An external conjugation is considered to be a conjugation in which the centers of mating circles (arcs) O 1 (radius R 1) and O 2 (radius R 2) are located behind the mating arc of radius R. An example is used to consider the external conjugation of arcs (Fig. 5). First we find the center of conjugation. The center of conjugation is the point of intersection of arcs of circles with radii R+R 1 and R+R 2, constructed from the centers of circles O 1 (R 1) and O 2 (R 2), respectively. Then we connect the centers of circles O 1 and O 2 with straight lines to the center of the conjugation, point O, and at the intersection of the lines with the circles O 1 and O 2 we obtain the conjugation points A and B. After this, from the center of the conjugation we build an arc of a given conjugation radius R and connect it points A and B.
Figure 5. External mate of circular arcs
Internal mate of circular arcs
An internal conjugation is a conjugation in which the centers of the mating arcs O 1, radius R 1, and O 2, radius R 2, are located inside the conjugate arc of a given radius R. Figure 6 shows an example of constructing an internal conjugation of circles (arcs). First, we find the center of conjugation, which is point O, the point of intersection of arcs of circles with radii R-R 1 and R-R 2 drawn from the centers of circles O 1 and O 2, respectively. Then we connect the centers of circles O 1 and O 2 with straight lines to the mate center and at the intersection of the lines with the circles O 1 and O 2 we obtain the mate points A and B. Then from the mate center we construct a mate arc of radius R and construct a mate.
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Figure 6. Internal mate of circular arcs |
Figure 7. Mixed mate of circular arcs |
Mixed mate of circular arcs
A mixed conjugation of arcs is a conjugation in which the center of one of the mating arcs (O 1) lies outside the conjugate arc of radius R, and the center of the other circle (O 2) lies inside it. Figure 7 shows an example of a mixed conjugation of circles. First, we find the center of the mate, point O. To find the center of the mate, we build arcs of circles with radii R+ R 1, from the center of a circle of radius R 1 of the point O 1, and R-R 2, from the center of a circle of radius R 2 of the point O 2. Then we connect the conjugation center point O with the centers of circles O 1 and O 2 by straight lines and at the intersection with the lines of the corresponding circles we obtain the conjugation points A and B. Then we build the conjugation.
Cam construction
The construction of the outline of the cam in each variant should begin with drawing the coordinate axes Oh And OU. Then the pattern curves are constructed according to their specified parameters and the areas included in the outline of the cam are selected. After this, you can draw smooth transitions between pattern curves. It should be taken into account that in all variants through the point D is tangent to the ellipse.
Designation Rx shows that the magnitude of the radius is determined by construction. On the drawing instead Rx You must enter the corresponding number with the “*” sign.
Pattern called a curve that cannot be constructed using a compass. It is built point by point using a special tool called a pattern. Pattern curves include ellipse, parabola, hyperbola, Archimedes' spiral, etc.
Among the regular curves, the ones of greatest interest for engineering graphics are second-order curves: ellipse, parabola and hyperbola, with the help of which surfaces that limit technical details are formed.
Ellipse- second order curve. One of the ways to construct an ellipse is the method of constructing an ellipse along two axes in Fig. 8. When constructing, we draw circles of radii r and R from one center O and an arbitrary secant OA. From intersection points 1 and 2 we draw straight lines parallel to the axes of the ellipse. At their intersection we mark point M of the ellipse. We construct the remaining points in the same way.
Parabola called a plane curve, each point of which is located at the same distance from a given straight line, called the directrix, and a point called the focus of the parabola, located in the same plane.
Figure 9 shows one way to construct a parabola. Given is the vertex of the parabola O, one of the points of the parabola A and the direction of the axis – OS. A rectangle is built on the segment OS and CA, the sides of this rectangle in the task are A1 and B1, they are divided into an arbitrary equal number of equal parts and the division points are numbered 1, 2, 3, 4... 10. Vertex O is connected to the division points on A1, and from points of division of segment B1 are drawn in straight lines parallel to the OS axis. The intersection of lines passing through points with the same numbers determines a number of points of the parabola.
Sine wave called a flat curve depicting the change in sine depending on the change in its angle. To construct a sinusoid (Fig. 10), you need to divide the circle into equal parts and divide the straight line segment into the same number of equal parts AB = 2lR. From the dividing points of the same name, draw mutually perpendicular lines, at the intersection of which we obtain points belonging to the sinusoid.
