Kinematic pair movable coupling of two solid links, imposing restrictions on their relative movement by connection conditions. Each connection condition eliminates one Degree of Freedom ,
that is, the possibility of one of 6 independent relative movements in space. In a rectangular coordinate system, 3 translational movements (in the direction of 3 coordinate axes) and 3 rotational movements (around these axes) are possible. According to the number of communication conditions S K. items are divided into 5 classes. Number of degrees of freedom K. p. W=6-S. Within each class, gears are divided into types according to the remaining possible relative movements of the links. Based on the nature of contact between the links, lower ones are distinguished - with contact along surfaces, and higher ones - with contact along lines or at points. Higher K. points are possible in all 5 classes and many types; lower - only 3 classes and 6 types ( Fig.1
). A distinction is also made between geometrically closed and non-closed CPs. In the former, constant contact of surfaces is ensured by the shape of their elements (for example, all CPs on rice. 1
), secondly, closing requires a pressing force, the so-called. force closure (for example, in a cam mechanism). Conventionally, mechanical joints include some movable connections with several intermediate rolling elements (for example, ball and roller bearings) and with intermediate deformable elements (for example, the so-called backlash-free hinges of devices with flat springs; rice. 2
). N. Ya. Nyberg.
Great Soviet Encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .
See what a “Kinematic pair” is in other dictionaries:
The connection of 2 links of a mechanism, allowing their relative movement. The kinematic pair in which the links are in contact along the surface is called the lowest (for example, a rotary joint, a translational slide and a guide). Kinematic pair... ... Big encyclopedic Dictionary
kinematic pair- pair A connection of two contacting links, allowing their relative movement. [Collection of recommended terms. Issue 99. Theory of mechanisms and machines. Academy of Sciences of the USSR. Committee of Scientific and Technical Terminology. 1984] Topics theory... ... Technical Translator's Guide
kinematic pair- kinematic pair; pair A connection of two contacting links that allows relative movement ...
The connection of 2 links of a mechanism, allowing their relative movement. A kinematic pair in which the links are in contact along the surface is called the lowest (for example, a rotary joint, a translational slide and a guide). Kinematic... ... encyclopedic Dictionary
- ... Wikipedia
kinematic pair- kinematinė pora statusas T sritis fizika atitikmenys: engl. kinematic pair vok. kinematisches Elementenpaar, n rus. kinematic pair, f pranc. paire cinématique, f … Fizikos terminų žodynas
The connection of two contacting links, allowing them to relate. movement. Surfaces, lines, points, to which a link can come into contact with another link, are called. elements of the link. K. p. are divided into lower (contact with surfaces) and higher... ... Big Encyclopedic Polytechnic Dictionary
kinematic pair- kinematic pair A connection between two rigid bodies of a mechanism, allowing their specified relative motion. IFToMM code: 1.2.3 Section: GENERAL CONCEPTS OF THE THEORY OF MECHANISMS AND MACHINES... Theory of mechanisms and machines
pair- kinematic pair; pair A connection of two contacting links that allows relative movement. couple of forces; pair A system of two parallel forces, equal in magnitude and directed in opposite directions... Polytechnic terminology Dictionary
top pair- A kinematic pair in which the required relative movement of the links can be obtained only by contact of its elements along lines and at points... Polytechnic terminological explanatory dictionary
rotational;
progressive;
screw;
spherical.
Symbols of links and kinematic pairs on kinematic diagrams.
The kinematic diagram of the mechanism is called graphic image on the selected scale of the relative position of the links included in the kinematic pairs, using symbols in accordance with GOST 2770-68. In capital letters The Latin alphabet on the diagrams indicates the centers of the hinges and other characteristic points. The directions of movement of the input links are indicated by arrows. The kinematic diagram must have all the parameters necessary for the kinematic study of the mechanism: dimensions of links, number of teeth of gear wheels, profiles of elements of higher kinematic pairs. The scale of the diagram is characterized by the length scale factor Kl, which is equal to the ratio of the length AB l of the link in meters to the length of the segment AB representing this link in the diagram, in millimeters: Kl = l AB / AB
A kinematic scheme is essentially a model that replaces a real mechanism to solve problems of its structural and kinematic analysis. Let us note the main assumptions that are implied in this schematization:
a) the links of the mechanism are absolutely rigid;
b) there are no gaps in kinematic pairs
Kinematic chains and their classification.
Kinematic chains, based on the nature of the relative movement of the links, are divided into flat and spatial. A kinematic chain is called flat if the points of its links describe trajectories lying in parallel planes. A kinematic chain is called spatial if the points of its links describe non-planar trajectories or trajectories lying in intersecting planes.
Classification of kinematic chains:
Flat - when one link is secured, the remaining links perform a flat movement, parallel to some fixed plane.
Spatial - when one link is secured, the remaining links move in different planes.
Simple - each link includes no more than two kinematic pairs.
Complex - at least one link has more than two kinematic pairs.
Closed - no more than two kinematic pairs are included, and these links form one or more closed contours
Open – the links do not form a closed loop.
The number of degrees of freedom of the kinematic chain, the mobility of the mechanism.
The number of input links to transform a kinematic chain into a mechanism must be equal to the number of degrees of freedom of this kinematic chain.
