Reversibility of the drawing, i.e. determination of a point in space by its projections can be determined by projection onto three projection planes. (Figure 2.1) Plane p 1 , is called horizontal, p 2 - frontal, p 3 – profile. The intersection lines of the projection planes form the coordinate axes (x, y, z). The point of intersection of the coordinate axes is taken as the origin of coordinates and is designated by the letter ABOUT. The positive direction of the coordinate axes is considered for the axis X- to the left of the origin, for the axis at- towards the observer from the plane p 2 , axis z- up from plane p 1 .
Let a point be given A in space (Figure 2.1). Point position A determined by three coordinates ( X, at, z), showing the distances at which the point is removed from the projection planes.
Figure 2.1
Points A¢, A¢¢, A¢¢¢ at which perpendicular lines drawn from this point intersect are called orthogonal projections of the point A.
A¢ – horizontal projection of the point A;
A¢¢ – frontal projection of the point A;
A¢¢¢ – profile projection of a point A.
Straight ( AA¢), ( AA¢¢), ( AA¢¢¢) are called projecting direct or projecting rays. In this case, the straight line ( AA¢) is called a horizontally projecting straight line, ( AA¢¢) – front projecting, ( AA¢¢¢) – a profile projecting straight line.
Two projection lines passing through a point A, form a plane, which is called projecting.
It is inconvenient to use the spatial layout shown in Figure 2.1 to display orthogonal projections of geometric figures due to its bulkiness, and also because the shape and size of the projected figure are distorted on the p 1 and p 3 planes. Therefore, instead of an image on a drawing of a spatial layout, they use a diagram, i.e. a drawing composed of two or more interconnected orthogonal projections of a geometric figure.
The transformation of the spatial layout into diagrams is carried out by combining the planes p 1 and p 3 with the frontal plane of projections p 2. To align plane p 1 with p 2, it is rotated 90° around the axis X clockwise, and to align the plane p 3 with p 2 it is rotated around the axis z counterclockwise (Figure 2.1). After the transformation, the spatial layout will take the form shown in Figure 2.2.
Since the planes do not have boundaries, then in the combined position (on the diagram) these boundaries are not shown, there is no need to leave inscriptions indicating the name of the projection planes. Then, in the final form of the diagrams, replacing the drawing of the spatial layout (Figure 2.1) will take the form shown in Figure 2.3.
On the diagram, straight lines perpendicular to the axes of the projections and connecting opposite projections of points are called projection connection lines. Note that the horizontal projection of a point A determined by abscissa X and ordinate at; its frontal projection is an abscissa X and fingering z, and the profile projection is the ordinate at and fingering z, i.e. A¢ ( X, at), A¢¢
(X, z), A¢¢¢
(y, z).
Figure 2.2 Figure 2.3
A point in a system of two projection planes.
To obtain projections of a point in a system of two projection planes, it is necessary to lower perpendiculars from a given point onto the corresponding projection planes; the bases of these perpendiculars will be the projections of the point on the corresponding projection planes.
Figure 7. Projections of a point in a system of two projection planes.
Point A' - projection onto the plane π 1 - is called the horizontal projection of point A. Point A'' - projection of point A onto the plane π 2 - frontal projection of point A. Similarly, the projection of point A onto the profile plane of projections (π 3 ) can be constructed. we obtain a profile projection of point A – A'''.
The segments AA’ and AA’’ are perpendicular to the projection planes π 1 and π 2, respectively, belonging to a certain plane α intersecting the projection axis at a certain point Ax. The α plane is perpendicular to the projection planes π 1 and π 2 and to the projection axis X, intersecting it at point Ax.
If the position of the planes π 1 and π 2 is fixed in space, then each point in space corresponds to an ordered pair of points on the projection planes. The converse statement is also true - an ordered pair of points on projection planes corresponds to a single point in space.
Projecting on two and three planes of projections
Epure Monge.
The considered image of a point in a system of two projection planes is not entirely convenient for drawing.
With the development of technology, the question of using a method that ensures the accuracy and convenience of images, that is, the ability to accurately determine the location of each point of the image relative to other points or planes and, using simple techniques, determine the sizes of segments of lines and figures, has become of paramount importance. Gradually, the accumulated individual rules and techniques for constructing such images were brought into a system and developed in the work of the French scientist Gaspard Monge, published in 1799 under the title “Geometrie descriptive”.
As noted earlier, the segments AA’ and AA’’ are perpendicular to the projection planes π 1 and π 2, respectively, belonging to a certain plane α intersecting the projection axis at a certain point Ax. The α plane is perpendicular to the projection planes π 1 and π 2 and to the projection axis X, intersecting it at point Ax.
The α plane intersects the projection planes π 1 and π 2 (segments A’Ax and A’’Ax). The segments A’Ax and A’’Ax are perpendicular to the projection axis X. The projections of a certain point are located on straight lines perpendicular to the projection axis and intersecting this axis at the same point (in our example, at the point
Gasprard Monge proposed a method for transforming a drawing by rotating the horizontal projection plane π 1 around the projection axis X until aligned with the frontal plane of projections π 2 (Fig. 9.).
