Tolerances and landing lectures for a technical school. Tolerances and landings. Basic definitions. Control questions and tasks
Dimensions on drawings
Introduction
In a mass production environment, it is important to ensure interchangeability the same details. Interchangeability allows you to replace a spare part that has broken during the operation of the mechanism. The new part must exactly match the replaced part in size and shape.
The main condition for interchangeability is the manufacture of a part with a certain accuracy. What should be the accuracy of the manufacture of the part, indicate on the drawings the permissible limit deviations.
The surfaces along which parts are connected are called conjugated . In the connection of two parts that are included one into the other, a female surface and a covered one are distinguished. The most common in mechanical engineering are connections with cylindrical and flat parallel surfaces. In a cylindrical connection, the surface of the hole covers the surface of the shaft (Fig. 1, a). The enclosing surface is called hole covering - shaft . These same terms hole and shaft conventionally used to refer to any other non-cylindrical enclosing and covered surfaces (Fig. 1, b).
Rice. 1. Explanation of terms hole and shaft
Landing
Any assembly operation of parts consists in the need to connect or, as they say, plant one detail to another. Hence, in technology, the expression landing to indicate the nature of the connection of parts.
Under the term landing understand the degree of mobility of the assembled parts relative to each other.
There are three groups of landings: with a gap, with an interference fit and transitional.
Landings with clearance
gap call the difference between the sizes of the hole D and the shaft d, if the size of the hole over size shaft (Fig. 2, a). The gap ensures free movement (rotation) of the shaft in the hole. Therefore, landings with a gap are called mobile landings. The larger the gap, the greater the freedom of movement. However, in reality, when designing machines with moving landings, such a gap is chosen that will minimize the friction coefficient of the shaft and hole.
Rice. 2. Landings
Interference landings
For these fits, the hole diameter D is less than the shaft diameter d (Fig. 2, b). .In reality, this connection can be made under pressure, when the female part (hole) is heated and (or) the male part (shaft) is cooled.
Interference landings are called fixed landings , since the mutual movement of the connected parts is excluded.
transitional landings
These landings are called transitional because before assembling the shaft and the hole, it is impossible to say what will be in the connection - a gap or an interference fit. This means that in transitional fits, the hole diameter D can be less, greater or equal to the shaft diameter d (Fig. 2, c).
Size tolerance. Tolerance field. Quality of accuracy Basic concepts
Dimensions in part drawings quantify the magnitude of the part's geometric shapes. Dimensions are divided into nominal, actual and limit (Fig. 3).
Nominal size - this is the main calculated size of the part, taking into account its purpose and the required accuracy.
Nominal connection size – this is the common (same) size for the hole and shaft that make up the joint. The nominal dimensions of parts and connections are not chosen arbitrarily, but according to GOST 6636-69 "Normal linear dimensions". In real production, in the manufacture of parts, nominal dimensions cannot be maintained and therefore the concept of actual dimensions is introduced.
actual size - this is the size obtained during the manufacture of the part. It always differs from the nominal up or down. The permissible limits of these deviations are established by means of limiting dimensions.
Limit dimensions two boundary values are called, between which the actual size must be. The larger of these values is called greatest size limit , smaller - smallest size limit. In everyday practice, on the drawings of parts, it is customary to indicate the limiting dimensions by means of deviations from the nominal.
Limit deviation - this is the algebraic difference between the limiting and nominal sizes. Distinguish between upper and lower deviations. Upper deviation is the algebraic difference between the largest size limit and the nominal size. lower deviation is the algebraic difference between the smallest size limit and the nominal size.
The nominal size serves as the starting point for deviations. Deviations can be positive, negative and zero. In standards tables, deviations are given in micrometers (µm). In the drawings, deviations are usually indicated in millimeters (mm).
Actual deviation - this is the algebraic difference between the actual and nominal sizes. The part is considered fit if the valid deviation of the checked size is between the upper and lower deviations.
Size tolerance - this is the difference between the largest and smallest limit sizes or the absolute value of the algebraic difference between the upper and lower deviations.
