Construction of axonometric projections

5.5.1. General provisions. Orthogonal projections of an object give a complete picture of its shape and size. However, the obvious disadvantage of such images is their low visibility - the figurative form is composed of several images made on different projection planes. Only as a result of experience does the ability to imagine the shape of an object develop—“read drawings.”

Difficulties in reading images in orthogonal projections led to the emergence of another method, which was supposed to combine the simplicity and accuracy of orthogonal projections with the clarity of the image - the method of axonometric projections.

Axonometric projection is a visual image obtained as a result of parallel projection of an object along with the axes of rectangular coordinates to which it is related in space onto any plane.

The rules for performing axonometric projections are established by GOST 2.317-69.

Axonometry (from the Greek axon - axis, metreo - measure) is a construction process based on reproducing the dimensions of an object in the directions of its three axes - length, width, height. The result is a three-dimensional image that is perceived as a tangible thing (Fig. 56b), in contrast to several flat images that do not give a figurative form of the object (Fig. 56a).

Rice. 56. Visual representation of axonometry

In practical work, axonometric images are used for various purposes, so various types of them have been created. What is common to all types of axonometry is that one or another arrangement of axes is taken as the basis for the image of any object. OX, OY, OZ, in the direction of which the dimensions of an object are determined - length, width, height.

Depending on the direction of the projecting rays in relation to the picture plane, axonometric projections are divided into:

A) rectangular– projecting rays are perpendicular to the picture plane (Fig. 57a);

b) oblique– the projecting rays are inclined to the picture plane (Fig. 57b).

Rice. 57. Rectangular and oblique axonometry

Depending on the position of the object and the coordinate axes relative to the projection planes, as well as depending on the direction of projection, units of measurement are generally projected with distortion. The sizes of projected objects are also distorted.

The ratio of the length of an axonometric unit to its true value is called coefficient distortion for a given axis.

Axonometric projections are called: isometric, if the distortion coefficients on all axes are equal ( x=y=z); dimetric, if the distortion coefficients are equal along two axes( x=z);trimetric, if the distortion coefficients are different.

For axonometric images of objects, five types of axonometric projections established by GOST 2.317 - 69 are used:

rectangular – isometric And dimetric;

oblique– frontal dimetric, frontalisometric, horizontal isometric.

Having orthogonal projections of any object, you can build its axonometric image.

It is always necessary to choose from all types the best view of a given image - the one that provides good clarity and ease of constructing axonometry.

5.5.2. General order of construction. The general procedure for constructing any type of axonometry comes down to the following:

a) select coordinate axes on the orthogonal projection of the part;

b) construct these axes in an axonometric projection;

c) build an axonometry of the complete image of the object, and then its elements;

d) draw the contours of the section of the part and remove the image of the cut-off part;

d) circle the remaining part and put down the dimensions.

5.5.3. Rectangular isometric projection. This type of axonometric projection is widespread due to the good clarity of the images and the simplicity of construction. In rectangular isometry, axonometric axes OX, OY, OZ located at angles of 120 0 to one another. Axis OZ vertical. Axles OX And OY It is convenient to build by setting aside angles of 30 0 from the horizontal using a square. The position of the axes can also be determined by setting aside five arbitrary equal units from the origin in both directions. Through the fifth divisions, vertical lines are drawn down and 3 of the same units are laid on them. The actual distortion coefficients along the axes are 0.82. To simplify the construction, a reduced coefficient of 1 is used. In this case, when constructing axonometric images, measurements of objects parallel to the directions of the axonometric axes are laid aside without abbreviations. The location of the axonometric axes and the construction of a rectangular isometry of a cube, into the visible faces of which circles are inscribed, are shown in Fig. 58, a, b.

Rice. 58. Location of axes of rectangular isometry

Circles inscribed in the rectangular isometry of squares - the three visible faces of the cube - are ellipses. The major axis of the ellipse is 1.22 D, and small – 0.71 D, Where D– diameter of the depicted circle. The major axes of the ellipses are perpendicular to the corresponding axonometric axes, and the minor axes coincide with these axes and with the direction perpendicular to the plane of the cube face (thickened strokes in Fig. 58b).

When constructing a rectangular axonometry of circles lying in coordinate or parallel planes, they are guided by the rule: The major axis of the ellipse is perpendicular to the coordinate axis that is absent in the plane of the circle.

Knowing the dimensions of the ellipse axes and the projections of diameters parallel to the coordinate axes, you can construct an ellipse from all points, connecting them using a pattern.

The construction of an oval using four points - the ends of the conjugate diameters of the ellipse, located on the axonometric axes, is shown in Fig. 59.

Rice. 59. Constructing an oval

Through the point ABOUT the intersection of the conjugate diameters of the ellipse draw horizontal and vertical lines and from it describe a circle with a radius equal to half the conjugate diameters AB=SD. This circle will intersect the vertical line at points 1 And 2 (centers of two arcs). From points 1, 2 draw arcs of circles with radius R=2-A (2-D) or R=1-C (1-B). Radius OE make notches on the horizontal line and get two more centers of mating arcs 3 And 4 . Next, connect the centers 1 And 2 with centers 3 And 4 lines that intersect with arcs of radius R give junction points K, N, P, M. The extreme arcs are drawn from the centers 3 And 4 radius R 1 =3-M (4-N).

The construction of a rectangular isometry of a part, specified by its projections, is carried out in the following order (Fig. 60, 61).

1. Select coordinate axes X, Y, Z on orthogonal projections.

2. Construct axonometric axes in isometry.

3. Build the base of the part - a parallelepiped. To do this, from the origin along the axis X lay down the segments OA And OB, respectively equal to the segments O 1 A 1 And About 1 In 1, taken from the horizontal projection of the part, and get the points A And IN, through which straight lines parallel to the axes are drawn Y, and lay down segments equal to half the width of the parallelepiped.

Get points C, D, J, V, which are isometric projections of the vertices of the lower rectangle, and connect them with straight lines parallel to the axis X. From the origin ABOUT along the axis Z set aside a segment OO 1, equal to the height of the parallelepiped O 2 O 2´; through the point O 1 draw axes X 1, Y 1 and construct an isometry of the upper rectangle. The vertices of the rectangles are connected by straight lines parallel to the axis Z.

4. Construct an axonometry of the cylinder. Axis Z from O 1 set aside a segment O 1 O 2, equal to the segment О 2 ´О 2 ´´, i.e. height of the cylinder, and through the point O 2 draw axes X 2,Y2. The upper and lower bases of the cylinder are circles located in horizontal planes X 1 O 1 Y 1 And X 2 O 2 Y 2; construct their axonometric images - ellipses. The outlines of the cylinder are drawn tangentially to both ellipses (parallel to the axis Z). The construction of ellipses for a cylindrical hole is carried out similarly.

5. Construct an isometric image of the stiffener. From point O 1 along the axis X 1 set aside a segment O 1 E=O 1 E 1. Through the point E draw a straight line parallel to the axis Y, and lay on both sides segments equal to half the width of the edge E 1 K 1 And E 1 F 1. From the obtained points K, E, F parallel to the axis X 1 draw straight lines until they meet an ellipse (points P, N, M). Next, draw straight lines parallel to the axes Z(the lines of intersection of the rib planes with the surface of the cylinder), and segments are laid on them RT, MQ And N.S., equal to the segments R 2 T 2, M 2 Q 2, And N 2 S 2. Points Q, S, T connect and trace along the pattern, and the points K, T And F,Q connected by straight lines.

6. Construct a cutout of a part of a given part, for which two cutting planes are drawn: one through the axes Z And X, and the other – through the axes Z And Y.

The first cutting plane will cut the lower rectangle of the parallelepiped along the axis X(line segment OA), top – along the axis X 1, and the edge – along the lines EN And ES, cylinders - along the generatrices, the upper base of the cylinder - along the axis X 2.

Similarly, the second cutting plane will cut the upper and lower rectangles along the axes Y And Y 1, and the cylinders - along the generatrices, the upper base of the cylinder - along the axis Y2.

The flat figures obtained from the section are shaded. To determine the direction of hatching, it is necessary to plot equal segments on the axonometric axes from the origin of coordinates, and then connect their ends.

Rice. 60. Construction of three projections of a part

Rice. 61. Performing rectangular isometry of a part

Hatch lines for a section located in a plane XOZ, will be parallel to the segment 1-2 , and for a section lying in the plane ZOY, – parallel to the segment 2-3 . Remove all invisible lines and trace the contour lines. Isometric projection is used in cases where it is necessary to construct circles in two or three planes parallel to the coordinate axes.

5.5.4. Rectangular dimetric projection. Axonometric images constructed with rectangular dimensions have the best clarity, but constructing images is more difficult than in isometry. The location of the axonometric axes in dimetry is as follows: axis OZ is directed vertically, and the axes OH And OY are made up with a horizontal line drawn through the origin of coordinates (point ABOUT), the angles are 7º10´ and 41º25´, respectively. The position of the axes can also be determined by laying eight equal segments from the origin in both directions; Through the eighth divisions, lines are drawn down and one segment is laid on the left vertical, and seven segments on the right. By connecting the obtained points with the origin of coordinates, the direction of the axes is determined OH And OU(Fig. 62).