Projecting on two and three planes of projections
Rice. 10. Converting a drawing using the Monge method.
After such a transformation, the plane π 1 in the drawing is combined with the plane π 2 and as a result we obtain a drawing in the form of planes π 1 and π 2 superimposed on each other. This method of depiction was called “Epure Monge” (from the French Épure - drawing, project).
Rice. 11. Position of point projections on the Monge diagram.
When considering the transformed drawing, it is necessary to take into account that the projection planes π 1 and π 2 occupy the entire space, and we see the overlap of the two planes.
On the Monge diagram, the projections of points A - A’ and A’’ on the projection planes π 1 and π 2 are located on one straight line perpendicular to the projection axis X. The segment
A'A'' is called communication line. Thus, two projections of a point are always located on the same connection line perpendicular to the projection axis.
If you carefully analyze the initial drawing of the position of a point in the system of two projection planes and Monge diagrams, you can see that the value of the segment Ax A'= AA'' determines the distance of point A from the projection plane π 2, and the value of the segment Ax A''= AA ' - determines the distance of point A from the plane π 1.
Projecting on two and three planes of projections
Two mutually perpendicular planes π 1 and π 2 divide the entire space into four quadrants (remember how two perpendicular axes X and Y on a plane divide this plane into four quarters).
Rice. 12. Dividing space by two planes into 4 quadrants.
Depending on which quadrant of space a point is located, its projections occupy a certain position on Monge’s diagram.
E'=Ex |
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Rice. 13. Position of points on Monge diagram.
Using the Monge diagram, we can determine that the points occupy the following positions in space:
Point A is located in the first quadrant; Point B is located in the second quadrant; Point C is located in the third quadrant; Point D is located in the fourth quadrant;
Point E is located directly in the π 2 plane.
Projecting on two and three planes of projections
A point in a system of three projection planes.
Along with projection onto two projection planes, a system of three planes is used. The position of any point in space, and therefore any geometric figure, can be determined in any coordinate system.
The most convenient is the Cartesian coordinate system in space, consisting of three mutually perpendicular axes. This system can be obtained as the intersection lines of three mutually perpendicular planes - horizontal π 1, frontal π 2 and profile π 3.
The intersection lines of these three planes form a system of three mutually perpendicular axes in space: the abscissa is the X axis, the ordinate is the Y axis and the applicate is the Z axis. The point of intersection of the three axes is the point “O” from the Latin “origo” - the beginning, is the origin of reference for all coordinate axes (Fig. 14), arrows indicate the positive direction of coordinate values. The X, Y and Z axes are called the projection axes.
A''Az
A A'''
Rice. 14. Position of a point in a system of three projection planes.
The size of the segment AA’ = A’’Ax is the distance from point A to the plane π 1. The size of the segment AA'' = A'Ax is the distance from point A to the plane π 2. The size of the segment AA’’’ = A’Ay is the distance from point A to the plane π 3.
Projecting on two and three planes of projections
Three intersecting planes divide all space into eight octants.
Rice. 15. Dividing space into eight octants. |
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By the signs of the coordinates of a point, you can determine in which octant of space it is located.
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Descriptive geometry. Engineering graphics. Levchenko S.V. |
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Projecting on two and three planes of projections
Converting a drawing into a system of three projection planes.
As in the case of projection in a two-plane system, in a three-plane system, the drawing transformation method proposed by Gaspard Monge is used.
This is due to the fact that in this form the drawing turns out to be bulky and distortion of the shapes and sizes of the figures occurs on the planes π 1 and π 2.
Rice. 16. Transformation of planes in a system of three projection planes.
In Figure 16, arrows indicate the direction of rotation of the planes around the projection axes.
During the transformation, the π 2 plane remains in place, the π 1 plane is rotated around the X axis until aligned with the π 2 plane, the π 3 plane is rotated around the Z axis until aligned with the π 2 plane. After such a transformation, all three planes are superimposed on each other (Fig.
Projecting on two and three planes of projections
Rice. 17. View of the drawing after conversion.
The π 1 plane contains the X and Y axes. The π 2 plane contains the X and Z axes. The π 3 plane contains the Y and Z axes.
In π 1, the X axis remains in place, and the Y axis in the drawing is directed downward.
IN result of plane transformationπ 3 the Z axis remains in place, and the Y axis in the drawing is directed to the right.
Thus, after transforming the drawing, the Y axis occupies two positions in the drawing: the downward Y axis belongs to the π 1 plane; The Y axis directed to the left belongs to the π 3 plane.
The position of the projections of a point in the drawing depends on the octant of space in which it is located.
Projections of any point can be constructed directly on the drawing: the position of the horizontal projection is determined by a pair of X,Y coordinates (the Y axis is directed down); the position of the frontal projection is determined by a pair of X, Z coordinates; the position of the profile projection is determined by a pair of Y, Z coordinates (the Y axis is directed to the right).
