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 an oval?

8 How is an ellipse constructed?

Notebook entries are purple, yellow background - explanations

We read and understand that the font is black

We do what is not done in the notebook, if it is not there, then on A4 formats so that we can paste it into the notebook

Subject. Pairings.

The meaning of mates in technical drawing

Graphic work No. 5. Drawing of a technical part using mates. A4 format (210 × 297).

The smooth transition of one line to another is called conjugation. The point common to the mating lines is called the junction point, or transition point. To construct mates, you need to find the mate center and mate points. Let's look at the different types of mates.

Right angle conjugation. Let it be necessary to mate a right angle with a mate radius equal to segment AB (R=AB). Let's find the connecting points. To do this, we place the leg of the compass at the top of the angle and with a compass opening equal to segment AB, we make notches on the sides of the angle. The resulting points a and b are the conjugation points. Let's find the center of the junction - a point equidistant from the sides of the angle. Using a compass opening equal to the conjugation radius, from points a and b, draw two arcs inside the corner until they intersect with each other. The resulting point O is the center of mate. From the center of conjugation we describe an arc of a given radius from point a to point b. First we draw an arc, and then straight lines.

Conjugation of acute and obtuse angles.

To construct the conjugation of an acute angle, take a compass opening equal to the given radius R=AB. Let us alternately place the leg of the compass at two arbitrary points on each side of the acute angle. Let's draw four arcs inside the corner, the pattern is shown in the drawing. 71, a. We draw two tangents to them until they intersect at point O - the center of the junction (Fig. 71, b) - From the center of the junction we lower perpendiculars to the sides of the angle. The resulting points a and b will be the conjugation points (Fig. 71, b). Having placed the leg of the compass at the center of the mate (O), with a compass opening equal to the given radius of the mate (R=AB), we draw a mate arc.

Conjugation of two parallel lines.

Given two parallel lines and a point d lying on one of them (Fig. 72). Let's consider the sequence of constructing the conjugation of two straight lines. At point d we erect a perpendicular until it intersects with another line. Points d and e are conjugation points. Dividing the segment de in half, we find the center of conjugation. From it, with a conjugation radius, we draw an arc conjugating the straight lines.

Conjugation of arcs of two circles with an arc of a given radius.

There are several types of conjugation of arcs of two circles with an arc of a given radius: external, internal and mixed.

Construction internal interface.

A). radii of mating circles R1 and R2;

b). distance l1 and l2 between the centers of these arcs;

V). radius R of the connecting arc.

Required:

b).find the connecting points s1 and s2;

c).draw a mating arc.

According to the given distances between the centers l1 and l2, centers O and O1 are marked in the drawing, from which conjugate arcs of radii R1 and R2 are described. An auxiliary arc of a circle is drawn from the center O1 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 connecting points, point O2 is connected to points O and O1 by straight lines. The points of intersection of the continuation of lines O2O and O2O1 with the mating arcs are the required conjugation points (points s and s1).

With radius R from the center O2, draw a conjugating arc between the conjugating points s and s1.

Construction external pairing.

b).distance l1 and l2 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;

c).find the connecting points s and s1;

c).draw a mating arc.

The construction of an external interface is shown in Fig. 18, b. Using the given distances between the centers l1 and l2, points O and O1 are found in the drawing, from which conjugate arcs of radii R1 and R2 are described. From the center O, draw an auxiliary arc of a circle 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 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 O2O2. These two lines intersect the conjugate arcs at the conjugation points s and s1. From the center O2 with radius R, a conjugating arc is drawn, limiting it to the conjugating points s1 and s.

Building a mixed mate.

a).radii R1 and R2 of mating circular arcs;

b).distances l1 and l2 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 l1 and l2, centers O and O1 are marked in the drawing, from which conjugate arcs of radii R1 and R2 are described. An auxiliary arc of a circle is drawn from the center O 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.

When drawing the contour of a part, you need to figure out where there are smooth transitions and imagine where certain types of connections need to be made.

To acquire the skills of constructing interfaces, perform exercises on drawing the contours of complex parts. Before the exercise, you need to review the task, outline the order of constructing the interfaces, and only after that start making constructions.

