Parallelism of planes: sign, condition. The relative position of two planes in space. Signs of parallelism of two planes Deviation from the parallelism of the axes of the holes
TEXT EXPLANATION OF THE LESSON:
Let us introduce the concept of parallel planes
According to axiom A3, if two planes have a common point, then they intersect in a straight line.
It follows from this that the planes either intersect in a straight line, or do not intersect, that is, they do not have a single common point.
Definition. Two planes are called parallel if they do not intersect.
If the planes are parallel, write:.
Theorem (a sign of parallelism of planes).
If two intersecting lines of one plane are respectively parallel to two intersecting lines of another plane, then these planes are parallel.
Proof.
Consider two planes: .
The intersecting lines a1 and b1 lie in the plane, and the intersecting lines a2 and b2 parallel to them lie in the plane.
Let's prove that.
Proof. We argue by contradiction.
Assume that the planes are not parallel. Then there is a line c, they intersect in some way.
Since the line a1 is parallel to the line a2 lying in the plane, the line a1 is parallel to the plane.
Similarly, the line b1 is parallel to the plane.
Now you can use the property of a straight line parallel to a plane.
Since the plane passes through the line a1 parallel to another plane and intersects this plane, the line of intersection of the planes c will be parallel to the line a1, i.e.
But the plane also passes through the line b1 parallel to the plane, therefore.
Thus, two lines a1 and b1 pass through point O1 and are parallel to line c.
But this is impossible, only one straight line parallel to c can pass through O1.
Assuming that we have arrived at a contradiction. Hence, .
The theorem has been proven.
Problem 1. Three segments A1A2, B1B2 and C1C2, not lying in the same plane, have a common midpoint. Prove that the planes A1B1C1 and A2B2C2 are parallel.
Segments A1A2, B1B2 and C1C2 do not lie in the same plane
O - common midpoint of segments
Prove: Plane A1B1C1 plane A2B2C2
In the plane A1B1C1 we take the intersecting segments A1B1 and A1C1 , and in the plane A2B2C2 - the segments A2B2 and A2C2. Let us prove that they are respectively parallel.
Consider the quadrilateral A1B1A2B2.
Since its diagonals are bisected at the point of intersection, it is a parallelogram.
Therefore A1B1 A2B2
Similarly, from the quadrilateral A1C1A2C2 we get that A1C1 A2C2.
On the basis of the parallelism of the planes,
Everyone who has ever studied or is currently studying at school has had to face various difficulties in studying the disciplines that are included in the program developed by the Ministry of Education.
What difficulties do you face
The study of languages is accompanied by the memorization of existing grammatical rules and the main exceptions to them. Physical education requires from students a great calculation, good physical shape and great patience.
However, nothing compares to the difficulties that arise in the study of exact disciplines. Algebra, containing intricate ways of solving elementary problems. Physics with a rich set of formulas for physical laws. Geometry and its sections, which are based on complex theorems and axioms.
An example is the axioms that explain the theory of parallelism of planes, which must be remembered, since they underlie the entire course of the school curriculum on stereometry. Let's try to figure out how easier and faster this can be done.
Parallel planes by examples
The axiom, indicating the parallelism of the planes, is as follows: " Any two planes are considered parallel only if they do not contain common points.”, that is, they do not intersect with each other. To imagine this picture in more detail, as an elementary example, we can cite the ratio of the ceiling and floor or opposite walls in a building. It becomes immediately clear what is meant, and the fact is also confirmed that these planes in the usual case will never intersect.
Another example is a double-glazed window, where glass sheets act as planes. They also under no circumstances will form points of intersection with each other. In addition to this, you can add bookshelves, a Rubik's cube, where the planes are its opposite faces, and other elements of everyday life.
The considered planes are designated with a special sign in the form of two straight lines "||", which clearly illustrate the parallelism of the planes. Thus, by applying real examples, one can form a clearer perception of the topic, and, therefore, one can proceed further to the consideration of more complex concepts.
Where and how is the theory of parallel planes applied?
When studying a school geometry course, students have to deal with versatile tasks, where it is often necessary to determine the parallelism of straight lines, a straight line and a plane between themselves or the dependence of planes on each other. Analyzing the existing condition, each task can be related to the four main classes of stereometry.
The first class includes tasks in which it is necessary to determine the parallelism of a straight line and a plane between themselves. Its solution reduces to the proof of the theorem of the same name. To do this, you need to determine whether for a line that does not belong to the plane under consideration, there is a parallel line lying in this plane.
The second class of problems includes those in which the sign of parallel planes is used. It is used to simplify the proof process, thereby significantly reducing the time to find a solution.
The next class covers the spectrum of problems on the correspondence of lines to the main properties of parallelism of planes. The solution of problems of the fourth class is to determine whether the condition of parallel planes is met. Knowing exactly how the proof of a particular problem takes place, it becomes easier for students to navigate when applying the existing arsenal of geometric axioms.
Thus, the tasks, the condition of which requires defining and proving the parallelism of straight lines, a straight line and a plane or two planes with each other, are reduced to the correct selection of the theorem and the solution according to the existing set of rules.
On the parallelism of a straight line and a plane
The parallelism of a straight line and a plane is a special topic in stereometry, since it is precisely this that is the basic concept on which all subsequent properties of the parallelism of geometric figures are based.