If the point is located in the first octant, then the values of all three coordinates (X,Y,Z) are positive.
Projecting on two and three planes of projections
Construction of the missing projection in a system of three projection planes:
Rice. 18. The procedure for constructing the missing projection of a point.
Let horizontal (A’) and frontal (A’’) projections of point A be given.
It is necessary to construct the missing profile projection (A’’’). When constructing constructions, you must remember the following rules of descriptive geometry:
1. Horizontal and frontal projections of a point are always on the same connection line perpendicular to the X axis.
2. The frontal and profile projections of the point are always on the same connection line, perpendicular to the Z axis.
3. Horizontal and profile projections of a point are always on the same horizontal-vertical connection line perpendicular to the Y axis.
Construction order:
Projecting on two and three planes of projections
Let’s draw a line perpendicular to the Z axis through point A’’. The desired profile projection should be on this line.
To construct a horizontal-vertical connection line perpendicular to the Y axis, we will use the constant straight line of the drawing.
A. The constant line of a drawing is the bisector of the angle formed by the Y axes. Usually denoted by the letter k.
Through the horizontal projection of the point, we draw a perpendicular to the vertical Y axis until it intersects with the constant straight line of the drawing (point 1), then from point 1 we draw a perpendicular to the vertical Y axis until it intersects with a connection line perpendicular to the Z axis.
The point of intersection of the connection line perpendicular to the Z axis and horizontal
vertical connection line perpendicular to the Y axis and is a profile projection of point A.
Let us note once again that the horizontal projection of a point is determined by its abscissa and ordinate, the frontal projection by the abscissa and applicate, and the profile projection by its ordinate and applicate.
A point in space is removed from the plane:
π 1 by an amount equal to the value of the segment A’’Ax or A’’’Ay.
π 2 by an amount equal to the value of the segment A’Ax or A’’’Az.
π 3 by an amount equal to the value of the segment A’Ay or A’’Az.
The process of obtaining an image on a plane is called projection. How are projections made?
Let's take an arbitrary point in space A and some kind of plane N. Let's draw through the point A straight line until it intersects with the plane N, the resulting point A there are intersections of line and plane projection points A. The plane on which the projection is obtained is called projection plane. Straight Ahh called projecting beam(Fig. 35).
Rice. 35. Projecting a ray onto a plane
Consequently, in order to construct a projection of a figure on a plane, it is necessary to draw imaginary projecting rays through the points of this figure until they intersect with the plane. Word projection- Latin, translated into Russian means “to throw forward.”
Points taken on an object are indicated in capital letters A, B, C, and their projections are lowercase a, b, c.
If the projecting rays come from one point, then projection called central. The point S from which the rays emanate is called central (Fig. 36).
Rice. 36. Central projection
Examples of central projection are photographs, film frames, and shadows cast from an object by the rays of an electric light bulb.
If the projecting rays are parallel to each other, then projection called parallel, and the resulting projection – parallel. An example of parallel projection can be considered the sun's shadows from objects.
With parallel projection, all rays fall on the projection plane at the same angle. If it is any acute angle, then the projection is called oblique(Fig. 37).
Rice. 37. Parallel projection
In the case when the projecting rays are perpendicular to the projection plane, projection called rectangular. The resulting projection is called rectangular (Fig. 38).
Rice. 38. Rectangular projection
Of all the projection methods considered, the basis for constructing an image is rectangular projection method, since the resulting image is projected onto the plane without distortion.
In space, the projection plane can be located anywhere: vertical, horizontal, oblique.
To obtain a projection of an object on a plane, it is placed parallel to this plane and rays are drawn through each vertex perpendicular to this projection plane.
Let's consider constructing a projection of the object shown in Fig. 39 per plane.
Rice. 39. Projection onto the frontal plane of projections
Let's choose a vertical projection plane located in front of the viewer. This plane is called frontal(from the French word « frontal», what does it mean « facing the viewer» and denoted by the letter V(ve).
Mentally consider the object parallel to the frontal plane and draw projecting rays through all points perpendicular to plane V. Mark the points of intersection of the rays with the plane and connect them with straight lines, and the points of the circle with a curved line. We get a projection of the object on a plane, which is called frontal projection(Fig. 40).
Rice. 40. Frontal projection
Based on the resulting projection, one can judge only two dimensions - height, length and diameter of the hole.
What is the width of the object? Using the resulting projection, we cannot say this. This means that one projection does not reveal the third dimension of an object; in addition, one projection does not always determine geometric the shape of the object (Fig. 41).
Rice. 41. Ambiguity in identifying the shape of an object with one projection:
A– frontal projection; b, c– possible shape of an object
Frontal projection shown in Fig. 42, matches all details.
Rice. 42. Projections on the frontal and horizontal planes of projections
In order to determine the shape of an object, it is necessary to construct a second projection onto the plane, which is called horizontal plane and is designated by the letter N (ash). The projection of an object onto this plane is called horizontal projection.
