Course on the theory of mechanisms and machines

Basic concepts of the theory of mechanisms and machines

Introduction

The course on the theory of mechanisms and machines is a transitional step in the chain of mechanical training of an engineer - it is based on the fundamental knowledge acquired by the student while studying mathematics, physics, theoretical mechanics and is the basis for studying subsequent practical (special) disciplines of the mechanical cycle (primarily for the course “Details”) machines and design principles").

The purpose of studying the discipline “Theory of Mechanisms and Machines” is to form the necessary initial knowledge base on general methods of analysis and synthesis of mechanical systems that form the basis of technological equipment used in the field of future professional activity of graduates of higher technical educational institutions.

Car

A machine is a device that performs mechanical movements to transform energy, materials and information in order to replace or facilitate human physical and mental labor.

From the point of view of the functions performed, machines can be divided into the following classes:

Energy machines (engine machines, generator machines).

Working machines (transport and technological).

Information machines (for receiving and converting information).

Cybernetic machines (replacing or imitating various mechanical, physiological or biological processes inherent in humans and living nature, and possessing elements of artificial intelligence - robots, automata).

A developed machine device, consisting of an engine, transmission mechanisms and a working machine (and in some cases, control and computing devices) is called a machine unit.

Basic concepts of machine elements

A part is an integral part of a mechanical device, made without the use of assembly operations (for example: a bolt, nut, shaft, machine bed produced by casting, etc.).

A link is a part or a group of parts that, from a kinematic point of view, represents a single whole (i.e., a group of parts rigidly connected to each other and moving as a single solid body).

A kinematic diagram is a conventional representation of the links and the entire mechanism, made strictly to scale.

When drawing up a kinematic diagram, the main elements of the link are identified, with which it is connected to other links of the mechanism (holes, guides, etc.). These elements are depicted conventionally (for example, holes - in the form of circles of arbitrary radius) and are connected by rigid rods.

In the theory of mechanisms and machines, scale is understood as the “price” of one millimeter. This understanding of scale (sometimes called a scale factor) is very convenient when analyzing the operation of a mechanism, because is universal and allows you to represent any physical quantity in the form of a segment, which is very important when using graphical and graphic-analytical research methods.

Similarly, you can represent any quantities (displacement of links, speed, acceleration, time, force, etc.) in the form of segments on plans, diagrams, various graphs, etc.

Depending on the nature of the movement, the links may have their own names, for example:

Crank - a link that performs a rotational movement around a fixed axis and makes a full revolution;

Rocker arm - a link that performs a reciprocating rotational movement;

Slider – a link moving forward;

Connecting rod is a link that performs a complex plane-parallel movement;

Rocker - a rocker arm (or sometimes a crank) along which the slider moves;

A stand is a link taken as a fixed one (by definition of a link, there can be only one stand in a mechanism - all fixed parts are necessarily attached to a certain frame, body, crankcase, base and represent one rigid structure, i.e. one link).

On a kinematic diagram, the rack is usually depicted in the form of separate fragments in those places where other links of the mechanism are attached to it, which greatly simplifies this diagram.

A kinematic pair is a movable connection of two links.

Kinematic pairs are classified according to various criteria:

1) by the number of connections imposed on the relative motion of the links connected in a kinematic pair. Based on this feature, kinematic pairs are divided into classes. The following notations are accepted:

W – number of degrees of freedom

S is the number of connections imposed on the relative movement of the links.

A free link in space has six degrees of freedom. When links are connected, some of these degrees of freedom are taken away ("bonds are superimposed"). The relationship between the number of superimposed connections and the remaining number of degrees of freedom in the relative movement of the links is obvious:

W=6–S or S=6–W,

Thus, there are five classes of kinematic pairs (if you subtract all six degrees of freedom, you get a fixed connection).

Examples of kinematic pairs:

The ball relative to the plane, without breaking away from it, can perform rotational movements around all three coordinate axes, as well as move along the “X” and “Y” axes. When moving along the "Z" axis, the ball will break away from the plane, i.e. there will be two free links - the kinematic pair will cease to exist. Thus, one connection is superimposed on the relative movement of the links - this is a kinematic pair of class I.

The cylinder is relative to the plane; without breaking the nature of the contact, the cylinder cannot be moved along the “Z” axis and rotated around the “Y” axis, i.e. the number of bonds is two – a class II pair.

