Kinematic couple. Types of kinematic pairs and their brief description Links of the higher kinematic pair
| Number of communication conditions S | Number of degrees of freedom H | Kinematic pair designation | Kinematic pair class | Couple name | Picture | Symbol |
| I | Five-movable ball-plane |  |  |
|||
| II | Four-movable cylinder-plane |  |  |
|||
| III | Tri-movable planar |  |  |
|||
| III | Tri-movable spherical |  |  |
|||
| IV | Two-movable spherical with a finger |  |  |
|||
| IV | Two-movable cylindrical |  |  |
|||
| V | Single-movable screw |  |  |
|||
| V | Single-movable rotary |  |  |
|||
| V | Single-moving translational |  |  |
The system of links that form kinematic pairs with each other is called kinematic chain.
mechanism such a kinematic chain is called in which, for a given movement of one or more links, usually called input or leading, relative to any of them (for example, racks), all the others perform uniquely defined movements.
A mechanism is called flat if all points of the links forming it describe trajectories lying in parallel planes.
Kinematic scheme mechanism is a graphic representation of the mechanism, made to scale by means of symbols of links and kinematic pairs. It gives a complete picture of the structure of the mechanism and the dimensions of the links necessary for kinematic analysis.
Structural scheme mechanism, in contrast to the kinematic diagram, can be performed without observing the scale and gives an idea only of the structure of the mechanism.
The number of degrees of freedom of the mechanism called the number of independent coordinates that determine the position of all links relative to the rack. Each of these coordinates is called generalized. That is, the number of degrees of freedom of the mechanism is equal to the number of generalized coordinates.
To determine the number of degrees of freedom of spatial mechanisms, the Somov-Malyshev structural formula is used:
W = 6n - 5p 1 - 4p 2 - 3p 3 - 2p 4 - 1p 5 , (1.1)
where: W - number of degrees of freedom of the mechanism;
n is the number of moving links;
p 1, p 2, p 3, p 4, p 5 - respectively, the number of one-, two-, three-, four and
five-moving kinematic pairs;
6 - the number of degrees of freedom of a single body in space;
5, 4, 3, 2, 1 - the number of communication conditions imposed respectively
for one-, two-, three-, four- and five-moving pairs.
To determine the number of degrees of freedom of a flat mechanism, the Chebyshev structural formula is used:
W = 3n - 2p 1 , - 1p 2 , (1.2)
where: W is the number of degrees of freedom of the flat mechanism;
n is the number of moving links;
p 1 - the number of single-moving kinematic pairs that are in
planes by lower kinematic pairs;
p 2 - the number of doubly moving kinematic pairs that are in the plane
are the highest;
3 - the number of degrees of freedom of the body on the plane;
2 - the number of bonds superimposed on the lowest kinematic
1 is the number of bonds imposed on the highest kinematic pair.
The degree of mobility determines the number of input links of the mechanism. When calculating the degree of mobility equal to 0 or greater than 1, it is necessary to check whether the mechanism has passive constraints or extra degrees of freedom.
The Somov-Malyshev and Chebyshev formulas are called structural, since they relate the number of degrees of freedom of the mechanism with the number of its links and the number and type of kinematic pairs.
When deriving these formulas, it was assumed that all superimposed bonds are independent, i.e. none of them can be obtained as a consequence of the others. In some mechanisms, this condition is not met; the total number of superimposed bonds may include a certain number q of redundant (repeated, passive) bonds that duplicate other bonds without changing the mobility of the mechanism, but only turning it into a statically indeterminate system. In this case, when using the Somov-Malyshev and Chebyshev formulas, these repeated bonds must be subtracted from the number of superimposed bonds:
W \u003d 6n - (5p 1 + 4p 2 + Zr 3 + 2p 4 + p 5 - q),
W \u003d 3n - (2p 1 + p 2 - q),
whence q \u003d W - 6n + 5p 1 + 4p 2 + Zp 3 + 2p 4 + p 5,
or q \u003d W - 3n + 2p 1 + p 2.
In the general case, there are two unknowns (W and q) in the last equations, and finding them is a difficult task.
However, in some cases, W can be found from geometric considerations, which allows us to determine q using the last equations.
Rice. 1.1 a) Crank-slider mechanism with redundant
connections (when the hinge axes are not parallel).
b) the same mechanism without redundant bonds (replaced
kinematic pairs B and C).
and the mechanism becomes spatial. In this case, the Somov-Malyshev formula gives the following result:
W \u003d 6n - 5p 1, \u003d 6 3-5 4 \u003d -2,
those. it turns out not a mechanism, but a farm, statically indeterminate. The number of redundant connections will be (because in reality W=l): q=l-(-2) = 3.