Figure 10. Construction of a sinusoid
Involute called a flat curve, which is the trajectory of any point on a straight line that rolls around a circle without sliding. The involute is constructed in the following order (Fig. 11): the circle is divided into equal parts; draw tangents to the circle, directed in one direction and passing through each division point; on the tangent drawn through the last point of dividing the circle, lay a segment equal to the length of the circle 2 l R, which is divided into as many equal parts. One division is laid on the first tangent 2 l R/n, on the second - two, etc.
Archimedes spiral– a flat curve, which is described by a point moving uniformly progressively from the center O along a uniformly rotating radius (Fig. 12).
To construct an Archimedes spiral, the spiral pitch is set - a, and the center O. From the center O, a circle of radius P = a (0-8) is described. Divide the circle into several equal parts, for example, into eight (points 1, 2, ..., 8). The segment O8 is divided into the same number of parts. From the center O with radii O1, O2, etc. draw arcs of circles, the points of intersection of which with the corresponding radius vectors belong to the spiral (I, II, ..., YIII)
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When constructing the conjugation of two circular arcs with a third arc of a given radius, three cases can be considered: when the conjugating arc of radius R touches given arcs of radii R 1 And R 2 from the outside (Figure 36, a); when she creates an internal touch (Figure 36, b); when internal and external touches are combined (Figure 36, c).
Building a center ABOUT conjugate arc radius R when touching externally, it is carried out in the following order: from the center O 1 radius equal to R + R 1, draw an auxiliary arc, and from the center O2 draw a pilot arc with a radius R + R 2 . At the intersection of the arcs the center is obtained ABOUT conjugate arc radius R, and at the intersection with radius R + R 1 And R + R 2 s arcs of circles are used to obtain connecting points A And A 1.
Building a center ABOUT when touching internally, it differs in that from the center O 1 R- R 1 a from the center O 2 radius R- R2. When combining internal and external touch from the center O 1 draw an auxiliary circle with a radius equal to R- R1, and from the center O 2- radius equal to R + R 2 .
Figure 36 – Conjugation of circles with an arc of a given radius
Conjugation of a circle and a straight line with an arc of a given radius
Two cases can be considered here: external coupling (Figure 37, A) and internal (Figure 37, b). In both cases, when constructing a conjugate arc of radius R mate center ABOUT lies at the intersection of the locus of points equidistant from a straight line and an arc of radius R by the amount R1.
When constructing an external fillet parallel to a given straight line at a distance R 1 draw an auxiliary line towards the circle, and from the center ABOUT radius equal to R + R 1,- an auxiliary circle, and at their intersection a point is obtained O 1- center of the conjugate circle. From this center with a radius R draw a conjugate arc between points A And A 1, the construction of which can be seen from the drawing.
Figure 37 - Conjugation of a circle and a straight line with a second arc
The construction of an internal conjugation differs in that from the center ABOUT draw an auxiliary arc with a radius equal to R- R1.
Ovals
Smooth convex curves outlined by circular arcs of different radii are called ovals. Ovals consist of two support circles with internal mates between them.
There are three-center and multi-center ovals. When drawing many parts, such as cams, flanges, covers and others, their contours are outlined with ovals. Let's consider an example of constructing an oval along given axes. Let for a four-center oval outlined by two supporting arcs of radius R and two conjugate arcs of radius r , major axis is specified AB and minor axis CD. The size of the radii R u r must be determined by construction (Figure 38). Connect the ends of the major and minor axis with segment A WITH, on which we plot the difference SE major and minor semi-axes of the oval. Draw a perpendicular to the middle of the segment AF, which will intersect the major and minor axes of the oval at points O 1 And O 2. These points will be the centers of the conjugating arcs of the oval, and the conjugating point will lie on the perpendicular itself.
Figure 38 – Constructing an oval
Pattern curves
Patterned are called flat curves drawn using patterns from previously constructed points. Pattern curves include: ellipse, parabola, hyperbola, cycloid, sinusoid, involute, etc.
Ellipse is a closed plane curve of the second order. It is characterized by the fact that the sum of the distances from any of its points to two focal points is a constant value equal to the major axis of the ellipse. There are several ways to construct an ellipse. For example, you can construct an ellipse from its largest AB and small CD axes (Figure 39, A). On the axes of the ellipse, as on diameters, two circles are constructed, which can be divided by radii into several parts. Through the division points of the great circle, straight lines are drawn parallel to the minor axis of the ellipse, and through the division points of the small circle, straight lines are drawn parallel to the major axis of the ellipse. The intersection points of these lines are the points of the ellipse.