Rice. 62. Arrangement of axes in rectangular diameter

Axis distortion coefficients OH, OZ are equal to 0.94, and along the axis OY– 0.47. To simplify in practice, the following distortion coefficients are used: along the axes OX And OZ the coefficient is equal to 1, along the axis OY– 0,5.

The construction of a rectangular cube with circles inscribed in its three visible faces is shown in Fig. 62b. Circles inscribed in faces are two types of ellipses. Axes of an ellipse located on a face that is parallel to the coordinate plane XOZ, are equal: major axis – 1.06 D; small – 0.94 D, Where D– the diameter of a circle inscribed in the face of a cube. In the other two ellipses the major axes are 1.06 D, and small ones - 0.35 D.

To simplify constructions, you can replace ellipses with ovals. In Fig. 63 provides techniques for constructing four center ovals that replace ellipses. An oval in the front face of a cube (rhombus) is constructed as follows. Perpendiculars are drawn from the middle of each side of the rhombus (Fig. 63a) until they intersect with the diagonals. Received points 1-2-3-4 will be the centers of the connecting arcs. The junction points of the arcs are located in the middle of the sides of the rhombus. The construction can be done in another way. From the midpoints of the vertical sides (points N And M) draw horizontal straight lines until they intersect with the diagonals of the rhombus. The intersection points will be the desired centers. From the centers 4 And 2 draw arcs with a radius R, and from the centers 3 And 1 – radius R 1.

Rice. 63. Constructing a circle in rectangular dimensions

An oval replacing the other two ellipses is made as follows (Fig. 63b). Direct LP And MN drawn through the midpoints of opposite sides of a parallelogram intersect at a point S. Through the point S draw horizontal and vertical lines. Direct LN, connecting the midpoints of adjacent sides of the parallelogram, is divided in half, and a perpendicular is drawn through its midpoint until it intersects the vertical line at the point 1 .

lay a segment on a vertical line S-2 = S-1.Direct 2-M And 1-N intersect a horizontal line at points 3 And 4 . Received points 1 , 2, 3 And 4 will be the centers of the oval. Direct 1-3 And 2-4 determine the junction points T And Q.

from centers 1 And 2 describe arcs of circles TLN And QPM, and from the centers 3 And 4 – arcs M.T. And NQ. The principle of constructing the rectangular dimetry of a part (Fig. 64) is similar to the principle of constructing the rectangular isometry shown in Fig. 61.

When choosing one or another type of rectangular axonometric projection, you should keep in mind that in rectangular isometry the rotation of the sides of the object is the same and therefore the image is sometimes not clear. In addition, often the diagonal edges of an object in the image merge into one line (Fig. 65b). These shortcomings are absent in images made in rectangular dimetry (Fig. 65c).

Rice. 64. Construction of a part in rectangular dimensions

Rice. 65. Comparison of different types of axonometry

5.5.5. Oblique frontal isometric projection.

The axonometric axes are located as follows. Axis OZ- vertical, axis OH– horizontal, axis OU relative to the horizontal line is located above an angle of 45 0 (30 0, 60 0) (Fig. 66a). On all axes, dimensions are plotted without abbreviations, in true size. In Fig. Figure 66b shows the frontal isometry of the cube.

Rice. 66. Construction of oblique frontal isometry

Circles located in planes parallel to the frontal plane are depicted in full size. Circles located in planes parallel to the horizontal and profile planes are depicted as ellipses.

Rice. 67. Detail in oblique frontal isometry

The direction of the ellipse axes coincides with the diagonals of the cube faces. For planes XOY And ZОY the major axis is 1.3 D, and small – 0.54 D (D– diameter of the circle).

An example of frontal isometry of a part is shown in Fig. 67.

To perform an isometric projection of any part, you need to know the rules for constructing isometric projections of flat and three-dimensional geometric shapes.

Rules for constructing isometric projections of geometric figures. The construction of any flat figure should begin with drawing the axes of isometric projections.

When constructing an isometric projection of a square (Fig. 109), from point O along the axonometric axes, half the length of the side of the square is laid out in both directions. Straight lines parallel to the axes are drawn through the resulting notches.

When constructing an isometric projection of a triangle (Fig. 110), segments equal to half the side of the triangle are laid along the X axis from point 0 in both directions. The height of the triangle is plotted along the Y axis from point O. Connect the resulting serifs with straight segments.

Rice. 109. Rectangular and isometric projections of a square

Rice. 110. Rectangular and isometric projections of a triangle

When constructing an isometric projection of a hexagon (Fig. 111), from point O the radius of the circumscribed circle is plotted (in both directions) along one of the axes, and H/2 along the other. Straight lines parallel to one of the axes are drawn through the resulting serifs, and the length of the side of the hexagon is plotted on them. Connect the resulting serifs with straight segments.

Rice. 111. Rectangular and isometric projections of a hexagon

Rice. 112. Rectangular and isometric projections of a circle

When constructing an isometric projection of a circle (Fig. 112), segments equal to its radius are laid out along the coordinate axes from point O. Straight lines parallel to the axes are drawn through the resulting serifs, obtaining an axonometric projection of the square. From vertices 1, 3 arcs CD and KL are drawn with a radius of 3C. Connect points 2 with 4, 3 with C and 3 with D. At the intersections of straight lines, centers a and b of small arcs are obtained, drawing which produces an oval, replacing the axonometric projection of a circle.

Using the described constructions, it is possible to perform axonometric projections of simple geometric bodies (Table 10).

10. Isometric projections of simple geometric bodies

Methods for constructing an isometric projection of a part:

1. The method of constructing an isometric projection of a part from a forming face is used for parts whose shape has a flat face, called a forming face; The width (thickness) of the part is the same throughout; there are no grooves, holes or other elements on the side surfaces. The sequence of constructing an isometric projection is as follows:

1) construction of isometric projection axes;

2) construction of an isometric projection of the formative face;

3) constructing projections of the remaining faces by depicting the edges of the model;

Rice. 113. Construction of an isometric projection of a part, starting from the formative face

4) outline of the isometric projection (Fig. 113).

1. The method of constructing an isometric projection based on the sequential removal of volumes is used in cases where the displayed form is obtained as a result of removing any volumes from the original form (Fig. 114).
2. The method of constructing an isometric projection based on sequential increment (adding) of volumes is used to create an isometric image of a part, the shape of which is obtained from several volumes connected in a certain way to each other (Fig. 115).
3. A combined method for constructing an isometric projection. An isometric projection of a part, the shape of which is obtained as a result of a combination of various shaping methods, is performed using a combined construction method (Fig. 116).

An axonometric projection of a part can be performed with an image (Fig. 117, a) and without an image (Fig. 117, b) of invisible parts of the form.

Rice. 114. Construction of an isometric projection of a part based on sequential removal of volumes

Rice. 115 Construction of an isometric projection of a part based on sequential increments of volumes

Rice. 116. Using a combined method of constructing an isometric projection of a part

Rice. 117. Options for depicting isometric projections of a part: a - with the image of invisible parts;
b - without images of invisible parts

The standard establishes the following views obtained on the main projection planes (Fig. 1.2): front view (main), top view, left view, right view, bottom view, rear view.

The main view is taken to be the one that gives the most complete idea of ​​the shape and size of the object.

The number of images should be the smallest, but providing a complete picture of the shape and size of the item.

If the main views are located in a projection relationship, then their names are not indicated. For the best use of the drawing field, views can be placed outside the projection connection (Fig. 2.2). In this case, the image of the view is accompanied by a type designation:

1) the direction of view is indicated

2) above the image of the view a designation is applied A, as in Fig. 2.1.

Types are designated in capital letters of the Russian alphabet in a font 1...2 sizes larger than the font of the dimensional numbers.

Figure 2.1 shows a part that requires four views. If these views are placed in a projection relationship, then they will take up a lot of space on the drawing field. You can arrange the necessary views as shown in Fig. 2.1. The drawing format is reduced, but the projection relationship is broken, so you need to designate the view on the right ().

2.2. Local species.

A local view is an image of a separate limited area of ​​the surface of an object.

It can be limited by the cliff line (Fig. 2.3 a) or not limited (Fig. 2.3 b).

In general, local species are designed in the same way as the main species.

2.3. Additional types.

If any part of an object cannot be shown in the main views without distorting the shape and size, then additional views are used.

An additional view is an image of the visible part of the surface of an object, obtained on a plane not parallel to any of the main projection planes.

If an additional view is performed in projection connection with the corresponding image (Fig. 2.4 a), then it is not designated.

If the image of an additional type is placed in free space (Fig. 2.4 b), i.e. If the projection connection is broken, then the direction of view is indicated by an arrow located perpendicular to the depicted part of the part and is indicated by a letter of the Russian alphabet, and the letter remains parallel to the main inscription of the drawing and does not turn behind the arrow.

If necessary, the image of an additional type can be rotated, then a letter and a rotation sign are placed above the image (this is a 5...6 mm circle with an arrow, between the wings of which there is an angle of 90°) (Fig. 2.4 c).

An additional type is most often performed as a local one.

3.Cuts.

A cut is an image of an object mentally dissected by one or more planes. The section shows what lies in the secant plane and what is located behind it.

In this case, the part of the object located between the observer and the cutting plane is mentally removed, as a result of which all surfaces covered by this part become visible.