Excessive connections in most cases should be eliminated by changing the mobility of the kinematic pairs.
For example, for the mechanism under consideration (Fig. 1.1, b), replacing hinge B with a two-moving kinematic pair (p 2 \u003d 1), and hinge C with a three-moving one (p 3 \u003d 1), we get:
q = 1 - 6 3 + 5 2 + 4 1 + 3 1 = 0,
those. there are no redundant connections, and the mechanism is statically determinable.
Sometimes redundant bonds are deliberately introduced into the composition of the mechanism, for example, to increase its rigidity. The performance of such mechanisms is ensured when certain geometric relationships are met. As an example, consider the mechanism of a hinged parallelogram (Fig. 1.2, a), in which AB / / CD, BC / / AD; n = 3, p 1 = 4, W = 1 and q = 0.
 |
 |
Rice. 1.2. Articulated parallelogram:
a) without passive connections,
b) with passive connections
To increase the rigidity of the mechanism (Fig. 1.2, b), an additional link EF is introduced, and with EF / / BC no new geometric constraints are introduced, the movement of the mechanism does not change and in reality still W \u003d 1, although according to the Chebyshev formula we have: W \u003d 3 4 – 2 6 = 0, i.e. formally, the mechanism is statically indeterminate. However, if EF is not parallel to BC, movement becomes impossible, i.e. W is indeed 0.
In accordance with the ideas of L.V. Assura, any mechanism is formed by sequentially connecting to a mechanical system with a certain movement (input links and rack) kinematic chains that satisfy the condition that the degree of their mobility is 0. Such chains, including only the lowest kinematic pairs of the 5th class, are called Assyrian groups.
The Assur group cannot be decomposed into smaller groups that have a zero degree of mobility.
Assur groups are subdivided into classes depending on their structure.
The input link, which forms the lowest kinematic pair with the rack, is called the first class mechanism (Figure 1.3). The degree of mobility of this mechanism is 1.
Fig 1.3. First class mechanisms
The degree of mobility of the Assur group is 0
From this condition, one can determine the relationship between the number of lower kinematic pairs of the fifth class and the number of links included in the Assur group.
Hence it is obvious that the number of links in the group must be even, and the number of pairs of the fifth class is always a multiple of 3.
Assur groups are subdivided into classes and orders. When n=2 and p 5 =3 are combined, Assur groups of the second class are formed.
In addition, groups are divided into orders. The order of the Assur group is determined by the number of elements (external kinematic pairs) by which the group is attached to the mechanism.
There are 5 types of Assur groups of the second class (Table 1.3).
The class of the Assur group above the second is determined by the number of internal kinematic pairs that form the most complex closed contour.
With a combination of n \u003d 4 p 5 \u003d 6, Assur groups of the third and fourth classes are formed (Table 1.3). These groups do not differ by type.
The general class of a mechanism is determined by the highest class of the Assur groups included in the given mechanism.
The formula for the structure of a mechanism shows the order in which Assur groups are attached to a mechanism of the first class.
For example, if the formula for the structure of a mechanism is
1 (1) 2 (2,3) 3 (4,5,6,7) ,
then this means that the Assur group of the second class, including links 2 and 3, and the Assur group of the third class, including links 4, 5, 6, 7, are attached to the mechanism of the first class (link 1 with a rack). The highest class of the group included in the mechanism, is the third class. Therefore, we have a mechanism of the third class.
A kinematic pair is a movable connection of two contiguous links that provides them with a certain relative movement. The elements of a kinematic pair are a set of Surfaces of lines or points along which a 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.
Share work on social networks
If this work does not suit you, there is a list of similar works at the bottom of the page. You can also use the search button
Lecture 2
Whatever the mechanism of the machine, it always consists only of links and kinematic pairs.
The connection conditions imposed in the mechanisms on the moving links, in the theory of machines and mechanisms It is customary to call kinematic pairs.
Kinematic couplecalled a movable connection of two contiguous links, providing them with a certain relative movement.
In table. 2.1 shows the names, drawings, symbols of the most common kinematic pairs in practice, as well as their classification.
The links, when combined into a kinematic pair, can come into contact with each other along surfaces, lines and points.
Elements of a kinematic pairthey call a set of Surfaces, lines or points along which a movable connection of two links occurs and which form a kinematic Pair. Depending on the type of contact of the elements of kinematic pairs, there are higher and lower kinematic pairs.
Kinematic pairs formed by elements in the form of a line or a point are called higher .
Kinematic pairs formed by elements in the form of surfaces are called lower.