Under quality understand a set of tolerances that vary depending on the size of the nominal size. 19 qualifications have been established, corresponding to different levels of accuracy in the manufacture of a part. For each qualification, rows of tolerance fields are built
Tolerance field is a field bounded by upper and lower deviations. All tolerance fields for holes and shafts are indicated by Latin letters: for holes - in capital letters (H, K, F, G, etc.); for shafts - lowercase (h, k, f, g, etc.).
Rice. 3. Explanation of terms
test questions
Plan
Standardization
Lecture notes
at the rate:
"Interchangeability,
Donetsk 2008
Lecture No. 1 “The concept of interchangeability and standardization. Fundamentals of the principle of interchangeability. 3
Lecture No. 2 "Tolerance and fit systems for elements of cylindrical and flat joints." ten
Lecture No. 3 "Calculation and selection of landings for the GCC." 17
Lecture No. 4 "Calculation and design of gauges for the control of parts of smooth joints." 28
Lecture No. 5 "Tolerances and fittings of rolling bearings." 36
Lecture No. 6 "Normalization and designation of surface roughness." 42
Lecture No. 7 "Tolerances for the shape and location of surfaces." 47
Lecture No. 8 "Dimensional chains." 56
Lecture No. 9 "Interchangeability, methods and means of measurement and control gears". 68
Lecture No. 10 "Interchangeability threaded connections". 77
Lecture No. 11 "Interchangeability of keyed and splined connections." 82
Lecture No. 12 “Angle tolerances. Interchangeability of conical connections. 86
Lecture No. 13 "The concept of metrology and technical measurements." 91
Lecture No. 1 “The concept of interchangeability and standardization. Fundamentals of the principle of interchangeability.
Modern mechanical engineering is characterized by:
Continuous increase in capacity and productivity of machines;
Continuous improvement of machine designs and other products;
Increasing requirements for the accuracy of manufacturing machines;
The growth of mechanization and automation of production.
For the successful development of mechanical engineering in these areas great importance has the organization of production of machines and other products on the basis of interchangeability and standardization.
The purpose of the discipline: familiarity with the methods of ensuring interchangeability,
standardization, as well as measurement and control methods
applied to modern products engineering.
From the history of the development of interchangeability and standardization.
Elements of interchangeability and standardization appeared a very long time ago.
So, for example, the plumbing, built by the slaves of Rome, was made of pipes of a strictly defined diameter. To build the pyramids Ancient Egypt unified stone blocks were used.
In the 18th century, by decree of Peter 1, a series of warships with the same size, armament, and anchors was built. In the metalworking industry, interchangeability and standardization were first applied in 1761 at the Tula and then Izhevsk arms factories.
The concept of interchangeability and its types.
Interchangeability is the ability to assemble independently manufactured parts into a unit, and units into a machine without additional processing and fitting operations. In this case, the normal operation of the mechanism must be ensured.
To ensure the interchangeability of parts and assembly units, they must be manufactured with a given accuracy, i.e. so that their dimensions, surface shape and other parameters are within the limits specified in the design of the product.
The complex of scientific and technical initial provisions, the implementation of which during the design, production and operation ensures the interchangeability of parts, assembly units and products, is called the principle of interchangeability.
Distinguish between complete and incomplete interchangeability of parts assembled into assembly units.
Full interchangeability provides the possibility of free assembly (or replacement during repair) of any independently manufactured parts of the same type with a given accuracy into an assembly unit. (For example, bolts, nuts, washers, bushings, gears).
Partially interchangeable are such parts, during assembly or change of which a group selection of parts (selective assembly), the use of compensators, adjustment of the position of parts, fitting may be required. (For example, assembling a gearbox, rolling bearings).
The level of interchangeability in the production of a product is characterized by an interchangeability coefficient equal to the ratio of the labor intensity of manufacturing interchangeable parts to the total labor intensity of manufacturing a product.
There are also external and internal interchangeability.
External - this is the interchangeability of purchased or cooperative products (mounted in other more complex products) and assembly units in terms of performance, size and shape of connecting surfaces. (For example, in electric motors, external interchangeability is provided by the shaft speed, power, and also by the shaft diameter; in rolling bearings - by the outer diameter of the outer ring and the inner diameter of the inner ring, as well as by rotation accuracy).
Internal interchangeability applies to parts, assembly units and mechanisms included in the product. (For example, in a rolling bearing, rolling elements and rings have internal group interchangeability).