A plane relative to another plane without disturbing the nature of the contact can move translationally along the “X” and “Y” axes, as well as rotate around the “Z” axis. Translational movement along the "Z" axis and rotational movements around the "X" and "Y" axes are impossible. Thus, the number of connections is three – a kinematic pair of class III.

W=5 W=4 W=3

S = 1 => I class. S = 2 => II class. S = 3 => III class.

Examples of kinematic pairs

For example, a bolt and a nut make up a fifth-class kinematic pair. In this case, there are two movements of the nut with a stationary bolt - a rotational movement around the axis of the bolt and a translational movement along this axis, but you cannot move the nut along the axis without rotating it, or rotate the nut so that it does not move along the axis. These two movements form one complex (in this case, a screw) movement. It determines one degree of freedom in the relative movement of these links, i.e. the number of connections is five;

2) by the nature of the contact of the links connected into a kinematic pair. On this basis, kinematic pairs are divided into higher and lower. Higher pairs have point or linear contact of the links that make up this kinematic pair. In the lower pair, the links are in contact with each other along some surface (in the particular case, along a plane).

Lower kinematic pairs have greater load-bearing capacity, because have a large contact area (in the highest pair, the contact area is theoretically zero, but in reality it is obtained due to the deformation of the elements of the kinematic pair - the “contact patch”). But in the lower pairs, during operation, one surface slides relative to the other, while in the higher pairs Both sliding and rolling may occur.

As a rule, the resistance to sliding is greater than the resistance to rolling of one surface relative to another, i.e. friction losses in the higher pair (if only rolling bearings are used) are less compared to the lower pair (therefore, to increase the efficiency, rolling bearings are usually installed instead of plain bearings).

Kinematic pairs: ball and plane, cylinder and plane are the highest, and the pair plane and plane are the lowest.

3) along the trajectory of movement of points belonging to the links that make up the kinematic pair. Based on this feature, spatial and plane kinematic pairs are distinguished.

In a flat kinematic pair, all points move in the same or parallel planes, and the trajectories of their movement are flat curves. In spatial pairs, points move in different planes and have trajectories in the form of spatial curves.

A significant number of mechanisms used in practice are flat mechanisms, so it is necessary to consider flat kinematic pairs in more detail.

A free link placed in a plane has three degrees of freedom (translational movements along the coordinate axes and rotational movements around an axis perpendicular to the given plane). Thus, placing a link in a plane takes away three degrees of freedom from it (imposes three connections). But the connection of a given link with another into a kinematic pair imposes additional connections on the relative motion (minimum number - 1). As a result, only kinematic pairs that have two or one degree of freedom in relative motion can exist on the plane.

According to the general classification, these are couples of the fourth and fifth classes. The simplest pairs of the fifth class provide only one movement - rotational or translational (a rotational kinematic pair in technology is called a hinge; a translational pair, by analogy with a translationally moving link, is sometimes also called a slider).

Two degrees of freedom in relative motion on a plane are usually provided by two contacting profiles (in a kinematic diagram, contact is at a point; in a real mechanism, this is possibly a line that is projected into a point). Thus, flat kinematic pairs of the fifth class (hinges and sliders) are simultaneously lower pairs, and kinematic pairs of the fourth class are higher pairs.

Examples of kinematic pairs:

4) by the nature of the closure of the links connected into a kinematic pair. There are two types of kinematic pairs that differ from each other in this respect. Kinematic pairs with geometric closure and kinematic pairs with force closure.

In pairs with positive closure, the configuration of the links prevents their separation during operation. For example, connecting the connecting rod to the crankshaft using a connecting rod cap, or any other hinges (door with jamb, window with window frame, etc.).

In pairs with force closure, the contact of the links during operation is ensured by a constantly acting force. Weight acts as a closing force. If the weight is not enough, then various elastic elements (most often springs) are usually used to create pressing force.

A kinematic chain is a combination of links connected into kinematic pairs.

There is a certain classification of kinematic chains - chains can be simple and complex, closed (closed) and open (open), spatial and flat.

A mechanism is a kinematic chain that has a stand (i.e., a link taken as a fixed one), in which the movement of one or more links completely determines the nature of the movement of the remaining links of this chain.

