Construct lines of intersection of cylindrical surfaces. Projecting the line of intersection of two surfaces of rotation of the second order onto a plane parallel to their common plane of symmetry

4. Intersection of polyhedra

1 MUTUAL INTERSECTION OF CURVED SURFACES

1.1 General provisions

Curved surfaces generally intersect along a spatial curved line, the projections of which are usually constructed point by point. To find these points, the given surfaces are intersected by a third auxiliary secant surface, the lines of intersection of the auxiliary surface with each of the given ones are determined, then the common points of the constructed intersection lines are found. By repeating such constructions many times, the required number of points is obtained to determine the intersection line.

General algorithm for constructing the line of intersection of surfaces:

1) Select the type of auxiliary surfaces. When choosing an auxiliary secant surface, you should choose surfaces that would intersect the given surfaces along the easiest lines to construct - straight lines or circles. Planes and spheres are most often used as auxiliary intermediary surfaces.

2) Construct lines of intersection of auxiliary surfaces with given surfaces.

3) Find the intersection points of the resulting lines and connect them to each other.

4) Determine the visibility of the intersection line relative to the surfaces and projection planes under consideration.

Constructions begin with the definition characteristic (reference) points(points located on the outline generating surfaces, which usually divide the intersection line into visible and invisible parts (visibility boundaries), the highest and lowest points of the intersection line, extreme points (right and left).

When constructing, methods of transforming the drawing are used if this simplifies and refines the construction.

1.2 Constructing a line of intersection of surfaces using auxiliary cutting planes

Task. Draw the line of intersection of the cone and the cylinder of rotation (Fig. 186).

First of all, we determine characteristic points intersection lines:

Projections of the highest and lowest points A2 and E2 defined using an auxiliary frontal plane Q, which intersects the surface of the cylinder and cone along the outermost generatrices. Horizontal projections of points are on a horizontal trace Qπ2 auxiliary plane.

Points C and C are found using a horizontal plane S drawn through the axis of the cylinder. The plane S intersects the surface of the cylinder along the outer generatrices (front and back), and the surface of the cone - along the circumference. The intersections of horizontal projections of the extreme generators and the circle give points C 1 and C 1 - horizontal projections of points C and C. The frontal projections of these points lie on the frontal trace of the plane S.

Intermediate points the intersection lines are found using horizontal planes P and R.

Figure 186

Figure 187

In the example considered, the points of the intersection line are found using auxiliary planes of particular position. Sometimes the introduction of planes of particular position does not give the desired effect and it is more expedient to use planes of general position.

1.3 Constructing a line of intersection of surfaces using auxiliary secant spheres with a constant center

It is known that if the axis of the surface of rotation passes through

the center of the sphere and the sphere intersects this surface, then the line of intersection of the sphere and the surface of revolution is a circle, the plane of which is perpendicular to the axis of the surface of revolution. Moreover, if the axis of the surface of rotation is parallel to the projection plane, then the line of intersection onto this plane is projected into a straight line segment.

In Fig. 187 shows the frontal projection of the intersection of a sphere of radius R and surfaces of revolution - a cone, torus, cylinder, sphere, the axes of which pass through the center of the sphere

radius R and parallel to the plane π 2. The circles along which the indicated surfaces of revolution intersect with the surface of the sphere are projected onto the plane in the form of straight segments. This property is used to construct a line of mutual intersection of two surfaces of revolution using auxiliary spheres.

The method of secant spheres with a constant center is used under the following conditions:

1) both surfaces are surfaces of revolution;

2) both surfaces of revolution intersect; the intersection point is taken as the center of the auxiliary (concentric) spheres;

3) the plane formed by the axes of the surfaces (the plane of symmetry) must be parallel to the projection plane. If this condition is not met, they resort to methods of converting the drawing.

Radius sphere (R min )

Figure 188

Example. Draw the line of intersection of the cone of rotation and the cylinder of rotation (Fig. 188).

The axes of the given surfaces of revolution intersect (point O) and are parallel to the projection plane π 2, therefore, the conditions necessary for applying the method of spheres exist.

We define the frontal projections of the reference points 1 2 and 2 2 as the intersection points of the frontal projections of the outlines of the cylinder and cone. The horizontal projections of these points are determined using projection communication lines.

Radius of sphere of maximum radius (Rmax)

equal to the distance from the frontal projection of the center of the spheres O 2 to the most distant point of the projection of the point of intersection of the outlines (point 1 2 ).

minimum

This is a sphere that can be inscribed in one geometric body and intersecting another.