Topic Pattern curves.

General information. Rules for using the pattern. Construction of pattern curves: ellipse, parabola, hyperbola, cycloids, sinusoids, involutes, Archimedes Spirals. Practical work. Exercise on constructing pattern curves

Box curved lines.

Some machine parts and tools for metal processing have contours limited by closed curved lines consisting of mutually mating arcs of circles of different diameters.

Box curves are curves formed by conjugating circular arcs. Such curves include ovals, ovoids, and curls.

Construction of an oval.

An oval is a closed box curve with two axes of symmetry.

The sequence of constructing an oval according to a given size of the major axis of the oval AB is carried out as follows (Fig. ,a). The AB axis is divided into three equal parts (AO1, O1O2, O2B). With a radius equal to O1O2, circles are drawn from the division points O1 and O2, intersecting at points m and n.

By connecting points n and m with points O1 and O2, we obtain straight lines nO1, nO2, mO1, mO2, which continue until they intersect with the circles. The resulting points 1,2,3, and 4 are the conjugation points of the arcs. From points m and n, as from centers, with radius R1 equal to n2 and m3, draw the upper arc 12 and the lower arc 34.

The AB and CD axes are drawn. From the point of their intersection with radius OS (half of the minor axis of the oval), an arc is drawn until it intersects with the major axis of the oval AB at point N. Point A is connected by a straight line to point C and on it a segment NB is laid off from point C, point N is obtained. In the middle of the segment AN1 restore the perpendicular and continue it until it intersects with the major and minor axes of the oval at points O1 and n. The distance OO1 is laid along the major axis of the oval to the right of point O, and the distance on from point O is laid along the minor axis of the oval upward, resulting in points n1 and O2. Points n and n1 are the centers of the upper arc 12 and the lower arc 34 of the oval, and points O1 and O2 are the centers of arcs 13 and 24. The desired oval is obtained.

Construction of curls.

A curl is a flat spiral curve drawn with a compass by connecting arcs of circles.

The construction of curls is carried out by drawing out parts such as springs and spiral guides.

Construction of an ovoid.

An ovoid is a closed box curve that has only one axis of symmetry. The radii R and R1 of circular arcs whose centers lie on the axis of symmetry of the ovoid are not equal to each other.

The construction of an ovoid along a given axis AB is performed in the following sequence.

Draw a circle with a diameter equal to the AB axis of the ovoid. Straight lines are drawn from points A and B through point O1 (the point of intersection of a circle of radius R with the axis of symmetry). From points A and B, as from centers, with radius R2 equal to the axis AB, draw arcs An and Bm, and from center O1 with radius R1 draw a small arc of the ovoid nm.

The construction of curls is made from two, three or more centers and depends on the shape and size of the “eye”, which can be a circle, a regular triangle, a hexagon, etc. The sequence of constructing a curl is as follows.

The outline of the “eye” is drawn in thin lines, for example a circle with a diameter of O1O2. From points O1 and O2, as from centers, draw two interconnected semicircles. Upper semicircle O21 from center O1, lower semicircle 12 from center O2. The desired curl is obtained.

Pattern curves.

When making drawings, you often have to resort to drawing curves consisting of a number of mating parts that cannot be drawn with a compass. Such curves are usually constructed from a number of points belonging to them, which are then connected with a smooth line, first by hand with a pencil, and then outlined using patterns.

The pattern curves under consideration are located in the same plane and are therefore called flat.

Pattern curves are widely used in mechanical engineering to outline various technical parts, for example: brackets, stiffeners, cams, gears, shaped tools, etc.

Pattern curves include ellipse, parabola, hyperbola, cycloid, epicycloid, involute, sinusoid, Archimedes spiral, etc.

Below we consider the methods for constructing curves most often found in technology.

Construction of an ellipse.

An ellipse is a closed plane curve, the sum of the distances of each point of which to two given points (foci) lying on the major axis is a constant value and equal to the length of the major axis.

A method widely used in technology for constructing an ellipse along the major (AB) and minor (CD) axes.

