Independent is a tolerance of location or shape, the value of which is constant for all parts manufactured according to a given drawing and does not depend on the actual dimensions of the surfaces in question.
Dependent is a variable location tolerance (the minimum value is indicated in the drawing), which can be exceeded by an amount corresponding to the deviation of the actual size of the part surface from the throughput limit.
Passage limit - largest shaft size or smallest size holes.
A dependent tolerance is preferable and is placed where it is necessary to ensure the assembleability of the part. Tolerance is controlled by complex gauges (prototype of mating parts).
The maximum value of the dependent tolerance is defined as:
where is the constant part of the dependent tolerance;
Additional, variable part of the dependent tolerance.
Below is the calculation of the dependent positional tolerance for the location of the hole axis and the dependent alignment tolerance.
Calculation of the dependent positional tolerance of the hole axis(Fig. 32)
Rice. 32. Minimum positional deviation of the axis.
Minimum positional deviation of the hole axis
where is the minimum gap in the connection.
The minimum value of the positional tolerance of the hole axis in radius terms is defined as:
Calculation of dependent alignment tolerance:
Deviation from the alignment of two holes, according to Fig. 34 is equal to:
where are the minimum gaps in the first and second connections.
Rice. 33. Dependent deviation from the alignment of two holes.
The calculation of the dependent tolerance for the distance between the axes of two holes when connecting parts with bolts (type A connection) is given below.
According to GOST 14140-86 “Tolerances for the location of the axes of holes for fasteners,” we will determine the deviation by the distance between the axes of two holes L (Fig. 35).
Rice. 35. Dependent tolerance for the location of the hole axes
Let's assume that . Then
_______________________________ ,
where and are the limit values of the distance between holes in the first part;
And - maximum distance values between holes in the second part;
Deviation of the hole axes from the nominal position.
Provided that ,
where is the tolerance for the distance between the axes of two holes.
The first method of specifying the accuracy of the location of the axes of holes for fasteners is shown in Fig. 36.
Rice. 36. The first method of specifying the accuracy of the location of the hole axes
The second way to indicate the accuracy of the location of the axes of holes for fasteners (preferred) is shown in Fig. 37.
Rice. 37. The second method of specifying the accuracy of the location of the hole axes
For a type A connection, the positional tolerance in diametrical terms is:
in radius expression:
The dependent tolerance for the distance L between the axes of two holes when connecting parts with screws or studs (type B connections) is determined according to Fig. 38.
Rice. 38. Accuracy of location of the axes of holes for fasteners
To calculate the dependent tolerance, we assume that , then
______________________,
If , then , , .
The first way to indicate the accuracy of the location of the hole axes for type B connections is shown in Fig. 39.
Rice. 39. The first way to indicate dependent tolerances.
The second method, preferable, is shown in Fig. 40.
Rice. 40. The second way to indicate dependent tolerances.
For a type B connection, the positional tolerance in radius terms is:
In diametrical terms:
The accuracy of the location of the axes of holes for fasteners can be specified in two ways.
1. Limit deviations of coordinating dimensions (Fig. 41).
2. Positional deviation of the hole axes (preferred) (Fig. 42).
Rice. 41. Limit deviations of coordinating dimensions
Rice. 42. Positional tolerance of hole axes
Dimensional chains
Dimensional chain– a set of interrelated dimensions that form closed loop and directly involved in solving the assigned problem.
Types of dimensional chains.
1. Design chain – a dimensional chain with the help of which the problem of ensuring accuracy in the design of products is solved. There are two types of design chains:
Assembly;
Detailed.
2. Technological chain - a dimensional chain with the help of which the problem of ensuring accuracy in the manufacture of parts is solved.
3. Measuring chain - a dimensional chain with the help of which the problem of measuring parameters characterizing the accuracy of the product is solved.