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, with the help of its own mass, springs, etc..
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 along the surface, and therefore, with the same load, lower specific pressures arise in it than in the higher one. Wear, ceteris paribus, is proportional to the specific pressure, and therefore the lower Pairs wear out Slower than the higher ones. Therefore, in order to reduce wear in machines, it is preferable to use lower pairs, however, 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 problem.
Kinematic Pairs are also divided bynumber of degrees of freedom(mobility), which it makes available to the links connected through it, orthe number of link conditions(pair class), imposed by the pair on the relative motion of the connected links. When using such a classification, machine developers receive information about the possible relative movements of the links and about the nature of the interaction of force factors between the elements of a pair.
A free link that is in the general case in M - dimensional space, allowing P types of the simplest movements, has a number of degrees of freedom! ( H) or W - movable.
So, if the link is in a three-dimensional space, allowing six types of simple movements - three rotational and three translational around and along the axes X, V, Z , then we 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 rotation 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 lose their degrees of freedom. This means that kinematic pairs impose on the links they connect by a number S.
Depending on the number of degrees of freedom that the links combined into a kinematic pair have in relative motion, determine the mobility of the pair ( W = H ). If H is the number of degrees of freedom of the links of the kinematic pair in relative motion, to pair mobility is determined as follows:
where P - the mobility of the space in which the pair under consideration exists; S - the number of bonds imposed by the pair.
It should be noted that the mobility of a pair W , defined by (2.1), does not depend on the type of space in which it is implemented, but only on the construction.
For example, a rotational (translational) (see Table 2.1) pair, both in six- and three-movable space, will still remain single-movable, in the first case 5 bonds will be imposed on it, and in the second case - 2 bonds, and, so we will have, respectively:
for six-movable space:
for a three-movable space:
As you 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 frequent division of kinematic pairs into classes suffers from the fact that the class of a 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 choose such a form of the elements of a pair, so that with one independent elementary movement, a second one arises - a dependent (derivative). 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 attributed to a single-moving one, 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 all the simplest relative movements of links necessary for creating mechanisms. However, when creating machines and mechanisms, they are rarely used. This is due to the fact that large friction forces usually arise at the points of contact of the links that form a pair. This leads to significant wear of the elements of the pair, and hence 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 motion of the links as the kinematic pair being replaced.
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-plane kinematic pairs.
From Table. 2.1 it can be seen that the simplest analogue of a rotational kinematic pair is a bearing with rolling elements. Likewise, roller guides replace the linear pair, and so on.
Kinematic connections are more convenient and reliable in operation, they withstand much greater forces (moments) and allow the mechanisms to operate at high relative speeds of the links.
The main types of mechanisms.
Mechanism It can be considered as a special case of a kinematic chain, in which at least one link is turned into a rack, and the movement of the remaining links is determined by the specified movement of the input links.
Distinctive features of the kinematic chain, representing the mechanism, are the mobility and certainty of the movement of its links relative to the rack.
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.
Mechanisms are divided intoguides and transmission.
transmission mechanismcalled 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 that forms kinematic pairs with only moving links coincides with a given curve.
Consider the main types of mechanisms that have found wide application in technology.
Mechanisms, the links of which form only the 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 the articulated four-link (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 is used in machine tool building, instrument making, as well as in agricultural, food, snowplow and other machines.
If we replace a rotational pair in a hinged four-link, for example D , to translational, then 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 has found wide application in compressors, pumps, internal combustion engines and other machines.
Replacing a rotational pair in a hinged four-link FROM to translational, we get a rocker mechanism (Fig. 2.3).
On p and c .2.3, in the rocker mechanism is obtained from a hinged four-link by replacing rotational pairs in it C and O for progressive.
Rocker mechanisms have found wide application in planing machines due to their inherent property of asymmetry of working and idling. Usually they have a long working stroke and a fast idle stroke that ensures the return of the cutter to its original position.
Rice. 2.3. Various types of rocker mechanisms:
1 - crank; 2 - stone; 3 - backstage.
Hinge-lever mechanisms have found great use in robotics (Fig. 2.4).
A feature of these mechanisms is that they have a large number of degrees of freedom, which means that 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 place in the surrounding space.
Widespread application in engineeringcam mechanisms. With the help of cam mechanisms, it is structurally the easiest way to get 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) an appropriate shape. The cam can perform rotational (Fig. 2.5, a, b ), translational (Fig. 2.5, c, g ) or complex movement. The output link, if it makes a translational movement (Fig. 2.5, a, in ), called a pusher, and if rocking (Fig. 2.5, G ) - rocker. To reduce friction losses in the higher kinematic pair AT use an additional link-roller (Fig. 2.5, G ).
