sizing and tolerancing - isothai.com

MENG 204 - Mechanical Drawing Lecture Notes by: Dr. Ala Hijazi

Sizing and Tolerancing of 21

Sizing and Tolerancing

Choosing standard sizes and applying proper tolerances to dimensions are among the important aspects in the design of mechanical systems. When two parts need to fit together in order to perform a certain role in a mechanical system, the sizing and tolerancing of such components becomes of prime importance.

Sizes and Preferred Numbers

The purpose of preferred numbers is to limit the number of choices available for sizing parts or items. Though preferred numbers limit the number of choices, but at the same time they cover a wide range of sizes with reasonable increments between the different sizes.

When manufacturers use the preferred numbers for sizing their products, this enhances the compatibility and interchangeability of the different types of parts or items. With this, there are higher possibilities to find different types of parts or items as readily available off-the-shelf components.

The Table gives the complete set of Preferred Sizes in millimeters (when a

choice can be made, use one of these sizes; however, not all parts or items are

available in all the sizes shown in the table):

Renard Numbers

Objects are often manufactured in a series of sizes of increasing magnitude. The manufacturer must decide what those sizes should be. Renard numbers provide a very limited number of choices that cover a wide range of sizes. The sizes provided by the Renard numbers are not far apart such that the difference between any two adjacent sizes is not very significant. Also, Renard numbers may be divided in different series providing different levels of preference.

Which series provides more evenly distributed sizes?



Renard series of numbers are defined as:

𝑅𝑥 ; 𝑎, ( √10𝑥

)𝑎, ( √10𝑥

)2

𝑎, ( √10𝑥

)3

𝑎, … .. , ( √10𝑥

)𝑥

𝑎

where, 𝑥 = 5, 10, 20, 40

Thus, the different series are as follows:

- R5: 10, 16, 25, 40, 63, 100.

- R10: 10, 12.5, 16, 20, 25, 31.5, 40, 50, 63, 80, 100.

- R20: 10, 11.2, 12.5, 14, 16, 18, 20, 22.4, 25, 28, 31.5, 35.5, 40, 45, 50, 56,

63, 71, 80, 90, 100.

- R40: 10, 10.6, 11.2, 11.8, 12.5, 13.2, 14, 15, 16, 17, 18, 19, 20, 21.2, 22.4,

23.6, 25, 26.5, 28, 30, 31.5, 33.5, 35.5, 37.5, 40, 42.5, 45, 47.5, 50, 53, 56,

60, 63, 67, 71, 75, 80, 85, 90, 95, 100.

The Table gives the Renard (R-Series) Numbers (ISO 3)divided in different

choice preference levels:

In addition to the Renard Numbers, there is also the Rounded Renard (R'-

Series) Numbers (ISO 497):

- R′10: 10, 12.5, 16, 20, 25, 32, 40, 50, 63, 80, 100.

- R′20: 10, 11, 12.5, 14, 16, 18, 20, 22, 25, 28, 32,36, 40, 45, 50, 56, 63, 71,

80, 90, 100.

- R′40:10, 10.5, 11, 12, 12.5, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26,

28, 30, 32, 34, 36, 38, 40, 42, 45, 48, 50, 53, 56, 60, 63, 67, 71, 75, 80, 85,

90, 95, 100.



Dimensions and Tolerances

Tolerance is the maximum allowable variation in a dimension or in the size of a part.

The tolerance in dimensions can be represented as Direct Limits or

more commonly as plus/minus tolerance values applied directly to

a dimension.

- The difference between the upper and lower limits of a

dimension is called the Tolerance Zone.

There are two ways for specifying plus/minus tolerance values for dimensions:

Bilateral tolerance: the variation in both directions from the basic size (the

basic size is the exact desired theoretical size).

Example: 100 ± 0.2

- Usually used for parts that fit besides each other

Unilateral tolerance: the basic size is taken as one of the limits and the

variation is only in one direction.

Example: 40−0.1+0 𝑜𝑟 40−0

+0.1

- Usually used for parts that fit inside each other (e.g., shaft & hole, key &

keyway)

Why do we specify tolerance? Because parts cannot be manufactured to the exact geometry and dimensions, so we specify the acceptable range of variation.

In general, the variations in dimensions of manufactured parts (the actual size) occur at random (within the tolerance zone) and, usually, they follow a normal distribution.