4. Linear chain – a chain whose constituent links are linear dimensions.
5. Angular chain - a chain whose links are angular dimensions.
6. Flat chain – a chain whose links are located in the same plane.
7. Spatial chain – a chain whose links are located in non-parallel planes.
Dependent tolerance according to GOST R 50056-92 - a variable tolerance of shape, location or coordinating size, the minimum value of which is indicated on the drawing or in the technical requirements and which can be exceeded by an amount corresponding to the deviation of the actual size of the considered and (or) base element details from maximum limit material. According to GOST 25346-89, the maximum material limit is a term referring to that of the maximum dimensions to which the largest volume of material corresponds, i.e. the largest maximum shaft size d max or the smallest maximum hole size D min.
The following permissions can be assigned to dependents:
- shape tolerances:
- - tolerance for straightness of the axis of the cylindrical surface;
- - symmetry surface flatness tolerance flat elements;
- location tolerances (orientation and location):
- - tolerance of perpendicularity of the axis or plane of symmetry relative to the plane or axis;
- - tolerance for inclination of the axis or plane of symmetry relative to the plane or axis;
- - alignment tolerance;
- - symmetry tolerance;
- - axis intersection tolerance;
- - positional tolerance of the axis or plane of symmetry;
- tolerances of coordinating dimensions:
- - tolerance of the distance between the plane and the axis or plane of symmetry of the element;
- - tolerance of the distance between the axes or planes of symmetry of two elements.
Full dependent tolerance value:
Where T t in - minimum dependent tolerance value specified
in the drawing, mm;
Gdop - permissible excess of the minimum value of the dependent tolerance, mm.
It is recommended to assign dependent tolerances, as a rule, to those elements of parts for which requirements are imposed assembly in connections with guaranteed clearance. Tolerance T t[P are calculated based on the smallest connection gap, and the permissible excess of the minimum value of the dependent tolerance is determined as follows:
For shaft 
For hole
Where d a and /) d - actual dimensions of the shaft and hole, respectively, mm.
The value of G add can vary from zero to the maximum value. d
If the shaft has a valid size dmin, and the hole is D max, then
For shaft
For hole
Where TdwTD- size tolerance of the shaft and hole, respectively, mm.
In this case, the dependent tolerance has a maximum value:
For shaft
For hole
If the dependent tolerance is related to the actual dimensions of the considered and base elements, then
where Gd 0P.r and Gd 0P.b are the permissible excesses of the minimum value of the dependent tolerance, depending on the actual dimensions of the considered and basic elements of the part, respectively, mm.
Examples of the use of dependent tolerances include:
- - positional tolerance for the location of through holes for fasteners (Fig. 2.17, A);
- - alignment tolerances of stepped bushings and shafts (see Fig. 2.17, b, V), assembled with a gap;
- - tolerance for the symmetry of the location of grooves, for example, keyways (see Fig. 2.17, d);
- - tolerance for perpendicularity of the axes of holes and end surfaces of body parts for glasses, plugs, lids.
Rice. 2.17.A - positional tolerance of holes for fasteners; b, c - coaxiality of the surfaces of the stepped bushing and shaft; G - symmetry keyway relative to the shaft axis
Dependent location tolerances are more economical and beneficial for production than independent ones, since they expand the tolerance value and allow the use of less precise and labor-intensive technologies for manufacturing parts, as well as reducing losses from defects. Control of parts with dependent location tolerances is carried out, as a rule, using complex pass-through gauges.
A dependent tolerance of shape or location is indicated in the drawing by a sign, which is placed in accordance with GOST 2.308-2011:
- - after the numerical value of the tolerance (Fig. 2.17, A), if the dependent tolerance is related to the actual dimensions of the element in question;
- - after letter designation base or without a letter designation in the third field of the frame (see Fig. 2.17, b), if the dependent tolerance is related to the actual dimensions of the base element;
- - after the numerical value of the tolerance and the letter designation of the base (see Fig. 2.17, G) or without a letter designation (see.
rice. 2.17, V), if the dependent tolerance is related to the actual dimensions of the considered and base elements.
On January 1, 2011, GOST R 53090-2008 (ISO 2692:2006) came into force. This GOST partially duplicates GOST R 50056-92, in force since January 1, 1994, in terms of standardization and indication on drawings of maximum material requirements (MMR - maximum material reguirement) in cases where it is necessary to ensure the assembly of parts in connections with a guaranteed gap. Minimum material requirements (LMR - least material reguirement), due to the need to limit the minimum wall thickness of parts, have not previously been presented.