The horizontal plane is located at an angle of 90 0 to the frontal one. The V and H planes intersect along the OX axis (O is the point of intersection of the axes), which is called the projection axis. From the horizontal projection you can determine the length and width of the part.
Images of an object are made in one plane, therefore, to obtain a drawing of an object, both planes are combined into one, rotating the horizontal plane around the OX axis downward by 90 0 so that it coincides with the frontal plane (see Fig. 42).
The boundaries of the plane are not shown in the drawing, as well as the axis of projections, if this is not necessary (Fig. 43).
Rice. 43. Location of frontal and horizontal projections in the drawing
The horizontal projection is located strictly under the frontal projection. The location between the projections is chosen arbitrarily, while providing space for applying dimensions.
2.2. Projection onto three projection planes. Types.
Arrangement of views in the drawing
Often, even two projections of a part do not give a complete picture of its geometric shape (Fig. 44).
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Rice. 44. Examples of ambiguous identification of the shape of a part using two projections
This drawing corresponds to several parts, so it becomes necessary to construct a third projection onto the plane. This plane is positioned perpendicular to the projection plane V and H.
The third projection plane is called profile, and the projection obtained on it is profile projection subject.
The profile plane is designated by the letter W (double - ve). The profile plane of projections is vertical; at the intersection with the H plane it forms the OY axis, and with the V plane it forms the OZ axis. The profile projection is located to the right of the frontal projection at the same height
(Fig. 45 A, b) Planes V,H,W form triangular angle. We place the projected object in the space of a trihedral angle and draw projecting rays through all points of the object until they intersect with the projection planes. Let us connect the intersection points with straight or curved lines, the resulting figures will be projections of the object on the planes V, H, W (Fig. 45, b).
Rice. 45. Projections of an object onto three planes of projections V, H, W
The projected object is placed in the space of a trihedral angle A) projections of an object on planes V, H, W.
To obtain a drawing of an object, the V, H, W planes are combined into one plane, turning the W plane 90 0 to the right, and H – 90 0 down (Fig. 46, b). The boundaries of the planes, projection axes and projecting rays are not shown in the drawing (Fig. 46, c, d).
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Rice. 46. Location of projection planes and axes on the plane:
A– trihedral angle formed by planes V, H, W; b– process of combining planes
3-sided angle with the plane of the drawing sheet; V- location of projection planes on the plane of the drawing sheet; G– location of the axes on the plane of the drawing sheet
Having examined the process of projection onto three projection planes, we can conclude that projection is carried out in the following sequence:
Object in the system of projection planes V, H, W;
The projecting rays are perpendicular to V and directed from the front, resulting in a frontal projection;
The rays are perpendicular to H and directed from above, resulting in a horizontal projection;
The rays are perpendicular to W and directed from the left, resulting in a profile projection;
We combine V, H, W into one plane.
A drawing consisting of several rectangular projections is called complex drawing or a drawing in a system of rectangular projections.
If a drawing is constructed with coordinate axes, it is called main drawing, and if without axes, it is called axleless. All projections in the drawing are in a projection connection, which is carried out through communication lines(Fig. 47).
Rice. 47. Construction of a profile projection of an object based on two data
You already know that the rules for the design and construction of drawings are established by ESKD standards. One of the standards of this system sets rules for depicting objects on the drawings, it provides definitions of the various images used in the execution of the drawings.
In technical drawings, projections on planes are called species.
View - This is an image of the visible part of an object facing the observer. The same standard states that the object is positioned relative to the frontal plane so that the image on it gives the most complete idea of the shape and size of the object. Therefore, the image on the frontal plane is called main view or front view.
The image on the horizontal plane is called top view.
The image on the profile plane is called left view(Fig. 48).
Rice. 48. Location of part views on projection planes
The top view is located below the main view, and to the right of the main view and at the same height as it is the left view.
Invisible parts of an object in views are shown with dashed lines.
The number of views in the drawing should be minimal, but sufficient to understand the shape of the depicted object. Views, as well as projections, are located in the same projection relationship with each other.
2.3. Geometric bodies and their projections.
Projections of vertices, edges, faces on a plane.
Projections of a group of geometric bodies
The shapes of parts found in technology are a combination of different geometric bodies or their parts.
To learn how to represent the shape of an object from a drawing, you need to know how geometric bodies are depicted in drawings.
Geometric body- this is a closed part of space, limited by planes or curved surfaces.
All geometric bodies are divided into polyhedra(cube, parallelepiped, prisms, pyramids) and bodies of rotation(cylinder, ball, cone).
Geometric bodies consist of certain elements - vertices, edges, faces(Fig. 49).
Rice. 49. Elements of geometric bodies
Edges located perpendicular to the projection planes are projected onto them in point.
Edges located parallel to the projection planes are projected onto them in natural size.