A sphere of minimum radius only touches the surface of the cone and, therefore, intersects it but the circle, the frontal projection of which is straight line A 2 B 2. Cylinder surface

the sphere R min also intersects along a circle, the frontal projection of which is straight line C 2 D 2. The intersection of these lines - point 4 2 is the frontal projection of one of the points of the desired intersection line.

In a similar way, using a sphere of intermediate radius R i, a frontal projection 3 2 of another point belonging to the intersection line was constructed. Horizontal projections of the found points can be constructed as projections of points lying on the surface of the cone.

2 SPECIAL CASES OF INTERSECTION OF SURFACES

1 Coaxial surfaces of rotation

Coaxial surfaces of rotation intersect along a circle, so the lines of intersection of a cone and a cylinder (Fig. 189) are two circles that are projected onto a horizontal plane in full size, and onto the π 2 plane - into straight segments.

Figure 189

2 Intersection of surfaces circumscribed around one sphere

As noted earlier, the line of intersection of two curved surfaces is generally a space curve. However, in some special cases this line can break up into flat curves.

Monge's theorem: two second-order surfaces described around a third second-order surface (or inscribed in it) intersect each other along two second-order curves

Figure 188

3 INTERSECTION OF A CURVED SURFACE WITH A POLYHEDON SURFACE

Each face of a polyhedron generally intersects a curved surface along a plane curve. These curves intersect each other at the points where the edges of the polyhedron meet the surface. Thus, the task of constructing the line of intersection of a curved surface with a polyhedron comes down to finding the line of intersection of the surface with a plane and the meeting points of the straight line with the surface.

Example. Construction of the line of intersection of the surfaces of the hemisphere

We carry out using the method of auxiliary cutting planes.

Each face of the prism intersects the surface of the hemisphere along semicircles, which intersect each other at the points where the edges of the prism meet the surface of the hemisphere.

In the example given, one of the faces of the prism is located parallel to the frontal plane of projections, therefore the circle along which this face intersects the surface of the hemisphere is projected onto the frontal plane of projections without distortion. The frontal projections of the remaining two arcs of semicircles will obviously be arcs of semi-ellipses. Constructing them on the diagram should begin with finding the reference points. To do this, frontal planes (P and Q) are drawn through each edge of the prism, which intersect the surface of the hemisphere along the circles.

Points of intersection of the frontal projections of the ribs with the corresponding

semicircles are frontal projections of the meeting points of the edges of the prism with the hemisphere (points 1, 2, 3).

Points 4 and 5, dividing the curves into visible and invisible parts, were obtained using the frontal plane S drawn through the center of the hemisphere.

Intermediate points were found by a similar construction (using the frontal planes R and T).

4 MUTUAL INTERSECTION OF POLYHEDRONS

The line of intersection of the surfaces of two polyhedra is a closed spatial broken line (or two closed broken lines) that passes through the points of intersection of the edges of one of the polyhedra with the faces of the other and the edges of the other with the faces of the first.

The construction of the line of intersection of polyhedra can be done in two ways, combining or selecting from them the one that, depending on the conditions, gives simpler constructions:

1 way. Determine the points at which the edges of one of the polyhedra intersect the faces of another and the edges of the second intersect the faces of the first. A broken line is drawn through the obtained points in a certain sequence, which is the line of intersection of the given surfaces. In this case, it is possible to connect by straight lines the projections of only those points that lie on the same face.

Method 2. Determine the straight line segments along which the faces of one of the polyhedra intersect the faces of the other; these segments are links of the broken line obtained by intersecting the polyhedra.

	Example. Construction of the line of intersection of the surfaces of the prism and
pyramids (Fig. 189)
	As can be seen from Fig. 189,
	surface		pyramids
intersects only
the front edge of the prism. So
perpendicular to the plane
π1,		horizontal
projections
output (points 1 and 2)
are noted		directly
but on the diagram.
		finding
frontal			projections
through the top of the pyramid and
front
carried out		auxiliary
horizontally
quoting		plane
She crossed the surface						Figure 189
pyramids in straight lines				SD and

SE, at the intersection of the frontal projections of which with the frontal projection of the front edge of the prism, the frontal projections 1 2 , 2 2 of entry and exit points 1 and 2 are marked. Since the faces of the prism are horizontal

Projecting planes, then constructing the meeting points of the edges of the pyramid with the faces of the prism (points 3, 4, 5, 6) does not present any difficulties and is clear from the drawing. By connecting the frontal projections of the found points in series, we obtain the frontal projection of the intersection line. Its horizontal projection coincides with the horizontal projection of the prism.