You can give an example of constructing an ellipse using two conjugate diameters (Figure 39, b) MN and KL. Two diameters are called conjugate if each of them bisects chords parallel to the other diameter. A parallelogram is constructed on conjugate diameters. One of the diameters MN divided into equal parts; The sides of the parallelogram parallel to the other diameter are also divided into the same parts, numbering them as shown in the drawing. From the ends of the second conjugate diameter KL Rays are passed through the division points. At the intersection of rays of the same name, ellipse points are obtained.
Figure 39 – Construction of an ellipse
Parabola called an open curve of the second order, all points of which are equally distant from one point - the focus and from a given straight line - the directrix.
Let's consider an example of constructing a parabola from its vertex ABOUT and any point IN(Figure 40, A). WITH for this purpose a rectangle is built OABC and divide its sides into equal parts, drawing rays from the division points. At the intersection of rays of the same name, parabola points are obtained.
You can give an example of constructing a parabola in the form of a curve tangent to a straight line with points given on them A And IN(Figure 40, b). The sides of the angle formed by these straight lines are divided into equal parts and the division points are numbered. Points of the same name are connected by straight lines. The parabola is drawn as the envelope of these lines.
Figure 40 – Construction of a parabola
Hyperbole called a flat, open curve of the second order, consisting of two branches, the ends of which move away to infinity, tending to their asymptotes. A hyperbola is distinguished by the fact that each point has a special property: the difference in its distances from two given focal points is a constant value equal to the distance between the vertices of the curve. If the asymptotes of a hyperbola are mutually perpendicular, it is called isosceles. An equilateral hyperbola is widely used to construct various diagrams when one point is given its coordinates M(Figure 40, V). In this case, lines are drawn through a given point AB And KL parallel to the coordinate axes. From the obtained intersection points, lines are drawn parallel to the coordinate axes. At their intersection, hyperbolic points are obtained.
Cycloid called a curved line representing the trajectory of a point A when rolling a circle (Figure 41). To construct a cycloid from the initial position of a point A set aside a segment AA], mark the intermediate position of the point A. So, at the intersection of a line passing through point 1 with a circle described from the center O 1, get the first point of the cycloid. By connecting the constructed points with a smooth straight line, a cycloid is obtained.
Figure 41 – Construction of a cycloid
Sine wave called a flat curve depicting the change in sine depending on the change in its angle. To construct a sinusoid (Figure 42), you need to divide the circle into equal parts and divide the straight line segment into the same number of equal parts AB = 2lR. From the dividing points of the same name, draw mutually perpendicular lines, at the intersection of which we obtain points belonging to the sinusoid.
Figure 42 – Construction of a sinusoid
Involute called a flat curve, which is the trajectory of any point on a straight line that rolls around a circle without sliding. The involute is constructed in the following order (Figure 43): the circle is divided into equal parts; draw tangents to the circle, directed in one direction and passing through each division point; on the tangent drawn through the last point of dividing the circle, lay a segment equal to the length of the circle 2 l R, which is divided into as many equal parts. One division is laid on the first tangent 2 l R/n, on the second - two, etc.
The resulting points are connected by a smooth curve and the involute of the circle is obtained.
Figure 43 – Construction of an involute
Self-test questions
1 How to divide a segment into any equal number of parts?
2 How to divide an angle in half?
3 How to divide a circle into five equal parts?
4 How to construct a tangent from a given point to a given circle?
5 What is called pairing?
6 How to connect two circles with an arc of a given radius from the outside?
7 What is called an oval?
8 How is an ellipse constructed?
Chapter 3. SOME GEOMETRIC CONSTRUCTIONS
§ 14. General information
When performing graphic work, you have to solve many construction problems. The most common tasks in this case are dividing line segments, angles and circles into equal parts, constructing various connections of lines with arcs of circles and arcs of circles with each other. Conjugation is the smooth transition of a circular arc into a straight line or into the arc of another circle.
The most common tasks involve constructing the following conjugations: two straight lines with a circular arc (rounding corners); two arcs of circles in a straight line; two arcs of circles with a third arc; arc and a straight second arc.
The construction of mates is associated with the graphic determination of centers and points of mate. When constructing a conjugation, geometric locations of points are widely used (straight lines tangent to a circle; circles tangent to each other). This is because they are based on the principles and theorems of geometry.