According to the available axioms, in the case when two points of a straight line belong to a certain plane, we can conclude that the given straight line also lies in it. In this situation, it becomes clear that there are three options for the location of the line relative to the plane in space:
- The line belongs to the plane.
- For a line and a plane there is one common point of intersection.
- There are no intersection points for a straight line and a plane.
We are, in particular, interested in the last variant, when there are no intersection points. Only then can we say that the line and the plane are parallel relative to each other. Thus, the condition of the main theorem on the sign of parallelism of a straight line and a plane is confirmed, which states that: "If a line not belonging to the plane in question is parallel to any line in that plane, then the line in question is also parallel to the given plane."
The need to use the sign of parallelism
The sign of parallelism of planes is usually used to find a simplified solution to problems about planes. The essence of this sign is as follows: If there are two intersecting lines lying in the same plane, parallel to two lines belonging to another plane, then such planes can be called parallel».
Additional theorems
In addition to using a feature that proves the parallelism of planes, in practice one can encounter the use of two other additional theorems. The first is presented in the following form: If one of the two parallel planes is parallel to the third, then the second plane is either also parallel to the third, or completely coincides with it».
Based on the use of the given theorems, it is always possible to prove the parallelism of the planes with respect to the space under consideration. The second theorem displays the dependence of planes on a perpendicular line and has the form: “ If two non-coincident planes are perpendicular to some straight line, then they are considered parallel to each other».
The concept of a necessary and sufficient condition
When repeatedly solving problems of proving the parallelism of planes, a necessary and sufficient condition for the parallelism of planes was derived. It is known that any plane is given by a parametric equation of the form: A 1 x+ B 1 y+ C 1 z+D 1 =0. Our condition is based on the use of a system of equations that specify the location of planes in space, and is represented by the following formulation: To prove the parallelism of two planes, it is necessary and sufficient that the system of equations describing these planes be inconsistent, that is, have no solution».
Basic properties
However, when solving geometric problems, using the sign of parallelism is not always enough. Sometimes a situation arises when it is necessary to prove the parallelism of two or more lines in different planes or the equality of the segments contained on these lines. To do this, use the properties of parallel planes. In geometry, there are only two of them.
The first property allows you to judge the parallelism of lines in certain planes and is presented in the following form: If two parallel planes are intersected by a third, then the lines formed by the lines of intersection will also be parallel to each other».
The meaning of the second property is to prove the equality of segments located on parallel lines. Its interpretation is presented below. " If we consider two parallel planes and enclose a region between them, then it can be argued that the length of the segments formed by this region will be the same».
This article will study the issues of parallelism of planes. Let's give a definition of planes that are parallel to each other; we denote the signs and sufficient conditions of parallelism; Let's look at theory through illustrations and practical examples.
Yandex.RTB R-A-339285-1 Definition 1
Parallel planes are planes that do not have common points.
To denote parallelism, the following symbol is used: ∥. If two planes are given: α and β , which are parallel, a short record about this will look like this: α ‖ β .
In the drawing, as a rule, planes parallel to each other are displayed as two equal parallelograms offset from each other.
In speech, parallelism can be denoted as follows: the planes α and β are parallel, and also - the plane α is parallel to the plane β or the plane β is parallel to the plane α.
Parallelism of planes: sign and conditions of parallelism
In the process of solving geometric problems, the question often arises: are the given planes parallel to each other? To answer this question, the sign of parallelism is used, which is also a sufficient condition for the parallelism of the planes. Let's write it down as a theorem.
Theorem 1
Planes are parallel if two intersecting lines of one plane are respectively parallel to two intersecting lines of another plane.
The proof of this theorem is given in the geometry program for grades 10 - 11.
In practice, to prove parallelism, among other things, the following two theorems are used.
Theorem 2
If one of the parallel planes is parallel to the third plane, then the other plane is either also parallel to this plane or coincides with it.
Theorem 3
If two non-coincident planes are perpendicular to some line, then they are parallel.
On the basis of these theorems and the sign of parallelism itself, the fact of parallelism of any two planes is proved.
Let us consider in more detail the necessary and sufficient condition for the parallelism of the planes α and β, given in a rectangular coordinate system of three-dimensional space.
Let us assume that in some rectangular coordinate system the plane α is given, which corresponds to the general equation A 1 x + B 1 y + C 1 z + D 1 = 0, and also the plane β is given, which is defined by the general equation of the form A 2 x + B 2 y + C 2 z + D 2 = 0 .
Theorem 4
For the given planes α and β to be parallel, it is necessary and sufficient that the system of linear equations A 1 x + B 1 y + C 1 z + D 1 = 0 A 2 x + B 2 y + C 2 z + D 2 = 0 has no solution (was incompatible).
Proof
Suppose that the given planes defined by the equations A 1 x + B 1 y + C 1 z + D 1 = 0 and A 2 x + B 2 y + C 2 z + D 2 = 0 are parallel, and therefore do not have common points . Thus, there is not a single point in the rectangular coordinate system of three-dimensional space, the coordinates of which would correspond to the conditions of both equations of the planes simultaneously, i.e. system A 1 x + B 1 y + C 1 z + D 1 = 0 A 2 x + B 2 y + C 2 z + D 2 = 0 has no solution. If the specified system has no solutions, then there is not a single point in the rectangular coordinate system of three-dimensional space, whose coordinates would simultaneously meet the conditions of both equations of the system. Therefore, the planes given by the equations A 1 x + B 1 y + C 1 z + D 1 = 0 and A 2 x + B 2 y + C 2 z + D 2 = 0 do not have any common points, i.e. they are parallel.