When determining the visibility of points, belonging to the intersection line are guided by the following rule: the projection of a point obtained by the intersection of two visible lines is visible. The point of intersection of two invisible lines or one visible and another invisible line is invisible.

Purpose of the task: Study methods for constructing the line of intersection of two cylinders.

Execution order: Draw a frame and main inscription. Draw three projections of intersecting cylinders according to the given dimensions. Find the projections of obvious points of the line of intersection of cylindrical surfaces. Determine the number of intermediate points and construct their projections. Mark the intersection line on all three projections and determine its visibility. Label the points. Trace the drawing, fill in the main inscription.

The values ​​of the parameters shown in Figure 10 are selected according to Table 4 according to the option. A sample is shown in Figure 12.

Guidelines for completing sheet 4

The common line of intersecting surfaces is called the line of intersection. In the drawings, the lines of intersection of surfaces are shown solid main line.

The method of constructing lines of intersection of surfaces of bodies is to draw auxiliary cutting planes and find individual points of the lines of intersection of these surfaces in these planes. As auxiliary secant planes, choose planes that intersect both given surfaces along simple lines - straight lines or circles, and the circles should be located in planes parallel to the projection planes.

Let us consider the construction of a line of intersection of the surfaces of two straight circular cylinders, the axes of which are perpendicular to the projection planes.

Before constructing a line of intersection of surfaces in a drawing, it is necessary to imagine this line in space (see Figure 11).

At the beginning of the construction, projections of obvious points 1, 4 are found (see Figure 10). The horizontal projection of the desired line of intersection of the surfaces coincides with the circle - the horizontal projection of one of the cylinders. The profile projection of the intersection line also coincides with the circle - the profile projection of the second cylinder. Thus, the frontal projection of the desired intersection line can be easily found using the general rule for constructing a curved line from points when two projections of points are known. For example, using horizontal projection 2 1 of point 2, profile projection 2 3 is found. Using two projections 2 1 and 2 3, the frontal projection 2 2 of point 2 belonging to the line of intersection of the cylinders is determined.

If intersecting cylindrical surfaces have axes located at an angle other than a right angle, then the line of their intersection is constructed using auxiliary cutting planes or other methods (for example, using the method of spheres).

Table 4 – Options for source data for sheet 4.

Designation, mm	Option No.
R
d
h
k

Figure 10 – Mutual intersection of cylinder surfaces.

Graphic work 5 “Technical drawing of the model”

Purpose of the task: Learn to draw technical details in simple forms by hand. Learn the rules for performing technical drawings.

Execution order: Draw a frame and main inscription. Analyze the shape of a part using its axonometric image. Complete technical drawing. The drawing is done by hand, by eye, first with thin lines, then shading and tracing are done to reveal the volume. Fill out the title block. The option is chosen according to Figure 13 a, b. A sample is shown in Figure 14.

Construction of a rectangular isometric projection of the cylinder in question, taking into account the previously completed binding of this figure to a rectangular coordinate system Oxyz(see Figure 3.2) let's start with the image of the axonometric axes (see Figure 2.4) on a separate sheet of Whatman paper in A3 or A4 format.

Next, we will construct an axonometric projection of the circle of the upper base of the cylinder. Such a projection is an ellipse having the following ratio of the major and minor axes: B.o. = 1.22 d, M.o. = 0.71 d, - Where d- diameter of the depicted circle. The minor axis of the ellipse is always located along the “free” coordinate axis. “Free” is the coordinate axis perpendicular to the plane in which the depicted circle is located. In the example under consideration, the circles of the cylinder bases are located in planes parallel to P 1 and the “free” axis is Oz.

First, we graphically determine the dimensions of the ellipse axes. It is known that in a rectangular isometric projection, the size of the minor axis of the ellipse is equal to the length of the side of the square inscribed in the depicted circle. Therefore, in the top view of the cylinder drawing, we will construct such a square (Figure 3.7) and determine the length of the segment t- half the side of the square. Subsequently, to simplify constructions when determining the length of a segment on an orthogonal drawing t Only a line located at an angle of 45° to the coordinate axes will be used (without depicting the entire square).

Further on the axonometric drawing (Figure 3.8), along the “free” axis O¢z¢, in both directions from the origin O¢ put aside the segment t and get the points B¢ And D¢, defining the minor axis of the ellipse. To find points A¢ And C¢, defining the major axis of the ellipse, from the found points B¢ And D¢, as from the centers, we will construct two arcs of radius R=2t before their mutual intersection. By connecting the found points to each other, we determine the major axis of the ellipse.