The links whose laws of motion are specified are called input links.

The links whose laws need to be determined are called outputs. The number of input links is determined by the number of degrees of freedom of the kinematic chain underlying this mechanism.

The concepts input and output (input and output) are a kinematic characteristic. This should not be confused with the concepts of leading link and driven link. The leading link is the link to which power is supplied; driven link - a link from which power is removed (to perform useful work).

Thus, the concepts of leading and driven link are a power (energy) characteristic. However, in the overwhelming majority of cases, the input link is also the leading link, and the output link is the driven link.

Main types of mechanisms

According to their functional purpose, mechanisms are usually divided into the following types:

Mechanisms of engines and converters (convert various types of energy into mechanical work or vice versa);

Transmission mechanisms (transmits movement from the engine to a technological machine or actuator, converting this movement into that necessary for the operation of a given technological machine or actuator);

Actuators (changes in the shape, state, position and properties of the processed environment or object);

Mechanisms for control, monitoring and regulation (to ensure and control the size of processed objects);

Mechanisms for feeding, transporting, feeding and sorting processed media and objects (mechanisms for screw augers, scraper and bucket elevators for transporting and feeding bulk materials, mechanisms for loading hoppers for piece workpieces, mechanisms for sorting finished products by size, weight, configuration, etc. );

Mechanisms for automatic counting, weighing and packaging of finished products (used in many machines, mainly producing mass piece products).

According to general methods of synthesis and analysis of work, the following types of mechanisms are distinguished:

Mechanisms with lower pairs (lever mechanisms)

Cam mechanisms

Gear mechanisms

Friction mechanisms

Mechanisms with flexible links

Mechanisms with deformable links (wave transmissions)

Hydraulic and pneumatic mechanisms.

Kinematics problems

Kinematic analysis is the study of the movement of mechanism links without taking into account the forces causing this movement. In kinematic analysis, the following problems are solved:

Determining the positions of the links that they occupy during operation of the mechanism, as well as constructing the trajectories of movement of individual points of the mechanism;

Determination of the velocities of characteristic points of the mechanism and determination of the angular velocities of its links;

Determination of accelerations of individual points of the mechanism and angular accelerations of its links.

When solving problems of kinematic analysis, all existing methods are used - graphic, graphic-analytical (method of velocity and acceleration plans) and analytical. In kinematic analysis, the input link (the link whose law of motion is specified) is taken as the initial link, i.e. the input link with the stand constitutes the initial mechanism - the solution to the problem begins with it.

Dynamics of mechanisms and machines

Dynamics problems

This section studies the movement of mechanism links taking into account the forces acting on them. In this case, the following main problems of dynamics are considered:

1) study of the forces acting on the links of the mechanism and determination of unknown forces for a given law of motion at the input;

2) the problem of the energy balance of the machine;

3) establishment of the true law of motion under the influence of given forces;

4) regulation of the speed of the machine;

5) balancing the forces of inertia;

6) drive dynamics.

Power calculation of mechanisms

The force calculation of mechanisms relates to the solution of the first problem of dynamics. As can be seen from the content of the dynamics problems given above, the first task includes two parts: the study of the forces acting on the links of the mechanism; determination of unknown forces for a given law of motion at the input (this second part is the task of force calculation).

In order to further understand the terminology and systematize the material, it is advisable to repeat the information about forces known from physics and theoretical mechanics, as well as introduce some new (used in the theory of mechanisms and machines) concepts. From the point of view of solving problems of force dynamics (in this case, force is understood as the generalized concept of a force factor - actual force or moment) can be classified as follows:

a) on the interaction of a mechanism link with other objects. On this basis, forces are divided into external and internal:

External forces are the forces of interaction of a mechanism link with some bodies or fields that are not part of the mechanism;

Internal forces are the forces of interaction between the links of the mechanism (reactions in kinematic pairs);

The driving force is the force that helps the link move and develops positive power;

b) according to the power developed by the force. Based on this feature, forces are divided into driving forces and resistance forces (Figure 16):

The resistance force prevents the movement of the link and develops negative power.