10. Self-test questions
SELF-TEST QUESTIONS
15. Which plane curve is called an involute?
15. Division of a line segment
§ 15. Division of a line segment
To divide a given segment AB into two equal parts, the points of its beginning and end are taken as the centers from which arcs are drawn with a radius exceeding half the segment AB. Arcs are drawn to mutual intersection, where points are obtained WITH And D. A line connecting these points will divide the segment at the point TO into two equal parts (Fig. 30, A).
To split a line AB for a given number of equal sections P, at any acute angle to AB draw an auxiliary straight line, on which they lay off from a common given straight point P equal sections of arbitrary length (Fig. 30, b). From the last point (sixth in the drawing) draw a straight line to the point IN and through points 5, 4, 3, 2, 1 draw straight lines parallel to the segment 6B. These straight lines will cut off on the segment AB a given number of equal segments (in this case 6).
Rice. 30 Dividing a given segment AB into two equal parts
Image:
16. Dividing a circle
§ 16. Division of a circle
To divide a circle into four equal parts, draw two mutually perpendicular diameters: at their intersection with the circle we get points dividing the circle into four equal parts (Fig. 31, a).
To divide a circle into eight equal parts, arcs equal to a quarter of the circle are divided in half. To do this, from two points limiting a quarter of the arc, as from the centers of the radii of a circle, notches are made beyond its boundaries. The resulting points are connected to the center of the circles and at their intersection with the line of the circle, points are obtained that divide the quarter sections in half, i.e., eight equal sections of the circle are obtained (Fig. 31, b).
The circle is divided into twelve equal parts as follows. Divide the circle into four parts with mutually perpendicular diameters. Taking the points of intersection of the diameters with the circle A, B, C, D beyond the centers, four arcs of the same radius are drawn until they intersect with the circle. Resulting points 1, 2, 3, 4, 5, 6, 7, 8 and points A, B, C, D divide the circle into twelve equal parts (Fig. 31, c).
Using the radius, it is not difficult to divide the circle into 3, 5, 6, 7 equal sections.
Rice. 31 Using the radius, it is easy to divide the circle into several equal sections.
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17. Rounding corners
§ 17. Rounding corners
The conjugation of two intersecting straight lines with an arc of a given radius is called corner rounding. It is performed as follows (Fig. 32). Parallel to the sides of the angle formed by the data
straight lines, draw auxiliary straight lines at a distance equal to the radius. The intersection point of the auxiliary lines is the center of the fillet arc.
From the received center ABOUT they lower perpendiculars to the sides of a given angle and at their intersection they obtain connecting points A a B. Between these points draw a conjugate arc with a radius R from the center ABOUT.
Rice. 32 The conjugation of two intersecting straight lines with an arc of a given radius is called rounding corners
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18. Conjugation of circular arcs with a straight line
§ 18. Conjugation of circular arcs with a straight line
When constructing the conjugation of circular arcs with a straight line, two problems can be considered: the conjugate straight line has an external or internal tangency. In the first problem (Fig. 33, A) from the center of the arc
smaller radius R1 draw a tangent to the auxiliary circle drawn by the radius R- R.I. Her point of contact Co. used to construct a junction point A on an arc of radius R.
To obtain the second mate point A 1 on an arc of radius R 1 draw an auxiliary line O 1 A 1 parallel O A. Points A and A 1 the section of the external tangent line will be limited.
The task of constructing an internal tangent line (Fig. 33, b) can be solved if an auxiliary circle is constructed with a radius equal to R + R 1,
Rice. 33 Conjugation of circular arcs with a straight line
Image:
19. Conjugation of two circular arcs with a third arc
§ 19. Conjugation of two arcs of circles with a third arc
When constructing the conjugation of two circular arcs with a third arc of a given radius, three cases can be considered: when the conjugating arc of radius R touches given arcs of radii R 1 And R 2 from the outside (Fig. 34, a); when it creates an internal touch (Fig. 34, b); when internal and external touches are combined (Fig. 34, c).
Building a center ABOUT conjugate arc radius R when touching externally, it is carried out in the following order: from the center O 1 radius equal to R + R 1, draw an auxiliary arc, and from the center O2 draw a pilot arc with a radius R + R 2 . At the intersection of the arcs the center is obtained ABOUT conjugate arc radius R, and at the intersection with radius R + R 1 And R + R 2 s arcs of circles are used to obtain connecting points A And A 1.