In this case, the number of degrees of freedom of the kinematic chain means the number of degrees of freedom of the moving links relative to the stand (the link taken as a fixed one). However, the stand itself can move in real space.
Let us introduce the following notation:
k – number of links in the kinematic chain
p1 – number of kinematic pairs of the first class in this chain
p2 – number of second class pairs
p3 – number of third class pairs
p4 – number of fourth class pairs
p5 – number of fifth class pairs.
The total number of degrees of freedom of k free links located in space is 6k. In a kinematic chain, they are connected into kinematic pairs (i.e., connections are superimposed on their relative motion).
In addition, a kinematic chain with a stand (a link taken as a fixed link) is used as a mechanism. Therefore, the number of degrees of freedom of the kinematic chain will be equal to the total number of degrees of freedom of all links minus the constraints imposed on their relative motion:
The number of connections imposed by all class I pairs is equal to their number, since each pair of the first class imposes one constraint on the relative movement of the links connected in such a pair; the number of bonds imposed by all pairs of the second class is equal to their double number (each pair of the second class imposes two bonds), etc.
All six degrees of freedom are taken away from a link that is taken to be stationary (six connections are superimposed on the rack). Thus:
S1=p1, S2=2p2, S3=3p3, S4=4p4, S5=5p5, Sracks=6,
and the sum of all connections
∑Si=p1+2p2+3p3+4p4+5p5+6.
The result is the following formula for determining the number of degrees of freedom of the spatial kinematic chain:
W=6k–p1–2p2–3p3–4p4–5p5–6.
Grouping the first and last terms of the equation, we get:
W=6(k–1)–p1–2p2–3p3–4p4–5p5,
or finally:
W=6n–p1–2p2–3p3–4p4–5p5,
Thus, the number of degrees of freedom of an open kinematic chain is equal to the sum of the mobility (degrees of freedom) of the kinematic pairs included in this chain. In addition to degrees of freedom on the quality of work of manipulators and industrial robots big influence affects their maneuverability.
Types of gear mechanisms, their structure and a brief description of.
A gear transmission is a three-link mechanism in which two moving links are gears, or a wheel and a rack with teeth that form a rotational or translational pair with a fixed link (housing).
A gear train consists of two wheels through which they engage with each other. A gear with a smaller number of teeth is called a gear, and a gear with a larger number of teeth is called a wheel.
The term "gear" is a general one. The gear parameters are assigned an index of 1, and the wheel parameters are assigned an index of 2.
The main advantages of gears are:
Constancy of the gear ratio (no slippage);
Compact compared to friction and belt drives;
High efficiency (up to 0.97...0.98 in one stage);
Greater durability and operational reliability (for example, for gearboxes for general use, a service life of 30,000 hours is established);
Possibility of application in a wide range of speeds (up to 150 m/s), powers (up to tens of thousands of kW).
Flaws:
Noise at high speeds;
Inability to continuously change the gear ratio;
The need for high precision manufacturing and installation;
Insecurity from overloads;
The presence of vibrations that arise as a result of inaccurate manufacturing and inaccurate assembly of gears.
Involute gears are widely used in all branches of mechanical engineering and instrument making. They are used in an exceptionally wide range of operating conditions. The powers transmitted by gears vary from negligible (instruments, clock mechanisms) to many thousands of kW (aircraft engine gearboxes). The most common are gears with cylindrical wheels, as they are the simplest to manufacture and operate, reliable and small-sized. Bevel, screw and worm gears are used only in cases where this is necessary according to the conditions of the machine layout.
Basic law of engagement.
To ensure constant transmission
relations: it is necessary that the profiles of the mating teeth be outlined by such curves that would satisfy the requirements of the basic gearing theorem
The basic law of engagement: the general normal N-N to the profiles, drawn at the point C of their contact, divides the interaxial distance a w into parts inversely proportional to the angular velocities. With a constant gear ratio ( = const) and fixed centers O 1 and O 2, point W will occupy a constant position on the line of centers. In this case, the velocity projections k 1 and k 2 are not equal. Their difference indicates the relative sliding of the profiles in the direction of the tangent K-K, which causes their wear. Equality of velocity projections is possible only in one position, when the contact point C of the profiles coincides with the intersection point W normals N-N and lines of centers O 1 O 2. Point W is called the engagement pole, and circles with diameters d w1 and d w2 that touch at the engagement pole and roll over each other without sliding are called initial.
To ensure a constant gear ratio, theoretically, one of the profiles can be chosen arbitrarily, but the shape of the profile of the mating tooth must be strictly defined to satisfy condition (1.82). The most technologically advanced to manufacture and operate are involute profiles. There are other types of gearing: cycloidal, lantern, Novikov gearing, which satisfy this requirement.
Types of kinematic pairs and their brief characteristics.
A kinematic pair is a connection of two contacting links that allows their relative movement.
The set of surfaces, lines, points of a link along which it can come into contact with another link, forming a kinematic pair, is called a link element (element of a kinematic pair).