Instead of an ellipse, let us construct an oval - a closed curve, representing four successively conjugate arcs of circles of radius R And r. To do this, we first determine the centers of these arcs (Figure 3.9). Centers O 1 And O 2 arc radius R define on the axis O¢z¢ at the points of intersection of it with a circle of radius equal to the semi-major axis of the ellipse, and the centers O 3 And O 4 arc radius r determined at the points of intersection of the major axis of the ellipse with a circle of radius equal to the semi-minor axis of the ellipse. After this, the radii of the arcs are determined:
R =О 1 В¢ = О 2 D¢; r = O 3 A¢ = O 4 C¢(Figure 3.10). Further from the found centers O 1, O 2, O 3, O 4 Using a compass we construct four conjugate arcs of the oval. Recall that the conjugation point of two arcs is located on a straight line passing through the centers of these arcs. For example, dot N conjugation of the lower arc radius R with left arc radius r is on a line passing through the centers

O 2 And O 3 arcs under consideration.

We construct the axonometry of the lower base of the cylinder by shifting downward by the amount h centers O 1, O 2, O 3, O 4 oval arcs of the upper base (Figure 3.11). Next, we build ¼ of the cylinder cutout and depict the frontal secondary projection of the prismatic hole formed by the planes a, b And g(Figure 3.12). Dimensions a, b And With, necessary for this, we transfer to the axonometric drawing from the orthogonal drawing (see Figure 3.2) parallel to the corresponding axonometric axes.

Let us denote by m¢ And n¢ axonometric outlines of the cylinder (Figure 3.13) and construct their frontal secondary projections m¢ 2 And n¢ 2(the sequence of constructions is shown by arrows). Next, mark the points 1 2 ¢, 2 2 ¢, 3 2 ¢, 4 2 ¢ - the intersection of the lines of the front secondary projection of the hole in the cylinder with the front secondary projections of the lines of the axonometric outline and find the points 1¢, 2¢, 3¢, 4¢ breaking lines m¢ And n¢ - axonometric outlines of the cone's boundary lines of the hole in it (Figure 3.14).

We construct the boundary lines of the hole in axonometry. To do this, first, on the secondary frontal projection of the hole, we find intermediate points (Figure 3.15), using the dimensions g And f, transferred from the orthogonal drawing (see the main view in Figure 3.2). Using the indicated secondary projections, we construct axonometric projections of intermediate points located on the boundary lines of the hole in the cylinder. The sequence of constructing these points is shown in Figure 3.16 by arrows. Segments whose lengths are used for structure of axonometric

projections of intermediate points are marked with dashes in Figures 3.2 and 3.16. By connecting the obtained points with a smooth curve, we obtain images of those boundary lines of the hole in the cylinder that are formed by the plane g. These lines are marked in Figure 3. 16 with arrows A and B. Similarly, you can construct points and an image of the boundary line of the hole formed by the plane b. However, the bulk of these points are not visible and therefore their construction is not required.

We build an oval defining the horizontal part of the prismatic hole in the cylinder, formed by the plane a(Figure 3.17). For this you can use arcs R And r oval of the upper base of the cone, finding new centers of these arcs. We save only those parts of the constructed oval that are visible in the axonometry.

To finalize the axonometric drawing of the cylinder, we apply shading to those elements of the cylinder cutout that are located in the planes xOz And yOz(Figure 3.18). You can determine the direction of the hatch lines in axonometry using the indicated coordinate planes as follows (Fig. 3.19). Let's construct a circle of arbitrary radius with a center at the origin and connect the intersection points of this circle with the coordinate axes defining the planes under consideration. The constructed segments will determine the directions of the hatching lines along the specified planes.

We emphasize that the final design of the axonometric drawing of the cylinder in question requires a smooth connection of all the obtained points when depicting the through hole and tracing all visible lines of the contour of the image of the cylinder.

3.4. Construction of orthogonal and axonometric drawings
cone of rotation

We proceed to consider in task 2 the construction of orthogonal and axonometric drawings of the cone of rotation.

Figure 3.20 shows images: the main view and partially the top view of a straight circular truncated cone, as well as an overall rectangle for subsequent construction of the view on the left.

The cone in question has a through hole formed by three planes: a horizontal plane a, dissecting the conical surface along the circumference, and two frontally projecting planes b And g, cutting its surface into ellipses.

To construct top and left views, as well as an axonometric image of this cone, we will bind this figure to a rectangular coordinate system Oxyz(Figure 3.21). We choose the plane of the lower base of the cone as the horizontal coordinate plane.

In the main view, we mark the characteristic and intermediate points of the boundary lines of the hole and construct them in the top view.