Cam mechanisms are used both in working machines and in various kinds of command devices.
Very often, in metal-cutting machines, presses, various instruments and measuring devices, screw mechanisms are used, 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 rotational motion into interdependent translational motion or vice versa. The interdependence of movements is established by the correct selection of the geometric parameters of the screw pair AT .
Wedge mechanisms (Fig. 2.7) are used in various types of clamping devices and devices in which it is required to create a large output force with limited input forces. A distinctive feature of these mechanisms is the simplicity and reliability of the design.
Mechanisms in which the transfer 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; in - 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 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 devices, tape drives, variators (mechanisms with smooth speed control).
To transfer rotational motion according to a given law between shafts with parallel, intersecting and crossing axes, various types of gears are used. mechanisms . With the help of gears, it is possible to transfer motion both between shafts withfixed axles, so with moving in space.
Gear mechanisms are used to change the frequency and direction of rotation of the output link, the summation or separation of movements.
On fig. 2.9 shows the main representatives of gears with fixed axles.
Fig 2.9. Gear drives with fixed axles:
a - cylindrical; b - conical; in - end; g - rack;
1 - gear; 2 - gear; 2 * rail
The smaller of the two meshing gears is called gear, and more - gear wheel.
The rack is a special case of a gear wheel in which the radius of curvature is equal to infinity.
If the gear train has gears with movable axles, then they are called planetary (Fig. 2.10):
Planetary gears, however, compared to fixed axle gears, allow the transfer of greater power and gear ratios with a smaller number of gears. They are also widely used in the creation of summing and differential mechanisms.
The transmission of movements between intersecting axes is carried out using a worm gear (Fig. 2.11).
A worm gear is obtained from a screw-nut transmission by cutting the nut longitudinally and folding it twice in mutually perpendicular planes. Worm gear has the property of self-braking and allows you to implement large gear ratios in one stage.
Rice. 2.11. Worm-gear:
1 - worm, 2 - worm wheel.
Intermittent motion gear mechanisms also include the Maltese cross mechanism. On fig. З-Л "2. shows the mechanism of the four-blade "Maltese cross".
The mechanism of the "Maltese cross" converts the continuous rotation of the leading even - crank 1 with a lantern 3 into the intermittent rotation of the "cross" 2 , lantern 3 enters the radial groove of the "cross" without impact 2 and turns it to the corner 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 swinging the rocker 1 rocking dog 3 imparts rotation to the ratchet wheel 2 only when moving the rocker arm counterclockwise. To hold the wheel 2 from spontaneous clockwise rotation when the rocker arm moves against the clock, a locking pawl is used 4 .
Maltese and ratchet mechanisms are widely used in machine tools and instruments,
If it is necessary to transfer mechanical energy from one point of space to another over a relatively long distance, then mechanisms with flexible links are used.
Belts, ropes, chains, threads, ribbons, balls, etc. are used as flexible links that transmit movement from one even of the mechanism to another,
On fig. 2.14 shows a block diagram of the simplest 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 have been considered above. mechanisms are also given in special Literature, pa-certificates and reference books, for example, such as.
Structural formulas of mechanisms.
There are general patterns in the structure (structure) of various mechanisms that relate 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 the structural formulas of mechanisms.
For spatial mechanisms, Malyshev's formula is currently the most common, the derivation of which is as follows.
Let in a mechanism with m links (including the rack), - 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, the total number of 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 bonds, each two-moving pair IV class - 4 bonds, etc. Therefore, the total number of degrees of freedom, equal to six, will be reduced by the amount
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, which is equal to the number of degrees of freedom of its moving kinematic chain relative to the rack, is determined by the following Malyshev formula:
or in shorthand
(2.2)
at , the mechanism is a statically determinate system; at , a statically indeterminate system.
In the general case, the solution of equation (2.2) is a difficult problem, since the unknown W and q ; the available solutions are complex and are not considered in this lecture. However, in a particular case, if W , equal to the number of generalized coordinates of the mechanism, found from geometric considerations, from this formula you can find the number of redundant connections (see Reshetov L. N. Designing rational mechanisms. M., 1972)
(2.3)
and solve the problem of the static determinability of the mechanism; or, knowing that the mechanism is statically determined, find (or check) W.
It is important to note that the structural formulas do not include the sizes of links, therefore, in the structural analysis of mechanisms, one can assume them 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-adjust; 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, the 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. At present, the Chebyshev formula is extended to any flat mechanisms and is derived taking into account excess constraints as follows
Let in a flat mechanism with m links (including the rack), - the number of movable links, - the number of lower pairs and - the number of higher pairs. If all the moving links were free bodies making a plane motion, the total number of degrees of freedom would be equal to 3 n . However, each lower pair imposes two bonds on the relative movement of the links that form the pair, leaving one degree of freedom, and each higher pair imposes one bond, leaving 2 degrees of freedom.