Kinematic pairs (KP) are classified according to the following criteria:
by type of contact point (connection point) of the link surfaces:
lower ones, in which the contact of the links is carried out along a plane or surface (sliding pairs);
higher ones, in which the contact of the links is carried out along lines or points (pairs that allow sliding with rolling).
by the relative motion of the links forming a pair:
rotational;
progressive;
screw;
spherical.
according to the method of closure (ensuring contact of the links of a pair):
force (due to the action of weight forces or the elastic force of a spring);
geometric (due to the design of the working surfaces of the pair).
Kinematic pair, as stated above, this is the connection of two contacting links, allowing their relative movement. Models of these movements are shown in Fig. 1.16. Links, when combined into a kinematic pair, can come into contact with each other along surfaces, lines and points. Elements of a kinematic pair call a set of surfaces, lines or points along which the movable connection of two links occurs and which form a kinematic pair. More precisely, the elements of a kinematic pair are the surfaces, lines or points common to the connected links, with which the links come into contact with each other, forming a kinematic pair. Thus, a kinematic pair cannot be formed by bodies that are not in contact. The degree of restriction of the freedom of movement of one link of a kinematic pair relative to another can depend only on geometric shapes places of contact, that is, from the elements of the kinematic pair. Neither the materials from which the links are made, nor the shape of those parts that are not in contact with each other, can impose restrictions on the relative mobility of the links, and therefore they are not considered in the theory of mechanisms and machines.
Rice. 1.16. Models of kinematic pairs, from left to right: top row - ball on a plane, cylinder on a plane, ball in a cylinder, plane pair, spherical pair and bottom row - spherical with a finger, cylindrical, translational, helical
Kinematic pairs are classified according to several criteria. In order for a pair to exist, the elements of the links included in it must be closed, that is, be in constant contact.
Classification of kinematic pairs
Table 1.2
|
Type of pair and degree of freedom |
Semi-constructed image |
Pair mobility w, number of bonds |
Conditional designation |
|
rotational |
 |
» € and and ^ „ |
 |
|
screw [ShZh00] |
 |
 |
|
|
cylindrical |
 |
 |
|
|
spherical |
 |
 |
|
|
planar |
 |
 |
|
|
linear; |
 |
 |
|
|
w = 4 5=2 |
|||
|
spot |
 |
 |
By geometric view connection of surfaces and method of closure
kinematic pairs are divided into lower and higher, with force or geometric closure. By closing pair is called ensuring constant contact of the corresponding elements of the pair. U lower contact of links, connection of surfaces is carried out along one or more surfaces. These are sliding pairs (their relative motion is always sliding), and such pairs are characterized by geometric closure due to the structural shape of the elements of the pair. U higher kinematic pairs of links touch along a line or at a point. Therefore, not only relative sliding is possible, but also rolling and spinning. Such pairs are often characterized by force closure, that is, the elements are pressed against each other by weight forces, elastic forces, etc. In Fig. 1.16, the higher pairs include a ball on a plane (contacting at a point), a cylinder on a plane (contacting along a straight segment) and a ball in a cylinder (contacting along a circle). All other pairs are inferior.
According to the relative movement of the links pairs are divided into rotational (B) (English, a revolute joint (R)), translational (English, a prismatic joint (P)), screw (English, a helical joint (H) or screw pair) , flat or plane (Pl) (English, planar joint (E)), cylindrical (English, a cylindrical joint (C)), spherical (English, a spherical or ball joint (S)), linear (L) and dotted (T).
According to the number of mobilityw(number of degrees of freedom) in the relative motion of the links of a pair, they are divided into one-, two-, three-, four- and five-movable.
By number of connectionss, superimposed on the relative movement of the links, kinematic pairs are divided into classes: 1-, 2-, 3-, 4-, 5-connected pairs form pairs of classes 1, 11, III, IV and V, respectively. Higher kinematic pairs can be of all classes and many types, and lower ones - only III, IV and V classes and 6 types. Table 1.2 shows different types kinematic pairs, their semi-constructive and schematic images, as well as the mobility of the pair w and the number of connections s.
The mobility of a pair w is determined by the formula
where P is the mobility of the space in which the couple is constructively realized, s- the number of connections imposed by a pair.
Let us recall that in three-dimensional space an absolutely rigid body (and therefore the links that are modeled by it) has six degrees of freedom. These are three degrees of freedom of translational motion, for example, along coordinate axes. And three degrees of freedom rotational movement, for example, rotation around the same coordinate axes.
Table 1.3
Kinematic connections equivalent to kinematic pairs
|
Link contact |
Types of couple |
Mobility |
Types of kinematic pairs |
Image |
Equivalent kinematic compound |
|
On the surface |
Lowest kinematic pair |
 |
 |
 |
|
 |
 |
 |
|||
 |
 |
 |
|||
 |
 |
 |
|||
 |
 |
 |
|||
 |
 |
 |
|||
|
Higher kinematic pair |
w = 4 5 = 2 |
 |
 |
 |
|
 |
 |
 |
Table 1.4
Symbols of kinematic pairs according to GOST 2.770-68
|
degrees |
Name |
Conditional designation |
|||
|
ball-plane |
 |
 |
|||
|
ball-cylinder |
 |
 |
|||
|
spherical |
 |
 |
|||
|
planar |
 |
 |
|||
|
cylindrical |
 |
 |
|||
|
spherical with finger |
 |
 |
|||
|
progressive |
 |
 |
|||
|
rotational |
 |
 |
|||
|
screw |
 |
 |
In plane motion, an absolutely rigid body has three degrees of freedom - two degrees of translational motion and one degree of rotational motion. Therefore, three-dimensional space is six-movable, and two-dimensional space is three-movable. The data in Table 1.2 should be viewed with this in mind. For example, a rotational pair and a translational one, both in 6-movable and in 3-movable space, will be unimovable, that is w- 1. In the first case, 5 connections will be applied to it (s = 5), and in the second - 2 connections (s = 2).