Faces perpendicular to the projection planes are projected into straight segments.
Faces parallel to projection planes are projected life size.
Faces and edges inclined to projection planes are projected onto them with distortion.
When constructing a drawing, you need to clearly imagine how each vertex, edge and face of the object will be depicted on it. It should be remembered that each view is an image of the entire object, and not just one side of it. The only difference is that some faces are projected into a true figure, others into straight segments (Fig. 50).
Rice. 50. Projecting faces and edges of geometric bodies onto projection planes
The projections of geometric bodies are flat geometric shapes.
Let's consider the basic geometric bodies and their projections.
Projections Cuba are three equal squares, prisms– two rectangles and a polygon; pyramids- two triangles and a polygon; truncated pyramid– two trapezoids and a polygon; cone– two triangles and a circle; truncated cone- two trapezoids and a circle; ball– three circles, a cylinder – two rectangles and a circle (Fig. 51).
A- tetrahedral prism b- triangular prism V- tetrahedral pyramid
G- 4-sided truncated pyramid d- cone
e- cone and- ball
Rice. 51. Projections of geometric bodies on projection planes
Let's consider a drawing of a group of geometric bodies (Fig. 52).
Rice. 52. Projection of a group of geometric bodies onto three projection planes
The group consists of three geometric bodies. The first geometric body on the planes V and W is depicted as a triangle, and on the plane N - all around. Such projections are only cone. The second geometric body on the H and W planes is represented by two rectangles, and on the frontal plane - circumference. Such projections have cylinder. The third geometric body on all planes is represented by rectangles, which means parallelepiped.
Thus, we can conclude that the drawing represents a group geometric bodies, consisting of cone, cylinder And parallelepiped. To determine which of the geometric bodies is closer to us, we need to consider top view. Based on the analysis, we come to the conclusion that there are closer to us parallelepiped And cylinder.
2.4. Analysis of the geometric shape of an object.
Projections of points lying on the surface of geometric bodies and objects
You already know that the objects around us, parts of machines and mechanisms have the shape of geometric bodies or their combinations.
Let's look at Fig. 53. Various details are depicted here, some of simple shapes, others of more complex shapes.
How to determine the shape of an object from a drawing? For this purpose, a complex-shaped part mentally dismember into separate parts shaped like geometric bodies.
Rice. 53. Parts consisting of a combination of simple geometric bodies
For example in Fig. 54. An image of the part is given. It is made up of parallelepiped, two half cylinders And truncated cone. The details include cylindrical hole.
Rice. 54. Analysis of the geometric shape of the support:
A– image of the support; b- components of the support
The mental division of an object into its constituent geometric bodies is called geometric shape analysis.
Any point on the image of geometric bodies is a projection of one or another element - vertices, edges, faces, curved surfaces.
This means that the image of any geometric body is reduced to the image of its vertices, edges, faces and curved surfaces.
Let's consider the process of constructing projections of points on drawings of geometric bodies and parts.
The work is carried out in the following sequence:
Set the face of the polyhedron or part of the surface of revolution on which the projection of the point is specified, and determine the visibility of this part of the geometric body in all views (Fig. 55, A);
Through a given projection of a point, draw a projection of an auxiliary straight line, construct it and the projection of the point in the view where the projection of the geometric body is combined with the projection of its base (Fig. 55, b);
Construct a projection of the auxiliary line and find on it the desired projection of the given point (Fig. 55, V).
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Rice. 55. An example of constructing a projection of a point on a given surface of geometric bodies
If you need to construct projections of points on the surface of an object represented by a drawing, then:
Analyze the geometric shape;
Set up geometric bodies with specified points on their surfaces;
Determine the projection of points one by one on each geometric body.
On the part, the points are indicated in capitals letters A, B, C, and their projections are lowercase, for example projections point A on the planes Н-а, V-а ′, W-а″, invisible points are included in brackets, for example, V-(a′), H-(a), W-(a″).
2.5. The procedure for reading and constructing a drawing of a part.
Construction of the third type based on two given
To get acquainted with the structure of any product, you need to read its drawing.
The drawing is read in the following sequence:
Determine what types of parts are given in the drawing;
Determine the geometric shape of the part;
Determine the overall dimensions of the part and its elements;
Let's look at an example of reading a drawing of a part (Fig. 56).
Rice. 56. Guide drawing
Questions about the drawing
1. What is the name of the part?
2. What material is it made from?
3. At what scale is the drawing made?
4. What types are shown in the drawing?
5. The combination of what geometric bodies determines the shape of the part?
6. What are the overall dimensions?
Answers to questions
1. The part is called a “guide”.
2. The part is made of steel.
3. Scale 1:1.
4. The drawing shows two views; main view and left view.
5. Having selected the parts of the part, we consider them from left to right, comparing both views.
The leftmost part in the main view is shaped like a rectangle, while in the left view it is a circle. So it's a cylinder.