Let us analyze the use of the necessary and sufficient condition for the parallelism of the planes.
Example 1
Given two planes: 2 x + 3 y + z - 1 = 0 and 2 3 x + y + 1 3 z + 4 = 0 . You need to determine if they are parallel.
Decision
We write down the system of equations from the given conditions:
2 x + 3 y + z - 1 = 0 2 3 x + y + 1 3 z + 4 = 0
Let's check whether it is possible to solve the resulting system of linear equations.
The rank of the matrix 2 3 1 2 3 1 1 3 is equal to one, since the second-order minors are equal to zero. The rank of the matrix 2 3 1 1 2 3 1 1 3 - 4 is equal to two, since the minor of 2 1 2 3 - 4 is non-zero. Thus, the rank of the main matrix of the system of equations is less than the rank of the extended matrix of the system.
Together with this, it follows from the Kronecker-Capelli theorem: the system of equations 2 x + 3 y + z - 1 = 0 2 3 x + y + 1 3 z + 4 = 0 has no solutions. This fact proves that the planes 2 x + 3 y + z - 1 = 0 and 2 3 x + y + 1 3 z + 4 = 0 are parallel.
Note that if we applied the Gauss method to solve a system of linear equations, this would give the same result.
Answer: given planes are parallel.
The necessary and sufficient condition for the planes to be parallel can be described in another way.
Theorem 5
For two non-coincident planes α and β to be parallel to each other, it is necessary and sufficient that the normal vectors of the planes α and β are collinear.
The proof of the formulated condition is based on the definition of the normal vector of the plane.
Assume that n 1 → = (A 1 , B 1 , C 1) and n 2 → = (A 2 , B 2 , C 2) are the normal vectors of the planes α and β, respectively. Let's write the condition of collinearity of these vectors:
n 1 → = t n 2 ⇀ ⇔ A 1 = t A 2 B 1 = t B 2 C 1 = t C 2, where t is some real number.
Thus, for non-coincident planes α and β with the normal vectors given above to be parallel, it is necessary and sufficient that a real number t takes place, for which the equality is true:
n 1 → = t n 2 ⇀ ⇔ A 1 = t A 2 B 1 = t B 2 C 1 = t C 2
Example 2
Planes α and β are given in a rectangular coordinate system of three-dimensional space. The plane α passes through the points: A (0 , 1 , 0) , B (- 3 , 1 , 1) , C (- 2 , 2 , - 2) . The plane β is described by the equation x 12 + y 3 2 + z 4 = 1 It is necessary to prove the parallelism of the given planes.
Decision
Let's make sure that the given planes do not coincide. Indeed, it is, since the coordinates of the point A do not correspond to the equation of the plane β.
The next step is to determine the coordinates of the normal vectors n 1 → and n 2 → corresponding to the planes α and β . We also check the condition of collinearity of these vectors.
The vector n 1 → can be specified by taking the cross product of vectors A B → and A C → . Their coordinates are respectively: (- 3 , 0 , 1) and (- 2 , 2 , - 2) . Then:
n 1 → = A B → × A C → = i → j → k → - 3 0 1 - 2 1 - 2 = - i → - 8 j → - 3 k → ⇔ n 1 → = (- 1 , - 8 , - 3)
To obtain the coordinates of the normal vector of the plane x 12 + y 3 2 + z 4 = 1, we reduce this equation to the general equation of the plane:
x 12 + y 3 2 + z 4 = 1 ⇔ 1 12 x + 2 3 y + 1 4 z - 1 = 0
Thus: n 2 → = 1 12 , 2 3 , 1 4 .
Let's check whether the condition of collinarity of vectors n 1 → = (- 1 , - 8 , - 3) and n 2 → = 1 12 , 2 3 , 1 4
Since - 1 \u003d t 1 12 - 8 \u003d t 2 3 - 3 \u003d t 1 4 ⇔ t \u003d - 12, then the vectors n 1 → and n 2 → are related by the equality n 1 → = - 12 n 2 → , i.e. are collinear.
Answer: planes α and β do not coincide; their normal vectors are collinear. Thus, the planes α and β are parallel.
If you notice a mistake in the text, please highlight it and press Ctrl+Enter
Lecture number 4.
Deviations in the shape and location of surfaces.
GOST 2.308-79
When analyzing the accuracy of the geometric parameters of parts, nominal and real surfaces, profiles are distinguished; nominal and real arrangement of surfaces and profiles. Nominal surfaces, profiles and surface arrangements are determined by nominal dimensions: linear and angular.
Real surfaces, profiles and surface arrangements are obtained as a result of manufacturing. They always have deviations from the nominal.
Form tolerances.
The basis for the formation and quantitative assessment of deviations in the shape of surfaces is adjoining principle.
adjoining element, this is an element in contact with the real surface and located outside the material of the part, so that the distance from it at the most remote point of the real surface within the normalized area would have a minimum value.