The number of superimposed bonds may include a certain number of redundant (repeated) bonds, 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 rack, is determined by the following Chebyshev formula:
(2.4)
If known, from here you can find the number of redundant connections
(2.5)
The index "p" reminds us that we are talking about a perfectly flat mechanism, or more precisely, about its flat scheme, since due to manufacturing inaccuracies, a flat mechanism is to some extent spatial.
According to formulas (2.2)-(2.5), a structural analysis of existing mechanisms and a synthesis of structural diagrams of new mechanisms are carried out.
Structural analysis and synthesis of mechanisms.
Influence of redundant connections on the performance and reliability of machines.
As mentioned above, with arbitrary (within certain limits) sizes of links, a mechanism with redundant links () cannot be assembled without deforming the links. Therefore, such mechanisms require increased manufacturing accuracy, otherwise, during the assembly process, the links of the mechanism are deformed, which causes the loading of kinematic pairs and links with significant additional forces (in addition to those main external forces for which the mechanism is intended to be transmitted). With insufficient accuracy in the manufacture of a mechanism with excessive links, friction in kinematic pairs can increase greatly and lead to jamming of the links, therefore, from this point of view, excessive links in mechanisms are undesirable.
As for redundant links in the kinematic chains of the mechanism, when designing machines, they should be eliminated or left to 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 the general case, the optimal solution should be sought, taking into account the availability of the necessary technological equipment, the cost of manufacturing, the required service life and the reliability of the machine. Therefore, this is a very difficult task for each specific case.
We will consider the methodology for determining and eliminating redundant links in the kinematic chains of mechanisms using examples.
Let a flat four-link mechanism with four single-moving rotational pairs (Fig. 2.15, a ) due to manufacturing inaccuracies (for example, due to the 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 placed on the axis X . However, to assemble a rotational pair AT , formed by links 1 and 2 , it will be possible only by combining the coordinate systems Bxyz and B ’ x ’ y ’ z ’ , which requires a linear displacement (deformation) of the point B ’ link 2 along the x-axis and angular deformations of the link 2 around the x and z axes (shown by arrows). This means that there are three redundant bonds in the mechanism, which is also confirmed by formula (2.3): . In order for this spatial mechanism to be statically determinable, its other structural scheme is needed, for example, shown in Fig. 2.15, b , where The assembly of such a mechanism will take place without tightness, since the alignment of the points B and B' will be possible by moving the point FROM in a cylindrical pair.
A variant of the mechanism is possible (Fig. 2.15, in ) with two spherical pairs (); In this case, apart frombasic mobilitymechanism appearslocal mobility- the ability to rotate the connecting rod 2 around its axis Sun ; this mobility does not affect the basic law of movement of the mechanism and can even be useful in terms of leveling the wear of the hinges: connecting rod 2 during the operation of the mechanism, it can rotate around its axis due to dynamic loads. The Malyshev formula confirms that such a mechanism will be statically determinate:
Rice. 2.15
The simplest and most effective way to eliminate redundant connections in the mechanisms of devices is to use a higher pair with a 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):
On fig. 2.16, a, b, c an example of eliminating redundant links in a cam mechanism with a progressively moving roller pusher is given. Mechanism (Fig. 2.16, a ) - four-link (); except for the main mobility (cam rotation 1
) there is local mobility (independent rotation of a round cylindrical roller 3
around its axis) Consequently, . The flat scheme has no redundant connections (the mechanism is assembled without interference). If, due to inaccuracies in manufacturing, the mechanism is considered spatial, then with linear contact of the roller 3 with cam 1 according to Malyshev's formula at , we obtain, but under a certain condition. Kinematic pair cylinder - cylinder (Fig. 2.16, 6
) when the relative rotation of the links is impossible 1 , 3 around the z-axis would be a tripartite pair. If such a rotation, due to inaccuracies in manufacturing, takes place, but is small, and linear contact is practically preserved (under loading, the contact patch is close to a rectangle in shape), then this 
the kinematic pair will be four-movable, therefore, and
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, in ), 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 you to transfer greater loads than point contact.
In Fig. 2.16, d, e another example is given of eliminating redundant connections in a four-link gear (, contact of the teeth of the wheels 1, 2 and 2, 3 - linear). In this case, according to the Chebyshev formula, - the flat scheme has no redundant connections; according to the Malyshev formula, the mechanism is statically indeterminate, therefore, high manufacturing accuracy will be required, in particular, to ensure the parallelism of the geometric axes of all three wheels.