The second part from the left in the main view is a trapezoid, in the left view it is two o circles, this truncated cone. The third part is shown as a rectangle in the main view, and in the left view - circle, that means cylinder. The fourth part on the main view – rectangle, and in the left view – hexagon, Means this is a hexagonal prism. The leftmost part in the main view is rectangle, and in the view on the left - circle, This cylinder. Dashed lines on the main view and circle ø 20 in the left view indicates that the part has through cylindrical hole.
6. Overall dimensions of the part 160x90x90.
Many technical parts have various technological and structural elements, which have their own names (Fig. 57).
Holes
Rice. 57. Name of structural elements of parts
Hole– a through or blind element of a part, having the shape of a geometric body.
Groove- a narrow slot or recess.
Cutout– removal of part of a part by two or more planes.
Slice– removal of part of a part using one plane.
Rib (stiffening rib)– a thin wall designed to enhance the rigidity of the structure.
Before you start constructing images, you need to clearly imagine the geometric shape of the part.
Let's consider the sequence of constructing views in the drawing (Fig. 58).
Rice. 58. Visual representation of the support
The general shape of the object shown in Fig. 58 – parallelepiped. It has rectangular cutouts and a triangular prism cutout. Let's start depicting the part with its general shape - a parallelepiped (Fig. 59).
Rice. 59. An example of the sequence of constructing views of a part:
A– image of general views of the part; b– construction of cutouts; V– drawing dimensions
By projecting the parallelepiped onto the planes V,H,W, we obtain rectangles on all three planes (Fig. 59, A).
All constructions are done first with thin lines. Since the part is symmetrical, we will plot the axes of symmetry in the main view and top view.
Now let's show the cutouts. It makes more sense to show them first in the main view.
To do this, you need to set aside 12 mm to the left and to the right from the axis of symmetry and draw vertical lines through the resulting points. Then, at a distance of 14 mm from the upper border, we draw segments of horizontal straight lines (Fig. 59, b).
Let's construct projections of these cutouts on other views. This can be done using communication lines. After this, in the top and left views you need to show the segments that limit the projections of the views.
In conclusion, the drawing is outlined and dimensions are applied (Fig. 59, V).
In drawing, quite often there are problems related to the construction of a third one using two given types.
Let us consider the sequence of construction of the third type based on two given ones (Fig. 60).
Rice. 60. Drawing of a block with a cutout
In Fig. 60 you see an image of a block with a cutout. Two views are given: front and top; you need to build a view on the left. To do this, you must first imagine the shape of the depicted part. Having compared the types, we determine that the block has the shape of a parallelepiped measuring 10x35x20 mm. A rectangular cutout measuring 12x12x10 mm is made in the parallelepiped.
In the front view, using communication lines, we draw two horizontal lines, one at the level of the lower base of the parallelepiped, the other at the level of the upper base. These lines limit the height of the view on the left. Draw a vertical line anywhere between the horizontal lines (Fig. 61).
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Rice. 61. Sequence of constructing the third projection
It will be a projection of the back face of the block onto the profile plane of projections (Fig. 61, A). From it to the right we will set aside a segment equal to 20 mm, i.e. the width of the block, and draw another vertical line - the projection of the front edge (Fig. 61, b).
Let us now show in the view on the left the cutout in the part. To do this, put a 12 mm segment to the left of the right vertical line, which is the projection of the front edge of the block, and draw another vertical line (Fig. 61, V).
After this, we delete all auxiliary construction lines and outline the drawing (Fig. 61, G).
There are many parts whose shape information cannot be conveyed by two drawing projections (Fig. 75).
In order for information about the complex shape of a part to be presented sufficiently fully, projection is used on three mutually perpendicular projection planes: frontal - V, horizontal - H and profile - W (read “double ve”).
The system of projection planes is a trihedral angle with its vertex at point O. The intersections of the trihedral angle planes form straight lines - the projection axes (OX, OY, OZ) (Fig. 76).
An object is placed in a trihedral corner so that its formative edge and base are parallel to the frontal and horizontal projection planes, respectively. Then, projection rays are passed through all points of the object, perpendicular to all three projection planes, on which frontal, horizontal and profile projections of the object are obtained. After projection, the object is removed from the trihedral angle, and then the horizontal and profile projection planes are rotated by 90*, respectively, around the OX and OZ axes until aligned with the frontal projection plane and a part drawing containing three projections is obtained.
Rice. 75. Projecting onto two projection planes does not always give
a complete understanding of the shape of the object
Rice. 76. Projection onto three mutually perpendicular
projection planes
The three projections of the drawing are interconnected with each other. Frontal and horizontal projections preserve the projection connection of images, i.e. projection connections are established between frontal and horizontal, frontal and profile, as well as horizontal and profile projections (see Fig. 76). Projection lines define the location of each projection on the drawing field.
In many countries of the world, another system of rectangular projection onto three mutually perpendicular projection planes has been adopted, which is conventionally called “American” (see Appendix 3). Its main difference is that the trihedral angle is located in space differently, relative to the projected object, and the projection planes unfold in other directions. Therefore, the horizontal projection appears above the frontal one, and the profile projection appears to the right of the frontal one.