In turn, resistance forces can be divided into forces of useful resistance and forces of harmful resistance:

The forces of useful resistance are the forces to overcome which the mechanism was created. By overcoming the forces of useful resistance, the mechanism creates useful work (for example, overcoming cutting resistance on a machine, they achieve the necessary change in the shape of a part; or, overcoming air resistance in a compressor, they compress it to the required pressure, etc.);

Forces of harmful resistance are forces to overcome which power is expended and this power is lost irretrievably. Typically, the harmful forces of resistance are the forces of friction, hydraulic and aerodynamic resistance. The work to overcome these forces is converted into heat and dissipated into space, so the efficiency of any mechanism is always less than one;

c) weight forces are the forces of interaction between the links of the mechanism and the gravitational field of the earth;

d) friction forces - forces that resist the relative movement of contacting surfaces;

e) inertial forces - forces that arise during uneven movement of a link and resist its acceleration (deceleration). The force of inertia acts on the body that causes the given link to accelerate (slow down). In the general case, with uneven movement, an inertial force and a moment of inertial force arise:

Fin=-m. as , Min=-Is . e,

Fin is the main vector of inertia forces applied at the center of mass of the link;

Min – main moment of inertia forces;

m – link mass;

Is – moment of inertia of the link relative to the center of mass;

as – acceleration of the center of mass of the link;

e is the angular acceleration of the link.

The minus sign in the formulas shows that the inertia force is directed opposite to the acceleration of the center of mass of the link, and the moment of inertia forces is directed opposite to the angular acceleration of the link. The sign of the force or moment is taken into account only when establishing the true direction of the force or moment in the design diagram, and in analytical calculations their absolute values ​​are used.

During the force analysis of mechanisms, various cases may arise when one or both force inertial factors may have a zero value. Figure 17 above shows some cases of the occurrence of forces and moments of inertial forces during the movement of mechanism links.

The force calculation itself comes down to determining the unknown forces acting on the links of the mechanism. As is known from theoretical mechanics, static equations are used to determine unknown forces.

The mechanism is a nonequilibrium system, because most of its links have uneven movement, and the points belonging to these links move along complex curvilinear trajectories (remember: the state of equilibrium is a state of rest or rectilinear uniform motion).

Therefore, to solve the problem, the kinetostatic method is used. The kinetostatics method is based on d'Alembert's principle: if inertial forces and moments of inertial forces are added to all external forces acting on the links of the mechanism, then this mechanism will be in a state of static equilibrium. That is, this is an artificial technique that brings a nonequilibrium system to a state of equilibrium.

The artificiality of the technique lies in the fact that inertial forces are applied not to those bodies that make the links move faster (slowly), but to the links themselves.

Having applied this technique, in the future it is possible to carry out force calculations using static equations. However, to solve a problem using only equilibrium equations, the system must be statically determinate.

Condition for static definability of a plane kinematic chain:

For each link located in a plane, three independent static equations can be compiled. If there are “n” moving links in a kinematic chain, then in total 3n independent statics (equilibrium) equations can be written for this chain. These equations are used to determine reactions in kinematic pairs and unknown external forces.

On the plane there are only kinematic pairs of the fifth and fourth classes. Pairs of the fifth class are represented by a rotational kinematic pair (hinge) and a translational pair (connection of the slider with the guide). In a hinge, the force between the links can be transmitted in any direction, therefore the magnitude and direction (two components) of the reaction at the hinge are unknown, i.e. To determine the total reaction in a rotational pair, two static equations must be used.

As a first approximation, the calculation is carried out without taking into account friction forces. In this case, nothing prevents the slider from moving along the guide. The slider cannot move across the guide and rotate, so in the translational pair the reaction is directed perpendicular to the guide and a reactive moment arises that prevents the slider from turning.

In force calculations, the reactive torque is usually not determined, but the conditional point of application of the reaction is found (the product of the reaction and the distance to its conditional point of application is the reactive torque). To determine the reaction in a translational pair, it is also necessary to use two static equations (to determine two components - the magnitude and the point of application). Thus, to determine the complete reaction in a kinematic pair of the fifth class, it is necessary to spend two static equations.

Pairs of the fourth class (highest pairs) on the plane represent profiles in contact with each other. In the highest pair, the force between the links is transmitted along the common normal to the touching profiles (without taking into account friction forces). Therefore, in the highest pair of the fourth class, the reaction is unknown only in magnitude (the point of application of the reaction at the point of contact of the profiles, the direction along the common normal to these profiles).