It is possible to select such a shape for the elements of a pair so that with one independent simple movement, a second, dependent one arises. An example of such a kinematic pair is a screw. In this pair, the rotational movement of the screw (nut) causes its (her) translational movement along the axis. Such a pair should be classified as single-moving (w = 1), since only one independent simple movement is realized in it.
The role of a kinematic pair can also be kinematic connection- a compact structure made of several moving parts with surface, linear or point contact of elements, providing the possibility of relative movement of the corresponding type, equivalent to a given kinematic pair. That is, kinematic connection called a kinematic chain designed to replace a kinematic pair. An example of such a kinematic connection is bearings. Kinematic connections most often have a large number of redundant local connections, but due to the structural design this does not affect the basic mobility of the kinematic pairs. Each pair in the mechanism can correspond to different options for kinematic connections in the form of several parts with local mobility that do not affect the final mobility of the pair (a roller bearing is equivalent to a two-moving cylindrical pair, a thrust ball bearing with a spherical outer surface mounted on a conical surface is equivalent to a five-moving point pair ). Table 1.3 shows kinematic pairs and equivalent kinematic connections.
At the end of this paragraph, we present the symbols of kinematic pairs according to GOST 2770-68 (Table 1.4).
A kinematic pair is a movable connection of two contacting links that provides them with a certain relative movement. The elements of a kinematic pair are the collection of surfaces of lines or points along which the movable connection of two links occurs and which form a kinematic pair. For a pair to exist, the elements of its constituent links must be in constant contact T.
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Lecture No. 2
Whatever the mechanism of the machine, it always consists only of links and kinematic pairs.
The connection conditions imposed in mechanisms on moving links, in the theory of machines and mechanisms, are usually called kinematic pairs.
Kinematic paircalled a movable connection of two contacting links, providing them with a certain relative movement.
In table 2.1 shows the names, pictures, symbols of the most common kinematic pairs in practice, and also their classification.
Links, when combined into a kinematic pair, can come into contact with each other along surfaces, lines and points.
Elements of a kinematic paircall a set of Surfaces, lines or points along which two links are movably connected and which form a kinematic Pair. Depending on the type of contact of the elements of kinematic pairs, they are distinguished high and low kinematic pairs.
Kinematic pairs formed by elements in the form of a line or point are called highest.
Kinematic pairs formed by elements in the form of surfaces are called inferior.
For a pair to exist, the elements of its constituent links must be in constant contact, i.e. be closed. The closure of kinematic pairs can begeometrically or forcefully, For example, using its own mass, springs, etc.
The strength, wear resistance and durability of kinematic pairs depend on their type and design. The lower pairs are more wear-resistant than the higher ones. This is explained by the fact that in the lower pairs the contact of the elements of the pairs occurs on the surface, and therefore, with the same load, lower specific pressures arise in it than in the higher one. Wear, other things being equal, is proportional to the specific pressure, and therefore lower pairs wear out more slowly than higher ones. Therefore, in order to reduce wear in machines, it is preferable to use lower pairs, but often the use of higher kinematic pairs makes it possible to significantly simplify the structural diagrams of machines, which reduces their dimensions and simplifies the design. Therefore, the correct choice of kinematic pairs is a complex engineering task.
Kinematic Pairs are also divided according tonumber of degrees of freedom(mobility) which it provides to the links connected through it, or bynumber of connection conditions(pair class), imposed by the pair on the relative motion of the connected links. When using this classification, machine developers receive information about the possible relative movements of the links and the nature of the interaction of force factors between the elements of the pair.
A free link, generally located in M - dimensional space Allowing P types of simple movements, has a number of degrees of freedom! ( N) or W - movable.
So, if the link is in three-dimensional space, allowing six types of simplest movements - three rotational and three translational around and along the axes X, V, Z , then they say that it has six degrees of freedom or has six generalized coordinates, or is six-movable. If the link is in a two-dimensional space that allows three types of simple movements - one rotating around Z and two translational along the axes X and Y , then they say that it has three degrees of freedom, or three generalized coordinates, or it is three-movable, etc.
Table 2.1
When links are combined using kinematic pairs, they are deprived of their degrees of freedom. This means that kinematic pairs impose a number of connections on the links they connect S.
Depending on the number of degrees of freedom that the links combined into a kinematic pair have in relative motion, they determine the mobility of the pair ( W =H ). If N is the number of degrees of freedom of the links of a kinematic pair in relative motion, to The mobility of the pair is determined as follows:
where P - mobility of the space in which the pair in question exists; S - the number of connections imposed by a pair.