Draw two perpendicular center lines. Then, from the center O, segments equal to the length of the minor semi-axis are laid up and down along the vertical axis, and segments equal to the length of the semi-major axis are laid to the left and right along the horizontal axis.

From the center O with radii OA and OS, two concentric circles and a series of rays-diameters are drawn. From the points of intersection of the rays with the circles, lines are drawn parallel to the axes of the ellipse until they intersect each other at points belonging to the ellipse. The resulting points are connected by hand and traced along the pattern.

Construction of a parabola.

A parabola is a flat curve, each point of which is equidistant from the directrix DD1 of a straight line perpendicular to the axis of symmetry of the parabola, and from the focus of the F-point located on the axis of symmetry of the parabola.

The distance KF between the directrix and the focus is called the parameter p of the parabola. Point O, lying on the axis of symmetry, is called the vertex of the parabola and divides the parameter p in half.

To construct a parabola based on a given value of the parameter p, draw the axis of symmetry of the parabola (vertical in the figure) and plot the segment KF=p. Directrix DD1 is drawn through point K perpendicular to the axis of symmetry. The segment KF is divided in half and the vertex O of the parabola is obtained. From vertex O down on the axis of symmetry, a series of arbitrary points I-IV are marked with a gradually increasing distance between them. Through these points, auxiliary straight lines are drawn perpendicular to the axis of symmetry. On auxiliary lines from the focus F, serifs are made with a radius equal to the distance from the line to the directrix. For example, from point F on the auxiliary line passing through points V, a notch is made with the arc R1=KV; the resulting point 5 belongs to the parabola.

In machine tool building and other branches of mechanical engineering, parts are often used whose contour outlines are made along a parabola, for example, the stand and sleeve of a radial drilling machine.

Construction of a sinusoid.

A sine wave is a flat curve depicting a change in sine depending on a change in angle.

The quantity L is called the wavelength of the sinusoid, L=PR.

To construct a sinusoid, draw a horizontal axis and plot the given length AB on it (Fig. 24). The segment AB is divided into several equal parts, for example, 12. A circle is drawn on the left, the radius of which is equal to the amplitude, and it is also divided into 12 equal parts ; The division points are numbered and horizontal lines are drawn through them. From the division points of the segment AB, perpendiculars to the sine axis are constructed and at their intersection with the horizontal lines the sine points are found.

The resulting sinusoid points a1, a2, a3,... are connected according to the curve pattern.

When making drawings of parts or tools whose surfaces are outlined along a sinusoid, the value of the wavelength AB is usually chosen regardless of the size of the amplitude r. For example, when drawing a screw, the wavelength L is less than the size 2Pr. Such a sinusoid is called compressed. If the wavelength is greater than 2Pr, then the sinusoid is called elongated.

Construction of a hyperbola.

A hyperbola is a flat curve consisting of two open, symmetrically located branches (Fig. 25). The difference in distances from each point of the hyperbola to two given points (foci F and F1) is a constant value and equal to the distance between the vertices of the hyperbola A and B.

Let's consider the technique of constructing a hyperbola using given vertices A and B and focal length FF1

Dividing the focal length FF1 in half, we get point O, from which half of the specified distance between vertices A and B is laid out in both directions. Down from the focus F, a number of arbitrary points 1,2,3,4... are marked with a gradually increasing distance between them . From focus F, describe an arc of an auxiliary circle with radius R equal, for example, to the distance from the vertex of hyperbola B to point 3. From focus F1, draw a second arc of an auxiliary circle with radius r equal to the distance from vertex A to point 3. At the intersection of these arcs, points C are found and C1, belonging to the hyperbole. The remaining points of the hyperbola are found in the same way.

Conjugation of two parallel lines

Given two parallel lines and one of them has a conjugate point M(Fig. 2.19, A). You need to build a pairing.

• 1) find the center of mate and the radius of the arc (Fig. 2.19, b). To do this from the point M restore the perpendicular to the intersection with the line at the point N. Line segment MN divided in half (see Fig. 2.7);
• 2) from a point ABOUT– center of mate with radius OM = ON describe an arc from the connecting points M And N(Fig. 2.19, V).