Replacing idler teeth 2 on barrel-shaped (Fig. 2.16, d ), we obtain a statically determinate mechanism.
1.2.1. Conditions for the existence of kinematic pairs
Kinematic pairs (KP) largely determine the performance of the machine, since forces are transmitted through them from one link to another. Due to friction, the elements of the pair are in a stressed state and are subject to wear. Therefore, when designing a mechanism, the correct choice of the type of kinematic pair, its geometric shape, dimensions, structural materials and lubricants is of great importance.
Three conditions are necessary for the existence of a kinematic pair:
The presence of two links;
The possibility of their relative movement;
The constant contact of these links.
In order to facilitate the correct choice of a kinematic pair, they are classified depending on the number of connection conditions, according to the type of relative movement of the links, according to the nature of the contact of the elements of the kinematic pairs and the method of closing the pair.
1.2.2. Classification of kinematic pairs
depending on the number of communication conditions
A rigid body moving freely in space has 6 degrees of freedom. Its possible movements can be represented as rotation around three coordinate axes and translational movement along the same axes (Fig. 2).
Rice. 2 . The number of degrees of freedom of any body in space
Links connected by kinematic pairs receive, to one degree or another, restrictions in their relative movement.
The restrictions imposed on the independent movements of the links forming a kinematic pair are called the connection conditions S.
H = 6 – S ,
where H is the number of degrees of freedom of the links;
S is the number of connection conditions.
If the link is not included in the kinematic pair, i.e., is not connected to another link, then it has no movement restrictions: S= 0.
If 6 conditions of connection are imposed on material bodies, they will lose their mutual mobility and a rigid connection will result, i.e. there will be no kinematic pair: S = 6.
Thus, the number of communication conditions imposed on the relative motion of each link can vary from 1 to 5.
The number of connection conditions for a kinematic pair determines its class (Fig. 3).
Rice. 3. Classes of kinematic pairs
1.2.3. Classification of kinematic pairs
by the nature of the relative movement of the links
By the nature of the relative motion of the links, kinematic pairs are distinguished:
Translational;
Rotational;
Screw.
If one link moves progressively relative to the other, then such a pair is called progressive . On the diagram, translational pairs can be depicted as follows:
If the links forming a pair rotate relative to each other, then such a kinematic pair is called rotational , and it is shown like this:
The symbol of a screw kinematic pair in the diagram is as follows:
1.2.4. Classification of kinematic pairs
by the nature of the contact of the elements of the pair
According to the nature of the contact of the elements of kinematic pairs, pairs of lower and higher are distinguished.
Lower kinematic pairs are pairs in which the elements touch each other along surfaces of finite dimensions.
These include: translational (Fig. 4), rotational (Fig. 5) and screw (Fig. 6) pairs. The lower pairs are reversible, that is, the nature of the movement does not change depending on which link included in the pair is fixed.
Rice. 4. Translational kinematic pair
Higher kinematic pairs are pairs whose elements touch each other along a line or at a point (Fig. 7).
a) 
b) 
Rice. 7. Mechanisms with higher kinematic pair:
a) contact along a line or at a point (cam with a pusher);
b) two teeth are in contact in a line (gearing)
Higher pairs are irreversible. The points of contact describe different curves depending on which link in the pair is fixed.
1.2.5. Classification of kinematic pairs according to the method of closure
According to the method of closure (ensuring the contact of the links of the pair), kinematic pairs are distinguished with power and geometric closures.
Power closure occurs due to the action of weight forces or spring elasticity (Fig. 8); geometric - due to the design of the working surfaces of the pair (Fig. 9).
Rice. 8. Power closure of a kinematic pair
Rice. 9. Geometric closure of a kinematic pair
The main types of mechanisms
The following classification of mechanisms has been adopted:
a) by type of motion transformation:
Reducers (the angular velocity of the driving link is greater than the angular velocity of the driven link);
Multipliers (the angular velocity of the leading link is less than the angular velocity of the driven link);
Couplings (the angular velocity of the driving link is equal to the angular velocity of the driven link).
b) according to the movement and arrangement of links in space:
Spatial (all links move in different, non-parallel planes);
Flat (all links move in the same plane).
in) according to the number of degrees of freedom of the mechanism:
With one degree of mobility;
With several degrees of mobility (integral - summing, differential - separating).
G) by type of kinematic pairs:
With lower kinematic pairs (all kinematic pairs of the mechanism are lower);
With higher kinematic pairs (at least one kinematic pair is higher).
