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 the geometric shapes of the points of contact, that is, on 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

According to the geometric type of connection between surfaces and the 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 of 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 of rotational motion, 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).

rotational;

progressive;

screw;

spherical.

Symbols of links and kinematic pairs on kinematic diagrams.

A kinematic diagram of a mechanism is a graphic representation on a selected scale of the relative position of the links included in kinematic pairs, using symbols in accordance with GOST 2770-68. Large letters of the Latin alphabet in the diagrams indicate 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, the quality of operation of manipulators and industrial robots is greatly influenced by their maneuverability.

Types of gear mechanisms, their structure and brief characteristics.

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 point C of the contact of the profiles coincides with the point W of the intersection of the normal N-N and the line 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 ( abbreviated as a pair) is a movable connection of two contacting links. The restriction imposed on the motion of a rigid body is called condition of connection.

Thus, the kinematic pair imposes a coupling condition on the relative motion of the two connected links. It is obvious that the largest number of coupling conditions imposed by a kinematic pair is five (5).

The different number of coupling conditions imposed on the relative motion of links by kinematic pairs allows us to divide the latter into five classes, so that the k-th class pair imposes k coupling conditions, where k is from (1,2,3,4,5). It follows that the kinematic pair of the kth class allows 6k degrees of mobility in the relative movement of the 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. articulated lever mechanism kinematic

Top pairs- these are pairs in which, when connecting two links, contact is made on curves and points.

Low pairs- these are pairs in which, when connecting two links, contact is made along the surfaces.

This mechanism consists of 6 links (Figure 2).

• A) 1- The crank, the moving link, makes a rotational movement;
• B) 2.4 connecting rods, moving links, perform complex movements;
• B) 3.5 - sliders, movable links, perform translational movement;
• D) 6- rack, fixed link;

Number of moving parts=5.

Determination of the degree of mobility of the mechanism.

The mechanism under consideration has seven (7) kinematic pairs, of which five (5) are rotational and two (2) translational.

The degree of mobility of mechanisms is determined by the formula:

W=3(n-1)-2P5-P4;

n - Number of links;

P5 - number of kinematic pairs of class 5;

P4 - number of kinematic pairs of class 4;

w=3(6-1)-2*7=1; w=1;

Assura group and entry level groups.

Let's divide the mechanism into asura groups. To do this, we will select initial-level groups. Since the degree of mobility of the mechanism is w=1, then the group of the initial link should be w=1. The group includes a stand (6) and a moving link (1).

The simplest groups of links, the addition of which to other links of the mechanism does not change the number of its degrees of freedom, are called asura groups. Since the only fixed link was included in the group of initial links, the assura group contains only movable links.

The degree of mobility of the asura group is w=0 and can be defined as the number of degrees of freedom of the group relative to the fixed link. Assura groups are classified according to the number of kinematic pairs with which they are attached to the main mechanism. This number determines the order of the group. In addition, the assur group has a class determined by the number of kinematic pairs that form the most complex closed circuit.

One on one with the enemy [Russian school of hand-to-hand combat] Kadochnikov Alexey Alekseevich

Kinematic pairs in the human body

Kinematic pairs used in technology and common in nature have a fundamentally important difference.

In technical mechanisms, kinematic pairs are usually arranged in such a way that only well-defined, predetermined plane movements are possible.

Kinematic pairs in the human body are movable connections of two bone links, ensuring their arbitrary spatial movements. The possibilities of movement of kinematic joints are determined by the skeletal structure of the body and the control action of the muscles.

Kinematic pairs in the human body are usually called biokinematic. Of all the biokinematic pairs, when studying human motor actions, specialists are primarily interested in the upper and lower extremities of the body, which, according to the accepted classification, are the lowest rotational kinematic pairs.

Rice. 17

In Fig. Figure 17 shows a kinematic model of the human upper limb. A ball joint 1 biokinematic pair is connected to the body; The links of the pair are connected to each other by a cylindrical hinge 2. Spatial biokinematic pairs of limbs can be closed or open. They have permanent and temporary connections, which determine how many and what degrees of freedom this pair in question has. Thus, the movements of the arm as an open biokinematic pair (Fig. 18a) are limited by the shoulder joint, which excludes linear movements of shoulder 1 relative to the body.

The orientation of the hand at any moment of its spatial movement relative to the body can be described by five parameters. The coordinates x A, y A, z A (Fig. 18b) determine the position of shoulder 1, the position of forearm 2 relative to the shoulder is specified by the angle? 2, rotation of the forearm around its own axis - angle? 2.

Rotating the forearm at an angle? 2 can be ignored, since it does not affect the orientation of the hand as a whole. With the accepted assumption, it is obvious that the human hand generally has four degrees of freedom.

The actual number of degrees of freedom of the hand depends on its orientation in space and is limited by the limits of mobility of the shoulder and elbow joints.

Rice. 18

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STRUCTURE OF MECHANISMS

Basic concepts and definitions.

The system of terms provides a uniform approach to describing any knowledge system. Therefore, let's start by clarifying the meaning and significance of the formulations used.

Mechanism - a system of bodies designed to convert the movement of one or more solid bodies and (or) forces acting on them into the required movements of other bodies and (or) forces. In the theory of mechanisms and machines, solid bodies are understood as both absolutely solid and deformable bodies.

Car– a device that performs mechanical movements to transform energy, materials and information. Materials mean objects of labor: processed products, transported loads, etc.