Building a center ABOUT when touching internally, it differs in that from the center O 1 R- R 1 a from the center O 2 radius R- R2. When combining internal and external touch from the center O 1 draw an auxiliary circle with a radius equal to R- R1, and from the center O 2- radius equal to R + R 2 .
20. Conjugation of a circular arc and a straight line with a second arc
§ 20. Conjugation of a circular arc and a straight line with a second arc
Here two cases can be considered: external coupling (Fig. 35, a) and internal (Fig. 35, b). In both cases, when constructing a conjugate arc of radius R mate center ABOUT lies at the intersection of the locus of points equidistant from a straight line and an arc of radius R by the amount R1.
When constructing an external fillet parallel to a given straight line at a distance R 1 draw an auxiliary line towards the circle, and from the center ABOUT radius equal to R + R 1,- an auxiliary circle, and at their intersection a point is obtained O 1- center of the conjugate circle. From this center with a radius R draw a conjugate arc between points A And A 1, the construction of which can be seen from the drawing.
The construction of an internal conjugation differs in that from the center ABOUT draw an auxiliary arc with a radius equal to R- R1.
Fig 34 External conjugation of a circular arc and a straight line with a second arc
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Fig. 35 Internal conjugation of a circular arc and a straight line with a second arc
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21. Ovals
§21. Ovals
Smooth convex curves outlined by circular arcs of different radii are called ovals. Ovals consist of two support circles with internal mates between them.
There are three-center and multi-center ovals. When drawing many parts, such as cams, flanges, covers and others, their contours are outlined with ovals. Let's consider an example of constructing an oval along given axes. Let for a four-center oval outlined by two supporting arcs of radius R and two conjugate arcs of radius r , major axis is specified AB and minor axis CD. The size of the radii R u r must be determined by construction (Fig. 36). Connect the ends of the major and minor axis with segment A WITH, on which we plot the difference SE major and minor semi-axes of the oval. Draw a perpendicular to the middle of the segment AF, which will intersect the major and minor axes of the oval at points O 1 And O 2. These points will be the centers of the conjugating arcs of the oval, and the conjugating point will lie on the perpendicular itself.
Rice. 36 Smooth convex curves outlined by arcs of circles of different radii are called ovals
22. Pattern curves
§ 22. Pattern curves
Patterned are called flat curves drawn using patterns from previously constructed points. Pattern curves include: ellipse, parabola, hyperbola, cycloid, sinusoid, involute, etc.
Ellipse is a closed plane curve of the second order. It is characterized by the fact that the sum of distances from any of its
Rice. 37
points up to two focal points is a constant value equal to the major axis of the ellipse. There are several ways to construct an ellipse. For example, you can construct an ellipse from its largest AB and small CD axes (Fig. 37, a). On the axes of the ellipse, as on diameters, two circles are constructed, which can be divided by radii into several parts. Through the division points of the great circle, straight lines are drawn parallel to the minor axis of the ellipse, and through the division points of the small circle, straight lines are drawn parallel to the major axis of the ellipse. The intersection points of these lines are the points of the ellipse.
You can give an example of constructing an ellipse using two conjugate diameters (Fig. 37, b ) MN and KL. Two diameters are called conjugate if each of them bisects chords parallel to the other diameter. A parallelogram is constructed on conjugate diameters. One of the diameters MN divided into equal parts; The sides of the parallelogram parallel to the other diameter are also divided into the same parts, numbering them as shown in the drawing. From the ends of the second conjugate diameter KL Rays are passed through the division points. At the intersection of rays of the same name, ellipse points are obtained.
Parabola called an open curve of the second order, all points of which are equally distant from one point - the focus and from a given straight line - the directrix.
Let's consider an example of constructing a parabola from its vertex ABOUT and any point IN(Fig. 38, A). WITH for this purpose a rectangle is built OABC and divide its sides into equal parts, drawing rays from the division points. At the intersection of rays of the same name, parabola points are obtained.
You can give an example of constructing a parabola in the form of a curve tangent to a straight line with points given on them A And IN(Fig. 38, b). The sides of the angle formed by these straight lines are divided into equal parts and
division points are measured. Points of the same name are connected by straight lines. The parabola is drawn as the envelope of these lines.