Rice. 2.19.

Given a circle with center ABOUT and point A. It is required to draw from point A tangent to the circle.

1. Point A connect a straight line to a given center O of a circle.

Construct an auxiliary circle with a diameter equal to OA(Fig. 2.20, A). To find the center ABOUT 1, divide the segment OA in half (see Fig. 2.7).

2. Points M And N intersection of the auxiliary circle with the given one - the required points of tangency. Full stop A connect straight lines to points M or N(Fig. 2.20, b). Straight A.M. will be perpendicular to the line OM, since the angle AMO based on diameter.

Rice. 2.20.

Drawing a line tangent to two circles

Given two circles of radii R And R 1. It is required to construct a straight line tangent to them.

There are two cases of touch: external (Fig. 2.21, b) and internal (Fig. 2.21, V).

At external touch construction is performed as follows:

• 1) from the center ABOUT draw an auxiliary circle with a radius equal to the difference between the radii of the given circles, i.e. R–R 1 (Fig. 2.21, A). A tangent line is drawn to this circle from center O1 Ο 1Ν. The construction of the tangent is shown in Fig. 2.20;
• 2) radius drawn from point O to point Ν, continue until they intersect at the point M with a given circle radius R. Parallel to the radius OM draw radius Ο 1Ρ smaller circumference. Straight line connecting junction points M And R,– tangent to given circles (Fig. 2.21, b).

Rice. 2.21.

At inner touch the construction is carried out in a similar way, but the auxiliary circle is drawn with a radius equal to the sum of the radii R+R 1 (Fig. 2.21, V). Then from the center ABOUT 1 draw a tangent to the auxiliary circle (see Fig. 2.20). Full stop N connect with a radius to the center ABOUT. Parallel to the radius ON draw radius O1 R smaller circumference. The required tangent passes through the connecting points M And R.

Conjugation of an arc and a straight arc of a given radius

Given an arc of a circle of radius R and straight. It is required to connect them with an arc of radius R 1.

• 1. Find the center of mating (Fig. 2.22, A), which should be at a distance R 1 from the arc and from the straight line. Therefore, an auxiliary straight line is drawn parallel to the given straight line at a distance equal to the radius of the mating arc R1) (Fig. 2.22, A). Compass opening equal to the sum of the given radii R+R 1 describe an arc from center O until it intersects with the auxiliary line. The resulting point O1 is the center of mate.
• 2. According to the general rule, the connecting points are found (Fig. 2.22, b): connect the straight centers of the mating arcs O1 and O and lower them from the center of the mating Ο 1 perpendicular to a given line.
• 3. From the mate center Οχ between junction points Μ And Ν draw an arc whose radius R 1 (Fig. 2.22, b).

Rice. 2.22.

Conjugation of two arcs with an arc of a given radius

Given two arcs whose radii are R 1 and R 2. It is required to construct a mate with an arc whose radius is specified.

There are three cases of touch: external (Fig. 2.23, a, b), internal (Fig. 2.23, V) and mixed (see Fig. 2.25). In all cases, the centers of mates must be located from the given arcs at a distance from the radius of the mate arc.

Rice. 2.23.

The construction is carried out as follows:

For external touch:

• 1) from centers Ο 1 and O2, using a compass solution equal to the sum of the radii of the given and mating arcs, draw auxiliary arcs (Fig. 2.23, A); radius of an arc drawn from the center Ο 1, equal R 1 + R 3; and the radius of the arc drawn from the center O2 is equal to R 2 + R 3. At the intersection of the auxiliary arcs, the center of mate is located – point O3;
• 2) connecting point Ο1 with point 03 and point O2 with point O3 by straight lines, find the connecting points M And N(Fig. 2.23, b);
• 3) from point 03 with a compass solution equal to R 3, between points Μ And Ν describe the conjugate arc.

For inner touch perform the same constructions, but the radii of the arcs are taken equal to the difference between the radii of the given and mating arcs, i.e. R 4 – R 1 and R 4 – R 2. Connection points R And TO lie on the continuation of the lines connecting point O4 with points O1 and O2 (Fig. 2.23, V).