3.1. Construction of sections.

Figure 3.1 shows three types of objects (without a cut). In the main view, the internal surfaces: a rectangular groove and a cylindrical stepped hole are shown with dashed lines.

In Fig. 3.2 shows a section obtained as follows.

Using a secant plane parallel to the frontal plane of projections, the object was mentally dissected along its axis passing through a rectangular groove and a cylindrical stepped hole located in the center of the object. Then the front half of the object, located between the observer and the secant plane, was mentally removed. Since the object is symmetrical, there is no point in giving a full cut. It is performed on the right, and the left view is left.

The view and the section are separated by a dash-dotted line. The section shows what happened in the cutting plane and what is behind it.

When examining the drawing you will notice the following:

1) the dashed lines, which in the main view indicate a rectangular groove and a cylindrical stepped hole, are outlined in the section with solid main lines, since they became visible as a result of mental dissection of the object;

2) in the section, the solid main line running along the main view, indicating the cut, has disappeared altogether, since the front half of the object is not depicted. The section located on the depicted half of the object is not marked, since it is not recommended to show invisible elements of the object with dashed lines on sections;

3) in the section, a flat figure located in the secant plane is highlighted by shading; shading is applied only in the place where the secant plane cuts the material of the object. For this reason, the back surface of the cylindrical stepped hole is not shaded, as well as the rectangular groove (when mentally dissecting the object, the cutting plane did not affect these surfaces);

4) when depicting a cylindrical stepped hole, a solid main line is drawn, depicting a horizontal plane formed by a change in diameters on the frontal plane of projections;

5) a section placed in the place of the main image does not change the images of the top and left views in any way.

When making cuts in drawings, you must follow the following rules:

1) make only useful cuts in the drawing (cuts chosen for reasons of necessity and sufficiency are called “useful”);

2) previously invisible internal outlines, depicted by dashed lines, should be outlined with solid main lines;

3) hatch the section figure included in the section;

4) mental dissection of an object should relate only to this cut and not affect the change in other images of the same object;

5) In all images, dashed lines are removed, since the internal contour is clearly readable in the section.

3.2 Designation of cuts

In order to know where the object has the shape shown in the cut image, the place where the cutting plane passed and the cut itself are indicated. The line indicating the cutting plane is called the cutting line. It is depicted as an open line.

In this case, select the initial letters of the alphabet ( A B C D E etc.). Above the section obtained using this cutting plane, an inscription is made according to the type A-A, i.e. two paired letters separated by a dash (Fig. 3.3).

Letters near section lines and letters indicating a section must be larger than the dimensional numbers in the same drawing (by one or two font numbers)

In cases where the cutting plane coincides with the plane of symmetry of a given object and the corresponding images are located on the same sheet in direct projection connection and are not separated by any other images, it is recommended not to mark the position of the cutting plane and not to accompany the cut image with an inscription.

Figure 3.3 shows a drawing of an object on which two cuts are made.

1. In the main view, the section is made by a plane, the location of which coincides with the plane of symmetry for a given object. It runs along the horizontal axis in the top view. Therefore this section is not marked.

2. Cutting plane A-A does not coincide with the plane of symmetry of this part, therefore the corresponding section is marked.

The letter designation of cutting planes and sections is placed parallel to the main inscription, regardless of the angle of inclination of the cutting plane.

3.3 Hatching materials in sections and sections.

In sections and sections, the figure obtained in the secant plane is hatched.

GOST 2.306-68 establishes graphic designations for various materials (Fig. 3.4)

Hatching for metals is applied in thin lines at an angle of 45° to the contour lines of the image, or to its axis, or to the lines of the drawing frame, and the distance between the lines should be the same.

The shading on all sections and sections for a given object is the same in direction and pitch (distance between strokes).

3.4. Classification of cuts.

Incisions have several classifications:

1. Classification, depending on the number of cutting planes;

2. Classification, depending on the position of the cutting plane relative to the projection planes;

3. Classification, depending on the position of the cutting planes relative to each other.

Rice. 3.5

3.4.1 Simple cuts

A simple cut is a cut made by one cutting plane.

The position of the cutting plane can be different: vertical, horizontal, inclined. It is chosen depending on the shape of the object whose internal structure needs to be shown.

Depending on the position of the cutting plane relative to the horizontal plane of projections, sections are divided into vertical, horizontal and inclined.

Vertical is a section with a cutting plane perpendicular to the horizontal plane of projections.

A vertically located cutting plane can be parallel to the frontal plane of projections or the profile, thus forming, respectively, frontal (Fig. 3.6) or profile sections (Fig. 3.7).

A horizontal section is a section with a secant plane parallel to the horizontal plane of projections (Fig. 3.8).

An inclined cut is a cut with a cutting plane that makes an angle with one of the main projection planes that is different from a straight line (Fig. 3.9).

1. Based on the axonometric image of the part and the given dimensions, draw three of its views - the main one, the top and the left. Do not redraw the visual image.

7.2. Task 2

2. Make the necessary cuts.

3. Construct lines of intersection of surfaces.

4. Draw dimension lines and enter size numbers.

5. Outline the drawing and fill in the title block.

7.3. Task 3

1. Draw the given two types of object according to size and construct a third type.

2. Make the necessary cuts.

3. Construct lines of intersection of surfaces.

4. Draw dimension lines and enter size numbers.

5. Outline the drawing and fill in the title block.

For all tasks, draw views only in projection connection.

7.1. Task 1.

Let's look at examples of completing tasks.

Problem 1. Based on the visual image, construct three types of parts and make the necessary cuts.

7.2 Problem 2

Problem 2. Using two views, construct a third view and make the necessary cuts.

Task 2. Stage III.

1. Make the necessary cuts. The number of cuts should be minimal, but sufficient to read the internal contour.

1. Cutting plane A opens internal coaxial surfaces. This plane is parallel to the frontal plane of projections, so the section A-A combined with the main view.

2. The view on the left shows a sectional view exposing a Æ32 cylindrical hole.

3. Dimensions are applied on those images where the surface is readable better, i.e. diameter, length, etc., for example Æ52 and length 114.

4. If possible, do not cross extension lines. If the main view is selected correctly, then the largest number of dimensions will be on the main view.

Check:

1. So that each element of the part has a sufficient number of dimensions.
2. So that all protrusions and holes are dimensioned to other elements of the part (size 55, 46, and 50).
3. Dimensions.
4. Outline the drawing, removing all the lines of the invisible contour. Fill out the title block.

7.3. Task 3.

Construct three types of parts and make the necessary cuts.

8. Information about surfaces.

Constructing lines belonging to surfaces.

Surfaces.

In order to construct lines of intersection of surfaces, you need to be able to construct not only surfaces, but also points located on them. This section covers the most commonly encountered surfaces.

8.1. Prism.

A triangular prism is specified (Fig. 8.1), truncated by a frontally projecting plane (2GPZ, 1 algorithm, module No. 3). S Ç L= t (1234)

Since the prism projects relatively P 1, then the horizontal projection of the intersection line is already in the drawing, it coincides with the main projection of the given prism.

Cutting plane projecting relative to P 2, which means that the frontal projection of the intersection line is in the drawing, it coincides with the frontal projection of this plane.

The profile projection of the intersection line is constructed using two specified projections.

8.2. Pyramid

A truncated trihedral pyramid is given Ф(S,АВС)(Fig.8.2).

This pyramid F intersected by planes S, D And G .

2 GPZ, 2 algorithm (Module No. 3).

F Ç S=123

S ^P 2 Þ S 2 = 1 2 2 2 3 2

1 1 2 1 3 1 And 1 3 2 3 3 3 F .

F Ç D=345

D ^P 2 Þ = 3 2 4 2 5 2

3 1 4 1 5 1 And 3 3 4 3 5 3 are built according to their belonging to the surface F .

F Ç G = 456

G SP 2 Þ Г 2 = 4 2 5 6

4 1 5 1 6 1 And 4 3 5 3 6 3 are built according to their belonging to the surface F .

8.3. Bodies bounded by surfaces of revolution.

Bodies of revolution are geometric figures bounded by surfaces of revolution (ball, ellipsoid of revolution, ring) or a surface of revolution and one or more planes (cone of revolution, cylinder of revolution, etc.). Images on projection planes parallel to the axis of rotation are limited by outline lines. These sketch lines are the boundary between the visible and invisible parts of geometric bodies. Therefore, when constructing projections of lines belonging to surfaces of revolution, it is necessary to construct points located on the outlines.

8.3.1. Rotation cylinder.

P 1, then the cylinder will be projected onto this plane in the form of a circle, and onto the other two projection planes in the form of rectangles, the width of which is equal to the diameter of this circle. Such a cylinder projects to P 1 .

If the axis of rotation is perpendicular P 2, then on P 2 it will be projected as a circle, and on P 1 And P 3 in the form of rectangles.

Similar reasoning for the position of the rotation axis perpendicular to P 3(Fig.8.3).

Cylinder F intersects with planes R, S, L And G(Fig.8.3).

2 GPZ, 1 algorithm (Module No. 3)

F ^P 3

R, S, L, G ^P 2

F Ç R = A(6 5 and )

F ^P 3 Þ Ф 3 = а 3 (6 3 =5 3 и = )

a 2 And a 1 are built according to their belonging to the surface F .

F Ç S = b (5 4 3 )

F Ç S = c (2 3 ) The reasoning is similar to the previous one.