A hyperbola is a flat, unclosed curve of the second order, consisting of two branches, the ends of which move off to infinity, tending to their asymptotes. A hyperbola is distinguished by the fact that each point has a special property: the difference in its distances from two given focal points is a constant value equal to the distance between the vertices of the curve. If the asymptotes of a hyperbola are mutually perpendicular, it is called isosceles. An equilateral hyperbola is widely used to construct various diagrams when one point is given its coordinates M(Fig. 38, V). In this case, lines are drawn through a given point AB And KL parallel to the coordinate axes. From the obtained intersection points, lines are drawn parallel to the coordinate axes. At their intersection, hyperbolic points are obtained.
The center of the mating arc must be equidistant (located at the same distance) from each of the two mating (given) lines. Any of the junction points (entry points) represents the intersection of a perpendicular dropped from the junction center to the corresponding straight line.
The algorithm for constructing the conjugation of two straight lines with an arc of a given radius (Fig. 13.39, a, b) is as follows:
1. At a distance ( R), equal to the radius of the mating arc, draw two straight lines parallel to the mating straight lines.
2. Determine their point of intersection, which is the center of mating ( ABOUT).
3. From point ( ABOUT) draw perpendiculars to the given straight lines and find the connecting points ( A) And ( IN).
4. From point ( A) to point ( IN) construct a conjugation arc of a given radius ( R).
Figure 13.49
Typical examples of mates are the contours of the parts shown in Fig. 13.40.
In AutoCAD, the pairing of two straight segments (Fig. XX a) is performed by the “Mate” command (Fillet, Key, Fillet) from the “Modification” menu. After selecting the command, use the “Radius” parameter to set the conjugation radius (for example, 10 mm), then successively mark both segments with the mouse pointer (see Fig. XX b).
Current settings: Mode = TRIM, Radius = 5.0000
radius
Specify fillet radius<5.0000>: 10
Select first object or :
Select second object:
The resulting element consists of two initial segments and a mating arc R=10mm (see Fig. XX c).
Rice. XX a) Fig. XX b) Fig. XX century)
1.2. Radius Circle Arc Fillet R and straight A with an arc of a given radius R1
To perform this conjugation (Fig. 3.31), first determine the set of centers of arcs of radius R 1. To do this at a distance R 1 from the straight line A draw a line parallel to it m, and from the center ABOUT radius ( R + R 1) – arcs of a concentric circle. Dot O 1 will be the center of the mating arc. Mating point WITH obtained on a perpendicular dropped from a point O 1 directly A, and point IN– on a straight line connecting points ABOUT And O 1.
Figure 3.31
In Fig. Figure 3.32 shows an example of an image of a bearing contour, in the construction of which the considered type of interfaces was used.
Figure 3.32
Conjugating a line and a circle in AutoCAD makes sense when constructing a line segment to a circle that is tangent to this circle. To do this, when constructing a segment, the starting point of the segment is set by coordinates or an object snap, the end point is set by the “Tangent” snap (Jump to tangent) relative to the circle (working with snapping is described in Appendix XXXXXXXXXXX).
1.3. Conjugation of arcs of two circles with radii R1 And R2, arc of conjugation of radius R
There are external (Fig. 13.42, a), internal (Fig. 13.42, b) and mixed (Fig. 13.42, c) conjugations. In the first case, the center of mate is the intersection point of the arc of circles of radii R 1 +R And R 2 + R, in the second - at the intersection of circles of radii R-R 1 And R-R 2, in the third - at the intersection of arcs of circles of radii R+R 1 And R-R 2. Mating points A 1 And A 2 lie on straight lines connecting the center of conjugation with the center of the corresponding circle.
Let's consider the case of external conjugation of two circles in AutoCAD. In Fig. XX.a shows two reference circles with radii R 1 and R 2, the centers of which lie at the ends of the dotted line. From the center of the circle R 1, an auxiliary circle with radius R 1 + R is constructed, and from the center of the circle R 2, a circle R 2 + R is built, as shown in Fig. XX.b (auxiliary circles are shown with a dashed line). Then, from the intersection point of the auxiliary circles, a circle with radius R is constructed (in Fig. XX c it is shown as a dashed-dotted line). The final constructions are performed using the “Crop” command from the “Modification” menu. Support circles are selected as secant objects and the upper part of circle R is cut off, then auxiliary circles are removed (the result of the construction is shown in Fig. XX.d).
Figure XX.a Figure XX.b
Figure XX.c Figure XX.d
Now let's look at the case of the internal conjugation of two circles in AutoCAD. Similar to the previous case, support circles with radii R 1 and R 2 are constructed. From the center of the circle R 1, an auxiliary circle with radius R–R 1 is built, and from the center of the circle R 2, a circle R–R 2 is built. Then, from the intersection point of the auxiliary circles, a circle with radius R is constructed (see Fig. XXX.a). Excess elements are removed similarly to the previous case (the result is shown in Fig. XXX.b).