Thus, to determine the reaction in a fourth-class pair, it is necessary to use one statics equation (to determine one component - the magnitude of the reaction).

If in a kinematic chain the number of pairs of the fifth class is equal to P5, then 2P5 static equations must be spent to determine the reactions in all these pairs. To determine reactions in all pairs of the fourth class, a number of equations is used equal to the number of these pairs P4.

Thus, out of 3n independent static equations, 2P5 equations are used to determine reactions in fifth-class pairs and P4 are used to determine reactions in fourth-class pairs. The remaining equations are used to determine the unknown external forces acting on the links of the mechanism.

Let X be the number of equations remaining to determine the unknown external forces, then

X=3n–2Р5–Р4,

but this formula coincides with the Chebyshev formula for determining the number of degrees of freedom of a plane kinematic chain. As a result, we can formulate the condition for the static definability of a kinematic chain as follows: a kinematic chain is statically determinate in the case when the number of unknown external forces acting on its links does not exceed the number of degrees of freedom of this chain.

Since solution methods have been developed for Assur groups, it is necessary to formulate a condition for the static definability of the Assur group. The Assur group is a kinematic chain that has its own degree of freedom equal to zero. Therefore, the Assur group is statically determinate if its links are not acted upon by unknown external forces. The equations in the Assur group are sufficient only to determine reactions in kinematic pairs. This circumstance predetermines the order of power calculation of the mechanism:

They divide the mechanism into Assur groups, taking as the initial link the link on which an unknown external force acts;

The solution begins with the last joined group and ends with the initial link.

With this approach, only known external forces will always act on Assur groups and from considering their equilibrium the reactions in kinematic pairs will be determined, and from considering the equilibrium conditions of the initial links the remaining reactions and unknown external forces will be determined.

Since the solution is based on Assur groups, the principle of force calculation of groups is considered below using the example of groups of the second class.

Group 1 species

Compose: ∑ mB(2)=0; ∑ mB(3)=0; ∑ F(2,3)=0; ∑ F(2)=0

Define: R12t ; R43t; R12n; R43n; R32

Replace reaction R12 with components R12n II AB and R12t⊥ AB

Group 2 types

Compose: ∑ mB(2)=0; ∑ F(2,3)=0; ∑ mB(3)=0; ∑ F(2)=0

Define: R12t ; R12n; R43; R43; R32

Replace reaction R12 with components R12n II AC and R12t⊥ AC

Group 3 types

Compose: ∑ mC(2,3)=0; ∑ F(2)=0; ∑ mC(3)=0; ∑ F(3)=0

Define: R12t ; R12n; R32n; h23 ; R43

Group 4 species

Compose: ∑ F(2,3)=0; ∑ mB(2)=0; ∑ mB(3)=0; ∑ F(2)=0

Define: R12; R43; h12 ; h43 ; R32

Group 5 species

Compose: ∑ F(3)=0; ∑ mA(2)=0; ∑ mA(2,3)=0; ∑ F(2)=0

Define: R23; R43; h32; h43 ; R12

The following notations and simplifications are used in the table:

The links of the study group are designated by numbers 2 and 3;

Link 1 is disconnected from link 2, therefore reaction R12 is applied (the action of disconnected link 1 on link 2 in question);

Link 4 is disconnected from link 3, so reaction R43 is applied to link 3;

The line above the reaction designation means that at this point the reaction is determined both in magnitude and in direction (i.e., there is an image of this vector on the force plan);

In order to reduce the clutter of the drawing and improve clarity, the external forces applied to the links of the group under consideration are not shown in the figure (you just need to keep in mind that all external forces acting on the links of the Assur group are known - this is determined by the order of force calculation of the mechanism).

Accounting for friction in mechanisms

According to physical characteristics, friction is distinguished between internal and external.

Internal friction is the processes occurring in solid, liquid and gaseous bodies during their deformation and leading to irreversible dissipation of mechanical energy. Internal friction manifests itself in the damping of free vibrations.

External friction is the resistance to relative movement that occurs between two bodies in areas of contact of surfaces, that is, in kinematic pairs. Based on kinematic characteristics, they are distinguished: sliding friction, which occurs when one body slides over the surface of another, and rolling friction, which occurs when one body rolls over the surface of another.