Detail – a product made from a uniform material by name and brand, without the use of assembly operations.

Link– a rigid body participating in a given transformation of motion. A link can consist of several parts that do not have relative movement among themselves.

Stand - a link that is conventionally accepted as stationary.

Input link- a link to which movement is communicated, converted by the mechanism into the required movements of other links.

Output link- a link that makes the movement that the mechanism is designed to perform.

Initial link - a link to which one or more generalized coordinates of the mechanism is assigned.

Generalized mechanism coordinate- each of the mutually independent coordinates that determine the position of all links of the mechanism relative to the rack.

Number of degrees of freedom of the mechanism– number of generalized coordinates of the mechanism.

Connection– any condition that reduces the number of degrees of freedom of the mechanism. Any connection can be discarded by replacing its action with a reaction.

Redundant coupling– a connection, the elimination of which does not change the number of degrees of freedom of the mechanism.

Kinematic pair– a connection of two rigid bodies of a mechanism, allowing their specified relative motion. The condition for the existence of a pair is: the presence of two links, their contact and the relative movement of the links.

Kinematic chain– a system of links and (or) solid elements of a mechanism that form kinematic pairs with each other. There are kinematic chains open And closed. Unclosed This is called a kinematic chain that has at least one link included in only one kinematic pair. U closed there are no links in the chain that have free elements of kinematic pairs. Each link of such a chain is included in at least two pairs.

Mechanism element- a solid, liquid or gas component of a mechanism that ensures the interaction of its parts that are not in direct contact with each other.

Kinematic pair coupling element- a common surface, line or point formed by the mating elements of two other bodies.

Number of degrees of freedom (mobility) of a kinematic pair (N)– the number of independent coordinates necessary to describe the relative position of the links of kinematic pairs.

It is known that a freely moving body in space has six degrees of freedom. Number of communication conditions S, superimposed on the relative movement of a link of a kinematic pair can vary within . There are one-, two-, three-, four- and five-movable kinematic pairs. Consequently, the relation holds H = 6 – S.

Single moving pair– a kinematic pair with one degree of freedom in the relative motion of the connected rigid bodies.

Double-moving pair– a kinematic pair with two degrees of freedom in the relative motion of the connected rigid bodies.

Three-moving pair– a kinematic pair with three degrees of freedom in the relative motion of the connected rigid bodies.

Quadruple pair– a kinematic pair with four degrees of freedom in the relative motion of the connected rigid bodies.

Five-movable pair– a kinematic pair with five degrees of freedom in the relative motion of the connected rigid bodies.

Structural formula– an algebraic expression that establishes the connection between the number of degrees of freedom of the mechanism, the number of moving links, the number and mobility of kinematic pairs.

Assur Group– a kinematic chain, the attachment of which to a mechanism or its disconnection forms a mechanism that has mobility equal to the mobility of the original mechanism, not divided into other chains with the same properties.

Scale factor– the ratio of the numerical value of a physical quantity in its inherent units to the length of the segment (mm) depicting this quantity (on a diagram, graph, etc.).

Scale– the reciprocal of the scale factor.

Classification of kinematic pairs

1. Depending on the number N differentiate one-, two-, three-, four-, and five-moving kinematic pairs. The number of constraint equations is taken as the class number.

2. According to the nature of contact of the elements of the links (more precisely, the type of elements), pairs are divided into inferior and in higher(proposal by F. Reuleaux). TO to the lowest include kinematic pairs, the elements of which are surfaces (Figure 1.2). Elements higher pairs are lines or points (Figure 1.2).

3. Based on the nature of the coupling, a distinction is made between kinematic pairs with force closure (contact of the links is ensured by the action of some force, for example, weight or spring) and kinematic (constant contact of the links is achieved due to the structural shape of the elements).

4. Depending on the nature of the relative movement of the links, kinematic pairs are divided into translational, rotational, helical, cylindrical, spherical, and planar.

In Fig. Figure 1.1 shows single-moving pairs (class V kinematic pairs); let’s look at them in more detail.

		V
		b
	A

	Figure 1.1. Single-moving kinematic pairs.

Single-moving pair:

1) Rotational(Fig. 1.1. a) – cylindrical hinge. Five coupling conditions are imposed: all movements except rotational are excluded.

2) Progressive(Fig. 1.1. b) – five connection conditions are imposed: all movements except one translational are excluded.

3) Screw(Fig. 1.1. c) – five connection conditions are imposed: all movements except translational are excluded. (Rotation does not introduce degrees of freedom, since in this case the translational and rotational motions are not independent).

In Fig. 1.2 shows pairs two-, three-, four-, and five-moving(kinematic pairs of IV, III, II and I classes) we will consider them in more detail.

	A		V		G
		b

	Figure 1.2. Kinematic pairs

Double mobile pair(Fig. 1.2.a) - bushing on the roller. Four connection conditions are imposed, translational and rotational movements along the O X and O Z axes are excluded.

Three-movable pair(Fig. 1.2.b) - spherical cylinder. Three connection conditions are imposed: translational movements along all three axes are excluded.

Four-movable pair(Fig. 1.2.c) - a cylinder on a plane. Two connection conditions are imposed: translational motion along the O Z axis and rotational motion around the O X axis are excluded.

Five-movable pair(Fig. 1.2.d) - a ball on a plane. One connection condition is imposed: translational motion along the O Z axis is excluded.