The basis for the implementation of interchangeability in modern industrial production is standardization.
Concepts about standardization. Categories of standards
The largest international organization in the field of standardization is ISO (until 1941 it was called ISA, organized in 1926) supreme body ISO is the General Assembly, which meets every 3 years, decides on the most important issues and elect the President of the organization. The organization consists of a large number clients. The Constitution states the main purpose of ISO - “to promote the favorable development of standardization throughout the world in order to facilitate the international exchange of goods and to develop mutual cooperation in various fields of activity.
Basic terms and definitions in the field of standardization are established by the ISO Committee for the Study of the Scientific Principles of Standardization (STACO).
Standardization is a planned activity to establish mandatory rules, norms and requirements, the implementation of which improves product quality and labor productivity.
Standard is normative white paper, which establishes requirements for groups of homogeneous products and rules that ensure its development, production and use.
Specifications(TU) - a regulatory and technical document that establishes requirements for specific products, materials, their manufacture and control.
To strengthen the role of standardization, a state (sovereign) DSS standardization system has been developed and put into action. It defines the goals and objectives of standardization, the structure of standardization bodies and services, the procedure for developing, formalizing, approving, publishing and implementing standards.
The main goals of standardization are:
Improving the quality of products;
Export development;
Development of specialization;
Development of cooperation.
Depending on the scope of the DSS, the following categories of standards are provided:
GOST (DST) - state;
OST - industry;
STP - enterprises.
Basic terms and definitions of the principle of interchangeability
Basic terms and definitions are established in GOST 25346 - 82.
A connection is two or more parts that are movably or fixedly mated to each other.
Figure 1 - Connection examples
The nominal size is the common size for the connection parts, obtained as a result of the calculation and rounded in accordance with the series of normal linear dimensions established by GOST 6636 - 69 and distributed on the basis of the series of preferred numbers GOST 8032 - 56.
Rows of preferred numbers (Renard series) are geometric progressions.
R5: \u003d 1.6 - 10; sixteen; 25; 40; 63; 100…
R10: = 1.25 - 10; 12.5; sixteen; 20; 25…
The actual size is the size obtained as a result of the processing of the part and measured with an allowable error.
When making drawings, it is most convenient to put down the size in the form of a nominal size with deviations.
Limit dimensions are two maximum allowable dimensions, between which the actual size of a suitable part must be. ()
Figure 2 - Limiting dimensions of the hole, shaft
Size tolerance is the difference between the largest and smallest limit sizes (T - Tolerance)
Tolerance is a measure of dimensional accuracy and determines the complexity of manufacturing a part. The larger the tolerance, the easier and cheaper it is to manufacture the part.
The concept of nominal size and deviations simplifies the graphical representation of tolerances in the form of tolerance field layouts.
Third lecture
2. Basic concepts of landings (conjugations)
Lecture plan
Concepts of clearance and tension.
Landing types.
The formation of landings in the hole system and in the shaft system.
Previously, the concepts shaft and holes as, respectively, the outer male and inner female elements. When pairing these elements belonging to two different parts, one or another fit is obtained.
Landing - the nature of the connection of two parts, determined by the values \u200b\u200bof the gaps and interferences resulting in this connection.
Clearance - the difference between the dimensions of the hole and the shaft before assembly:
The gap characterizes the freedom of relative movement of the parts to be joined. The larger the gap, the greater the freedom of relative movement of the interface elements. You can also remember the term backlash(German - Luft), indicating the gap between the mating surfaces of the assembly parts.
If the shaft size is larger than the hole size, a positive interference is obtained in the connection. Preload - the difference between the dimensions of the shaft and the hole before assembly:
Both the gap and the tightness can, generally speaking, be considered as algebraic quantities, assuming that S \u003d - N.
The concept of "landing" refers to a set of pairs of mating elements, the size of each of which is a random variable. The scattering field of a given random variable is limited by the specified limit deviations. Therefore, the resulting gaps (preloads) during assembly are also random variables.