F G = d (12 and

The problems in Figures 8.4, 8.5, 8.6 are solved similarly to the problem in Figure 8.3, since the cylinder

profile-projecting everywhere, and the holes are surfaces projecting relatively

P 1- 2GPZ, 1 algorithm (Module No. 3).

If both cylinders have the same diameters (Fig. 8.7), then their intersection lines will be two ellipses (Monge’s theorem, module No. 3). If the axes of rotation of these cylinders lie in a plane parallel to one of the projection planes, then the ellipses will be projected onto this plane in the form of intersecting line segments.

8.3.2. Cone of rotation

The problems in Figures 8.8, 8.9, 8.10, 8.11, 8.12 -2 GPZ (module No. 3) are solved using algorithm 2, since the surface of the cone cannot be projecting, and the cutting planes are always front-projecting.

Figure 8.13 shows a cone of rotation (body) intersected by two frontally projecting planes G And L. The intersection lines are constructed using algorithm 2.

In Figure 8.14, the surface of the cone of revolution intersects with the surface of the profile-projecting cylinder.

2 GPZ, 2 solution algorithm (module No. 3), that is, the profile projection of the intersection line is in the drawing, it coincides with the profile projection of the cylinder. The other two projections of the intersection line are constructed according to their belonging to the cone of rotation.

Fig.8.14

8.3.3. Sphere.

The surface of the sphere intersects with the plane and with all surfaces of revolution with it, along circles. If these circles are parallel to the projection planes, then they are projected onto them into a circle of natural size, and if they are not parallel, then in the form of an ellipse.

If the axes of rotation of the surfaces intersect and are parallel to one of the projection planes, then all intersection lines - circles - are projected onto this plane in the form of straight segments.

In Fig. 8.15 - sphere, G- plane, L- cylinder, F- frustum.

S Ç G = A- circle;

S Ç L=b- circle;

S Ç Ф =с- circle.

Since the axes of rotation of all intersecting surfaces are parallel P 2, then all intersection lines are circles on P 2 are projected onto line segments.

On P 1: circumference "A" is projected into the true value because it is parallel to it; circle "b" is projected onto a line segment, since it is parallel P 3; circle "With" is projected in the form of an ellipse, which is constructed according to its belonging to the sphere.

First the points are plotted 1, 7 And 4, which define the minor and major axes of the ellipse. Then builds a point 5 , as if lying on the equator of a sphere.

For other points (arbitrary), circles (parallels) are drawn on the surface of the sphere and, based on their affiliation, the horizontal projections of the points lying on them are determined.

9. Examples of completing tasks.

Task 4. Construct three types of parts with the necessary cuts and apply dimensions.

Task 5. Construct three types of parts and make the necessary cuts.

10.Axonometry

10.1. Brief theoretical information about axonometric projections

A complex drawing, composed of two or three projections, having the properties of reversibility, simplicity, etc., at the same time has a significant drawback: it lacks clarity. Therefore, wanting to give a more visual idea of ​​the subject, along with a comprehensive drawing, an axonometric drawing is provided, which is widely used in describing product designs, in operating manuals, in assembly diagrams, to explain drawings of machines, mechanisms and their parts.

Compare two images - an orthogonal drawing and an axonometric drawing of the same model. Which image is easier to read the form? Of course, in an axonometric image. (Fig. 10.1)

The essence of axonometric projection is that a geometric figure, together with the axes of rectangular coordinates to which it is assigned in space, is parallelly projected onto a certain projection plane, called the axonometric projection plane, or picture plane.

If plotted on the coordinate axes x,y And z line segment l (lx,ly,lz) and project onto the plane P ¢ , then we get axonometric axes and segments on them l"x, l"y, l"z(Fig. 10.2)

lx, ly, lz- natural scale.

l = lx = ly = lz

l"x, l"y, l"z- axonometric scales.

The resulting set of projections on P¢ is called axonometry.

The ratio of the length of axonometric scale segments to the length of natural scale segments is called the indicator or coefficient of distortion along the axes, which are designated Kx, Ky, Kz.

Types of axonometric images depend on:

1. From the direction of the projecting rays (they can be perpendicular P"- then the axonometry will be called orthogonal (rectangular) or located at an angle not equal to 90° - oblique axonometry).

2. From the position of the coordinate axes to the axonometric plane.

Three cases are possible here: when all three coordinate axes make some acute angles (equal and unequal) with the axonometric plane of projections and when one or two axes are parallel to it.

In the first case, only rectangular projection is used, (s ^P") in the second and third - only oblique projection (s P") .

If the coordinate axes OX, OY, OZ not parallel to the axonometric plane of projections P", then will they be projected onto it in life-size? Of course not. In general, the image of straight lines is always smaller than actual size.

Consider an orthogonal drawing of a point A and its axonometric image.

The position of a point is determined by three coordinates - X A, Y A, Z A, obtained by measuring the links of a natural broken line OA X - A X A 1 – A 1 A(Fig. 10.3).

A"- main axonometric projection of a point A ;

A- secondary projection of the point A(projection of the projection of a point).

Distortion coefficients along the axes X", Y" and Z" will be:

k x = ; k y = ; k y =

In orthogonal axonometry, these indicators are equal to the cosines of the angles of inclination of the coordinate axes to the axonometric plane, and therefore they are always less than one.

They are connected by the formula

k 2 x + k 2 y + k 2 z= 2 (I)

In oblique axonometry, distortion indicators are related by the formula

k x + k y + k z = 2+ctg a(III)

those. any of them can be less than, equal to or greater than one (here a is the angle of inclination of the projecting rays to the axonometric plane). Both formulas are a derivation from Polke's theorem.

Polke's theorem: the axonometric axes on the drawing plane (P¢) and the scales on them can be chosen completely arbitrarily.

(Hence, the axonometric system ( O" X" Y" Z") in the general case is determined by five independent parameters: three axonometric scales and two angles between the axonometric axes).

The angles of inclination of the natural coordinate axes to the axonometric plane of projections and the direction of projection can be chosen arbitrarily, therefore many types of orthogonal and oblique axonometries are possible.

They are divided into three groups:

1. All three distortion indicators are equal (k x = k y = k z). This type of axonometry is called isometric. 3k 2 =2; k= "0.82 - theoretical distortion coefficient. According to GOST 2.317-70, you can use K=1 - reduced distortion coefficient.

2. Any two indicators are equal (for example, kx=ky kz). This type of axonometry is called dimetry. k x = k z ; k y = 1/2k x 2 ; k x 2 +k z 2 + k y 2 /4 = 2; k = "0.94; k x = 0.94; ky = 0.47; kz = 0.94 - theoretical distortion coefficients. According to GOST 2.317-70, distortion coefficients can be given - k x =1; k y =0.5; k z =1.

3. 3. All three indicators are different (k x ¹ k y ¹ k z). This type of axonometry is called trimetry .

In practice, several types of both rectangular and oblique axonometry are used with the simplest relationships between distortion indicators.

From GOST 2.317-70 and various types of axonometric projections, we will consider orthogonal isometry and dimetry, as well as oblique dimetry, as the most frequently used.

10.2.1. Rectangular isometry

In isometry, all axes are inclined to the axonometric plane at the same angle, therefore the angle between the axes (120°) and the distortion coefficient will be the same. Select scale 1: 0.82=1.22; M 1.22:1.

For ease of construction, the given coefficients are used, and then natural dimensions are plotted on all axes and lines parallel to them. The images thus become larger, but this does not affect the clarity.

The choice of axonometry type depends on the shape of the part being depicted. It is easiest to build rectangular isometry, which is why such images are more common. However, when depicting details that include quadrangular prisms and pyramids, their clarity decreases. In these cases, it is better to perform rectangular dimetry.

Oblique diameter should be chosen for parts that have a large length with a small height and width (such as a shaft) or when one of the sides of the part contains the largest number of important features.

Axonometric projections retain all the properties of parallel projections.

Consider the construction of a flat figure ABCDE .

First of all, let's construct the axes in axonometry. Figure 10.4 shows two ways to construct axonometric axes in isometry. In Fig. 10.4 A shows the construction of axes using a compass, and in Fig. 10.4 b- construction using equal segments.

Fig.10.5

Figure ABCDE lies in the horizontal projection plane, which is limited by the axes OH And OY(Fig. 10.5a). We construct this figure in axonometry (Fig. 10.5b).

How many coordinates does each point lying in the projection plane have? Two.

A point lying in the horizontal plane - coordinates X And Y .

Let's consider the construction t.A. From what coordinate will we start the construction? From coordinates X A .

To do this, measure the value on the orthogonal drawing OA X and put it on the axis X", we get a point A X " . A X A 1 Which axis is parallel? Axles Y. So from t. A X " draw a straight line parallel to the axis Y" and plot the coordinate on it Y A. Received point A" and will be an axonometric projection t.A .

All other points are constructed similarly. Dot WITH lies on the axis OY, which means it has one coordinate.

Figure 10.6 shows a pentagonal pyramid whose base is the same pentagon ABCDE. What needs to be completed to make a pyramid? We need to complete the point S, which is its top.

Dot S- a point in space, therefore it has three coordinates X S, Y S and Z S. First, a secondary projection is constructed S (S 1), and then all three dimensions are transferred from the orthogonal drawing. Connecting S" c A", B", C", D" And E", we obtain an axonometric image of a three-dimensional figure - a pyramid.