Module: Graphic design of drawings.
Result 1: Be able to draw up formats of standard sheets in accordance with GOST 2.303 - 68. Have the skills to draw the contours of parts, be able to apply dimensions, be able to make inscriptions in accordance with GOST 2.303 - 68.
Result 2: Know the construction rules and have the skills to construct a pairing. Be able to explain the rules of construction.
1. Rules for formatting, rules for filling out the title block in accordance with the standard.
2. Rules for applying dimensions, types of lines.
3. Rules for making inscriptions in fonts in accordance with GOST 2.303 – 68.
4. Rules for drawing the contours of technical parts. Geometric constructions.
5. Rules for drawing and constructing connections.
Lesson topic: Rules for constructing mates.
Goals:
- Know the definition of a mate, types of mates.
- Be able to build connections and explain the construction process.
- Develop technical literacy.
- Develop skills in group work and independent work.
- Cultivate a respectful attitude towards the speaker and the ability to listen.
DURING THE CLASSES
1. Organizational and motivational stage –10 minutes.
1.1. Student motivation:
- connection with other objects;
- consideration of parts, geometric bodies from which parts are composed and connections between them (smooth transitions from one line to another);
1.2. Dividing the group into subgroups of 5-6 people (into four subgroups).
All students in the group are asked to choose one from four types of geometric shapes; after the choice is made, the students are united into subgroups to work independently in subgroups.
Students are told what topic they have to study, get acquainted with the rules for constructing conjugations, which will help them understand how smooth transitions (conjugations) are constructed. Each group is invited to study and present one of the types of pairing (the teacher distributes material on the topic of the lesson to each section in sections).
2. Organization of independent activities of students on the topic of the lesson – 25 minutes.
2.1. The concept of pairing.
2.2. General algorithm for constructing mates.
2.3. Types of pairing. Rules for their construction.
2.3.1. Conjugation between two straight lines.
2.3.2. Internal and external conjugation between a straight line and an arc of a circle.
2.3.3. Conjugation internally and externally between two arcs of circles.
2.3.4. Mixed pairing.
3. Summing up, group reports on the topic after independent work in subgroups - 25 minutes.
4. Checking the degree of mastery of the material – 10 minutes.
5. Filling out diaries (about the lesson) – 5 minutes.
6. Evaluation of student activities.
Conjugation is a smooth transition from one line to another.
3. Construct a conjugation (smooth transition from one line to another)
2. 3.1. Constructing a conjugation of two sides of an angle of a circle of a given radius.
The conjugation of two sides of an angle (acute and obtuse) with an arc of a given radius R is performed as follows:
Two auxiliary straight lines are drawn parallel to the sides of the angle at a distance equal to the radius of the arc R. The intersection point of these lines (point O) will be the center of an arc of radius R, that is, the center of conjugation. From point O they describe an arc that smoothly turns into straight lines - the sides of the angle. The arc ends at the connecting points n and n1, which are the bases of the perpendiculars drawn from the center O to the sides of the angle. When constructing a mating of the sides of a right angle, it is easier to find the center of the mating arc using a compass. From the vertex of angle A, an arc of radius R is drawn until mutual intersection at point O, which is the center of conjugation. From the center O, describe the conjugation arc. The construction of the pairing of two sides of the angle is shown in Fig. 1.
General algorithm for constructing a pairing:
1. It is necessary to find the junction point.
2. It is necessary to find the connecting points.
3. Construction of a conjugation (smooth transition from one line to another).
2.3.2 Construction of internal and external connections between a straight line and a circular arc.
The conjugation of a straight line with a circular arc can be performed using an arc with an internal tangency of the arc and an external tangency. Figure 2(a, b) shows the conjugation of a circular arc of radius R and a straight line AB by a circular arc of radius r with an external tangency. To construct such a conjugation, draw a circle of radius R and a straight line AB. A straight line ab is drawn parallel to a given straight line at a distance equal to the radius r (radius of the conjugate arc). From the center O, draw an arc of a circle with a radius equal to the sum of the radii R and r until it intersects the straight line ab at point O1. Point O1 is the center of the mating arc. The conjugation point c is found at the intersection of straight line OO1 with a circular arc of radius R. Conjugation point O1 to this straight line AB. Using similar constructions, points O2, c2, c3 can be found. Figure 2(a, b) shows a bracket, when drawing it it is necessary to carry out the construction described above.