The nature of the mating (that is, the fit) is conveniently represented on the diagram of the tolerance fields of the hole and shaft. In the geometric interpretation, the tolerance field is a part of the plane bounded above and below by lines of limiting dimensions (deviations). Deviations ES and EI (es and ei) on the tolerance field diagrams (Fig. 2.1) are plotted from the nominal size line - the zero line - in microns.
The specific content of the given tolerance field scheme can be better understood from Fig. 2.2, which shows the same nature of the connection.
Depending on the relative position of the tolerance fields of the mating landing elements, there are three types:
With guaranteed clearance, P(S > 0) = 1;
With guaranteed tightness, P(S< 0) = 1 или P(N > 0) = 1;
Transitional, i.e. 0< P(s) < 1.
Of course, P(S > 0) + P(N > 0) = 1.
A measure of the accuracy of the connection is the fit tolerance. Just as the size tolerance is the difference between its maximum and minimum limit values, the landing tolerance is found as the difference between the largest and smallest gaps:
TS \u003d S max - S min \u003d D max - d min - (D min - d max) \u003d T D + T d.
The relation obtained illustrates a simple idea: high connection accuracy can only be ensured with a correspondingly high dimensional accuracy of the mating elements.
Landings are appointed, as a rule, either in hole system either in shaft system.
The word "system" means order, regularity. The regularity, first of all, is expressed in the fact that the tolerance field of one of the mating parts has a well-defined constant location relative to the nominal size line. Such a detail is called the main one. The constant certainty of the location of the tolerance field of the main part is that it is in contact with the zero line and is overturned “into the material of the part” (the so-called principle of “saving metal”).
Landings in the hole system are obtained by a combination of different tolerance fields of the external male connection elements (shafts) with the tolerance field of the main hole (Fig. 2.3):
Here, the upper deviation of the hole for all mates is constant and equal to the hole size tolerance (ES = T D = const), and the lower deviation of the hole is zero (EI = 0). The maximum deviations of the shaft mating with this hole are selected according to the nature of the assigned interface.
Fits in the shaft system are obtained by combining different tolerance fields of internal female elements (holes) with the tolerance field of the main shaft (Fig. 2.4):
Here es = 0, ei = - T d ; depending on the required nature of the connection, the limit deviations of the hole (ES, EI) are selected.
It is preferable to use a hole system: the manufacture of an internal element (hole) is often more difficult and expensive; for processing holes, a measured cutting tool is usually used (for example, reamers, broaches), the range of which should be reduced.
In some cases more profitable system shaft:
The use of standardized components, the outer elements of which must be matched in different ways (that is, with the formation of different landings) with the holes of other parts;
Using the same shaft to obtain several different matings with female internals of other parts;
The use of standard calibrated bars for the manufacture of parts without their mechanical processing.
Literature
Belkin V.M. Tolerances and landings (Basic standards of interchangeability). - M .: Mashinostroenie, 1992. - 528 p.
Dunin-Barkovsky I.V. Interchangeability, standardization and technical measurements. - M.: Publishing house of standards, 1987. - 352 p.
Anukhin V.I. Tolerances and landings: Tutorial. - St. Petersburg: Peter, 2008. - 207 p.
The property of independently manufactured parts (or units) to take their place in the unit (or machine) without additional processing during assembly and perform their functions in accordance with technical requirements to the operation of this node (or machine)
Incomplete or limited interchangeability is determined by the selection or additional processing parts during assembly
Hole system
A set of fits in which different gaps and interferences are obtained by connecting different shafts to the main hole (hole, the lower deviation of which is zero)
Shaft system
A set of landings in which various gaps and interferences are obtained by connecting various holes with the main shaft (a shaft whose upper deviation is zero)
In order to increase the level of interchangeability of products, reduce the range of normal tools, tolerance fields for shafts and holes for preferred applications have been established.