Let's look at the points first 1, 2, 3 , located on the horizontal boundary lines of the hole formed by the plane a(See Figure 3.21). These points (there are six in total) are determined in the top view along the communication lines on a circle of radius R. We measure the specified radius in the main view, in the plane a from the axis of the cone to its outline generatrix.

Similarly, we determine the horizontal projections of points 4, 5 And 6 boundary lines of the hole located in the plane b(Figure 3.22). To do this, we construct circles of radius R1, R2 And R 3, located in intermediate horizontal planes a 1 , a 2 , a 3.

Similarly, in the top view we construct points of the boundary lines of the hole located in the plane g. We sequentially connect the found horizontal projections of the points with smooth curves. The final design of the top view is shown in Figure 3.23. Here the lines of the intersection of planes are shown by the lines of an invisible contour a And b, g And b, a And g.

The construction of profile projections of the points under consideration (see Figure 3.23) is carried out both along communication lines (points 3 3 And 6 3 ) on the lines of the profile outline of the cone, and by transferring ordinate segments of points from the top view to the left view. The transferred segments are shown with the same symbols both in the top view, where they are measured, and in the left view, where they are laid aside. We sequentially connect the found profile projections of the points

smooth curve, and also depict the lines of an invisible contour that define the lines of intersection of planes a And b,
g And b, a And g.

Next, we build horizontal and profile sections of the cone. Modeling of horizontal and profile sections of a cone with a through hole is shown in Figure 3.24. The horizontal section is depicted in the top view, and the profile section is shown in the left view (Figure 3.25). In both cases, we combine half of the corresponding view with half of the section, using the vertical center line as the boundary between these images. In the combined image, we place the sections to the right of the border, and the views to the left of it. We designate the horizontal section. After constructing the necessary sections in the drawing on all its images, we remove the lines of the invisible contour.

More detailed information about the rules for constructing and designating sections in accordance with GOST 2.305 - 68 is given in section 3.2.

Let's construct a rectangular isometric projection of the cone under consideration, using the binding of an orthogonal coordinate system to it Oxyz performed earlier (see Figure 3.21). On a separate sheet of Whatman paper A3 or A4 Let's depict the axonometric axes (see Figure 2.4).

Next, we will construct axonometric projections of the circles of the lower and upper bases of the cone. Such projections will be two ellipses, the centers of which are located on the coordinate axis O¢z¢ and are displaced relative to each other by a distance h(Figure 3.26). Ellipses have the following relationship between the major and minor axes: B.o. = 1.22 d, M.o. = 0.71 d, - Where d- diameter of the depicted circle. The minor axis of ellipses is located along the “free” coordinate axis O¢z¢, and its size is equal to the length of the side of the square inscribed in the depicted circle.

For ease of construction, we depict ovals instead of ellipses (see Figures 3.9 and 3.10). In this case, we use a graphical definition of both semi-minor axes of ellipses (see Figure 3.20, in the top view the segments t And t¢), and semi-major axes (see Figure 3.8).

Next we build straight lines m¢ And n¢, which are an axonometric outline of a conical surface (Figure 3.27). At the same time, we determine the points of contact of these lines with the ellipses, which are the bases of the cone. To do this, we lengthen the generators a¢ And b¢ to points A¢ And В¢ the intersection of these lines with the upper base of the cone. Generators a¢ And b¢ together with the center line of the drawing, they form three straight lines passing through the top of the cone. This vertex is not accessible in the drawing. These three straight lines intersect the ellipses (ovals) of the bases at six points. Connecting the intersection points with oval of adjacent straight lines cross on cross, and through the points of their intersections (see, for example, points C¢ And D¢) draw straight lines until they intersect with ellipses (see points E¢, F¢, Q¢, L¢). We connect the found points of the lower and upper bases of the cone with straight segments. These will be the lines of the axonometric outline of the cone.

Then we cut out ¼ of the cone and build a frontal secondary projection of the prismatic hole in the cone, i.e.

essentially constructing frontal secondary projections of planes a, b And g, forming a hole in the cone (Figure 3.28). In this case, the dimensions a, b And c from the orthogonal drawing (see the main view in Figure 3.23) we transfer it to the axonometric drawing parallel to the corresponding axonometric axes.

Next you need to construct points 1¢, 2¢, 3¢ And 4¢ interruption of the lines of the axonometric outline of the cone by the boundary lines of the hole in it. However, before this, we will first determine their frontal secondary projections 1 2¢, 2 2¢, 3 2¢, 4 2¢(Figure 3.29). To do this, we first build frontal secondary projections m 2 ¢, n 2 ¢ outlines of the cone's generatrices and find the points of intersection of these projections with the lines of the secondary projection of the hole. The sequence of these constructions is shown by arrows. At the same time, we emphasize that constructions begin not at the end points of the major axes of ellipses (ovals), but at the boundary points E¢, F¢, Q¢, L¢ axonometric outlines constructed earlier. Next we find the required points 1¢, 2¢, 3¢ And 4¢(Figure 3.30).