Geometric Dimensioning and Tolerancing (GD&T)

Traditional tolerances can be used for defining the size and location of features. Geometric dimensioning and tolerancing is a more advanced method for applying tolerances that can be used for defining the geometric attributes of features (e.g., location, orientation, form).

Basic size

Limits

Basic size

Limits



The geometric tolerancing method is defined by the standard ASME Y14.5 and it is used to state the maximum allowable variations in the form of a feature or its position from the perfect geometry implied on the drawing.

The different types of geometric tolerances are defined using standard Geometric Characteristic Symbols as shown in the table.

Geometric tolerances are indicated on drawings using a feature control frame as shown in the figure.

The datum is the reference relative to which the geometric tolerance is defined (multiple references can be used at the same time) and it is identified on drawings using the symbol shown in the figure.

The modifiers are used to specify the condition to which the tolerances are

applied. There are three modifiers that can be applied to the tolerance value:

- Maximum Material Condition (MMC) and its symbol is:

- Regardless of Feature Size (RFS) and it is the default, thus it has no symbol

- Least Material Condition (LMC) and its symbol is:

Examples of using geometric tolerances in drawings are shown in the figure.



Standard Tolerance Values

The values of tolerances to be assigned depend largely on the application. Commercial products for instance does not require tight tolerances, while on the contrast, for precision measuring tools, tight tolerances must be used.

In general, the value of the tolerance depend on the dimension where smaller tolerances are used for small dimensions and bigger tolerances are used for big dimensions.

The tighter the tolerances, the higher the COST. Thus, tight tolerances should not be specified unless such tight tolerances are needed.

The international standard ISO 286 defines 20 different tolerance grades (01, 0, 1, 2, ...., 18) where the bigger the tolerance grade number, the bigger the tolerance is.

The figure shows the practical use of the International Tolerance Grades (IT#):



The IT Grade number that can be obtained is related to the manufacturing process used for making a part. The table shows the range of the IT grades that can be obtained using the different manufacturing processes.

The Table gives the Total Tolerance values for IT Grades from IT1 to IT18:



It should be noted the tolerance values given in the table are the full

tolerance zone (full range), therefore the value should be divided by 2 when it

is written as a bilateral tolerance.

Example: Write the following dimensions with bilateral tolerances using

IT12: 6, 50, 150 mm

6 ± 0.06 , 50 ± 0.125 , 150 ± 0.2

Non-Toleranced Dimensions Standard

It is a common practice to apply tolerances even to the non-toleranced dimensions. The ISO standard (ISO 2768-1) defines general tolerance values divided in four categories (fine, medium, coarse and very course) to be used for non-toleranced dimensions on drawings where tolerances are dependent on the nominal sizes.

The tables below show the general tolerance values to be used for non-toleranced dimensions.



Tolerance Stack-up Analysis

When tolerances are applied to the dimensions of individual features of a part (such as in chain dimensioned parts), tolerances will stack-up and the overall dimension of the part will be influenced by all the individual tolerances.

The same tolerance stack-up effect also applies to assemblies where the size of an assembly will be influenced by the tolerances applied to the dimensions of the individual parts.

There are two methods used for tolerance stack-up analysis:

Worst Case Scenario Analysis: The upper and lower limits of the dimensions are added (or subtracted when needed) algebraically to find the upper and lower limits for the desired dimension.

𝑑𝑇𝑜𝑡𝑎𝑙 ± 𝑡𝑇𝑜𝑡𝑎𝑙 = (𝑑1 + 𝑑2+. . . ) ± (𝑡1 + 𝑡2+. . . )

For the example above that will be: x = 7.5 ± 0.15

- This method is not accurate, but it is easy and used more often. Statistical Analysis: Each of the dimensions is treated as an independent

random variable and its tolerance limits are taken as the ±3𝜎 limits (𝜎 is the standard deviation). Then the variances (𝜎2) of each of the dimensions are added (or subtracted when needed) algebraically.

𝑑𝑇𝑜𝑡𝑎𝑙 ± 𝑡𝑇𝑜𝑡𝑎𝑙 = (𝑑1 + 𝑑2+. . . ) ± 3√∑ 𝜎𝑖2𝑛

𝑖=1

For the example above that will be: ±3𝜎1 = ±0.04 → 𝜎1 = 0.0133 ±3𝜎2 = ±0.06 → 𝜎2 = 0.02 ±3𝜎3 = ±0.05 → 𝜎3 = 0.0167

Thus, x = 7.5 ± 3√0.01332 + 0.022 + 0.01672 = 7.5 ± 0.088

- This method is more accurate, however it is not used that often.