An adjacent element can be: a straight line, a plane, a circle, a cylinder, etc. (Fig. 1, 2).
1 - adjacent element;
2 - real surface;
L is the length of the normalized section;
Δ - shape deviation, determined from the adjacent element along the normal to the surface.
T - shape tolerance.
Fig 2. Fig. one
Tolerance field- an area in space bounded by two equidistant surfaces spaced from one another at a distance equal to the tolerance T, which is deposited from the adjacent element into the body of the part.
The quantitative deviation of the shape is estimated by the greatest distance from the points of the real surface (profile) to the adjacent surface (profile) along the normal to the latter (Fig. 2). Adjacent surfaces are: working surfaces of working plates, interference glasses, curved rulers, calibers, control mandrels, etc.
Form tolerance is called the largest permissible deviation Δ (Fig. 2).
Deviations in the shape of surfaces.
1. Deviation from straightness in the plane is the maximum from the points of the real profile to the adjacent straight line. (Fig. 3a).
Rice. 3
Designation on the drawing:
Straightness tolerance 0.1mm on base length 200mm
2. Flatness tolerance- this is the largest allowable distance () from the points of the real surface to the adjacent plane within the normalized area (Fig. 3b).
Designation on the drawing:
Flatness tolerance (not more than) 0.02 mm on the base surface 200 100 mm.
Control methods.
Measuring flatness with a rotary plane meter.
Figure 5a.
Fig 5b. Scheme for measuring non-flatness.
Control in scheme 6b
carried out in the light or
with a probe
(error 1-3μm)
Figure 6. Schemes for measuring non-straightness.
Flatness control is carried out:
By the method "On the paint" by the number of spots in the frame size 25 25mm
With the help of interference plates (for finished surfaces up to 120mm) (Fig. 7).
When a plate is applied with a slight inclination to the surface of a rectangular part to be checked, interference fringes appear, and interference rings appear on the surface of a round part.
When observed in white light, the distance between the fringes is in= 0.3 µm (half the wavelength of white light).
|
micron 
Straightness tolerance axes cylinder 0.01mm (shape tolerance arrow rests on the size arrow 20f 7). (Figure 8)
Measurement scheme
Surface straightness tolerances are set on guides; flatness - for flat end surfaces to ensure tightness (planes of parting of body parts); operating at high pressures (end distributors), etc.
Axis straightness tolerances - for long cylindrical surfaces (like rods) moving in a horizontal direction; cylindrical guides; for parts assembled with mating surfaces on several surfaces.
Tolerances and deviations of the shape of cylindrical surfaces.
1. roundness tolerance- the most permissible deviation from roundness, the largest distance i from the points of the real surface to the adjacent circle.
Tolerance field- an area bounded by two concentric circles on a plane perpendicular to the axis of the surface of revolution.
Surface roundness tolerance 0.01mm.
Round meters
Figure 9. Schemes for measuring the deviation from roundness.
Particular types of deviations from roundness are ovality and cutting (Fig. 10).
Ovality Cut
For different cuts, the indicator head is set at an angle (Fig. 9b).
2. Cylindricity tolerances- this is the largest permissible deviation of the real profile from the adjacent cylinder.
It consists of the deviation from roundness (measured at least at three points) and the deviation from the straightness of the axis.
3. Longitudinal section profile tolerance- this is the largest permissible deviation of the profile or shape of the real surface from the adjacent profile or surface (specified by the drawing) in a plane passing through the axis of the surface.
Longitudinal section profile tolerance 0.02mm.
Particular types of deviation of the profile of the longitudinal section:
Taper Barrel Saddle
| |
Fig. 11. Deviation of the profile of the longitudinal section a, b, c, d and measurement scheme e.
Tolerances of roundness and profile of the longitudinal section are set to ensure uniform clearance in individual sections and along the entire length of the part, for example, in plain bearings, for parts of a piston-cylinder pair, for spool pairs; cylindricity for surfaces that require complete contact of parts (connected by fits with interference and transitional), as well as for parts of great length such as "rods".
Location tolerances
Location tolerances- these are the largest allowable deviations of the actual location of the surface (profile), axis, plane of symmetry from its nominal location.
When evaluating deviations in the location, deviations in the form (considered surfaces and base ones) should be excluded from consideration (Fig. 12). In this case, real surfaces are replaced by adjacent ones, and axes, symmetry planes are taken as axes, symmetry planes and centers of adjacent elements.
Plane parallelism tolerances- this is the largest allowable difference between the largest and smallest distances between adjacent planes within the normalized area.
To normalize and measure tolerances and deviations of the location, base surfaces, axes, planes, etc. are introduced. These are surfaces, planes, axes, etc., which determine the position of the part during assembly (product operation) and relative to which the position of the elements under consideration is set. Basic elements on
the drawing are indicated by the sign ; capital letters of the Russian alphabet are used.
The designation of bases, sections (A-A) should not be duplicated. If the base is an axis or a plane of symmetry, the sign is placed on the continuation of the dimension line:
Parallelism tolerance 0.01mm relative to the base
surfaces A.
Surface alignment tolerance in
diametrically 0.02mm
relative to the base axis of the surface
In the event that the design, technological (determining the position of the part during manufacture) or measuring (determining the position of the part during measurement) do not match, you should recalculate the measurements performed.