10.2.2. Circle isometry

Circles are projected onto a life-size projection plane when they are parallel to that plane. And since all planes are inclined to the axonometric plane, the circles lying on them will be projected onto this plane in the form of ellipses. In all types of axonometry, ellipses are replaced by ovals.

When depicting ovals, you must first of all pay attention to the construction of the major and minor axis. You need to start by determining the position of the minor axis, and the major axis is always perpendicular to it.

There is a rule: the minor axis coincides with the perpendicular to this plane, and the major axis is perpendicular to it, or the direction of the minor axis coincides with an axis that does not exist in this plane, and the major axis is perpendicular to it (Fig. 10.7)

The major axis of the ellipse is perpendicular to the coordinate axis that is absent in the plane of the circle.

The major axis of the ellipse is 1.22 ´ d env; the minor axis of the ellipse is 0.71 ´ d env.

In Figure 10.8 there is no axis in the plane of the circle Z Z ".

In Figure 10.9 there is no axis in the plane of the circle X, so the major axis is perpendicular to the axis X ".

Now let’s look at how an oval is drawn in one of the planes, for example, in the horizontal plane XY. There are many ways to construct an oval, let's get acquainted with one of them.

The sequence of constructing the oval is as follows (Fig. 10.10):

1. The position of the minor and major axis is determined.

2.Through the intersection point of the minor and major axis we draw lines parallel to the axes X" And Y" .

3.On these lines, as well as on the minor axis, from the center with a radius equal to the radius of a given circle, plot the points 1 And 2, 3 And 4, 5 And 6 .

4. Connecting the dots 3 And 5, 4 And 6 and mark the points of their intersection with the major axis of the ellipse ( 01 And 02 ). From point 5 , radius 5-3 , and from the point 6 , radius 6-4 , draw arcs between points 3 And 2 and dots 4 And 1 .

5. Radius 01-3 draw an arc connecting the points 3 And 1 and radius 02-4 - points 2 And 4 . Ovals are constructed similarly in other planes (Fig. 10.11).

To simplify the construction of a visual image of the surface, the axis Z may coincide with the height of the surface, and the axis X And Y with axes of horizontal projection.

To plot a point A, belonging to the surface, we need to construct its three coordinates X A , Y A And Z A. A point on the surface of a cylinder and other surfaces is constructed similarly (Fig. 10.13).

The major axis of the oval is perpendicular to the axis Y ".

When constructing an axonometry of a part limited by several surfaces, the following sequence should be followed:

Option 1.

1. The part is mentally broken down into elementary geometric shapes.

2. The axonometry of each surface is drawn, the construction lines are saved.

3. A 1/4 cutout of the part is created to show the internal configuration of the part.

4. Hatching is applied in accordance with GOST 2.317-70.

Let's consider an example of constructing an axonometry of a part, the outer contour of which consists of several prisms, and inside the part there are cylindrical holes of different diameters.

Option 2. (Fig. 10.5)

1. A secondary projection of the part is constructed on the projection plane P.

2. The heights of all points are plotted.

3. A cutout of 1/4 of the part is constructed.

4. Hatching is applied.

For this part, option 1 will be more convenient for construction.

10.3. Stages of making a visual representation of a part.

1. The part fits into the surface of a quadrangular prism, the dimensions of which are equal to the overall dimensions of the part. This surface is called the wrapping surface.

An isometric image of this surface is performed. The wrapping surface is built according to overall dimensions (Fig. 10.15 A).

Rice. 10.15 A

2. Protrusions are cut out from this surface, located on the top of the part along the axis X and a prism 34 mm high is built, one of the bases of which will be the upper plane of the wrapping surface (Fig. 10.15 b).

Rice. 10.15 b

3. From the remaining prism, cut out a lower prism with a base of 45 ´35 and a height of 11 mm (Fig. 10.15 V).

Rice. 10.15 V

4. Two cylindrical holes are constructed, the axes of which lie on the axis Z. The upper base of the large cylinder lies on the upper base of the part, the second one is 26 mm lower. The lower base of the large cylinder and the upper base of the small one lie in the same plane. The lower base of the small cylinder is built on the lower base of the part (Fig. 10.15 G).

Rice. 10.15 G

5. A 1/4 part of the part is cut out to reveal its internal contour. The cut is made by two mutually perpendicular planes, that is, along the axes X And Y(Fig. 10.15 d).

Fig.10.15 d

6. The sections and the entire remaining part of the part are outlined, and the cut out part is removed. Invisible lines are erased and sections are shaded. The hatching density should be the same as in the orthogonal drawing. The direction of the dashed lines is shown in Fig.10.15 e in accordance with GOST 2.317-69.

The hatch lines will be lines parallel to the diagonals of the squares lying in each coordinate plane, the sides of which are parallel to the axonometric axes.

Fig.10.15 e

7. There is a peculiarity of shading of the stiffener in axonometry. According to the rules

GOST 2.305-68 in a longitudinal section, the stiffener in the orthogonal drawing is not

shaded, and shaded in axonometry. Figure 10.16 shows an example

shading of the stiffener.

10.4 Rectangular dimetry.

A rectangular dimetric projection can be obtained by rotating and tilting the coordinate axes relative to P ¢ so that the distortion indicators along the axes X" And Z" took equal value, and along the axis Y"- half as much. Distortion indicators" k x" And " k z" will be equal to 0.94, and " k y "- 0,47.

In practice, the given indicators are used, i.e. along the axes X" And Z" lay down the natural dimensions, and along the axis Y"- 2 times less than natural ones.

Axis Z" usually positioned vertically, axis X"- at an angle of 7°10¢ to the horizontal line, and the axis Y"-at an angle of 41°25¢ to the same line (Fig. 12.17).

1. A secondary projection of the truncated pyramid is constructed.

2. The heights of the points are constructed 1,2,3 And 4.

The easiest way to build an axis X ¢ , placing 8 equal parts on a horizontal line and 1 equal part down a vertical line.

To build an axis Y" at an angle of 41°25¢, you need to put 8 parts on a horizontal line, and 7 of the same parts on a vertical line (Fig. 10.17).

Figure 10.18 shows a truncated quadrangular pyramid. To make it easier to construct it in axonometry, the axis Z must coincide with the height, then the tops of the base ABCD will lie on the axes X And Y (A and S Î X ,IN And D Î y). How many coordinates do points 1 and have? Two. Which? X And Z .

These coordinates are plotted in natural size. The resulting points 1¢ and 3¢ are connected to points A¢ and C¢.

Points 2 and 4 have two Z coordinates and Y. Since they have the same height, the coordinate Z is deposited on the axis Z". Through the received point 0 ¢ draw a line parallel to the axis Y, on which the distance is plotted on both sides of the point 0 1 4 1 reduced by half.

Received points 2 ¢ And 4 ¢ connect to dots IN ¢ And D" .

10.4.1. Constructing circles in rectangular dimensions.

Circles lying on coordinate planes in rectangular dimetry, as well as in isometry, will be depicted as ellipses. Ellipses located on planes between axes X" And Y",Y" And Z" in the reduced dimetry will have a major axis equal to 1.06d, and a minor axis equal to 0.35d, and in the plane between the axes X" And Z"- the major axis is also 1.06d, and the minor axis is 0.95d (Fig. 10.19).

Ellipses are replaced by four-cent ovals, as in isometry.

10.5. Oblique dimetric projection (frontal)

If we place the coordinate axes X And Y parallel to the P¢ plane, then the distortion indicators along these axes will become equal to one (k = t=1). Axis distortion index Y usually taken equal to 0.5. Axonometric axes X" And Z" make a right angle, axis Y" usually drawn as the bisector of this angle. Axis X can be directed either to the right of the axis Z", and to the left.

It is preferable to use the right-hand system, since it is more convenient to depict objects in dissected form. In this type of axonometry, it is good to draw parts that have the shape of a cylinder or cone.

For the convenience of depicting this part, the axis Y must be aligned with the axis of rotation of the cylinder surfaces. Then all circles will be depicted in natural size, and the length of each surface will be halved (Fig. 10.21).

11. Inclined sections.

When making drawings of machine parts, it is often necessary to use inclined sections.

When solving such problems, it is necessary first of all to understand: how the cutting plane should be located and which surfaces are involved in the section in order for the part to be read better. Let's look at examples.

Given a tetrahedral pyramid, which is dissected by an inclined frontally projecting plane A-A(Fig. 11.1). The cross section will be a quadrilateral.

First we construct its projections onto P 1 and on P 2. The frontal projection coincides with the projection of the plane, and we construct the horizontal projection of the quadrangle according to its membership in the pyramid.

Then we construct the natural size of the section. To do this, an additional projection plane is introduced P 4, parallel to a given cutting plane A-A, we project a quadrilateral onto it, and then combine it with the drawing plane.

This is the fourth main task of transforming a complex drawing (module No. 4, p. 15 or task No. 117 from the workbook on descriptive geometry).

Constructions are carried out in the following sequence (Fig. 11.2):

1. 1.On a free space in the drawing, draw a center line parallel to the plane A-A .