The stack-up of tolerances in the dimensions of parts can be avoided by using baseline dimensions instead of chain dimensions.

Tolerance stack-up can have a significant effect on the gap or interference in assemblies since the gap or interference values are usually small by nature.



When several parts are assembled, the gap (or interference) depends on both the dimensions and tolerances of the individual parts.

Consider the shown assembly: 𝐺𝑎𝑝 = 𝑎 − 𝑏 + 𝑐 − 𝑑 + 𝑒 − 𝑓

Using 𝑥 for ( → ) and 𝑦 for ( ← ) we can write;

𝑤 = (𝑥1 + 𝑥2 + ⋯ ) − (𝑦1 + 𝑦2 + ⋯ )

𝑤 = ∑ 𝑥𝑖 − ∑ 𝑦𝑗

Using worst case scenario analysis we find:

- The largest gap 𝑤𝑚𝑎𝑥 occurs when the 𝑥 values are the largest possible & the 𝑦 values are the smallest possible.

If we call the bilateral tolerance as “ 𝑡 ” we get,

𝑤𝑚𝑎𝑥 = ∑(𝑥𝑖 + 𝑡𝑖) − ∑(𝑦𝑗 − 𝑡𝑗) = ∑ 𝑥𝑖 − ∑ 𝑦𝑗 + ∑ 𝑡𝑎𝑙𝑙

- The smallest gap 𝑤𝑚𝑖𝑛 occurs when the 𝑥 values are the smallest possible & the 𝑦 values are the largest possible.

𝑤𝑚𝑖𝑛 = ∑(𝑥𝑖 − 𝑡𝑖) − ∑(𝑦𝑗 + 𝑡𝑗) = ∑ 𝑥𝑖 − ∑ 𝑦𝑗 − ∑ 𝑡𝑎𝑙𝑙

Example: Find the maximum and minimum gap using worst case scenario analysis, then give the size of the gap using bilateral tolerance.

𝑎 = 500 ± 1 𝑚𝑚 𝑏 = 350 ± 0.7 𝑚𝑚 𝑐 = 120 ± 0.1 𝑚𝑚

Solution:

𝑤𝑚𝑎𝑥 = 500 − 350 − 120 + (1 + 0.7 + 0.1) = 31.8 𝑚𝑚

𝑤𝑚𝑖𝑛 = 500 − 350 − 120 − (1 + 0.7 + 0.1) = 28.2 𝑚𝑚

𝑤 = 30 ± 1.8 𝑚𝑚



Limits & Fits

The fit is the degree of tightness or looseness between two mating parts to perform a definite function, while the limits are the two extreme permissible sizes of a dimension between which actual size of dimension is contained.

When a shaft or pin needs to be inserted into a hole, the type of fit is decided based on the intended application.

- For example if the shaft and the hole are to form a pin-joint or a journal bearing , then the hole needs to be slightly bigger than the shaft to allow the shaft to rotate inside the hole (clearance is needed).

- Another example when the shaft is inserted into a rolling-contact bearing, the shaft needs to be slightly bigger than the hole of the inner-ring in order to prevent any slipping between the shaft and the inner-ring of the bearing (interference is needed).

Since tolerances are always present, fits are divided in three types:

Clearance fit: It occurs when two toleranced mating parts will always leave a space or clearance when assembled.

Interference fit: It occurs when two toleranced mating parts will always interfere when assembled.

Transition fit: It occurs when two toleranced mating parts will sometimes be an interference fit and sometimes be a clearance fit when assembled.

Important Terms

- Nominal size: a dimension used to describe the general size (the size we use in speaking of an element).

- Basic size: the theoretical size used as a starting point for the application of tolerances (the exact desired theoretical size).

- Actual size: the measured size of the finished part after machining.

Limits and fits are standardized according to the international standard (ISO 286) where there are two different systems that can be used for obtaining any desired type of fit:

Hole Basis System: The hole size is kept constant and the size of the shaft is varied.



Shaft Basis System: The shaft size is kept constant and the size of the hole is varied.

Because of considerations related to manufacturing, the hole basis system is more commonly used.

The figure shows the shaft and hole configurations for hole basis system under a clearance type of fit where the terms used for defining the fit are illustrated graphically.

- Capital letters refer to the hole and lower case letters refer to the shaft. - The Basic size (or nominal size) is the size to which limits are assigned and it

is the same for both members of the fit.