The nature of the connection (fit) is determined by the difference in the dimensions of the hole and the shaft
Terms and definitions according to GOST 25346
The size- numerical value of a linear quantity (diameter, length, etc.) in the selected units of measurement
actual size is the element size set by the measurement
Limit dimensions- two maximum allowable sizes of the element, between which there must be (or which may be equal to) the actual size
The largest (smallest) size limit- largest (smallest) allowable size element
Nominal size- the size relative to which deviations are determined
Deviation- algebraic difference between the size (actual or limit size) and the corresponding nominal size
Actual deviation- algebraic difference between the actual and the corresponding nominal dimensions
Limit deviation- algebraic difference between the limit and the corresponding nominal size. Distinguish between upper and lower limit deviations
Upper deviation ES, es- algebraic difference between the largest limit and the corresponding nominal size
ES- upper deviation of the hole; es- upper shaft deflection
Lower deviation EI, ei- algebraic difference between the smallest limit and the corresponding nominal size
EI- lower deviation of the hole; ei- lower shaft deflection
Basic deviation- one out of two limit deviations(upper or lower), which determines the position of the tolerance field relative to the zero line. In this system of tolerances and landings, the main deviation is the closest to the zero line
Zero line- a line corresponding to the nominal size, from which dimensional deviations are plotted in the graphic representation of tolerance and fit fields. If the zero line is horizontal, then positive deviations are plotted up from it, and negative deviations are plotted down.
Tolerance T- the difference between the largest and smallest limit sizes or the algebraic difference between the upper and lower deviations
Tolerance is an absolute value without a sign
Standard IT approval- any of the tolerances established by this system of tolerances and landings. (Hereinafter, the term "tolerance" means "standard tolerance")
Tolerance field- a field limited by the largest and smallest limit sizes and determined by the tolerance value and its position relative to the nominal size. With a graphical representation, the tolerance field is enclosed between two lines corresponding to the upper and lower deviations relative to the zero line
Quality (degree of accuracy)- a set of tolerances considered as corresponding to the same level of accuracy for all nominal sizes
Tolerance unit i, I- a multiplier in the tolerance formulas, which is a function of the nominal size and serves to determine the numerical value of the tolerance
i- tolerance unit for nominal sizes up to 500 mm, I- tolerance unit for nominal sizes of St. 500 mm
Shaft- a term conventionally used to refer to the external elements of parts, including non-cylindrical elements
Hole- a term commonly used to refer to internal elements parts, including non-cylindrical elements
main shaft- shaft, the upper deviation of which is equal to zero
Main hole- hole, the lower deviation of which is zero
Maximum (minimum) material limit- a term referring to that of the limiting dimensions, which corresponds to the largest (smallest) volume of material, i.e. the largest (smallest) limit size of the shaft or the smallest (largest) limit size of the hole
Landing- the nature of the connection of two parts, determined by the difference in their sizes before assembly
Nominal fit size- nominal size common to the hole and shaft that make up the connection
fit tolerance- the sum of the tolerances of the hole and the shaft that make up the connection
Gap- the difference between the dimensions of the hole and the shaft before assembly, if the size of the hole is larger than the size of the shaft
Preload- the difference between the dimensions of the shaft and the hole before assembly, if the size of the shaft is larger than the size of the hole
Preload can be defined as the negative difference between the dimensions of the hole and the shaft
Landing with clearance- landing, in which a gap is always formed in the connection, i.e. the smallest hole size limit is greater than or equal to the largest shaft size limit. In the graphical representation, the hole tolerance field is located above the shaft tolerance field
Landing with interference - fit, in which there is always an interference in the connection, i.e. the largest hole size limit is less than or equal to the smallest shaft size limit. In the graphical representation, the hole tolerance field is located under the shaft tolerance field
transition fit- landing, in which it is possible to obtain both a gap and an interference fit in the connection, depending on the actual dimensions of the hole and shaft. With a graphical representation of the tolerance field, the hole and the shaft overlap completely or partially
Landings in the hole system
- landings in which the required clearances and interferences are obtained by combining different shaft tolerance fields with the tolerance field of the main hole
Fits in the shaft system
- landings in which the required clearances and interferences are obtained by a combination of different tolerance fields of the holes with the tolerance field of the main shaft
normal temperature- tolerances and limit deviations established in this standard refer to the dimensions of parts at a temperature of 20 degrees C
When assembling 2 parts that are included one into the other, they distinguish covered and covering surfaces, the meaning of which is clear from the name.
The enclosing surface is called hole covered - shaft.
For example, the inner cylindrical surface of the sleeve and the surface of the keyway - female surfaces, holes; the outer cylindrical surface of the bushing and the surface of the key - male surfaces, shafts.