It should be noted that the mobility of the pair W , defined by (2.1), does not depend on the type of space in which it is realized, but only on the construction.
For example, a rotational (translational) (see, Table 2.1) pair, both in six- and three-moving space, will still remain single-moving, in the first case 5 connections will be imposed on it, and in the second case - 2 connections, and, This means we will have, respectively:
for six-moving space:
for three-movable space:
As we can see, the mobility of kinematic pairs does not depend on the characteristics of space, which is an advantage of this classification. On the contrary, the frequently encountered division of kinematic pairs into classes suffers from the fact that the class of the pair depends on the Characteristics of the space, which means that the same pair in different spaces has a different class. This is inconvenient for practical purposes, which means that such a Classification of kinematic pairs is irrational, so it is better not to use it.
It is possible to select such a form for the elements of the pair that with one independent simplest movement a second, dependent (derivative) movement arises. An example of such a kinematic pair is a screw (Table 2. 1) . In this pair, the rotational movement of the screw (nut) causes its (her) translational movement along the axis. Such a pair should be classified as single-moving, since only one independent, simplest Movement is realized in it.
Kinematic connections.
Kinematic pairs given in table. 2.1, simple and compact. They implement almost everything necessary when creating mechanisms - the simplest relative movements of links. However, when creating machines and mechanisms, they are rarely used. This is due to the fact that at the points of contact of the links forming a pair, large friction forces usually arise. This leads to significant wear of the elements of the pair, and therefore to its destruction. Therefore, the simplest two-link kinematic chain of a kinematic pair is often replaced by longer kinematic chains, which together implement the same relative movement of the links as the replaced kinematic pair.
A kinematic chain designed to replace a kinematic pair is called a kinematic connection.
Let us give examples of kinematic chains for the most common in practice rotational, translational, helical, spherical and plane-to-plane kinematic pairs.
From the table 2.1 it is clear that the simplest analogue of a rotational kinematic pair is a bearing with rolling elements. Likewise, roller guides replace the translational pair, etc.
Kinematic connections are more convenient and reliable in operation, withstand significantly greater forces (moments) and allow mechanisms to operate at high relative link speeds.
Main types of mechanisms.
The mechanism can be considered as special case a kinematic chain in which at least one link is facing the rack, and the movement of the remaining links is determined by the given movement of the input links.
The distinctive features of the kinematic chain representing the mechanism are the mobility and certainty of the movement of its links relative to the stand.
A mechanism can have several input and one output link, in which case it is called a summing mechanism, and, conversely, one input and several output links, then it is called a differentiating mechanism.
According to their purpose, the mechanisms are divided intoguides and transmissions.
Transmission mechanismis a device designed to reproduce a given functional relationship between the movements of the input and output links.
Guide mechanismthey call a mechanism in which the trajectory of a certain point of a link, forming kinematic pairs only with moving links, coincides with a given curve.
Let us consider the main types of mechanisms that are widely used in technology.
Mechanisms whose links form only lower kinematic pairs are calledarticulated-lever. These mechanisms are widely used due to the fact that they are durable, reliable and easy to operate. The main representative of such Mechanisms is a four-bar hinge (Fig. 2.1).
The names of mechanisms are usually determined by the names of their input and output links or the characteristic link included in their composition.
Depending on the laws of motion of the input and output links, this mechanism can be called crank-rocker, double crank, double rocker, rocker-crank.
The articulated four-link linkage is used in machine tool building, instrument making, as well as in agricultural, food, snow removal and other machines.
If you replace a rotation pair in a four-bar articulated linkage, for example D , to translational, we get the well-known crank-slider mechanism (Fig. 2.2).
Rice. 2.2. Various types of crank-slider mechanisms:
1 crank 2 - connecting rod; 3 - slider
The crank-slider (slider-crank) mechanism is widely used in compressors, pumps, internal combustion engines and other machines.
Replacing the rotation pair in a four-bar hinge WITH to translational, we get a rocker mechanism (Fig. 2.3).
On p and c .2.3, the rocker mechanism is obtained from a four-bar hinge by replacing the rotation pairs in it S and O to progressive ones.
Rocker mechanisms have found wide application in planing machines due to their inherent property of asymmetry of the working and idling strokes. Usually they have a long working stroke and a fast stroke, ensuring the return of the cutter to initial position idling.
Rice. 2.3. Various types of rocker mechanisms:
1 crank; 2 stone; 3 backstage.
Articulating-lever mechanisms are widely used in robotics (Fig. 2.4).
The peculiarity of these mechanisms is that they have a large number of degrees of freedom, which means they have many drives. The coordinated operation of the drives of the input links ensures the movement of the gripper along a rational trajectory and to a given location in the surrounding space.
Widely used in technologycam mechanisms. With the help of cam mechanisms, it is structurally most simple to obtain almost any movement of the driven link according to a given law,
Currently, there are a large number of varieties of cam mechanisms, some of which are shown in Fig. 2.5.
