35

0.0094

-2.75

0.0030

-3.15

-1.96

0.0250

-2.36

0.0091

-2.76

0.0029

-3.16

-1.97

0.0244

-2.37

0.0089

-2.77

0.0028

-3.17

131

Tolerance Analysis 0.0018 0.0017 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014 0.0013 5 0.0013 1 0.0012 6 0.0012 2 0.0011 8 0.0011 4 0.0011 1 0.0010 7 0.0010 4 0.0010 0 0.0009 7 0.0009 4 0.0009 0 0.0008 7 0.0008 5 0.0008 2 0.0007 9 0.0007 6

Design for Manufacture and Assembly -1.98

0.0239

-2.38

0.0087

-2.78

0.0027

-3.18

-1.99

0.0233

-2.39

0.0084

-2.79

0.0026

-3.19

Tolerance Analysis 0.0007 4 0.0007 1

Table 2.2 Free Dimensions Tolerance OVER

UPTO

TOLERANCE

0 3 10 18 30 50 80 120 250 315

3 10 18 30 50 80 120 250 315 500

0.1 0.15 0.20 0.25 0.30 0.35 0.40 0.50 0.60 0.70

References: Merhyle Spotts,F.,”Dimensioning and Tolerancing for Quantity Production”,PrenticeHall,Inc.,Englewood Cliffs,1983.

132