The difference between the dimensions of the female and male surfaces (between the dimensions of the hole and the shaft) determines connection nature details or landing, i.e. a greater or lesser degree of mobility of parts or a degree of strength of joints (for fixed joints).
If the size of the hole D is greater than the size of the shaft d, then the positive difference between them, characterizing the degree of mobility (freedom of relative movement) is called gap S:
S = D - d; Dd; S0. (3.8)
If the size of the shaft d is greater than the size of the hole D, then the positive difference between them, characterizing the degree of strength of the connection, is called interference N:
N = d – D; d D; N0. (3.9)
Preload (if necessary) can be expressed as a negative clearance and vice versa:
S=-N;N=-S. (3.10)
Nominal size - the main estimated size, rounded up to the standard. The nominal dimensions of the hole and the shaft in the fit are marked on the drawing and deviations are counted from it, which are given in the table of standards for tolerances.
Nominal dimensions (when rounded after calculation for strength, stiffness, stability ...) are selected according to GOST 6636-69 * “Normal linear dimensions”. The use of only standard linear dimensions leads to a reduction in the standard dimensions of workpieces, cutting, measuring tools and a reduction in the cost of production.
According to GOST, a range of sizes is provided from 0.001 to 20000 mm, built on the basis of preferred numbers. Four rows of sizes are established, increasing in geometric progression with significant =; 
;
;
. Rows are designated Ra5, Ra10, Ra20, Ra40. The largest number of sizes in the last row, the smallest - in the first. When choosing denominations, each previous row should be preferred to the next.
Actual Size called the size obtained as a result of measurement with an allowable error.
The dimensions between which the actual size of the good parts in the batch must be (or be equal) are called the limit - respectively maximum limit D max , d max and the smallest limit D min , d min .
To simplify, in the drawings and tables, instead of the limiting dimensions, the corresponding limiting deviations are set - upper and lower.
Upper deviation(ES, es) is the algebraic difference between the largest size limit and the nominal size of the connection.
ES = D max - d n s; (3.11)
es = d max - d n s, (3.12)
where d n s is the nominal diameter of the connection.
Lower Deviation(EI, ei) - algebraic difference between the smallest limit size and the nominal size of the connection:
EI = D min - d n s; (3.13)
ei = d min - d n s. (3.14)
Deviations can be positive, negative or zero.
The size tolerance T is the difference between the limit sizes:
T D \u003d D max - D min; (3.15)
T d \u003d d max - d min. (3.16)
Tolerance - the value is always positive, so it is indicated in documents without a sign.
Substituting into expressions (3.15) and (3.16) the values of the limiting dimensions, expressed in terms of deviations and face value, we determine:
T D \u003d (ES + d n s) - (EI + d n s) \u003d ES - EI; (3.17)
T d \u003d (es + d n s) - (ei + d n s) \u003d es - ei. (3.18)
The tolerance is equal to the difference in limit deviations (with its own sign!).
Tolerance characterizes the accuracy of the size. The smaller the tolerance, the higher the accuracy, the smaller the possible range of size changes in the batch and vice versa. The tolerance value affects the operational properties of the connection and the product, as well as the complexity of manufacturing and the cost of the part. The manufacture of parts with a lower tolerance requires the use of more accurate equipment, accurate measuring instruments, fixtures, appropriate processing modes, which increases the cost of the product.
When assembling parts (for example, a shaft is connected to a sleeve) manufactured within tolerance, depending on random combinations of hole and shaft sizes, various landings. They are usually divided into landings with clearance (S), interference (N), transitional (N-S).
clearance fit is called a fit in which gaps are provided in all joints on the assembly. The interference landings.
transitional called landing, in which some of the connections on the assembly get gaps, and the rest - tightness.
Each landing is characterized by limiting (largest, smallest) clearances or interferences, the value of which is determined by the limiting dimensions of the parts.
The smallest gap S min in the connection is formed if a shaft with a size d max is installed in a hole with a size D min:
S min = D min -d max (3.19)
S min \u003d (EI + d n s) - (es + d n s) \u003d EI - es. (3.20)
The largest gap S max in the connection will be obtained if a shaft with the smallest limit size d min is installed in the hole with the largest limit size D max:
S max = D max -d min (3.21)
S max \u003d (ES + d n s) - (ei + d n s) \u003d ES - ei. (3.22)
Likewise,
N min \u003d d min - D max \u003d ei - ES \u003d - S max; (3.23)
N max \u003d d max - D min \u003d eS - EI \u003d - S min. (3.24)
The average clearance or interference is equal to:
S c (N c) = 
.