Measurement of deviations from parallel planes.
(at two points on a given surface length)
The deviation is defined as the difference between the readings of the head at a given interval from each other (the heads are set to "0" according to the standard).
Tolerance of parallelism of the hole axis relative to the reference plane A on the length L.
Figure 14. (Measurement scheme)
Axis parallelism tolerance.
Deviation from parallelism of axes in space- the geometric sum of deviations from parallelism of the projections of the axes in two mutually perpendicular planes. One of these planes is a common plane of the axes (that is, it passes through one axis and a point on the other axis). Deviation from parallelism in the common plane- deviation from parallelism of the projections of the axes on their common plane. Axes misalignment- deviation from the projections of the axes onto a plane perpendicular to the common plane of the axes and passing through one of the axes.
Tolerance field- this is a rectangular parallelepiped with the sides of the section - , the side faces are parallel to the base axis. or cylinder
Fig 15. Measurement scheme 
Tolerance of parallelism of the axis of the hole 20H7 relative to the axis of the hole 30H7.
Alignment tolerance.
Deviation from coaxiality relative to a common axis is the greatest distance between the axis of the considered surface of revolution and the common axis of two or more surfaces.
Concentricity tolerance field is an area in space bounded by a cylinder whose diameter is equal to the alignment tolerance in diametric terms ( F = T) or twice the alignment tolerance in radial terms: R=T/2(Fig. 16)
Alignment tolerance in radial expression of surfaces and relative to the common axis of the holes A.
Figure 16. Alignment tolerance field and measurement scheme
(axis deviation relative to the base axis A-eccentricity); R-radius of the first hole (R+e) – distance to the base axis in the first measurement position; (R-e) - distance to the base axis in the second position after turning the part or indicator 180 degrees.
The indicator registers the difference in readings (R+e)-(R-e)=2e=2 - deviation from alignment in diametric terms.
Tolerance of coaxiality of the necks of the shaft in diametric terms 0.02 mm (20 μm) relative to the common axis of the AB. Shafts of this type are installed (based) on rolling or sliding bearings. The base is the axis passing through the middle of the shaft journals (hidden base).
Figure 17. Scheme of misalignment of the shaft journals.
The displacement of the axes of the shaft journals leads to a misalignment of the shaft and a violation of the performance of the entire product as a whole.
Figure 18. Scheme for measuring the misalignment of the shaft journals
The basing is made on knife supports, which are placed in the middle sections of the shaft necks. When measuring, the deviation is obtained in the diametric expression D Æ = 2e.
The misalignment relative to the base surface is usually determined by measuring the runout of the surface being checked in a given section or extreme sections - when the part rotates around the base surface. The result of the measurement depends on the non-circularity of the surface (which is about 4 times less than the misalignment).
Figure 19. Scheme for measuring the alignment of two holes
Accuracy depends on the accuracy of the fit of the mandrels to the hole.
Dependent tolerance can be measured using a gauge (Fig. 20).
Tolerance of surface alignment relative to the base axis of the surface in diametric terms 0.02 mm, dependent tolerance.
Symmetry tolerance
Symmetry tolerance with respect to the reference plane- the largest allowable distance between the considered plane of symmetry of the surface and the base plane of symmetry.
Figure 21. Symmetry tolerances, measurement schemes
The tolerance of symmetry in the radius expression is 0.01 mm relative to the base plane of symmetry A (Fig. 21b).
Deviation DR(in radius expression) is equal to the half-difference of distances A and B.
In diametric terms DT \u003d 2e \u003d A-B.
Alignment and symmetry tolerances are assigned to those surfaces that are responsible for the exact assembly and functioning of the product, where significant displacements of the axes and symmetry planes are not allowed.
Axis intersection tolerance.
Axis crossing tolerance- the largest allowable distance between the considered and the base axes. It is defined for axes that, in the nominal arrangement, must intersect. The tolerance is specified in a diametrical or radius expression (Fig. 22a).
Location tolerances- these are the largest allowable deviations of the actual location of the surface (profile), axis, plane of symmetry from its nominal location.
When evaluating deviations shape deviation locations (considered surfaces and base ones) should be excluded from consideration (Fig. 12). In this case, real surfaces are replaced by adjacent ones, and axes, symmetry planes are taken as axes, symmetry planes and centers of adjacent elements.
Plane parallelism tolerances- this is the largest allowable difference between the largest and smallest distances between adjacent planes within the normalized area.
For standardization and measurement tolerances and location deviations, base surfaces, axes, planes, etc. are introduced. These are surfaces, planes, axes, etc., which determine the position of the part during assembly (product operation) and relative to which the position of the elements under consideration is set. Basic elements in the drawing are indicated by the sign; capital letters of the Russian alphabet are used. The designation of bases, sections (A-A) should not be duplicated. If the base is an axis or a plane of symmetry, the sign is placed on the continuation of the dimension line:
Parallelism tolerance 0.01mm relative to the base
surfaces A.
Surface alignment tolerance in
diametrically 0.02mm
relative to the base axis of the surface
In the event that the design, technological (determining the position of the part during manufacture) or measuring (determining the position of the part during measurement) do not match, recalculate the measurements performed.
Measurement of deviations from parallel planes.