We construct axonometric projections of intermediate points of the boundary lines of the hole. To do this, first mark intermediate points on the lines of the frontal secondary projection of the hole (Figure 3.31). In this case we use the dimensions g And f, transferring them from the orthogonal drawing (see Figure 3.23). Next, through the found secondary projections we draw straight lines parallel to the axis Oh¢y¢, and putting on them in both directions the ordinates of the required points (Figure 3.32). The ordinates of intermediate points, marked with primes, are transferred from orthogonal drawing (see Figure 3.23) to an axonometric drawing. In this case, we depict only the points visible on the axonometric drawing. Consistently connecting the found points with smooth curves (arcs of ellipses), we construct visible sections of the boundary lines of the hole in the cone, formed by the plane b(see figure 3.32 lines A And B) and plane g(see line IN).

We build an oval that defines the boundary lines of the horizontal part of the hole in the cone and formed by the plane a(Figure 3.33). The visibility limits are conventionally shown by arrows. We draw a straight line, which is the line of intersection of planes a And g.

We hatch sections of the cone located in coordinate planes xOz And yOz. Determining the directions of hatch lines in rectangular isometry is shown in Figure 3.19.

The final design of the axonometric drawing of a cone with a through hole (Figure 3.34) requires careful tracing of all lines of the image: the arcs of the ovals are traced with a compass, and other curves are traced using a pattern.

4. Construction of orthogonal and axonometric drawings of the part
(third task)

The layout of the sheet and the construction of images of the part according to the dimensions applied to these images in an individual task are shown in Figure 4.1. Images include: main view, top view, and an outline rectangle for further construction of the left view.

To construct a left view and an axonometric drawing of the part, we will bind the part to the rectangular coordinate system O xyz(Figure 4.2) . For the horizontal coordinate plane we will take the plane of the upper base of a cylindrical slab, cut off on the sides by two frontal planes, and having two semi-oval cutouts. On this plate there is a cylinder of rotation, the axis of which coincides with the coordinate axis Oz. It is reinforced by two stiffening ribs - prismatic triangular elements. The internal shape of the part consists of a through stepped cylindrical hole.

When constructing the view on the left, of particular interest is the construction of the arc of the ellipse formed by the intersection of the cylinder with the inclined face of the stiffener. The construction is made using three points ( 1, 2 And 2 ) by transferring the ordinates of points from the top view to the left view 2 And 2 , equal to the half-width of the stiffener (see size b/2). Dot 1 in the adopted coordinate system has a zero ordinate.

In the third task, in addition to views, it is necessary to construct frontal and profile sections of the part. Since the part under consideration has two planes of symmetry: frontal and profile, and along these planes its dissection is performed, we do not indicate the position of the secant planes in the drawing, but combine the sections with the halves of the corresponding views (Figure 4.3). The boundary between these images is the axis of symmetry (dashed dotted line). We leave the view to the left of the center line, and place the section to the right of this line. When making cuts, we remove all lines depicting the external shape of the part, and replace the invisible contour lines (dashed lines) with solid main lines. In all views we remove dashed lines. The contours of the part located in secant planes are shaded with thin parallel lines located at an angle of 45° to the lines of the main inscription of the drawing. The direction of the hatching must be the same for all cuts made. It is recommended to maintain a hatching interval of 2.5 ... 3 mm.

Let us recall that the round base of any cylindrical or conical element of a part, located in the coordinate plane or parallel to such a plane, in rectangular isometry is represented by an ellipse having the following ratio of the major and minor axes: B.o. = 1.22 d, M.o. = 0.71 d, - Where d- diameter of the depicted circle. The minor axis of the ellipses is located along the “free” coordinate axis - the axis perpendicular to the plane in which the depicted circle is located, and the size of the minor axis is equal to the length of the side of the square inscribed in the depicted circle. For ease of construction and to obtain better image quality in the axonometric drawing, instead of ellipses, we construct ovals - circular curves (see Figures 3.9 and 3.10). Therefore, first we build ovals that define the horizontal secondary projections of all cylindrical elements of the part (Figure 4.4). To graphically determine the semi-minor axes of ellipses, we use the constructions shown in Figure 4.3 (see size A and segments marked with primes). Dimensions b, c, m And n, used for construction, are transferred from the orthogonal drawing (see Figure 4.2). Next, we build straight lines that define the horizontal secondary projections of the flat elements of the part (Figure 4.5). At the next stage of constructing the axonometry, we remove unnecessary drawing lines, taking into account the subsequent cutting of ¼ of the part (Figure 4.6).