Trunnion friction

First hypothesis. The specific pressure over the supporting surface is distributed evenly, i.e. q=const (Figure 25a).

Let us select an infinitesimal surface element, determined by the central angle dα, at a distance α from the vertical axis. This element is subject to a normal reaction dRN, which is determined through the specific pressure and area of ​​the selected element:

The sum of elementary normal reactions in projection onto the vertical axis balances the radial force acting on the axle:

An intermediate result is obtained that determines the value of the specific pressure:

However, this result is of great independent significance. It shows that the specific pressure (and in strength calculations this is the bearing stress on the surface of the contacting parts) is determined by dividing the radial force by the projection of the contact area onto the diametrical plane of the shaft (and not by the full value of the contact area). This provision is widely used in the calculations of machine parts.

Let us determine the magnitude of the elementary friction force acting on the selected element and the elementary friction moment from this force:

Having summed up the elementary moments from the friction force over the entire contact area, we obtain the value of the friction moment on the surface of the axle according to this hypothesis:

Here fI" is the reduced friction coefficient calculated according to the first hypothesis.

Second hypothesis. The calculation is carried out taking into account the wear of the contact surface. In this case, the following assumption is made - the bearing wears out, but the shaft remains unchanged. This assumption fully corresponds to the real situation, because the shaft takes all the loads from the gears, operates in heavy duty, is usually made of high-quality steel, the supporting surfaces are often subjected to thermal hardening.

In order to reduce friction losses (to form an antifriction pair), sliding bearings are made of softer materials that, when paired with a steel shaft, have reduced friction coefficients (bronze, babbitt, etc.). It is clear that it is the softer material that will wear out first.

As a result of bearing wear, the shaft will “sag” by a certain amount (Figure 25b). From the theory of wear it is known that the amount of wear is proportional to the specific pressure and relative speed of the rubbing surfaces. But in this case, the relative speed is the peripheral speed on the surface of the axle, which is the same at all points. Therefore, the amount of wear will be greater in those places where the specific pressure is greater, i.e. the amount of wear is proportional to the specific pressure.

Figure 25b shows two positions of the shaft - at the beginning of operation and after surface wear has occurred. The worn layer is a crescent-shaped figure. But since wear is proportional to the specific pressure, this crescent-shaped figure can be taken as a diagram of the specific pressure, made on a certain scale.

As can be seen, as a result of wear, a redistribution of the specific pressure on the friction surface occurs. The maximum pressure qmax is located on the line of action of the radial load acting on the shaft.

Since the shaft has dropped by a certain amount as a result of bearing wear, the vertical distance for any point on the shaft between its original and new positions is the same (and is equal to qmax). Therefore, the current value of the specific pressure on the selected element can be expressed approximately from a curvilinear right triangle (Figure 25 b):

The further course of solving the problem is no different from the solution according to the first hypothesis. As a result, the following dependencies are obtained for determining the moment of friction forces according to the second hypothesis:

Thus, the reduced friction coefficient decreases (by approximately 20%) and, accordingly, friction losses decrease and efficiency increases. That is why all new cars must be run-in at partial power.

As a result of running-in, primary wear of the surface occurs (smoothing of micro-irregularities), and surfaces are broken in (“grinding in” of surfaces to one another). Only then can the machine be used to its full capacity.

Heel friction

First hypothesis. Since in this case the supporting surface is a plane, the constant specific pressure (Figure 26a) is determined by simply dividing the axial force by the area of ​​the support ring:

Let us select a ring surface element of thickness dρ at a distance ρ from the center of the heel (Figure 26c). The elementary normal reaction acting on this element is determined by multiplying the specific pressure by its area:

Let us determine the elementary friction force and the moment from this friction force:

Integrating over the entire supporting surface, we obtain the total friction moment:

Substituting the value of q, we finally get:

Second hypothesis. As practice shows, over time, uniform wear of the supporting surface of the heel occurs, i.e. the product of specific pressure and relative speed is a constant value:

In this case, the speed at different points of the contact surface is different:

But since the angular velocity is the same for the shaft, wear will be proportional to the product q⋅ρ, in other words, this product is some constant k:

Thus, the specific pressure diagram is a hyperbolic dependence (Figure 26b). As a result of surface wear, the specific pressure is redistributed in such a way that when approaching the axis of rotation of the shaft, it increases sharply (theoretically increasing to infinity at the center of the supporting surface). That is why solid heels are practically not used in technology.