(at two points on a given surface length)
The deviation is defined as the difference between the readings of the head at a given interval from each other (the heads are set to "0" according to the standard).
Tolerance of parallelism of the hole axis relative to the reference plane A on the length L.
Figure 14. (Measurement scheme)
Axis parallelism tolerance.
Deviation from parallelism of axes in space - the geometric sum of deviations from parallelism of the projections of the axes in two mutually perpendicular planes. One of these planes is a common plane of the axes (that is, it passes through one axis and a point on the other axis). Deviation from parallelism in the common plane- deviation from parallelism of the projections of the axes on their common plane. Axes misalignment- deviation from the projections of the axes onto a plane perpendicular to the common plane of the axes and passing through one of the axes.
Tolerance field- This rectangular parallelepiped with section sides -, side faces are parallel to the base axis. or cylinder
Fig 15. Measurement scheme
Tolerance of parallelism of the axis of the hole 20H7 relative to the axis of the hole 30H7.
Alignment tolerance.
Misalignment relative to a common axis is the greatest distance between the axis of the considered surface of revolution and the common axis of two or more surfaces.
Concentricity tolerance field is an area in space bounded by a cylinder whose diameter is equal to the alignment tolerance in diametric terms ( F = T) or twice the alignment tolerance in radial terms: R=T/2(Fig. 16)
Alignment tolerance in radial expression of surfaces and relative to the common axis of the holes A.
Figure 16. Alignment tolerance field and measurement scheme
(axis deviation relative to the base axis A-eccentricity); R-radius of the first hole (R+e) - distance to the base axis in the first measurement position; (R-e) - distance to the base axis in the second position after turning the part or indicator 180 degrees.
The indicator registers the difference in readings (R+e)-(R-e)=2e=2 - deviation from alignment in diametric terms.
Shaft journal alignment tolerance in diametric terms, 0.02 mm (20 µm) relative to the common axis of the AB. Shafts of this type are installed (based) on rolling or sliding bearings. The base is the axis passing through the middle of the shaft journals (hidden base).
Figure 17. Scheme of misalignment of the shaft journals.
The displacement of the axes of the shaft journals leads to a misalignment of the shaft and a violation of the performance of the entire product as a whole.
Figure 18. Scheme for measuring the misalignment of the shaft journals
The basing is made on knife supports, which are placed in the middle sections of the shaft necks. When measuring, the deviation is obtained in the diametric expression D Æ = 2e.
Misalignment relative to the base surface is usually determined by measuring the runout of the surface being checked in a given section or extreme sections - when the part rotates around the base surface. The result of the measurement depends on the non-circularity of the surface (which is approximately 4 times less than the misalignment).
Figure 19. Scheme for measuring the alignment of two holes
Accuracy depends on the accuracy of the fit of the mandrels to the hole.
Rice. 20.
Dependent tolerance can be measured using a gauge (Fig. 20).
Tolerance of surface alignment relative to the base axis of the surface in diametric terms 0.02 mm, dependent tolerance.
Symmetry tolerance
Symmetry tolerance relative to the reference plane- the largest allowable distance between the considered plane of symmetry of the surface and the base plane of symmetry.
Figure 21. Symmetry tolerances, measurement schemes
The tolerance of symmetry in the radius expression is 0.01 mm relative to the base plane of symmetry A (Fig. 21b).
Deviation DR(in radius expression) is equal to the half-difference of distances A and B.
In diametric terms DT \u003d 2e \u003d A-B.
Alignment and symmetry tolerances are assigned to those surfaces that are responsible for the exact assembly and functioning of the product, where significant displacements of the axes and symmetry planes are not allowed.
Axis intersection tolerance.
Axis crossing tolerance - the largest allowable distance between the considered and the reference axes. It is defined for axes that, in the nominal arrangement, must intersect. The tolerance is specified in a diametrical or radius expression (Fig. 22a).
Figure 22. a)
The tolerance of the intersection of the axes of the holes Æ40H7 and Æ50H7 in radius terms is 0.02mm (20µm).
Fig. 22. b, c Scheme for measuring the deviation of the intersection of the axes
The mandrel is placed in 1 hole, measured R1- height (radius) above the axis.
The mandrel is placed in the 2nd hole, measured R2.
Measurement result DR = R1 - R2 is obtained in a radius expression, if the hole radii are different, to measure the deviation of the location, you need to subtract the actual dimensions and (or take into account the dimensions of the mandrels. The mandrel fits over the hole, contact by fit)
DR = R1 - R2- ( - ) - deviation is obtained in radius expression
Axes intersection tolerance is assigned to parts where failure to comply with this requirement leads to a violation of performance, for example: a bevel gear housing.
Perpendicularity tolerance
Perpendicularity tolerance for a surface relative to the reference surface.
Tolerance of perpendicularity of the side surface is 0.02 mm relative to the reference plane A. Squareness deviation is the deviation of the angle between the planes from the right angle (90°), expressed in linear units D along the length of the normalized section L.
Figure 23. Scheme for measuring perpendicularity deviation
The measurement can be carried out with several indicators set to "0" according to the standard.
Tolerance of perpendicularity of the hole axis relative to the surface in diametric terms 0.01 mm at the measurement radius R = 40 mm.