The table lists the specific types of Preferred Fits for both hole and shaft basis systems.

Once the type of fit is decided according to the intended use, the fit is described by the basic size and the ISO fit symbol.

Example: A joint with basic size of 40 mm having a Hole Basis Sliding fit The fit can be identified as: 40 H7/g6 where,

The hole 40 H 7

The shaft 40 g 6

It should be clear that for the Hole Basis system, the fundamental deviation is applied to the shaft dimension only.

International tolerance grade number (IT#)

Fundamental deviation letter



The table gives the Fundamental Deviations for Shafts (to be used for the hole basis system):

For the Hole Basis system, the maximum and minimum diameters for the hole and shaft are found as follows:

Hole: 𝐷𝑚𝑎𝑥 = 𝐷 + ∆𝐷 & 𝐷𝑚𝑖𝑛 = 𝐷

Shaft:

- For clearance fits: c, d, f, g, & h

𝑑𝑚𝑎𝑥 = 𝑑 + 𝛿𝐹 & 𝑑𝑚𝑖𝑛 = 𝑑 + 𝛿𝐹 − ∆𝑑

- For interference fits: k, n, p, s, & u

𝑑𝑚𝑎𝑥 = 𝑑 + 𝛿𝐹 + ∆𝑑 & 𝑑𝑚𝑖𝑛 = 𝑑 + 𝛿𝐹



where,

𝐷 : basic size of the hole

𝑑 : basic size of the shaft

𝛿𝐹: fundamental deviation

∆𝐷: tolerance grade for the hole

∆𝑑: tolerance grade for the shaft

Example: Find the shaft and hole dimensions for a sliding fit (Hole Basis system) with 25mm basic size then specify the dimensions of both members using unilateral tolerance.

Solution:

ISO symbol: 25 H7/g6

Hole: 25 H 7 𝐷 = 25

𝐷𝑚𝑎𝑥 = 𝐷 + ∆𝐷 = 25.021 𝑚𝑚

𝐷𝑚𝑖𝑛 = 𝐷 = 25 𝑚𝑚

Shaft: 25 g 6 𝑑 = 25

𝑑𝑚𝑎𝑥 = 𝑑 + 𝛿𝐹 = 25 + (−0.007) = 24.993 𝑚𝑚

𝑑𝑚𝑖𝑛 = 𝑑 + 𝛿𝐹 − ∆𝑑 = 25 + (−0.007) − 0.013 = 24.98 𝑚𝑚

Hole: 25−0+0.021 𝑚𝑚

Shaft: 24.993−0.013+0 𝑚𝑚

Sliding fit: g

IT 7 ∆𝐷 = 0.021

IT 6 ∆𝑑 = 0.013

g 𝛿𝐹 = −0.007

same value

found from the table according to the IT#

found from the table according to the fundamental deviation letter



Alternatively, instead of using the equations, the limit values of the shaft and hole dimensions for different types of fits and different basic sizes can be obtained directly from tables.

An example of such tables is given below where it gives the hole and shaft sizes for the Preferred Hole Basis Clearance Fits (for basic sizes from 7 to 19 mm divided in three preference levels: F, S & T):

Example: Find the shaft and hole dimensions for a close running fit (Hole Basis system) with 10 mm basic size and specify the dimensions using unilateral tolerance. Find the answer using the equations given earlier then compare the results with that given in the table above.

Solution: Hole: 10−0+0.022 𝑚𝑚 Shaft: 9.987−0.015

+0 𝑚𝑚



Surface Finish

The development of high speed machines has resulted in increasing the loads and speeds of moving parts. Such highly loaded surfaces running at high speeds require accurate control of surface quality in order to minimize friction and wear. Designers are responsible for specifying a surface that will give the maximum performance and surface life at the lowest cost. The required surface finish can be specified based on past experience with similar parts, on field service data, or on engineering tests. Generally speaking, the ideal finish is the roughest one that will do the job satisfactorily.

The two main reasons for surface finish control are:

Friction Reduction: When a film of lubricant must be maintained between two moving parts (such as in bearings, gears, etc.), the surface irregularities must be small enough so they will not penetrate the oil film under the most severe operating conditions.

Wear Control: Surface finish is also important to the wear service of certain parts that are subject to dry friction, such as dies, clutches, brakes, etc.