For mixed (external and internal) touch(1st case):

• 1) a compass solution equal to the sum of the radii R 1 and R 3, an arc is drawn from point O2, as from the center (Fig. 2.24, a);
• 2) a compass solution equal to the difference in radii R 2 and R 3, a second arc is drawn from point O2, intersecting with the first at point O3 (Fig. 2.24, b);
• 3) from point O1 draw a straight line to point O3, from the second center (point O2) draw a straight line through point O3 until it intersects with the arc at point M(Fig. 2.24, c).

Point O3 is the center of the mate, the point M And N – interface points;

4) placing the leg of the compass at point O3, with radius R 3 draw an arc between the connecting points Μ And Ν (Fig. 2.24, G).

Rice. 2.24.

For mixed touch(2nd case):

• 1) two conjugate arcs of circles of radii R 1 and R 2 (Fig. 2.25);
• 2) distance between centers About i and O2 of these two arcs;
• 3) radius R 3 mating arcs;

required:

• 1) determine the position of the center O3 of the mating arc;
• 2) find the connecting points on the mating arcs;
• 3) draw a mating arc

Construction sequence

Set aside specified distances between centers Ο 1 and O2. From the center ABOUT 1 draw an auxiliary arc with a radius equal to the sum of the radii of the mating arc of radius R 1 and conjugate arc radius R 3, and from the center O2 a second auxiliary arc is drawn with a radius equal to the difference in radii R 3 and R 2, until it intersects with the first auxiliary arc at point O3, which will be the desired center of the mating arc (Fig. 2.25).

Rice. 2.25.

Conjugation points are found according to the general rule, connecting the centers of arcs O3 and O1 with straight lines , O 3 and O2. At the intersection of these lines with the arcs of the corresponding circles, points are found M And N.

Pattern curves

In technology there are parts whose surfaces are limited by flat curves: an ellipse, an involute circle, an Archimedes spiral, etc. Such curved lines cannot be drawn with a compass.

They are built along points that are connected by smooth lines using patterns. Hence the name pattern curves.

Shown in Fig. 2.26. Each point of a straight line, if rolled without sliding along a circle, describes an involute.

Rice. 2.26.

The working surfaces of the teeth of most gears have involute gearing (Fig. 2.27).

Rice. 2.27.

Archimedes spiral shown in Fig. 2.28. This is a flat curve described by a point moving uniformly from the center ABOUT along a rotating radius.

Rice. 2.28.

A groove is cut along the Archimedes spiral, into which the protrusions of the cams of a self-centering three-jaw chuck of a lathe enter (Fig. 2.29). When the bevel gear, on the back of which has a spiral groove, rotates, the cams are compressed.

When making these (and other) pattern curves in the drawing, you can use the reference book to make your work easier.

The dimensions of the ellipse are determined by the size of its major AB and small CD axes (Fig. 2.30). Describe two concentric circles. The larger diameter is equal to the length of the ellipse (major axis AB), the diameter of the smaller one is the width of the ellipse (minor axis CD). Divide a large circle into equal parts, for example 12. The division points are connected by straight lines passing through the center of the circles. From the points of intersection of straight lines with circles, lines are drawn parallel to the axes of the ellipse, as shown in the figure. When these lines intersect each other, points belonging to the ellipse are obtained, which, having previously been connected by hand with a thin smooth curve, are outlined using a pattern.

Rice. 2.29.

Rice. 2.30.

Practical application of geometric constructions

Given the task: make a drawing of the key shown in Fig. 2.31. How to do it?

Before starting to draw, an analysis of the graphic composition of the image is carried out to determine which cases of geometric constructions need to be applied. In Fig. Figure 2.31 shows these constructions.

Rice. 2.31.

To draw a key, you need to draw mutually perpendicular straight lines, describe circles, build hexagons by connecting their upper and lower vertices with straight lines, and connect arcs and straight lines with arcs of a given radius.

What is the sequence of this work?