Figure 24. Scheme for measuring the deviation of the perpendicularity of the axis
The perpendicularity tolerance is assigned on the surface that determines the function of the product. For example: to ensure a uniform gap or a snug fit along the ends of the product, the perpendicularity of the axes and the plane of technological devices, the perpendicularity of the guides, etc.
Tilt tolerance
Deviation of the slope of the plane - the deviation of the angle between the plane and the base from the nominal angle a, expressed in linear units D over the length of the normalized section L.
To measure the deviation, templates and fixtures are used.
Position tolerance
Position tolerance- this is the largest permissible deviation of the actual location of the element, axis, plane of symmetry from its nominal position
Control can be carried out through the control of its individual elements, with the help of measuring machines, with - calibers.
Positional tolerance is assigned to the location of the centers of holes for fasteners, spheres of connecting rods, etc.
Total shape and location tolerances
Total flatness and parallelism tolerance
Assigned to flat surfaces that determine the position of the part (based) and provide a snug fit (tightness).
Total flatness and perpendicularity tolerance.
Assigned to flat side surfaces that determine the position of the part (based) and provide a snug fit.
Radial runout tolerance
Radial runout tolerance is the largest allowable difference between the largest and smallest distances from all points of the real surface of revolution to the base axis in a section perpendicular to the base axis.
Full radial runout tolerance.
Figure 26.
Tolerance of full radial runout within the normalized area.
radial runout is the sum of deviations from roundness and coaxiality in diametric terms, - the sum of deviations from cylindricity and coaxiality.
Tolerance of radial and full radial runout is assigned to critical rotating surfaces, where the requirement for the alignment of parts dominates, separate control of shape tolerances is not required. .
Runout tolerance
End runout tolerance is the largest allowable difference between the largest and smallest distances from points on any circle of the end surface to a plane perpendicular to the base axis. The deviation is made up of
deviations from perpendicularity and straightness (fluctuations of the surface of the circle).
Full run-out tolerance
Full end runout tolerance - this is the largest allowable difference between the largest and smallest distances from points of the entire end surface to a plane perpendicular to the base axis.
End runout tolerances are set on the surfaces of rotating parts that require minimal runout and impact on the parts in contact with them; for example: thrust surfaces for rolling bearings, plain bearings, gear wheels.
Tolerance of the shape of a given profile, a given surface
Tolerance of the shape of a given profile, the tolerance of the shape of a given surface - these are the largest deviations of the profile or shape of the real surface from the adjacent profile and surface specified by the drawing.
Tolerances are set on parts that have curved surfaces such as cams, templates; barrel profiles, etc.
Normalization of shape and location tolerances
Can be carried out:
by levels of relative geometric accuracy;
Based on the worst conditions of assembly or operation;
Based on the results of the calculation of dimensional chains.
Levels of relative geometric accuracy.
According to GOST 24643-81, 16 degrees of accuracy are established for each type of shape and location tolerance. The numerical values of the tolerances in the transition from one degree of accuracy to another change with an increase factor of 1.6.
Depending on the ratio between the size tolerance and the shape and location tolerance, there are 3 levels of relative geometric accuracy:
A - normal: set to 60% of tolerance T
B - increased - set 40%
C - high - 25%
For cylindrical surfaces:
Level A » 30% of T
Level B » 20% of T
By level C » 12.5% of T
Since the tolerance of the shape of the cylindrical surface limits the deviation of the radius, not the entire diameter.
For example: Æ 45 +0.062 in A:
In the drawings, the shape and location tolerance is indicated when they should be less than the size tolerances.
If there is no indication, then they are limited to the tolerance of the size itself.
Designations on the drawings
Shape and location tolerances are indicated in rectangular boxes; in the first part of which - a conventional sign, in the second - numerical values in mm; for location tolerances, the base is indicated in the third part.
The direction of the arrow is normal to the surface. The measurement length is indicated through the fraction sign "/". If it is not specified, control is carried out over the entire surface.
For location tolerances that determine the relative positions of surfaces, it is allowed not to specify the base surface:
It is allowed to indicate the base surface, axis, without designation with a letter:
Before the numerical value of the tolerance, the symbol T, Æ, R, sphere,
if the tolerance field is given in diametrical and radius terms, the sphere Æ, R will be used for ; (hole axis); .
If the sign is not specified, the tolerance is specified in the diametrical expression.
To allow symmetry, use the signs T (instead of Æ) or (instead of R).
Dependent tolerance, indicated by the sign.
After the tolerance value, a symbol can be indicated, and on the part this symbol indicates the area relative to which the deviation is determined.
Rationing of shape and location tolerances from the worst assembly conditions.
Consider a part that contacts simultaneously on several surfaces - a rod.
In that case, if there is a large misalignment between the axes of all three surfaces, the assembly of the product will be difficult. Let's take the worst option for assembly - the minimum gap in the connection.
Let's take for the base axis - the axis of the connection.
Then the axis offset .
In diametric terms, this is 0.025 mm.
If the base is the axis of the center holes, then proceeding from similar considerations.
Example 2
Let us consider a stepped shaft contacting on two surfaces, one of which is working, the second is subject only to the requirements of collection.
For the worst conditions for assembling parts: and.
Assume that the sleeve and shaft parts are perfectly aligned: In the presence of gaps and perfectly aligned parts, the gaps are distributed evenly on both sides and .