The necessary law of motion of the output link of the cam mechanism is achieved by giving the input link (cam) the appropriate shape. The cam can rotate (Fig. 2.5, a, b ), translational (Fig. 2.5, c, d ) or complex movement. The output link, if it makes translational motion (Fig. 2.5, a, c ), is called a pusher, and if it is rocking (Fig. 2.5, G ) - rocker arm. To reduce friction losses in the higher kinematic pair IN use an additional link-roller (Fig. 2.5, G ).
Cam mechanisms are used both in working machines and in various types of command devices.
Very often, screw mechanisms are used in metal-cutting machines, presses, various instruments and measuring devices, the simplest of which is shown in Fig. 2.6:
Rice. 2.6 Screw mechanism:
1 - screw; 2 - nut; A, B, C - kinematic pairs
Screw mechanisms are usually used where it is necessary to convert rotary motion into interdependent translational motion or vice versa. The interdependence of movements is established by the correct selection of geometric parameters of the screw pair IN .
Wedge mechanisms (Fig. 2.7) are used in various types clamping devices and devices in which it is necessary to create a large force at the output with limited forces acting at the input. Distinctive feature These mechanisms are simple and reliable in design.
Mechanisms in which the transmission of motion between contacting bodies is carried out due to friction forces are called frictional. The simplest three-link friction mechanisms are shown in Fig. 2.8
Rice. 2.7 Wedge mechanism:
1, 2 - links; L, V, C - kinematic feasts.
Rice. 2.8 Friction mechanisms:
A - friction mechanism with parallel axes; b - friction mechanism with intersecting axes; V - rack and pinion friction mechanism; 1 - input roller (wheel);
2 output roller (wheel); 2" - rail
Due to the fact that the links 1 and 2 are attached to each other, along the line of contact between them a friction force arises, which drags the driven link along with it 2 .
Friction gears are widely used in instruments, tape drive mechanisms, and variators (mechanisms with continuously variable speed control).
To transmit rotational motion according to a given law between shafts with parallel, intersecting and intersecting axes, various types of gears are used mechanisms . With the help of gears, it is possible to transmit motion between shafts withfixed axes, so with moving in space.
Gear mechanisms are used to change the frequency and direction of rotation of the output link, summing or dividing movements.
In Fig. Figure 2.9 shows the main representatives of gears with fixed axes.
Figure 2.9. Gear drives with fixed axes:
a - cylindrical; b - conical; c - end; g - rack and pinion;
1 - gear; 2 - gear; 2 * rack
The smaller of the two meshing gears is called gear, and more - gear wheel.
A rack is a special case of a gear whose radius of curvature is infinity.
If a gear transmission has gears with moving axes, then they are called planetary (Fig. 2.10):
Compared to transmissions with fixed axles, planetary gears allow the transmission of greater power and gear ratios with fewer gears. They are also widely used in the creation of summing and differential mechanisms.
The transmission of motion between intersecting axes is carried out using a worm gear (Fig. 2.11).
A worm gear is obtained from a screw-nut transmission by longitudinally cutting the nut and folding it twice in mutually perpendicular planes. The worm gear has the property of self-braking and allows large gear ratios to be realized in one stage.
Rice. 2.11. Worm-gear:
1 - worm, 2 - worm wheel.
TO gear mechanisms intermittent movements also include the Maltese cross mechanism. In Fig. Z-L"2. shows the mechanism of a four-bladed "Maltese cross".
The “Maltese cross” mechanism transforms the continuous rotation of the leading even - crank 1 with a lantern 3 in intermittent rotation of the "cross" 2, Tsevka 3 fits into the radial groove of the “cross” without impact 2 and turns it to the angle where z is the number of grooves.
To carry out movement in only one direction, ratchet mechanisms are used. Figure 2.13 shows a ratchet mechanism consisting of a rocker arm 1, a ratchet wheel 3 and pawls 3 and 4.
When the rocker arm swings 1 rocking dog 3 imparts rotation to the ratchet wheel 2 only when the rocker moves counterclockwise. To hold the wheel 2 to prevent spontaneous clockwise rotation when the rocker moves counterclockwise, a locking pawl serves 4 .
Maltese and ratchet mechanisms are widely used in machine tools and instruments,
If it is necessary to transfer mechanical energy over a relatively long distance from one point in space to another, then mechanisms with flexible links are used.
Belts, ropes, chains, threads, tapes, balls, etc. are used as flexible links that transmit movement from one even mechanism to another.
In Fig. Figure 2.14 shows a block diagram of a simple mechanism with a flexible link.
Gears with flexible links are widely used in mechanical engineering, instrument making and other industries.
The most typical simple mechanisms were discussed above. mechanisms are given in special Literature, certificates and reference books, for example, such as.
Structural formulas of mechanisms.
Exist general patterns in the structure (structure) of a wide variety of mechanisms, connecting the number of degrees of freedom W mechanism with the number of links and the number and type of its kinematic pairs. These patterns are called structural formulas of mechanisms.
For spatial mechanisms, the Malyshev formula is currently the most common, the derivation of which is carried out as follows.