Classification of kinematic pairs. There are several classifications of kinematic pairs
There are several classifications of kinematic pairs. Let's consider some of them.
By elements of the connection of links:
- higher(they are available, for example, in gear and cam mechanisms); in them, the links are connected to each other along a line or at a point:
- lower, in them the connection of links with each other occurs along the surface; they are:
- rotational
- translational
– cylindrical
– spherical
By the number of connections:
The body, being in space (in the Cartesian coordinate system X, Y, Z.) has 6 degrees of freedom, namely, to move along each of the three axes X, Y and Z, as well as rotate around each axis (Fig. 1.2). If a body (link) forms a kinematic pair with another body (link), then it loses one or more of these 6 degrees of freedom.
According to the number of degrees of freedom lost by the body (link), kinematic pairs are divided into 5 classes. For example, if the bodies (links) that formed a kinematic pair lost 5 degrees of freedom each, this pair is called a kinematic pair of the 5th class. If 4 degrees of freedom are lost - the 4th class, etc. Examples of kinematic pairs of different classes are shown in fig. 1.2.
Rice. 1.2. Examples of kinematic pairs of various classes
According to the structural and constructive feature, kinematic pairs can be divided into:
- rotational
- progressive
- spherical,
– cylindrical
Kinematic chain.
Several links interconnected by kinematic pairs form kinematic chain.
Kinematic chains are:
closed
open
To from the kinematic chain get gear, necessary:
a) make one link immovable - form a frame (rack),
b) set the law of motion for one or several links (make them leading) in such a way that all other links perform required purposeful movements.
Number of degrees of freedom of the mechanism- this is the number of degrees of freedom of the entire kinematic chain relative to the fixed link (rack).
For spatial kinematic chain in a general form, we conditionally denote:
number of moving links n,
the number of degrees of freedom of all these links is 6n,
number of kinematic pairs of the 5th class - P5,
the number of bonds imposed by kinematic pairs of the 5th class on the links included in them, - 5Р 5 ,
number of kinematic pairs of the 4th class - R 4,
the number of bonds imposed by kinematic pairs of the 4th class on the links included in them, - 4P 4,
The links of the kinematic chain, forming kinematic pairs with other links, lose some of the degrees of freedom. The remaining number of degrees of freedom of the kinematic chain relative to the rack can be calculated by the formula
This is the structural formula of a spatial kinematic chain, or Malyshev's formula. It was received by P.I. Somov in 1887 and developed by A.P. Malyshev in 1923.
the value W called the degree of mobility of the mechanism(if a mechanism is formed from a kinematic chain).
This formula is called P.L. Chebyshev (1869). It can be obtained from the Malyshev formula, provided that on the plane the body has not 6, but 3 degrees of freedom:
W \u003d (6 - 3)n - (5 - 3)P 5 - (4 - 3) P 4.
The value of W shows how many driving links the mechanism should have (if W= 1 - one, W= 2 - two leading links, etc.).
1.2. Classification of mechanisms
The number of types and types of mechanisms is in the thousands, so their classification is necessary to select one or another mechanism from a large number of existing ones, as well as to synthesize the mechanism.
There is no universal classification. The most common 3 types of classification:
1) functional/2/ - according to the principle of the technological process, namely the mechanisms:
Propulsion of the cutting tool;
Power supply, loading, removal of parts;
transportation;
2) structural and constructive/3/ - provides for the separation of mechanisms both by design features and by structural principles, namely the mechanisms:
Crank-slider;
rocker;
Lever-toothed;
Cam-lever, etc.
3) structural- this classification is simple, rational, closely related to the formation of the mechanism, its structure, methods of kinematic and force analysis.
It was proposed by L.V. Assur in 1916 and is based on the principle of constructing a mechanism by layering (attaching) kinematic chains (in the form of structural groups) to the initial mechanism.
According to this classification, any mechanism can be obtained from a simpler one by attaching kinematic chains to the latter with the number of degrees of freedom W= 0, which are called structural groups or Assur groups. The disadvantage of this classification is the inconvenience for choosing a mechanism with the required properties.
The connection of two contiguous links, allowing their relative movement, is called kinematic pair. In the diagrams, kinematic pairs are denoted by capital letters of the Latin alphabet.
The set of surfaces, lines and individual points of a link, along which it can come into contact with another link, forming a kinematic pair, is called elements of a kinematic pair.
Kinematic pairs (KP) are classified according to the following criteria:
1. By type of contact point (connection point) of the link surfaces:
- lower, in which the contact of the links is carried out along a plane or surface of finite dimensions (sliding pairs);
- higher, in which the contact of the links is carried out along lines or points (pairs that allow sliding with rolling).