The figure shows that the parts will be assembled even if the axes of the steps are shifted relative to each other by an amount.
For and , i.e. allowable displacement of the axes in radius terms. = e = 0.625mm, or = 2e = 0.125mm - in diametric terms.
Example 3
Consider the bolted connection of parts, when gaps are formed between each of the parts to be joined and the bolt (type A), while the gaps are located in opposite directions. The axis of the hole in part 1 is shifted from the axis of the bolt to the left, and the axis of part 2 is shifted to the right.
Holes for fasteners are performed with tolerance fields H12 or H14 in accordance with GOST 11284-75. For example, holes can be used under M10 (for precise connections) and mm (for non-critical connections). With a linear clearance Offset of the axes in diametric terms, the value of the positional tolerance = 0.5 mm, i.e. is equal to =.
Example 4
Consider the screw connection of parts, when the gap is formed only between one of the parts and the screw: (type B)
In practice, accuracy margin factors are introduced: k
Where k \u003d 0.8 ... 1, if the assembly is carried out without adjusting the position of the parts;
k \u003d 0.6 ... 0.8 (for studs k \u003d 0.4) - during adjustment.
Example 5
Two flat precision end surfaces are in contact, S=0.005mm. It is required to normalize the flatness tolerance. In the presence of end gaps due to non-flatness (the slopes of the parts are selected using springs), leakage of the working fluid or gas occurs, which reduces the volumetric efficiency of the machines.
The deviation value for each of the parts is defined as half =. Can be rounded up to integer values \u003d 0.003 mm, because the probability of worse combinations is rather negligible.
Rationing of location tolerances based on dimensional chains.
Example 6
It is required to normalize the alignment tolerance of the mounting axis 1 of the technological device, for which the tolerance of the entire device is set = 0.01.
Note: the tolerance of the entire fixture should not exceed 0.3 ... 0.5 of the tolerance of the product.
Consider the factors affecting the alignment of the entire fixture as a whole:
Misalignment of part surfaces 1;
Maximum clearance in the connection of parts 1 and 2;
Misalignment of the hole in 2 parts and the base (mounted in the machine) surface.
Because small chain of sizes (3 links) is used to calculate the method of complete interchangeability; according to which the tolerance of the closing link is equal to the sum of the tolerances of the constituent links.
The alignment tolerance of the entire fixture is equal to
To eliminate the influence when connecting 1 and 2 parts, you should take a transitional fit or an interference fit.
If accepted, then
The value is achieved in the operation of fine grinding. If the fixture has small dimensions, then it can be provided with assembly processing.
Example 7
Dimensioning with a ladder and a chain for holes for fasteners.
If the dimensions are elongated under one line, a chain is made.
.TL D 1 = TL 1 + TL 2
TL D 2 = TL 2 + TL 3
TL D 3 = TL 3 + TL 4, i.e.
The accuracy of the master link is always affected by only 2 links.
If a TL 1 = TL 2 =
For our example TL 1 = TL 2 = 0.5 (±0.25mm)
This setting allows you to increase the tolerances of the constituent links, reduce the complexity of processing.
Example 9
Calculation of the value of the dependent tolerance.
If for example 2 are indicated, then this means that the 0.125 mm alignment tolerance, determined for the worst assembly conditions, can be increased if the gaps formed in the connection are greater than the minimum.
For example, in the manufacture of the part, dimensions of -39.95 mm; - 59.85 mm were obtained, additional gaps arise S add1 = d 1max - d 1izg = 39.975 - 39.95 = 0.025mm, and S add2 = d 2max - d 2izg = 59, 9 - 59.85 \u003d 0.05 mm, the axes can additionally be shifted relative to each other by e add \u003d e 1 dop + e 2 dop \u003d (in diametric terms, by S 1 dop + S 2 dop \u003d 0.075 mm).
The misalignment in diametrical terms, taking into account additional clearances, will be: = 0.125 + S add1 + S add2 = 0.125 + 0.075 = 0.2 mm.
Example 10
You want to define a dependent alignment tolerance for a sleeve part.
Symbol: hole alignment tolerance Æ40H7 relative to the base axis Æ60p6, tolerance dependent only on hole dimensions.
Note: the dependence is indicated only on those surfaces where additional clearances are formed in the fits, for surfaces connected by fits with an interference fit or transition - additional axle slips are excluded.
During manufacture, the following dimensions were obtained: Æ40.02 and Æ60.04
T head \u003d 0.025 + S 1dop \u003d 0.025 + (D bend1 - D min1) \u003d 0.025 + (40.02 - 40) \u003d 0.045 mm(in diametric terms)
Example 11.
Determine the value of the center-to-center distance for the part, if the dimensions of the holes after manufacturing are equal: D 1izg \u003d 10.55 mm; D 2izg \u003d 10.6 mm.
For the first hole
T zav1 \u003d 0.5 + (D 1izg - D 1min) \u003d 0.5 + (10.55 - 10.5) \u003d 0.55 mm or ± 0.275 mm
For the second hole
T head2 \u003d 0.5 + (D 2bend - D 2min) \u003d 0.5 + (10.6 - 10.5) \u003d 0.6 mm or ± 0.3 mm
Deviations at the center distance.