2. 2. From the points of intersection of the edges of the pyramid with the plane, we draw projecting rays perpendicular to the cutting plane. Points 1 And 3 will lie on a line perpendicular to the axial one.

3. 3.Distance between points 2 And 4 transferred from horizontal projection.

4. Similarly, the true size of the section of the surface of revolution is constructed - an ellipse.

Distance between points 1 And 5 -major axis of the ellipse. The minor axis of the ellipse must be constructed by dividing the major axis in half ( 3-3 ).

Distance between points 2-2, 3-3, 4-4 transferred from horizontal projection.

Let's consider a more complex example, including polyhedral surfaces and surfaces of revolution (Fig. 11.3)

A tetrahedral prism is specified. There are two holes in it: a prismatic one, located horizontally, and a cylindrical one, the axis of which coincides with the height of the prism.

The cutting plane is front-projecting, so the frontal projection of the section coincides with the projection of this plane.

A quadrangular prism projects to the horizontal plane of projections, which means the horizontal projection of the section is also in the drawing, it coincides with the horizontal projection of the prism.

The actual size of the section into which both prisms and the cylinder fall is constructed on a plane parallel to the cutting plane A-A(Fig. 11.3).

Sequence of performing an inclined section:

1. The section axis is drawn parallel to the cutting plane on the free field of the drawing.

2. A cross-section of the external prism is constructed: its length is transferred from the frontal projection, and the distance between the points from the horizontal one.

3. A cross section of the cylinder is constructed - part of the ellipse. First, characteristic points are constructed that determine the length of the minor and major axis ( 5 4 , 2 4 -2 4 ) and points limiting the ellipse (1 4 -1 4 ) , then additional points (4 4 -4 4 And 3 4 -3 4).

4. A cross section of the prismatic hole is constructed.

5. Hatching is applied at an angle of 45° to the main inscription, if it does not coincide with the contour lines, and if it does, then the hatching angle can be 30° or 60°. The hatching density on the section is the same as on the orthogonal drawing.

The inclined section can be rotated. In this case, the designation is accompanied by the sign. It is also allowed to show half of the inclined section figure if it is symmetrical. A similar arrangement of an inclined section is shown in Fig. 13.4. The designations of points when constructing an inclined section can be omitted.

Figure 11.5 shows a visual representation of a given figure with a section by plane A-A .

Control questions

1. What is a species called?

2. How do you get an image of an object on a plane?

3.What names are assigned to the views on the main projection planes?

4.What is called the main species?

5.What is called an additional view?

6. What is called a local species?

7.What is a cut called?

8. What designations and inscriptions are installed for sections?

9. What is the difference between simple cuts and complex ones?

10.What conventions are followed when making broken cuts?

11. Which incision is called local?

12. Under what conditions is it permissible to combine half the view and half the section?

13. What is called a section?

14. How are the sections arranged in the drawings?

15. What is called a remote element?

16. How are repeating elements shown in a drawing in a simplified manner?

17. How do you conventionally shorten the image of long objects in a drawing?

18. How do axonometric projections differ from orthogonal ones?

19. What is the principle of formation of axonometric projections?

20. What types of axonometric projections are established?

21. What are the features of isometry?

22. What are the features of dimetry?

Bibliography

1. Suvorov, S.G. Mechanical engineering drawing in questions and answers: (reference book) / S.G. Suvorov, N.S. Suvorova. - 2nd ed. reworked and additional - M.: Mechanical Engineering, 1992.-366 p.

2. Fedorenko V.A. Handbook of mechanical engineering drawing / V.A. Fedorenko, A.I. Shoshin, - Ed. 16-ster.; m Reprint. from the 14th edition 1981-M.: Alliance, 2007.-416 p.

3. Bogolyubov, S.K. Engineering graphics: Textbook for environments. specialist. textbook establishments for special purposes tech. profile/ S.K. Bogolyubov.-3rd ed., revised. and additional - M.: Mechanical Engineering, 2000.-351 p.

4. Vyshnepolsky, I.S. Technical drawing e. Textbook. for the beginning prof. education / I.S. Vyshnepolsky. - 4th ed., revised. and additional; Grif MO.- M.: Higher. school: Academy, 2000.-219 p.

5. Levitsky, V.S. Mechanical engineering drawing and automation of drawings: textbook. for colleges/V.S.Levitsky.-6th ed., revised. and additional; Grif MO.-M.: Higher. school, 2004.-435p.

6. Pavlova, A.A. Descriptive geometry: textbook. for universities/ A.A. Pavlova-2nd ed., revised. and additional; Grif MO.- M.: Vlados, 2005.-301 p.

7. GOST 2.305-68*. Images: views, sections, sections/Unified system of design documentation. - M.: Standards Publishing House, 1968.

8. GOST 2.307-68. Application of dimensions and maximum deviations/Unified system

design documentation. - M.: Standards Publishing House, 1968.

Axonometric views of machine parts and assemblies are often used in design documentation in order to clearly show the design features of a part (assembly) and to imagine what the part (assembly) looks like in space. Depending on the angle at which the coordinate axes are located, axonometric projections are divided into rectangular and oblique.

You will need

• Drawing program, pencil, paper, eraser, protractor.

Instructions

Rectangular projections. Isometric projection. When constructing a rectangular isometric projection, take into account the distortion coefficient along the X, Y, Z axes, equal to 0.82, while , parallel to the projection planes, are projected onto the axonometric projection planes in the form of ellipses, the axis of which is equal to d, and the axis is 0.58d, where d – diameter of the original circle. For ease of calculations, isometric projection without distortion along the axes (distortion coefficient is 1). In this case, the projected circles will look like ellipses with an axis equal to 1.22d and a minor axis equal to 0.71d.

Dimetric projection. When constructing a rectangular dimetric projection, the distortion coefficient along the X and Z axes is equal to 0.94, and along the Y axis – 0.47. To dimetric projection in a simplified manner, they are performed without distortion along the X and Z axes and with a distortion coefficient along the Y axis = 0.5. A circle parallel to the frontal projection plane is projected onto it in the form of an ellipse with a major axis equal to 1.06d and a minor axis equal to 0.95d, where d is the diameter of the original circle. Circles parallel to two other axonometric planes are projected onto them in the form of ellipses with axes equal to 1.06d and 0.35d, respectively.

Oblique projections. Frontal isometric view. When constructing a frontal isometric projection, the standard establishes the optimal angle of inclination of the Y axis to the horizontal at 45 degrees. Allowed angles of inclination of the Y axis to the horizontal are 30 and 60 degrees. The distortion coefficient along the X, Y and Z axes is 1. Circle 1, located on the frontal projection plane, is projected onto it without distortion. Circles parallel to the horizontal and profile planes of projections are made in the form of ellipses 2 and 3 with a major axis equal to 1.3d and a minor axis equal to 0.54d, where d is the diameter of the original circle.

Horizontal isometric projection. A horizontal isometric projection of a part (assembly) is built on axonometric axes located as shown in Fig. 7. It is allowed to change the angle between the Y axis and the horizontal by 45 and 60 degrees, leaving unchanged the angle of 90 degrees between the Y and X axes. The distortion coefficient along the X, Y, Z axes is 1. A circle lying in a plane parallel to the horizontal projection plane is projected as circle 2 without distortion. Circles parallel to the frontal and profile planes of projections, type of ellipses 1 and 3. The dimensions of the axes of the ellipses are related to the diameter d of the original circle by the following dependencies:
ellipse 1 – major axis is 1.37d, minor axis is 0.37d; ellipse 3 – major axis is 1.22d, minor axis is 0.71d.

Frontal dimetric projection. An oblique frontal dimetric projection of a part (assembly) is built on axonometric axes similar to the axes of the frontal isometric projection, but from it by a distortion coefficient along the Y axis, which is equal to 0.5. On the X and Z axes, the distortion coefficient is 1. It is also possible to change the angle of the Y axis to the horizontal to values ​​of 30 and 60 degrees. A circle lying in a plane parallel to the frontal axonometric plane of projections is projected onto it without distortion. Circles parallel to the planes of horizontal and profile projections are drawn in the form of ellipses 2 and 3. The dimensions of the ellipses on the size of the diameter of the circle d are expressed by the dependence:
the major axis of ellipses 2 and 3 is 1.07d; the minor axis of ellipses 2 and 3 is 0.33d.

Video on the topic

note

Axonometric projection (from ancient Greek ἄξων “axis” and ancient Greek μετρέω “I measure”) is a method of depicting geometric objects in a drawing using parallel projections.

Helpful advice

The plane onto which the projection is made is called axonometric or picture. An axonometric projection is called rectangular if, during parallel projection, the projecting rays are perpendicular to the picture plane (=90) and oblique if the rays make an angle of 0 with the picture plane

Sources:

• Handbook of Drawing
• axonometric projection of a circle

The image of an object in the drawing should give a complete idea of ​​its shape and design features and can be done using rectangular projection, linear perspective and axonometric projection.

Instructions

Remember that dimetry is one of the types of axonometric projection of an object, in which the image is rigidly tied to the natural Oxyz coordinate system. Dimetry in that two distortion coefficients along the axes are equal and different from the third. Dimetry rectangular and frontal.

With a rectangular diameter, the z axis is vertical, the x axis with a horizontal line is at an angle of 7011`, and the y angle is 410 25`. The reduced distortion coefficient along the y-axis is ky = 0.5 (real 0.47), kx = kz = 1 (real 0.94). GOST 2.317–69 recommends using only the given coefficients when constructing images in a rectangular dimetric projection.