Next, we will create a three-dimensional image of the base of the part (Figure 4.7). To do this, from the points of the horizontal secondary projection of the base of the part located closer to the observer, we build auxiliary straight lines parallel to the axis O¢ z¢, and on them we lay down segments of length t, which determines the thickness of the base slab. Thus, we determine the contour points of the lower part of the base. We draw images of flat sections of the base only by their boundary points, and for cylindrical sections we also construct intermediate points. Length of the segment t determined on an orthogonal drawing (see Figure 4.2). By connecting the found points of the lower plane of the base with straight or smooth curves and removing unnecessary auxiliary vertical segments, we will construct the base of the part.

Similarly, using auxiliary vertical segments of length N, using horizontal secondary projections of cylindrical elements, it is possible to construct points of the upper base of these elements of the part (Figure 4.8). We connect the found points with smooth curves, and remove the auxiliary vertical segments and invisible lines of the drawing. To construct an image of stiffening ribs, we find points 1¢ And 2 ¢ (Figure 4.9). To do this, from the corresponding points of the horizontal secondary projections of the edges, we construct auxiliary vertical segments of length e And f. We measure the lengths of these segments on an orthogonal drawing (see Figure 4.2). We build only the visible elements of the edges, and remove the invisible ones.

Having removed all the invisible lines of the drawing, including the secondary projections of the cylinders and stiffeners constructed earlier, we proceed to depict the elements of the lower part of the stepped cylindrical hole (Figure 4.10). The construction of the lower visible part of the circle of a cylindrical hole of a smaller radius is carried out using auxiliary vertical segments of length h drawn down from five points of the upper base of this hole . Three of the five constructed points are connected by a smooth curve.

To depict in axonometry the visible part of a circle of radius r a cylindrical recess located in the lower part of the part, we build the generatrices of this cylindrical surface, falling into the cutout of ¼ of the part and an oval corresponding to the circle of the cylindrical recess located in the lower plane of the base of the part (see in Figure 4.10 the oval depicted by a dashed line). For the constructed oval we save only its visible part, shown in Figure 4.10 by an arrow.

In conclusion, we outline the drawing and apply shading (Figure 4.11). Determining the directions of hatch lines in axonometry is shown in Figure 3.19.

The final design of the axonometric drawing of the part requires a smooth (using patterns) connection of the constructed points of curved lines depicting both the elements of the through stepped cylindrical hole in the part and the elements of its external shape. The design of the drawing is completed by filling out its main inscription.

The finalized orthogonal and axonometric drawings of the part are shown in Figures 4.12 and 4.13, respectively.

Let us also note that in all the previously discussed constructions, measuring dimensions on an orthogonal drawing and transferring them to an axonometric drawing was carried out using a meter.

In images of orthogonal and axonometric drawings, it is recommended to save characteristic and auxiliary points of the constructed lines, without marking these points.

Literature

1. Unified system of design documentation. General rules for making drawings. M., 1991, 453 p.

2. Averin V.N., Kukoleva I.F. Drawing dimensions on drawings. Guidelines for practical exercises in engineering graphics. M.: MIIT, 2008. 37 p.

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Educational and methodological publication

The intersection of cylinders in this article is determined by the secant sphere method. But first you need to familiarize yourself with the task located below.

Having familiarized yourself with this task, you can begin drawing.

The procedure for performing work on the intersection of cylinders:

1.) Initially, the figures are drawn.

2.) After construction, the smallest radius of the auxiliary secant sphere is required (it is located from the intersection of the axes of the figures to the edge of the figure, which has a larger size in width). In this case, the smallest radius has the length from the junction of the axes to the edge of the vertically located cylinder.

3.) The constructed radius intersects each figure at two points (“1” is connected to “2”, “3” to “4”), which are connected to each other and the first point is formed at the intersection.

4.) Auxiliary spheres are also drawn (the radii are taken arbitrarily) and then the points are determined. The principle of determining points is described in paragraph “3”.

5.) Points 1 2 and 5 2 can be immediately shown because the figures are located on the same axis when viewed from above.

6.) The next step is to transfer all the found points in the upper image to the lower one. And for this, an auxiliary circle is constructed (located on the right), to which straight lines are drawn from the points (indicated in red, blue and green).