Let in a mechanism having m links (including the stand), - the number of one-, two-, three-, four- and five-moving pairs. Let us denote the number of moving links. If all moving links were free bodies, total number degrees of freedom would be 6 n . However, each single-moving pair V class imposes on the relative movement of the links forming a pair, 5 connections, each two-moving pair IV class - 4 connections, etc. Consequently, the total number of degrees of freedom, equal to six, will be reduced by
where is the mobility of a kinematic pair, is the number of pairs whose mobility is equal to i . The total number of superimposed connections may include a certain number q redundant (repeated) connections that duplicate other connections, without reducing the mobility of the mechanism, but only turning it into a statically indeterminate system. Therefore, the number of degrees of freedom of the spatial mechanism, equal to the number degrees of freedom of its movable kinematic chain relative to the rack is determined by the following Malyshev formula:
or in short form
(2.2)
when the mechanism is a statically determinate system, when - a statically indeterminate system.
In general, solving equation (2.2) is a difficult task, since unknown W and q ; the available solutions are complex and are not discussed in this lecture. However, in the special case if W , equal to the number of generalized coordinates of the mechanism, was found from geometric considerations; from this formula the number of redundant connections can be found (see Reshetov L.N. Design of rational mechanisms. M., 1972)
(2.3)
and solve the question of the static definability of the mechanism; or, knowing that the mechanism is statically determinate, find (or check) W.
It is important to note that the structural formulas do not include the dimensions of the links, therefore, in the structural analysis of the mechanisms, they can be assumed to be any (within certain limits). If there are no redundant connections (), the assembly of the mechanism occurs without deformation of the links, the latter seem to self-install; Therefore, such mechanisms are called self-aligning. If there are redundant connections (), then the assembly of the mechanism and the movement of its links become possible only when the latter are deformed.
For flat mechanisms without redundant connections structural formula bears the name of P. L. Chebyshev, who first proposed it in 1869 for lever mechanisms with rotational pairs and one degree of freedom. Currently, Chebyshev’s formula applies to any plane mechanisms and is derived taking into account redundant connections as follows
Let, in a flat mechanism having m links (including a stand), be the number of moving links, - the number of lower pairs and - the number of higher pairs. If all the moving links were free bodies performing plane motion, the total number of degrees of freedom would be equal to 3 n . However, each lower pair imposes two constraints on the relative motion of the links forming the pair, leaving one degree of freedom, and each higher pair imposes one constraint, leaving 2 degrees of freedom.
The number of superimposed connections may include a certain number of redundant (repeated) connections, the elimination of which does not increase the mobility of the mechanism. Consequently, the number of degrees of freedom of a flat mechanism, i.e. the number of degrees of freedom of its movable kinematic chain relative to the stand, is determined by the following Chebyshev formula:
(2.4)
If known, the number of redundant connections can be found from here
(2.5)
The index “p” reminds us that we are talking about a perfectly flat mechanism, or more precisely about its flat design, since due to manufacturing inaccuracies the flat mechanism is to some extent spatial.
Using formulas (2.2)-(2.5), a structural analysis of existing mechanisms and synthesis of structural diagrams of new mechanisms are carried out.
Structural analysis and synthesis of mechanisms.
The influence of redundant connections on the performance and reliability of machines.
As mentioned above, with arbitrary (within certain limits) link sizes, a mechanism with redundant connections () cannot be assembled without deforming the links. Therefore, such mechanisms require increased manufacturing precision; otherwise, during the assembly process, the mechanism links are deformed, which causes the kinematic pairs and links to be loaded with significant additional forces (in addition to those basic external forces that the mechanism is designed to transmit). If the manufacturing accuracy of a mechanism with excessive connections is insufficient, friction in kinematic pairs can greatly increase and lead to jamming of the links, therefore, from this point of view, excessive connections in mechanisms are undesirable.
As for redundant connections in the kinematic chains of a mechanism, when designing machines one should strive to eliminate them or leave a minimum amount if their complete elimination turns out to be unprofitable due to the complexity of the design or for some other reasons. In general, the optimal solution should be sought, taking into account the availability of the necessary technological equipment, manufacturing cost, required service life and machine reliability. Therefore, this is a very difficult task for each specific case.
We will consider the methodology for determining and eliminating redundant connections in the kinematic chains of mechanisms using examples.
Let a flat four-bar mechanism with four single-moving rotational pairs (Fig. 2.15, A ) due to manufacturing inaccuracies (for example, due to non-parallelism of the axes A and D ) turned out to be spatial. Assembly of kinematic chains 4, 3, 2 and separately 4, 1 does not cause difficulties, but points B , B can be positioned on the axis X . However, to assemble a rotational pair IN , formed by links 1 and 2 , it will be possible only by combining the coordinate systems Bxyz and B x y z , which will require linear movement (deformation) of the point B links 2 along the x axis and angular deformations of the link 2 around x and z axes (shown by arrows). This means the presence of three redundant connections in the mechanism, which is confirmed by formula (2.3): . For this spatial mechanism to be statically determinable, it needs another structural diagram, for example, shown in Fig. 2.15, b , where Assembly of such a mechanism will occur without interference, since the combination of points B and B will be possible by moving the point WITH in a cylindrical pair.