To draw a rectangular dimetric projection, mark the vertical Oz axis on the drawing. To construct the x-axis, draw in the drawing a rectangle with legs 1 and 8 units, the vertex of which is point O. The hypotenuse of the rectangle will become the x-axis, which deviates from the horizon at an angle of 7011`. To construct the y-axis, also draw a right triangle with its vertex at point O. The size of the legs in this case is 7 and 8 units. The resulting hypotenuse will be the y-axis, deviating from the horizon at an angle of 410 25`.

When constructing a dimetric projection, the size of the object is increased by 1.06 times. In this case, the image is projected into an ellipse in the xOy and yO coordinate planes with a major axis equal to 1.06d, where d is the diameter of the projected circle. The minor axis of the ellipse is 0.35 d.

Video on the topic

note

Many industries use drawings. The rules for depicting objects and drawing up drawings are regulated by the “Unified System of Design Documentation” (ESKD).

To make any part, you need to design it and produce drawings. The drawing should show the main and auxiliary views of the part, which, if read correctly, provide all the necessary information about the shape and dimensions of the product.

Instructions

How, designing new parts, studying state and industry standards according to which design documentation is carried out. Find all GOSTs and OSTs that will be needed when drawing a part. To do this, you need standards numbers by which you can find them on the Internet in electronic form or in the enterprise archive in paper form.

Before you start drawing, select the required sheet on which it will be located. Consider the number of projections of the part that you need to depict in the drawing. For parts of simple shape (especially for bodies of revolution), the main view and one projection are sufficient. If the designed part has a complex shape, a large number of through and blind holes, grooves, then it is advisable to make several projections, as well as provide additional local views.

Draw the main view of the part. Choose the view that will give the most complete idea of ​​the shape of the part. Make other views if necessary. Draw cuts and sections showing the internal holes and grooves of the part.

Apply dimensions in accordance with GOST 2.307-68. Overall dimensions are better than the size of the part, so put these dimensions so that they can be easily identified on the drawing. Enter all dimensions with tolerances or indicate the quality according to which the part should be manufactured. Remember that in real life, produce a part with exact dimensions. There will always be a deviation upward or downward, which should be within the tolerance range for the size.

Be sure to indicate the surface roughness of the part in accordance with GOST 2.309-73. This is very important, especially for precision instrument-making parts that are part of assembly units and are connected by fit.

Write down the technical requirements for the part. Indicate its manufacture, processing, coating, operation and storage. In the title block of the drawing, do not forget to indicate the material from which the part is made.

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When designing and practically debugging power supply systems, it is necessary to use various schemes. Sometimes they are given in ready-made form, attached to the technical system, but in some cases you have to draw the diagram yourself, restoring it based on installation and connections. How accessible it will be to understand depends on the correct drawing of the diagram.

Instructions

Use the Visio computer program to draw a power supply diagram. For accumulation, you can first diagram an abstract supply circuit, including an arbitrary set of elements. In accordance with the standards and requirements of the unified design system, the principal design is drawn in a single-line image.

Select Page Options settings. In the “File” menu, use the appropriate command, and in the window that opens, set the required format for the future image, for example, A3 or A4. Also select portrait or landscape drawing orientation. Set the scale to 1:1, and the unit of measurement to millimeters. Complete your selection by clicking on the “OK” button.

Using the "Open" menu, find the stencil library. Open the set of main inscriptions and transfer the frame, inscription shape and additional columns to the sheet of the future drawing. Fill in the necessary columns that explain the diagram.

Draw the actual supply circuit diagram using stencils from the program, or use other blanks at your disposal. It is convenient to use a specially designed kit for drawing electrical diagrams of various power circuits.

Since many components of the power supply circuit of individual groups are often of the same type, draw similar ones by copying already drawn elements, and then make adjustments. In this case, select the elements of the group with the mouse and move the copied fragment to the desired place in the diagram.

Finally, move the input circuit components from the stencil set. Carefully fill out the explanatory notes for the diagram. Save the changes under the required name. If necessary, print the finished power supply diagram.

Constructing an isometric projection of a part allows you to obtain the most detailed understanding of the spatial characteristics of the image object. Isometry with a cutout of part of a part, in addition to the appearance, shows the internal structure of the object.

You will need

• - a set of drawing pencils;
• - ruler;
• - squares;
• - protractor;
• - compass;
• - eraser.

Instructions

Draw the axes with thin lines so that the image is located in the center of the sheet. In a rectangular isometry The angles between the axes are one hundred degrees. In a horizontal oblique isometry the angles between the X and Y axes are ninety degrees. And between the X and Z axes; Y and Z - one hundred thirty-five degrees.

Start from the top surface of the part being depicted. Draw vertical lines down from the corners of the horizontal surfaces and mark the corresponding linear dimensions from the part drawing on these lines. IN isometry linear dimensions along all three axes remain unity. Consistently connect the resulting points on vertical lines. The outer contour of the part is ready. Draw images of holes, grooves, etc. on the edges of the part.

Remember that when depicting objects in isometry the visibility of curved elements will be distorted. Circumference in isometry is depicted as an ellipse. Distance between ellipse points along axes isometry equal to the diameter of the circle, and the axes of the ellipse do not coincide with the axes isometry.

All actions must be performed using drawing tools - ruler, pencil, compass and protractor. Use several pencils of different hardnesses. Hard - for thin lines, hard - for dotted and dash-dotted lines, soft - for main lines. Do not forget to draw and fill out the main inscription and frame in accordance with GOST. Also construction isometry can be performed in specialized software such as Compass, AutoCAD.

Sources:

• isometric drawing

There are not many people these days who have never had to draw or draw something on paper in their lives. The ability to make a simple drawing of any design is sometimes very useful. You can spend a lot of time explaining “on your fingers” how this or that thing is made, while one glance at its drawing is enough to understand it without any words.

You will need

• – sheet of whatman paper;
• – drawing accessories;
• - drawing board.

Instructions

Select the sheet format on which the drawing will be drawn - in accordance with GOST 9327-60. The format should be such that the main information can be placed on the sheet kinds details in the appropriate scale, as well as all necessary cuts and sections. For simple parts, choose A4 (210x297 mm) or A3 (297x420 mm) format. The first can be positioned with its long side only vertically, the second - vertically and horizontally.

Draw a frame for the drawing, 20 mm from the left edge of the sheet, and 5 mm from the other three. Draw the main inscription - a table in which all data about details and drawing. Its dimensions are determined by GOST 2.108-68. The width of the main inscription remains unchanged - 185 mm, the height varies from 15 to 55 mm depending on the purpose of the drawing and the type of institution for which it is being carried out.

Select the main image scale. Possible scales are determined by GOST 2.302-68. They should be chosen so that all the main elements are clearly visible in the drawing. details. If at the same time some places are not visible clearly enough, they can be taken out as a separate view, shown with the necessary magnification.

Select main image details. It should represent the direction of view of the part (direction of projection) from which its design is most fully revealed. In most cases, the main image is the position in which the part is on the machine during the main operation. Parts that have an axis of rotation are located on the main image, as a rule, so that the axis has a horizontal position. The main image is located at the top left of the drawing (if there are three projections) or close to the center (if there is no side projection).

Determine the location of the remaining images (side view, top view, sections, sections). Kinds details are formed by its projection onto three or two mutually perpendicular planes (Monge's method). In this case, the part must be positioned in such a way that most or all of its elements are projected without distortion. If any of these types is informationally redundant, do not perform it. The drawing should have only those images that are necessary.

Select the cuts and sections to be made. Their difference from each other is that it also shows what is located behind the cutting plane, while the section displays only what is located in the plane itself. The cutting plane can be stepped or broken.

Proceed directly to drawing. When drawing lines, follow GOST 2.303-68, which defines kinds lines and their parameters. Place the images at such a distance from each other that there is enough space for dimensioning. If the cutting planes pass along the monolith details, hatch the sections with lines running at an angle of 45°. If the hatch lines coincide with the main lines of the image, you can draw them at an angle of 30° or 60°.

Draw dimension lines and mark down the dimensions. In doing so, be guided by the following rules. The distance from the first dimension line to the outline of the image must be at least 10 mm, the distance between adjacent dimension lines must be at least 7 mm. The arrows should be about 5 mm long. Write numbers in accordance with GOST 2.304-68, take their height to be 3.5-5 mm. Place the numbers closer to the middle of the dimension line (but not on the image axis) with some offset relative to the numbers placed on adjacent dimension lines.

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Sources:

• Electronic textbook on engineering graphics

The ratio of angles and planes of any object visually changes depending on the object’s position in space. That is why the part in the drawing is usually performed in three orthogonal projections, to which a spatial image is added. Usually this . When performing it, vanishing points are not used, as when constructing a frontal perspective. Therefore, the dimensions do not change as they move away from the observer.

You will need

• - ruler;
• - compass;
• - paper.

Instructions

Define the axes. To do this, draw a circle of arbitrary radius from point O. Its central angle is 360º. Divide the circle into 3 equal ones, using the OZ axis as the base radius. In this case, the angle of each sector will be equal to 120º. The two radii represent the OX and OY axes you need.