The MMR and LMR requirements combine the constraints of dimensional tolerance and geometric tolerance into one comprehensive requirement that more closely matches the intended purpose of the parts. This complex requirement allows, without compromising the performance of the part’s functions, to increase the geometric tolerance of the normalized (considered) part element if the actual size of the element does not reach the limit value determined by the established size tolerance.
The maximum material requirement (as well as the dependent tolerance according to GOST R 50056-92) is indicated on the drawings with a sign, and the minimum material requirement is indicated with a sign (L), placed in a frame to indicate the geometric tolerance of the normalized element after numerical value this tolerance and/or symbol bases.
Calculation of geometric tolerance values T m, ensuring the maximum material requirement can be performed similarly to the calculation of dependent tolerances (see formulas 2.10-2.15).
Designating similarly to dependent tolerances T m, geometric tolerances, which are subject to minimum material requirements - T L , can be written:
Where T m in - the minimum value of the geometric tolerance specified
in the drawing, mm;
Tdop - permissible excess of the minimum value of the geometric tolerance, mm.
T add values are determined as follows:
For shaft
For hole
dmin, and the hole Dmax, That
If the shaft has a valid size d max , and the hole Z) min , then
For shaft
For hole
In this case, the geometric tolerance has a maximum value:
For shaft 
For hole
If the geometric tolerance is related to the actual dimensions of the normalized and base elements, then the value of G additional is found from dependence (2.15).
Examples of the application of maximum material requirements are examples of assigning dependent tolerances according to GOST R 50056-92 in Fig. 2.17. An example of the application of the minimum material requirement is shown in Fig. 2.18, A.
Both the maximum material requirements and the minimum material requirements can be supplemented by an interaction requirement (RPR - reciprocity requirement), which allows increasing the tolerance of the size of a part element if the actual geometric deviation (deviation of shape, orientation or location) of the normalized element does not fully use the restrictions imposed by the requirements MMR or LMR. An example of the application of minimum material requirements and the interaction of size 05 tolerance O_ o, oz9 and concentricity tolerance are shown in Fig. 2.18, b, and an example of applying the requirement for maximum material and the interaction of size 16_о,т and perpendicularity tolerance is in Fig. 2.18, V.
Example 2.2. A dependent tolerance is specified for the alignment of hole 016 +OD8 relative to the outer surface 04O_o.25 of the bushing shown in Fig. 2.19.
From the symbol it is clear that the alignment tolerance depends on the actual size of the element, the axis of which is the base axis, i.e. surfaces 04О_ о 25.
Rice. 2.18.A- minimum material; b - minimum material and interaction; V- maximum material and interaction
Rice. 2.19.
The minimum value of the alignment tolerance indicated in the drawing (7pcs = 0.1 mm) corresponds to the maximum limit of the outer surface material, in in this case size d a = d max = 40 mm, i.e. at d a = d max = 40 mm 
If the outer surface has an actual size d a = dmin, The alignment tolerance can be increased:
Intermediate size values d a and their corresponding tolerance values T m are given in table. 2.9, and in Fig. Figure 2.20 shows a graph of the dependence of the alignment tolerance on the actual size of the outer surface of the bushing.
Rice. 2.20.
Values of dependent alignment tolerance, mm(see Fig. 2.20)
The standards establish two types of location tolerances: dependent and independent.
Dependent tolerance has a variable value and depends on the actual sizes of the base and considered elements. Dependent tolerance is more technologically advanced.
The following tolerances for the location of surfaces can be dependent: positional tolerances, tolerances of coaxiality, symmetry, perpendicularity, intersection of axes.
Shape tolerances can be dependent: axis straightness tolerance and flatness tolerance for the symmetry plane.
Dependent tolerances must be indicated by a symbol or specified in text in the technical requirements.
Independent clearance has a constant numerical value for all parts and does not depend on their actual dimensions.
The parallelism and tilt tolerance can only be independent.