A variant of the mechanism is possible (Fig. 2.15, V ) with two spherical pairs (); in this case, in addition tobasic mobilitymechanism appearslocal mobility- possibility of connecting rod rotation 2 around its axis Sun ; this mobility does not affect the basic law of movement of the mechanism and can even be useful from the point of view of leveling out the wear of the hinges: connecting rod 2 can rotate around its axis during operation of the mechanism due to dynamic loads. Malyshev’s formula confirms that such a mechanism will be statically determinate:
Rice. 2.15
The simplest and effective method eliminating redundant connections in device mechanisms - using a higher pair with point contact instead of a link with two lower pairs; the degree of mobility of the flat mechanism in this case does not change, since, according to the Chebyshev formula (at):
In Fig. 2.16, a, b, c An example is given of eliminating redundant connections in a cam mechanism with a progressively moving roller follower. Mechanism (Fig. 2.16, A ) - four-link (); in addition to basic mobility (cam rotation 1
) there is local mobility (independent rotation of a round cylindrical roller 3
around its axis); hence, . The flat circuit has no redundant connections (the mechanism is assembled without interference). If, due to manufacturing inaccuracies, the mechanism is considered spatial, then with linear contact of the roller 3 with cam 1 according to Malyshev’s formula, we obtain, but under certain conditions. Kinematic pair cylinder - cylinder (Fig. 2.16, 6
) if relative rotation of the links is impossible 1 , 3 around the z axis would be a three-movable pair. If, due to manufacturing inaccuracy, such a rotation occurs, but is small, and linear contact is practically maintained (under loading, the contact patch is close in shape to a rectangle), then this 
the kinematic pair will be four-movable, therefore,
Fig.2.17
Reducing the class of the highest pair by using a barrel-shaped roller (five-moving pair with point contact, Fig. 2.16, V ), we obtain for and - the mechanism is statically determinate. However, it should be remembered that the linear contact of the links, although it requires increased manufacturing accuracy, allows for the transfer of greater loads than point contact.
In Fig. 2.16, d, e another example is given of eliminating redundant connections in a four-bar gear transmission (, contact of wheel teeth 1, 2 and 2, 3 - linear). In this case, according to Chebyshev’s formula, the flat circuit has no redundant connections; according to Malyshev’s formula, the mechanism is statically indeterminate, therefore, high manufacturing precision will be required, in particular to ensure parallelism of the geometric axes of all three wheels.
Replacing the teeth of the idler wheel 2 to barrel-shaped ones (Fig. 2.16, d ), we obtain a statically definable mechanism.
Any kinematic pair limits the movement of the connected links.
Restriction imposed on movement solid, called condition of connection .
Thus, kinematic pair imposes coupling conditions on the relative motion of two connected links. It's obvious that greatest number the connection conditions imposed by the kinematic pair are equal to five.
A different number of connection conditions imposed on the relative movement of links by kinematic pairs allows us to divide the latter into 5 classes, so that the k-th class pair imposes k connection conditions, where k is from (1,2,3,4,5). It follows that a kinematic pair of the kth class allows 6-k degrees of mobility in the relative movement of links.
It should be noted that the mechanisms use only kinematic pairs of the fifth, fourth and third classes. Kinematic pairs of the first and second classes have not found application in existing mechanisms.
Since the links are in contact with geometric elements, then, obviously, the kinematic pair is a set of such elements of the connected links. It follows that the nature of the relative movement of the connected links depends on the shape of the geometric elements. This relative movement of one link in relation to another can be obtained if one of the two connected links is made motionless, and the other is given the movement allowed by the connections imposed by the kinematic pair.
Any point of the moving link describes a trajectory in relative motion, which for brevity we will call relative motion trajectory. If the trajectories of the relative motion of such points are plane curves and are located in parallel planes, then the pair is called flat. When spatial kinematic pairs, the indicated trajectories of relative motion are spatial curves.
In addition to division into classes, kinematic pairs are also divided depending on the type geometric element couples:
- top pairs – these are pairs in which, when connecting two links, contact occurs only on curves or points;
- low pairs - these are pairs in which, when connecting two links, contact is made along surfaces.
Higher kinematic pairs are used to reduce friction in the elements of these pairs and are often implemented as rollers or bearings. But the peculiarities of the internal structure of such elements, in the general case, do not affect the relative movement of the links connected by a pair. There are also certain techniques that allow you to replace mechanisms with higher kinematic pairs with their analogues with lower pairs (which allows you to simplify the study of the kinematics of the mechanism in the future). Therefore, further we will consider only mechanisms with lower pairs.
Lower kinematic pairs are most often used in practice and have a simpler internal structure, compared to the higher pairs. The element of the lower kinematic pair consists of two surfaces sliding over each other, which, on the one hand, distributes the load in this element, and on the other hand, increases friction during the relative movement of the links. In this regard, the use of lower kinematic pairs makes it possible to transfer a significant load from one link to another, due precisely to the fact that in these pairs the links are in contact along the surface.
| Number of degrees of freedom | Number of links (pair class) | Pair name | Drawing | Symbol |
| 1 | 5 | Rotational |
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| 1 | 5 | Progressive |
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| 1 | 5 | Screw | ||
| 2 | 4 | Cylindrical |
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| 2 | 4 | Spherical with finger |
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| 3 | 3 | Spherical |
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| 3 | 3 | Flat |
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|
| 4 | 2 | Cylinder-plane |
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| 5 | 1 | Ball-plane |
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