Of the flat pairs, the lowest kinematic pairs include translational and rotational. (Lower kinematic pairs allow you to transfer greater forces, are more technologically advanced and wear out less than higher kinematic pairs).
2. According to the relative movement of the links forming a pair:
- rotational;
- progressive;
- screw;
- flat;
- spatial;
- spherical.
3. According to the method of closing (ensuring contact between the links of the pair):
- power (Fig. 2) (due to the action of weight forces or spring elasticity);
- geometric (Fig. 3.) (due to the design of the working surfaces of the pair).
On fig. 3. it can be seen that in rotational and translational kinematic pairs, the closure of the connected links is carried out geometrically. In kinematic pairs "cylinder-plane" and "ball-plane" (see Table 2) by force, i.e. due to the own mass of the cylinder and the ball or other design solutions (for example, in a spherical hinge, the ball can be pressed against the female surface due to the elastic forces of the spring additionally introduced into the design of the ball joint of the car). The elements of a geometrically closed pair cannot be separated from each other due to design features.
4. According to the number of communication conditions, superimposed on the relative motion of the links ( the number of connection conditions determines the class of the kinematic pair );
Depending on the method of connecting the links into a kinematic pair, the number of connection conditions can vary from one to five. Therefore, all kinematic pairs can be divided into five classes.
5. According to the number of movements in the relative motion of the links (the number of degrees of freedom determines the type of the kinematic pair);
Kinematic pairs are denoted by P i , where i =1 - 5 is the class of the kinematic pair. (A kinematic pair of the fifth class is a pair of the first kind).
The classification of CPs according to the number of mobilities and the number of bonds is shown in Table 2.
The table shows some types of kinematic pairs of all five classes. The arrows indicate the possible relative movements of the links. By the form of the simplest independent movements realized in kinematic pairs, notation is introduced (a cylindrical pair is denoted PV, spherical VVV etc., where P – progressive, AT – rotary motion).
Mobility of a kinematic pair is the number of degrees of freedom in the relative motion of its links. There are one-, two-, three-, four- and five-moving kinematic pairs.
Table 2. Classification of kinematic pairs
Single-moving ( class V pair) is a kinematic pair with one degree of freedom in the relative motion of its links and five imposed connection conditions. A single-moving pair can be rotational, translational or helical.
Rotary pair allows one rotational relative movement of its links around the X axis. The elements of the links of rotational pairs come into contact along the side surface of round cylinders. Therefore, these pairs are among the lowest.
Translational couple is called a single-moving pair that allows rectilinear-translational relative motion of its links. Translational pairs are also the lowest, since the contact of the elements of their links occurs along the surfaces.
screw pair is called a single-moving pair that allows helical (with a constant pitch) relative movement of its links and belongs to the number of lower pairs.
When a kinematic pair is formed, it is possible to choose the shape of the elements of the kinematic pairs in such a way that, with one independent simple displacement, another derivative motion arises, as, for example, in a screw pair. Such kinematic pairs are called trajectory .
Two-moving kinematic pair(pair IV class) is characterized by two degrees of freedom in the relative movement of its links and four conditions of connection. Such pairs can be either with one rotational and one translational relative movement of the links, or with two rotational movements.
The first type is the so-called cylindrical pair, those. the lowest kinematic pair, allowing independent rotational and oscillatory (along the axis of rotation) relative movements of its links.
An example of a pair of the second kind is spherical pair with a finger. This is the lowest geometrically closed pair that allows relative rotation of its links around the X and Y axes.
Three-movable pair is called a kinematic pair with three degrees of freedom in the relative motion of its links, which indicates the presence of three imposed connection conditions. Depending on the nature of the relative motion of the links, three types of pairs are distinguished: with three rotational movements; with two rotational and one translational movements; with one rotational and two translational.
The main representative of the first type is spherical pair. This is the lowest geometrically closed pair, allowing spherical relative motion of its links.
The third type is the so-called planar pair , i.e. the lowest kinematic pair, allowing plane-parallel relative motion of its links.
Four-moving pair(class II pair) is a kinematic pair with four degrees of freedom in the relative motion of its links, i.e. with two imposed communication conditions. All four-moving couples are the highest. An example is a pair that allows two rotational and two translational movements.
Five-moving couple(pair of class I) is called a kinematic pair with five degrees of freedom in the relative motion of its links, i.e. with one imposed link condition. Such a pair, composed of two spheres, allows three rotational and two translational movements and will always be the highest.
Kinematic connection- a kinematic pair with more than two links.