First, draw those lines whose position is determined by the given dimensions and do not require additional construction (Fig. 2.32, A), i.e. draw axial and center lines, describe four circles according to given dimensions and connect the ends of the vertical diameters of smaller circles with straight lines.

Rice. 2.32.

Further work on the execution of the drawing requires the use of the geometric constructions set out in paragraphs 2.2 and 2.3.

In this case, you need to build hexagons and pair arcs with straight lines (Fig. 2.32, b). This will be the second stage of work.

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.

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)

table 2

Cam Option No. R 1 R 2 R 3 d 1

Cam Option No. R 1 R 2 R 3 d 1

Cam Option No. R 1 R 2 R 3 d 1 y 1

Cam Option No. R 1 R 2 R 3 d 1

Cam Option No. S 1 a 1 b 1 y 1 R 1 R 2 R 3

Cam Option No. R 1 R 2 R 3 d 1 y 1

Cam Option No. R 1 R 2 R 3 a 1 b 1

Cam Option No. R 1 R 2 R 3 a 1 b 1

Cam Option No. R 1 R 2 R 3 d 1

Cam Option No. R 1 R 2 R 3 d 1

Cam Option No. R 1 R 2 R 3 d 1

Cam Option No. R 1 R 2 R 3 d 1

Cam Option No. R 1 R 2 R 3 d 1 y 1

Cam Option No. R 1 R 2 R 3 d 1

Cam Option No. S 1 a 1 b 1 y 1 R 1 R 2 R 3

Cam Option No. R 1 R 2 R 3 d 1 y 1

Cam Option No. R 1 R 2 R 3 a 1 b 1

Cam Option No. R 1 R 2 R 3 a 1 b 1

Conjugation of an arc and a straight arc of a circle of a given radius

There may be two cases of such conjugation: external contact of the mating arc with a given one and internal contact. In both cases, the task comes down to determining the center of the connecting arc and the points of contact.

When touching externally (Figure 52, a) from the center of a given arc - a point O 1 draw a pilot arc with a radius R + R with . At a distance equal to the radius R c conjugate arc, draw a straight line parallel to a given straight line. Dot ABOUT the intersection of the auxiliary arc and the straight line is the center of the conjugate arc. At the intersection of a line connecting points ABOUT And O 1 with a given arc, mark the point of contact A . Second touch point IN defined as the point of intersection of a given line with a perpendicular dropped onto it from the point ABOUT .

With an internal touch (Figure 52, b), the determination of the center of the mating arc and the points of contact are similar to the previous case, with the only difference being that the radius of the auxiliary arc is equal to R c – R .

Figure 52

There are three types of such coupling:

1) external conjugation when the conjugating arc externally touches two given ones;

2) internal conjugation when the conjugating arc internally touches two given ones;

3) mixed conjugation with an external contact of the mating arc with one given one and an internal contact with another.

At external interface (Figure 53, a) center of the connecting arc point O is located at the point of intersection of auxiliary arcs with radii r + R c And R + R c , drawn respectively from the centers of conjugate arcs - points O2 And O 1 . Touch points A And B are defined as the points of intersection of given arcs with straight lines OO 1 And OO 2 .

Internal pairing arc radii r And R arc radius R c shown in Figure 53, b. To determine the center of the connecting arc - point ABOUT draw auxiliary arcs with radii R c – r And R c – R respectively, from the centers of given arcs - points O2 And O 1 . Dot ABOUT the intersection of these arcs will be the center of the conjugate arc. From the point ABOUT through points O 1 And O2 draw straight lines until they intersect with given arcs and obtain, respectively, two points of tangency - A And B .

Figure 53

At mixed pairing center of the connecting arc – point ABOUT is defined as the point of intersection of two auxiliary arcs of radii R c +R And R with – r (Figure 53, c) or R with – R And R with + r , drawn respectively from the centers of given arcs - points O 1 And O2 . To determine the points of tangency of the mating arc with the given ones, draw two straight lines: one through the points ABOUT And O 1 , another through points ABOUT And O2 . The points of intersection of each of them with given arcs give the required points of tangency A And B .