In the absence of special symbols in the drawing, tolerances are understood as independent. For independent tolerances, a symbol may be used, although its indication is not required.
Independent tolerances are used for critical connections when their value is determined by the functional purpose of the part.
Independent tolerances are also used in small-scale and single-piece production, and their control is carried out by universal measuring instruments (see Table 3.13).
Dependent tolerances are established for parts mating simultaneously along two or more surfaces, for which interchangeability is reduced to ensuring assembly along all mating surfaces (connecting flanges using bolts).
Dependent tolerances are used in connections with guaranteed clearance in large-scale and mass production, their control is carried out by location gauges. The drawing indicates the minimum tolerance value ( Tr min), which corresponds to the flow limit (the smallest limit hole size or the largest limit shaft size). The actual value of the dependent location tolerance is determined by the actual dimensions of the parts being connected, i.e., it may be different in different assemblies. For slip fit connections Tp min = 0. The full value of the dependent tolerance is determined by adding to Tr min additional value T additional, depending on the actual dimensions of this part (GOST R 50056):
Tp manager = Tr min + T add.
Examples of calculating the tolerance expansion value for typical cases are given in Table 3.14. This table also provides formulas for converting location tolerances to positional tolerances when designing location gauges (GOST 16085).
The location of the axes of holes for fasteners (bolts, screws, studs, rivets) can be specified in two ways:
Coordinate, when maximum deviations are specified ± δ L coordinating sizes;
Positional, when positional tolerances are specified in diametrical terms - Tr.
Table 3.13 – Conditions for selecting dependent location tolerance
|
Connection conditions |
Location tolerance type |
|
Selection conditions: Large series, mass production It is only required to ensure collection under the condition full interchangeability Location gauge control Type of connections: Irrelevant connections Through holes for fasteners |
Dependent |
|
Selection conditions: Single and small-scale production It is necessary to ensure the correct functioning of the connection (centering, tightness, balancing and other requirements) Control by universal means Type of connections: Critical connections with interference fit or transitional fits Threaded holes for studs or holes for pins Seats for bearings, holes for gear shafts |
Independent |
Conversion of tolerances from one method to another is carried out using the formulas in Table 3.15 for the system of rectangular and polar coordinates.
The coordinate method is used in single, small-scale production, for unspecified location tolerances, as well as in cases where fitting of parts is required, if different tolerance values are specified in coordinate directions, if the number of elements in one group is less than three.
The positional method is more technologically advanced and is used in large-scale and mass production. Positional tolerances are most often used to specify the location of the axes of holes for fasteners. In this case, the coordinating dimensions are indicated only nominal values in square frames, since these dimensions are not covered by the concept of “general tolerance”.
Numerical values of positional tolerances do not have degrees of accuracy and are determined from the basic series of numerical values according to GOST 24643. The basic series consists of the following numbers: 0.1; 0.12; 0.16; 0.2; 0.25; 0.4; 0.5; 0.6; 0.8 µm, these values can be increased by 10 ÷ 10 5 times.
The numerical value of the positional tolerance depends on the type of connection A(with bolts, two through holes in the flanges) or IN(connection with studs, i.e. a gap in one part). Based on the known diameter of the fastener, a number of holes and their diameter ( D) and minimum clearance ( S min).
Table 3.14 – Recalculation of surface location tolerances to positional tolerances
|
Surface location tolerance |
Formulas for determining positional tolerance |
Maximum extension of tolerance Tdop |
|
|
Tolerance for alignment (symmetry) relative to the axis of the base surface |
|
For the base T P = 0 For con T rollable surface T And T P = T WITH |
T extra = Td 1 T extra = Td 2 |
|
Tolerance of coaxiality (symmetry) relative to the common axis |
|
T P1 = T C1 T P2 = T C2 |
T extra = Td 1 + Td 2 |
|
Tolerance for coaxiality (symmetry) of two surfaces Base not specified |
|
T P1 = T P2 = |
T extra = T.D. 1 + T.D. 2 |
|
Tolerance of perpendicularity of the surface axis relative to the plane |
|
T P = T |
T extra = T.D. |
In the drawing, the details indicate the value of the positional tolerance (see Table 3.7), resolving the issue of its dependence. For through holes the tolerance is assigned dependent, and for threaded holes it is independent, so it expands.