Also, surface finish must be controlled for the purpose of increasing the fatigue strength of highly stressed members which are subjected to load reversals (i.e., fatigue loading). A smooth surface eliminates the sharp irregularities which are the greatest potential source of fatigue cracks.

In some cases, a slightly rough surface finish is required. When boundary lubrication condition exist, or when two extremely hard surfaces running together, a slightly roughened surface will usually assist in lubrication. Also, most new moving parts do not attain a condition of complete lubrication as a result of imperfect geometry, running clearances, and thermal distortions. Therefore, the surfaces must wear in by a process of actual removal of metal.

Surface finish, also known as surface texture, has three components: lay, surface roughness, and waviness, as shown in the figure.

Lay: It is the direction of the predominant surface pattern and it is usually determined by the production method used.

Surface Roughness: (commonly shortened to roughness) It is a measure of the finely spaced surface irregularities.

Waviness: It is the measure of surface irregularities with a spacing greater than that of surface roughness. These usually occur due to warping, vibrations, or deflection during machining.



Surface roughness can be quantitatively measured using a device called Profilometer. A roughness parameter is used to give a numerical representation of the surface roughness. There are many different roughness parameters in use, but 𝑅𝑎 is by far the most commonly used. Other common parameters include 𝑅𝑞, 𝑅𝑧

and 𝑅𝑠𝑘.

The 𝑅𝑎 roughness parameter is the arithmetic average of the absolute values of the deviations of surface height readings from the surface roughness mean line. The 𝑅𝑎 is given in micrometers and it is mathematically defined as:

𝑅𝑎 =1

𝑛∑|𝑦𝑖|

𝑖=1

𝑛

According to older standards, Roughness Grade Numbers (N1, ... N12) were alternatively used instead of the roughness 𝑅𝑎 values. However, these roughness grade numbers are no longer used in the recent ISO standards.

The table shows the 𝑅𝑎values corresponding to the roughness grade numbers (for reference purposes) and the corresponding old Finish Marks symbols (also not being used anymore).



Surface Finish Symbols

Requirements for surface texture are indicated in technical drawings by several variants of standard graphical symbols (ISO 1302), each having its own significant meaning.

The simplest form of the surface finish symbols used in technical drawings is shown in the figure (the symbol may be omitted on views of parts when the finish quality of a surface is not important).

The symbol is always placed in upright position or it can be rotated 90⁰ CCW if needed (never at an angle or upside down) such that it can be read from the bottom side or the right side. The symbol is generally placed directly on the surface. An extension line or a leader can be used for placing the symbol when needed, as seen in the figure.

If additional surface characteristics need to be specified, the Complete Graphical Symbol (it has an additional horizontal line similar to the square root symbol) can be used. When the complete graphical symbol is used, additional complementary surface requirements may be indicated on the symbol and they are positioned as shown in the figure.

The complementary requirements that are shown at each position are as follows:

- Position a - Single surface texture requirement



Indicate the surface texture parameter designation, the numerical limit value and the transmission band or sampling length.

Example 1: 0.0025-0.8/Ra 3.2 (transmission band and sampling length are indicated)

Example 2: 0.8/Ra 3.2 (only sampling length is indicated)

- Position a and b - Two or more surface texture requirements

Indicate the first surface texture requirement at position “a” and indicate the second surface texture requirement at position “b”.

- Position c - Manufacturing method

Indicate the manufacturing method, treatment, coatings or other requirements for the manufacturing process etc. to produce the surface (for example, turned, ground, plated).

- Position d - Surface lay and orientation

Indicate the symbol of the required surface lay and the orientation, if any, of the surface lay, as shown in the figure:

- Position e - Machining allowance Indicate the required machining allowance, if any, as a numerical value given in millimeters.

When the same surface texture is required on all surfaces around a part outline, a circle shall be added to the complete graphical symbol as shown in the figure.



An example of a surface finish symbol showing a complete set of complementary requirements is shown in the figure.

The surface finish is highly dependent on the manufacturing process being used to produce the surface of a part as shown by the chart.



The surface finish 𝑅𝑎 value to be specified depends on the intended application. For instance, when stresses applied to a part are not that high and the appearance of the surface is not important, 𝑅𝑎 value of 6.3 micrometers will be acceptable. However, when the surface of a component is subjected to stress concentration, a smoother surface finish (𝑅𝑎 value of 0.8 micrometers) will be necessary.

The table provides some guidance on specifying the appropriate 𝑅𝑎 values for surfaces according to the intended application.

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