(3.25)
The gap or interference range determines the clearance, interference or fit tolerance (T S , T N).
fit tolerance(Т S, T N) - the difference between the limiting clearances or interference:
T S = (T N) = S max (N max) - S min (N min). (3.26)
In this expression, instead of S max , S min, we substitute their values according to (3.20), (3.22):
T S \u003d (ES - ei) - (EI - es) \u003d (ES - EI) + (es - ei) \u003d T D + T d. (3.27)
Thus, the fit tolerance is equal to the sum of the tolerances of the hole and the shaft.
Likewise,
T N \u003d N max - N min \u003d T D + T d. (3.28)
Imagine that there is a batch of bushings and shafts that need to be assembled. In this batch of bushings with the largest dimensions, D max will be very small (for example, 1 out of 100 pieces), similarly, in a batch of shafts with the smallest dimensions, d min will also be small (for example, 1 out of 100). It is natural to assume that the assembler, choosing parts and assembling connections without selection, is unlikely to simultaneously take parts with dimensions D max and d min (the probability of this event for our example is 1/1001/100 = 1/10 4). The probability of such an event is very small, so there will be practically no joints with a gap equal to S max in the assembly. For the same reasons, there will be practically no connections with a gap equal to S max in the assembly.
In order to determine the magnitude of the largest 
and least 
(probabilistic) gaps resulting from the assembly, we will approach this engineering problem from the standpoint of probability theory.
We assume that the distribution of the dimensions of the parts follows the normal law and the manufacturing tolerance is equal to the size range during manufacturing, i.e. T = 6. We also assume that there is no selection of parts during assembly (assembly is random).
It is known that the composition (combination) of two normal laws also gives a normal law. Therefore, the distribution of clearance (interference) values follows the normal law.
From the course of probability theory it is known that the mathematical expectation of the sum of random variables is equal to the sum of their mathematical expectations. The actual dimensions of the parts are random variables, the mathematical expectations of which will be close to the average sizes in the batch.
The mathematical expectation of the sum of random sizes is the mathematical expectation of the gap:
M S = M D + M -d . (3.29)
S c = D c - d c , (3.30)
where S c , D c , d c are the average values of the gap, hole and shaft dimensions.
The variance of the sum of independent random variables is equal to the sum of their variances. The variance D is the standard deviation squared:
D S = DD + D d; (3.31)
.
(3.32)
Then, taking T = 6, we get:
T S = 
.
(3.33)
With a probability P = 0.9973, the values of the actual gaps will be within:
Then the largest probabilistic gap will be equal to:
,
(3.35)
and the smallest probabilistic gap:
.
(3.36)
Expressions (3.35) and (3.36) are approximate (previously, the conditions for obtaining them were specified). More precisely, these values will be determined in the section “Dimensional chains”.
To simplify the calculations of tolerances and landings, the layout of the tolerance fields is used. Constructions on them are carried out relative to the nominal line, designated 0 - 0. The lines of limiting and nominal sizes are laid off from one border.
Therefore, lines of dimensions larger than the nominal will be located above the 0 - 0 line, and lines of dimensions smaller than the nominal will be below.
Up from the line 0 - 0 on the selected scale show positive deviations, down - negative. Two lines of maximum dimensions or maximum deviations of the hole and shaft form two tolerance fields, which are designated as rectangles (the scale of the rectangle is arbitrary along the length). The tolerance field is the zone of resizing, enclosed between the lines of the upper and lower deviations (or the corresponding dimensions). The tolerance field is a broader concept than tolerance. It is characterized not only by the value of the tolerance, but also by its location relative to the face value. Different (by location) tolerance fields may have the same tolerance.
In landings with a gap, the hole tolerance field is located above the shaft tolerance field, in interference fits, the hole tolerance field should be located below the shaft tolerance field. In transitional landings, the tolerance fields must overlap.