Determine the position. Divide the angles between the axes in half. Connect point O to these new points with thin lines. Center position circle depends on conditions. Mark it with a dot and draw a perpendicular to it in both directions. This line will determine the position of the large diameter.

Calculate the diameters. They depend on whether you apply a distortion factor or not. This coefficient for all axes is 0.82, but quite often it is rounded and taken as 1. Taking into account the distortion, the major and minor diameters of the ellipse are 1 and 0.58 of the original, respectively. Without applying the coefficient, these dimensions are 1.22 and 0.71 of the diameter of the original circle.

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note

To create a three-dimensional image, you can construct not only an isometric, but also a dimetric projection, as well as a frontal or linear perspective. Projections are used in drawing parts, while perspectives are used primarily in architecture. A circle in dimetry is also depicted as an ellipse, but there is a different arrangement of axes and different distortion coefficients. When performing various types of perspectives, changes in size with distance from the observer are taken into account.

For a visual representation of objects (products or their components), it is recommended to use axonometric projections, choosing the most suitable one in each individual case.

The essence of the axonometric projection method is that a given object, together with the coordinate system to which it is assigned in space, is projected onto a certain plane by a parallel beam of rays. The direction of projection onto the axonometric plane does not coincide with any of the coordinate axes and is not parallel to any of the coordinate planes.

All types of axonometric projections are characterized by two parameters: the direction of the axonometric axes and the distortion coefficients along these axes. The distortion coefficient is understood as the ratio of the image size in an axonometric projection to the image size in an orthogonal projection.

Depending on the ratio of distortion coefficients, axonometric projections are divided into:

Isometric, when all three distortion coefficients are the same (k x =k y =k z);

Dimetric, when the distortion coefficients are the same along two axes, and the third is not equal to them (k x = k z ≠k y);

Trimetric, when all three distortion coefficients are not equal to each other (k x ≠k y ≠k z).

Depending on the direction of the projecting rays, axonometric projections are divided into rectangular and oblique. If the projecting rays are perpendicular to the axonometric plane of projections, then such a projection is called rectangular. Rectangular axonometric projections include isometric and dimetric. If the projecting rays are directed at an angle to the axonometric plane of projections, then such a projection is called oblique. Oblique axonometric projections include frontal isometric, horizontal isometric and frontal dimetric projections.

In rectangular isometry, the angles between the axes are 120°. The actual coefficient of distortion along the axonometric axes is 0.82, but in practice, for ease of construction, the indicator is taken equal to 1. As a result, the axonometric image is enlarged by a factor of 1.

The isometric axes are shown in Figure 57.

Figure 57

The construction of isometric axes can be done using a compass (Figure 58). To do this, first draw a horizontal line and draw the Z axis perpendicular to it. From the point of intersection of the Z axis with the horizontal line (point O), draw an auxiliary circle with an arbitrary radius, which intersects the Z axis at point A. From point A, draw a second circle with the same radius to intersections with the first at points B and C. The resulting point B is connected to point O - the direction of the X axis is obtained. In the same way, point C is connected to point O - the direction of the Y axis is obtained.

Figure 58

The construction of an isometric projection of a hexagon is presented in Figure 59. To do this, it is necessary to plot the radius of the circumscribed circle of the hexagon on the X axis in both directions relative to the origin. Then, along the Y axis, set aside the size of the key, draw lines from the resulting points parallel to the X axis and set off along them the size of the side of the hexagon.

Figure 59

Constructing a circle in a rectangular isometric projection

The most difficult flat figure to draw in axonometry is a circle. As is known, a circle in isometry is projected into an ellipse, but constructing an ellipse is quite difficult, therefore GOST 2.317-69 recommends using ovals instead of ellipses. There are several ways to construct isometric ovals. Let's look at one of the most common ones.

The size of the major axis of the ellipse is 1.22d, minor 0.7d, where d is the diameter of the circle whose isometry is being constructed. Figure 60 shows a graphical method for determining the major and minor axes of an isometric ellipse. To determine the minor axis of the ellipse, points C and D are connected. From points C and D, as from centers, arcs of radii equal to CD are drawn until they intersect each other. Segment AB is the major axis of the ellipse.

Figure 60

Having established the direction of the major and minor axes of the oval depending on which coordinate plane the circle belongs to, two concentric circles are drawn along the dimensions of the major and minor axes, at the intersection of which with the axes points O 1, O 2, O 3, O 4 are marked, which are the centers oval arcs (Figure 61).

To determine the connecting points, draw center lines connecting O 1, O 2, O 3, O 4. from the resulting centers O 1, O 2, O 3, O 4, arcs of radii R and R 1 are drawn. the dimensions of the radii are visible in the drawing.

Figure 61

The direction of the axes of the ellipse or oval depends on the position of the projected circle. There is the following rule: the major axis of the ellipse is always perpendicular to the axonometric axis that is projected onto a given plane at a point, and the minor axis coincides with the direction of this axis (Figure 62).

Figure 62

Hatching and isometric projection

Hatch lines of sections in an isometric projection, according to GOST 2.317-69, must have a direction parallel either only to the large diagonals of the square, or only to the small ones.

Rectangular dimetry is an axonometric projection with equal distortion rates along the two axes X and Z, and along the Y axis the distortion rate is half as much.

According to GOST 2.317-69, in a rectangular diameter, the Z axis is used, located vertically, the X axis inclined at an angle of 7°, and the Y axis at an angle of 41° to the horizon line. The distortion indicators for the X and Z axes are 0.94, and for the Y axis - 0.47. Usually the given coefficients are used: k x =k z =1, k y =0.5, i.e. along the X and Z axes or in directions parallel to them, the actual dimensions are plotted, and along the Y axis the dimensions are halved.

To construct dimetric axes, use the method indicated in Figure 63, which is as follows:

On a horizontal line passing through point O, eight equal arbitrary segments are laid in both directions. From the end points of these segments, one similar segment is laid down vertically on the left, and seven on the right. The resulting points are connected to point O and the direction of the axonometric axes X and Y in rectangular dimetry is obtained.

Figure 63

Constructing a dimetric projection of a hexagon

Let's consider the construction in dimetry of a regular hexagon located in the plane P 1 (Figure 64).

Figure 64

On the X axis we plot a segment equal to the value b, to let him the middle was at point O, and along the Y axis there was a segment A, the size of which is halved. Through the obtained points 1 and 2 we draw straight lines parallel to the OX axis, on which we lay down segments equal to the side of the hexagon in full size with the middle at points 1 and 2. We connect the resulting vertices. Figure 65a shows a hexagon in dimetry, located parallel to the frontal plane, and in Figure 66b, parallel to the profile plane of projection.

Figure 65

Constructing a circle in dimetry

In rectangular dimetry, all circles are depicted as ellipses,

The length of the major axis for all ellipses is the same and equal to 1.06d. The magnitude of the minor axis is different: for the frontal plane it is 0.95d, for the horizontal and profile planes it is 0.35d.

In practice, the ellipse is replaced by a four-center oval. Let's consider the construction of an oval that replaces the projection of a circle lying in the horizontal and profile planes (Figure 66).

Through point O - the beginning of the axonometric axes, we draw two mutually perpendicular straight lines and plot on the horizontal line the value of the major axis AB = 1.06d, and on the vertical line the value of the minor axis CD = 0.35d. Up and down from O vertically we lay out the segments OO 1 and OO 2, equal in value to 1.06d. Points O 1 and O 2 are the center of the large oval arcs. To determine two more centers (O 3 and O 4), we lay off on a horizontal line from points A and B the segments AO 3 and BO 4, equal to ¼ of the minor axis of the ellipse, that is, d.

Figure 66

Then, from points O1 and O2 we draw arcs whose radius is equal to the distance to points C and D, and from points O3 and O4 - with a radius to points A and B (Figure 67).

Figure 67

We will consider the construction of an oval, replacing an ellipse, from a circle located in the P 2 plane in Figure 68. We draw the dimetric axes: X, Y, Z. The minor axis of the ellipse coincides with the direction of the Y axis, and the major one is perpendicular to it. On the X and Z axes, we plot the radius of the circle from the beginning and get points M, N, K, L, which are the conjugation points of the oval arcs. From points M and N we draw horizontal straight lines, which, at the intersection with the Y axis and the perpendicular to it, give points O 1, O 2, O 3, O 4 - the centers of the oval arcs (Figure 68).

From centers O 3 and O 4 they describe an arc of radius R 2 = O 3 M, and from centers O 1 and O 2 - arcs of radius R 1 = O 2 N

Figure 68

Hatching of rectangular diameter

The hatching lines of cuts and sections in axonometric projections are made parallel to one of the diagonals of the square, the sides of which are located in the corresponding planes parallel to the axonometric axes (Figure 69).

Figure 69

1. What types of axonometric projections do you know?
2. At what angle are the axes located in isometry?
3. What shape does the isometric projection of a circle represent?
4. How is the major axis of the ellipse located for a circle belonging to the profile plane of projections?
5. What are the accepted distortion coefficients along the X, Y, Z axes to construct a dimetric projection?
6. At what angles are the axes in dimetry located?
7. What figure will be the dimetric projection of the square?
8. How to construct a dimetric projection of a circle located in the frontal plane of the projections?
9. Basic rules for applying shading in axonometric projections.