7.) The thicker segments (indicated in red, blue and green) are measured from the axis, as shown in the figure. And from them we draw straight lines until they intersect with the lines lowered from the points.

It is necessary to construct a line of intersection of surfaces of revolution - a cone with a cylinder of revolution. The rotation axes of these surfaces are mutually perpendicular and project according to the projection planes.

To solve such a problem in descriptive geometry, you need to know:

— construction of surfaces of rotation on a complex drawing
at given point coordinates;

— special cases of intersection of a cone and a cylinder of rotation by a projecting plane;

— cutting plane method for constructing an intersection line
surfaces.

Procedure for solving the Problem

1. On the right side of a sheet of A3 paper, according to the assignment option, outlines of the surfaces of a cone and cylinder of rotation are constructed in horizontal and frontal projections.

Fig.8.1

Looking at the resulting drawing, it is easy to notice that the line of intersection of these surfaces already exists in the frontal plane of projections, i.e. it is specified by the original drawing, highlight it in red (the desired line). Thus, to solve the problem, it remains to project (transfer) it onto a horizontal plane.

2. We begin constructing the intersection line by marking the reference points. These are points above (below) which to the right (to the left) there is no intersection line; note, by the way, that the intersection line can only be located in places that simultaneously belong to both surfaces.

The reference points on the frontal projection will be 1’ and 6'. Finding them on a horizontal projection is not difficult. They will be located on the outermost generatrices of the cone, which are projected onto this plane by a straight line Sb. Transferring them along communication lines, we get 1 and 5 (Fig.8.2.a).

Fig.8.2

3. Next, we use the cutting plane method, which can be drawn at a certain interval or through characteristic points of the intersection line, draw the first cutting plane ’ through the point 2’ . From special cases it is known that if the secant plane in the frontal projection intersects the cone perpendicular to the axis of rotation, then in the horizontal plane the section will be in the form of a circle with a radius taken from the axis of rotation to the outline of the surface (the extreme right or left generatrix). Draw the indicated circle of given radius R a in a horizontal plane, placing the leg of the compass in the center of the conical surface. Since dot 2 simultaneously belongs to a conical and cylindrical surface and is located in a secant plane, then its horizontal projection must be at the intersection of horizontal projections from the secant plane along the cone and cylinder.

It has already been noted that the horizontal projection from the cutting plane, along a cone - a circle; and along the cylinder - a straight line, because the cutting plane runs parallel to the axis of rotation of the cylinder.

Then from the projection of the point 2’ draw a connection line (straight line of the section of the cylinder) of its intersection with the circle and obtain horizontal projections of the point 2 . Obviously, there will be two projections of the point: one on the front side of cone 2 (the lowest point in the horizontal projection plane), the second on the back side of the cone surface 2 1 (top point in the horizontal projection plane) ( Fig.8.2.b).

4. T In exactly the same way we find the horizontal projections of the remaining points 4 and 5, i.e. through their frontal projections we draw secant planes, in the horizontal plane of projections we draw the corresponding circles onto which we project the indicated points ( Fig.8.3 - b).

5. The resulting horizontal projections of the points are connected sequentially by a smooth line, taking into account visibility, which is determined relative to both surfaces. Visibility along the cone will be complete, since in a horizontal projection any point lying on its surface will be visible. Visibility along the cylinder is determined in such a way that all points located above the diameter of the cylinder on the frontal projection will be visible on the horizontal projection, and all points located below the diameter of the cylinder on the frontal projection will be invisible on the horizontal ( Fig.8.3-b).

So, in the horizontal plane of the point 1, 2, 3 will be visible, and the points 4, 5, 6 will be invisible at the point 3 (3; 3 1) there is a change in visibility. By connecting the visible points with a contour line and the invisible ones with a dotted line, we obtain the desired line of intersection of the given surfaces.

Fig.8.3

In conclusion, we note two comments:

1. In practice and in variants of tasks, there are so-called complete and incomplete intersections of surfaces. In case of incomplete intersection, when one surface does not completely intersect the other (in our case), the intersection line is one closed loop; at a complete intersection, when one surface completely intersects another, the intersection line breaks up into several closed branches and there will be as many of them as there are complete intersections of sections of given surfaces. The proposed variants of tasks consider problems with 2-3 loops of the intersection line. Their construction is the same as the construction considered ( Fig.8.4)

Fig.8.4

2. The proposed problems on the intersection of surfaces can be solved by the generatrix method, when a number of generators are drawn through a given line of intersection of surfaces, the points of intersection of these generators with the given intersection line are marked, then these generators, together with the points on them, are projected onto the conjugate projection plane.

Section: Descriptive Geometry /