The shape of most objects is a combination of various geometric bodies or their parts. Therefore, to read and execute drawings, you need to know how geometric bodies are depicted in the system of three projections in production (Table 7). (Drawings containing three views are called complex drawings.)
7. Complex and production drawings of simple geometric parts
Notes: 1. Depending on the characteristics of the production process, a certain number of projections are depicted in the drawing. 2. In drawings, it is customary to give the smallest but sufficient number of images to determine the shape of the object. The number of drawing images can be reduced using the symbols s, l, ? which you already know.
Let's consider the profile plane of projections. Projections onto two perpendicular planes usually determine the position of a figure and make it possible to find out its real size and shape. But there are times when two projections are not enough. Then the construction of the third projection is used.
The third projection plane is drawn so that it is perpendicular to both projection planes simultaneously (Fig. 15). The third plane is usually called profile.
In such constructions, the common straight line of the horizontal and frontal planes is called axis X , the common straight line of the horizontal and profile planes – axis at , and the common straight line of the frontal and profile planes is axis z . Dot ABOUT, which belongs to all three planes, is called the origin point.
Figure 15a shows the point A and three of its projections. Projection onto the profile plane ( A) are called profile projection and denote A.
To obtain a diagram of point A, which consists of three projections a, a, a, it is necessary to cut the trihedron formed by all the planes along the y-axis (Fig. 15b) and combine all these planes with the plane of the frontal projection. The horizontal plane must be rotated about the axis X, and the profile plane is about the axis z in the direction indicated by the arrow in Figure 15.
Figure 16 shows the position of the projections a, a And A points A, obtained by combining all three planes with the drawing plane.
As a result of the cut, the y-axis appears in two different places on the diagram. On a horizontal plane (Fig. 16) it takes a vertical position (perpendicular to the axis X), and on the profile plane – horizontal (perpendicular to the axis z).
There are three projections in Figure 16 a, a And A points A have a strictly defined position on the diagram and are subject to unambiguous conditions:
A And A should always be located on the same vertical line, perpendicular to the axis X;
A And A should always be located on the same horizontal straight line, perpendicular to the axis z;
3) when carried out through a horizontal projection and a horizontal straight line, and through a profile projection A– a vertical straight line, the constructed straight lines will necessarily intersect on the bisector of the angle between the projection axes, since the figure Oa at A 0 A n – square.
When constructing three projections of a point, you need to check whether all three conditions are met for each point.
Point coordinates
The position of a point in space can be determined using three numbers called its coordinates. Each coordinate corresponds to the distance of a point from some projection plane.
Determined point distance A to the profile plane is the coordinate X, while X = a˝A(Fig. 15), the distance to the frontal plane is coordinate y, and y = AA, and the distance to the horizontal plane is the coordinate z, while z = aA.
In Figure 15, point A occupies the width of a rectangular parallelepiped, and the measurements of this parallelepiped correspond to the coordinates of this point, i.e., each of the coordinates is represented in Figure 15 four times, i.e.:
x = a˝A = Oa x = a y a = a z á;
y = а́А = Оа y = а x а = а z а˝;
z = aA = Oa z = a x á = a y a˝.
In the diagram (Fig. 16), the x and z coordinates appear three times:
x = a z a ́= Oa x = a y a,
z = a x á = Oa z = a y a˝.
All segments that correspond to the coordinate X(or z), are parallel to each other. Coordinate at represented twice by an axis located vertically:
y = Oa y = a x a
and twice – located horizontally:
y = Oa y = a z a˝.
This difference appears due to the fact that the y-axis is present on the diagram in two different positions.
It should be taken into account that the position of each projection is determined on the diagram by only two coordinates, namely:
1) horizontal – coordinates X And at,
2) frontal – coordinates x And z,
3) profile – coordinates at And z.
Using coordinates x, y And z, you can construct projections of a point on a diagram.
If point A is given by coordinates, their recording is defined as follows: A ( X; y; z).
When constructing point projections A the following conditions must be checked:
1) horizontal and frontal projections A And A X X;
2) frontal and profile projections A And A must be located at the same perpendicular to the axis z, since they have a common coordinate z;
3) horizontal projection and also removed from the axis X, like profile projection A away from the axis z, since the projections á and a˝ have a common coordinate at.
If a point lies in any of the projection planes, then one of its coordinates is equal to zero.
When a point lies on the projection axis, two of its coordinates are equal to zero.
If a point lies at the origin, all three of its coordinates are zero.
Line projections
To define a straight line, two points are needed. A point is determined by two projections on the horizontal and frontal planes, i.e., a straight line is determined using the projections of its two points on the horizontal and frontal planes.
Figure 17 shows the projections ( A And a, b And b́) two points A and B. With their help, the position of a certain line is determined AB. When connecting the projections of these points with the same name (i.e. A And b, a And b́) projections can be obtained ab And ab straight AB.