When drawing a flywheel, an arc of radius R is paired with a straight arc AB of radius r with an internal tangency. The center of the conjugation arc O1 is located at the intersection of an auxiliary line drawn parallel to this line at a distance r with the arc of an auxiliary circle described from the center O with a radius equal to the difference R-r. The point of conjugation with 1 is the base of the perpendicular dropped from point O1 to this line. The mating point c is found at the intersection of straight line OO1 with the mating arc. An example of constructing a connection between a straight line and a circular arc is shown in Figure 3.
Conjugation is a smooth transition from one line to another.
General algorithm for constructing a pairing:
1. It is necessary to find the center of the mate.
2. It is necessary to find the connecting points.
3. Construction of a conjugation line (smooth transition from one line to another).
2.3.3. Constructing a conjugation between two arcs of circles.
The conjugation of two arcs of circles can be internal or external.
With internal conjugation, the centers O and O1 of the mating arcs are located inside the mating arc of radius R. With external conjugation, the centers O and O1 of the mating arcs of radii R1 and R2 are located outside the mating arc of radius R.
Constructing an external interface:
a) radii of mating circles R and R1;
Required:
Shown in Figure 4(b). According to the given distances between the centers, centers O and O1 are marked in the drawing, from which conjugate arcs of radii R and R1 are described. From the center O1, draw an auxiliary arc of a circle with a radius equal to the difference between the radii of the mating arc R and the mating arc R2, and from the center O - with a radius equal to the difference in the radii of the mating arc R and the mating arc R1. The auxiliary arcs will intersect at point O2, which will be the desired center of the connecting arc. To find the points of intersection of the continuation of straight lines O2O and O2O1 with the mating arcs, the required conjugation points (points s and s1) are used.
Construction of internal interface:
a) radii R and R1 of mating circular arcs;
b) the distances between the centers of these arcs;
c) radius R of the mating arc;
Required:
a) determine the position O2 of the mating arc;
b) find the connecting points s and s1;
c) draw a mating arc;
The construction of the external interface is shown in Figure 4(c). Using given distances in the drawing, points O and O1 are found, from which conjugate arcs of radii R1 and R2 are described. From center O, draw an auxiliary arc of a circle with a radius equal to the sum of the radii of the mating arc R2 and the mating arc R. The auxiliary arcs will intersect at point O2, which will be the desired center of the mating arc. To find the connecting points, the centers of the arcs are connected by straight lines OO2 and O1O2. These two lines intersect the conjugate arcs at the conjugation points s and s1. From the center O2 with radius R, a conjugate arc is drawn, limiting it to points S and S1.
2.3.4. Construction of mixed conjugation.
An example of mixed pairing is shown in Figure 5.
a) The radii R and R1 of the mating mating arcs are specified;
b) the distances between the centers of these arcs;
c) radius R of the mating arc;
Required:
a) determine the position of the center O2 of the mating arc;
b) find the connecting points s and s1;
c) draw a mating arc;
According to the given distances between the centers, centers O and O1 are marked in the drawing, from which conjugate arcs of radii R1 and R2 are described. From the center O, an auxiliary arc of a circle is drawn with a radius equal to the sum of the radii of the mating arc R1 and the mating arc R, and from the center O1 - with a radius equal to the difference between the radii R and R2. The auxiliary arcs will intersect at point O2, which will be the desired center of the connecting arc. By connecting points O and O2 with a straight line, we obtain the conjugation point s1; connecting points O1 and O2, find the conjugation point s. From the center O2, a conjugation arc is drawn from s to s1. Figure 5 shows an example of constructing a mixed mate.
3. Summing up the results of students’ independent work in groups. Students' reports on each section of the lesson topic at the blackboard.
4. Checking the degree of student knowledge acquisition. Students from each group ask questions from students from the other group.
5. Filling out diaries. Each student is asked to fill out a diary at the end of the lesson.
In order to gain a good amount of knowledge, it is important to record how successfully the lesson went. This journal allows you to record every detail of your work during the lesson during the module. If you are satisfied, satisfied, disappointed with how your lesson went, then indicate your attitude towards the elements of the lesson in the appropriate cell of the questionnaire.
| Lesson elements | Satisfied |
Satisfied |
Disappointed |