For connection type (A) T pos = S p , for connections like ( IN) for through holes T pos = 0.4 S p , and for threaded T pos =(0.5÷0.6) S p (Figure 3.4).
1, 2 – parts to be connected
Figure 3.4 – Types of connecting parts using fasteners:
A– type A, with bolts; b– type B, studs, pins
Design gap S p, necessary to compensate for the error in the location of the holes, is determined by the formula:
S p = S min,
where is the coefficient TO using the gap to compensate for deviations in the location of the axes of holes and bolts. It can take the following values:
TO= 1 – in connections without adjustment under normal assembly conditions;
TO = 0.8 – in connections with adjustment, as well as in connections without adjustment, but with recessed and countersunk screw heads;
TO= 0.6 – in connections with adjustment of the location of parts during assembly;
K = 0 – for a base element made using a sliding fit ( H/h), when the nominal positional tolerance of this element is zero.
If a positional tolerance is specified at a certain distance from the surface of the part, then it is specified as a protruding tolerance and is indicated by the symbol ( R). For example: the center of a drill, the end of a pin screwed into the body.
Table 3.15 – Recalculation of maximum deviations of dimensions coordinating the axes of holes to positional tolerances according to GOST 14140
|
Type of location |
Formulas for determining positional tolerance (in diametrical terms) |
|
|
Rectangular coordinate system |
||
|
One hole specified from the assembly base
|
T p = 2δ L δ L= ±0.5 T R T extra = T.D. |
|
|
Two holes coordinated relative to each other (no assembly base)
|
T p = δ L δ L = ± T R T extra = T.D. |
|
|
Three or more holes located in one row (no assembly base)
|
T p = 1.4δ L δ L=±0.7 T R T extra = T.D. δ L y = ±0.35 T R (δ L y – about T bowing about T wear T along the base axis) δ L forest = δ L∑∕2 (ladder) δ L chain = δ L∑ ∕(n–1) (chain) δ L∑ – largest dissipation T the glow between the axes of adjacent o T vers T th |
|
|
Two or more holes are located in one row (set from the assembly base)
|
T extra = T.D. T p = 2.8δ L 1 = 2.8 δ L 2 δ L 1 = δ L 2 = ±0.35 T R (O T axle declination T general plane T and – A or assembly base) |
|
|
The holes are arranged in two rows (no assembly base)
Holes are coordinated relative to two assembly bases |
Tр1.4δ L 1 1.4 δ L 2 δ L 1 = δ L 2 = ±0.7 T R T p = δ L d δ L d = ± T R (size is set to the diagonal) T extra = T.D. δ L 1 = δ L 2 = δ L Tр 2.8 δ L δ L= ±0.35 T R |
|
|
The holes are arranged in several rows (no assembly base)
|
δ L 1 = δ L 2 = … δ L Tр 2.8 δ L δ L= ±0.35 T R T p = δ L d δ L d = ± T R (size is set to the diagonal) T extra = T.D. |
|
|
Polar coordinate system |
||
|
Two holes, coordinated relative to the axis of the central element
|
T p = 2.8 δR δR = ±0.35 T R δα
= ± (corner mine T s) T extra = T.D. |
|
|
Three or more holes located in a circle (no assembly base)
Three or more holes are located around the circumference, central element is an assembly base |
T extra = T.D. T p = 1.4 δα δα = ±0.7 T R (corner mine T s) δα 1 = δα 2 = T extra = T.D. + TD bases |
|
3400
Table 3.16 – Diameters of through holes for fasteners and corresponding guaranteed clearances according to GOST 11284, mm
|
Fastener Diameter d | ||||||
|
Notes: 1 The 1st row is preferred and is used for connections of types A And IN(holes can be obtained by any method). 2 For connection types A And IN It is recommended to use the 2nd row when making holes by marking, punching with a high-precision stamp, in investment casting or under pressure. 3 Connection type A can be made along the 3rd row when arranged from the 6th to the 10th type, as well as connections like IN when located from the 1st to the 5th type (any processing method, except riveted joints). |
||||||
An independent tolerance for the location of the hole axes is a tolerance whose numerical value is constant for a large number of parts of the same name (for example, a batch of parts) and does not depend on the actual size (diameter) of the hole or (or perhaps “and”) on the size of the base. If there are no indications on the drawing, then the tolerance is considered independent.