Figure 18 shows the projections of both points, and Figure 19 shows the projections of a straight line passing through them.
If the projections of a line are determined by the projections of two of its points, then they are designated by two side by side Latin letters corresponding to the designations of the projections of points taken on the line: with strokes to indicate the frontal projection of the line or without strokes for a horizontal projection.
If we consider not individual points of a line, but its projections as a whole, then these projections are designated by numbers.
If some point WITH lies on a straight line AB, its projections с and с́ are on the same projections of the line ab And ab. This situation is illustrated by Figure 19.
Traces of a straight line
The trail is straight- this is the point of its intersection with a certain plane or surface (Fig. 20).
Horizontal trace of a straight line some point is called H, in which the straight line meets the horizontal plane, and frontal– point V, in which this straight line meets the frontal plane (Fig. 20).
Figure 21a shows the horizontal trace of a straight line, and its frontal trace is shown in Figure 21b.
Sometimes a profile trace of a straight line is also considered, W– the point of intersection of the straight line with the profile plane.
The horizontal trace is in the horizontal plane, i.e. its horizontal projection h coincides with this trace, and the frontal h́ lies on the x axis. The frontal trace lies in the frontal plane, therefore its frontal projection ν́ coincides with it, and the horizontal projection v lies on the x axis.
So, H = h, And V= ν́. Therefore, to designate traces of a straight line, letters can be used h and ν́.
Various straight positions
Direct is called general position, if it is neither parallel nor perpendicular to any projection plane. Projections of a straight line in general position are also not parallel and not perpendicular to the axes of the projections.
Straight lines that are parallel to one of the projection planes (perpendicular to one of the axes). Figure 22 shows a straight line that is parallel to the horizontal plane (perpendicular to the z axis), - a horizontal straight line; Figure 23 shows a straight line that is parallel to the frontal plane (perpendicular to the axis at), – frontal line; Figure 24 shows a straight line that is parallel to the profile plane (perpendicular to the axis X), – profile straight line. Despite the fact that each of these lines forms a right angle with one of the axes, they do not intersect it, but only intersect with it.
Due to the fact that the horizontal straight line (Fig. 22) is parallel to the horizontal plane, its frontal and profile projections will be parallel to the axes defining the horizontal plane, i.e. the axes X And at. Therefore the projections ab́|| X And a˝b˝|| at z. The horizontal projection ab can occupy any position on the diagram.
At the frontal straight line (Fig. 23) the projection ab|| x and a˝b˝ || z, i.e. they are perpendicular to the axis at, and therefore in this case the frontal projection ab the straight line can take any position.
At the profile straight line (Fig. 24) ab|| y, ab|| z, and both of them are perpendicular to the x-axis. Projection a˝b˝ can be placed on the diagram in any way.
When considering the plane that projects a horizontal straight line onto the frontal plane (Fig. 22), you can notice that it projects this straight line onto the profile plane, i.e., it is a plane that projects a straight line onto two projection planes at once - the frontal and profile. Based on this, it is called double projecting plane. In the same way, for the frontal straight line (Fig. 23), the doubly projecting plane projects it onto the planes of horizontal and profile projections, and for the profile line (Fig. 23), onto the planes of horizontal and frontal projections.
Two projections cannot define a straight line. Two projections 1 And 1 profile line (Fig. 25) without specifying the projections of two points of this line on them will not determine the position of this line in space.
In a plane that is perpendicular to two given planes of symmetry, there may be an infinite number of straight lines, for which the data on the diagram 1 And 1 are their projections.
If a point is on a line, then its projections in all cases lie on the same projections of this line. The opposite situation is not always true for a profile straight line. On its projections, you can arbitrarily indicate the projections of a certain point and not be sure that this point lies on this line.
In all three special cases (Fig. 22, 23 and 24) the position of the straight line in relation to the projection plane is an arbitrary segment of it AB, taken on each of the straight lines, is projected onto one of the projection planes without distortion, i.e. onto the plane to which it is parallel. Segment AB horizontal straight line (Fig. 22) gives a full-size projection onto a horizontal plane ( ab = AB); segment AB frontal straight line (Fig. 23) – in full size on the plane of the frontal plane V ( ab́ = AB) and a segment AB profile straight (Fig. 24) – in full size on the profile plane W (a˝b˝= AB), i.e. it seems possible to measure the actual size of the segment in the drawing.
In other words, using diagrams you can determine the natural dimensions of the angles that the straight line in question forms with the projection planes.
The angle that a straight line makes with a horizontal plane N, is usually denoted by the letter α, with the frontal plane - by the letter β, with the profile plane - by the letter γ.
Any of the straight lines under consideration has no trace on the plane parallel to it, i.e. the horizontal straight line has no horizontal trace (Fig. 22), the frontal straight line has no frontal trace (Fig. 23), and the profile straight line has no profile trace (Fig. 24 ).