The meaning of this concept comes down to the fact that with an independent tolerance during measurement, it is necessary to determine the location error in such a way that the value of the size (diameter) of the hole does not affect the value of the location deviation.
In the previous figures, the location tolerances are independent, i.e. center-to-center distances must be maintained within tolerances specified by positional deviations, or by maximum deviations and do not depend on what the actual diameters of the holes are (but, of course, the holes, in turn, must be made within their permissible dimensions).
Dependent location tolerance - a tolerance indicated on the drawing or in other technical documents in the form of a minimum value that can be exceeded by a value depending on the deviation of the actual size of the element (hole) and/or base in question from the maximum material limit, i.e. for a hole from the smallest limit hole size.
The dependent location tolerance is highlighted with the symbol M,
standing next to the location tolerance and/or the base.
The full value of the dependent location tolerance is determined by the formula:
,
where is the minimum tolerance value indicated in the drawing (the part of the dependent tolerance that is constant for all parts);
|
|
– additional tolerance value depending on the actual dimensions of the holes.
If the hole is made with maximum size(diameter), then will be maximum and determined as
, ,
where is the hole tolerance.
Interpreting the above, it can be argued that the minimum guaranteed clearance for the passage of a fastener can be increased (which occurs when the actual dimensions of the mating elements deviate from the passage limits), and a correspondingly increased position deviation allowed by the dependent tolerance becomes acceptable.
Let us explain the above using specific examples.
In Fig. 7, and the positional tolerance of the location is independent (there are no indications on the drawing). This means that the center of the ø10H12 hole must be within the circle with a diameter of 0.1mm and not go beyond, regardless of what the actual diameter of the hole is.
In Fig. 7, b the positional tolerance is dependent (this is indicated by the symbol M next to the location tolerance). This means that the minimum position tolerance value is 0.1 mm (for hole diameter ).
As the diameter of the hole increases, the location tolerance can be increased (due to the resulting gap in the connection). The maximum position tolerance value can be when the hole is made at the upper limit size, i.e. when = 10.15 mm. Eventually
,
and then, i.e. the center of the hole ø 10H12 can be in a circle with a diameter of 0.25 mm.
5.Numerical tolerance values
hole locations
For connection (Fig. 1, a, type A), both plates 1 and 2 to be connected are provided with through holes for the passage of fasteners. For connection type B – through holes only in the 1st plate. The diametrical gap between the fastener and the hole in the plate must ensure free passage of the bolt (rivet) into the hole to ensure assembly. The guarantee can be achieved when the actual hole size is obtained close to the minimum limit hole size, and the shaft (bolt, rivets) is close to the maximum limit size (usually, where d is the nominal bolt size). The difference between the sizes and is the minimum gap, which is guaranteed, since with a larger gap, the better the assembly will be ensured. The minimum diametrical clearance is taken as the positional tolerance for the location of the holes, and:
– for type A connections: ;
– for type B connections: (gap in only one plate).
Here T is the main positional tolerance in diametrical terms (twice the maximum displacement from the nominal location according to GOST 14140-81).
For standard fasteners, there are developed tables with the diameters of through holes for them and the corresponding smallest (guaranteed) clearances (GOST 11284-75). One of these tables is given in Appendix 1.
2. When setting dimensions, using a “ladder” with reference to the assembly base:
For type A connections – 
;
For type B connections – 
.
In Appendix 2 “Recalculation of positional tolerances for maximum deviations of dimensions coordinating the axes of holes. Rectangular coordinate system" according to GOST 14140-81, numerical values of maximum deviations are given depending on the specified positional tolerance for some sizing schemes.
Appendix 3 provides examples of converting positional tolerances into maximum deviations for some sizing schemes with tolerance symbols on the drawings.