The further solution is carried out similarly to the solution according to the first hypothesis. As a result, the following dependence is obtained for determining the moment from the friction forces on the supporting surface of the heel:

In this form, it is difficult to compare hypotheses with each other. Therefore, to evaluate the results, solid spots (d=0) are considered:

A comparison shows that by running in the surfaces of the heel, an effect similar to that which occurs in axles is achieved - the magnitude of friction forces is reduced by 20...25%

Friction of flexible bodies

Flexible tapes, belts, ropes and other similar materials that provide low bending resistance are widely used in machines in the form of belt and rope drives, as well as in the mechanisms of lifting machines, and in band brakes.

Introduction

The object and product of the theory of mechanisms and machines (TMM) is the kinematic or other diagram of the machine. The diagram reflects the most important, fundamental properties of the machine.

The theory of mechanisms and machines is the science of the most general methods of analysis and synthesis of mechanisms and machines. Analysis and synthesis are carried out at the level of circuits - kinematic and others.

Basic concepts of TMM

A machine is a device that transforms energy, materials and information through mechanical movements. Accordingly, they distinguish: a) energy machines, b) technological and transport machines, c) information machines.

A mechanism is a converter of the movement of some solid bodies into the required movements of others.

Usually the mechanism is seen as a kind of hinged chain, hence the components of the mechanism in its kinematic or other diagram are called

are divided into links.

LINK - a part or a group of parts rigidly connected to each other (solid link). In addition, there are flexible links (cables, belts, chains).

Figure 1 The fixed link of the mechanism is called the rack and is designated

number 0 (Fig. 1). The link to which movement is communicated is called the input link, usually designated – 1 (Fig. 1). The link from which the movement required from the mechanism is removed is called output; as a rule, its designation has the greatest algebraic weight (in Fig. 1 it is designated - 3).

2 Lecturer Sadovets V.Yu.

IN Depending on the nature of the movement relative to the rack, the moving links have the following names:

CRANK - a link in a lever mechanism that makes a complete

rotation around a fixed axis (in Fig. 1, a), b) and c) is designated – 1). ROCKER ARM - a link in a lever mechanism that makes partial

rotation around a fixed axis (intended to perform a rocking motion; in Fig. 1, c) indicated - 3).

CONNECTING ROD - a link of a lever mechanism that performs plane-parallel movement and forms kinematic pairs only with moving links (there are no pairs associated with the stand; in Fig. 1, a) and c) designated - 2).

SLIDER - a link of a lever mechanism, forming a translational pair with a stand (for example, a piston - cylinder in an internal combustion engine; in Fig. 1, a) it is designated - 3).

SLINGER - a link of a lever mechanism, rotating around a fixed axis and forming a translational pair with another moving link (in Fig. 1, b) indicated - 2).

ROCKET STONE - link of the lever mechanism, moving progressively along the rocker (in Fig. 1, b) indicated - 3).

CAM-link, the profile of which, having a variable curvature, determines the movement of the driven link (in Fig. 2, a) is designated - 1).

GEAR WHEEL - a link with a closed system of teeth that ensures continuous movement of another link (in Fig. 2, b) is indicated

Figure 2 A distinction is made between flat and spatial mechanisms. The mechanism is

is called flat if all its links move parallel to the same plane. Otherwise the mechanism is called spatial

nom.

Lecturer Sadovets V.Yu.

Planar mechanisms can be studied using both three-dimensional and two-dimensional models. 3D model– this is the mechanism itself with any simplifications that do not affect the number of dimensions. 2D model– this is the projection of the mechanism onto a plane parallel to which the links of the mechanism move.

Due to its simplicity, the two-dimensional model is used as the first stage of analysis and synthesis of mechanisms. Two-dimensional models can also be built for some spatial mechanisms.

A movable connection consisting of two directly contacting links is called kinematic pair. For example, the mechanisms presented in Figure 1 have four kinematic pairs. They are formed by links 0-1, 1-2, 2-3, 3-0.

According to the nature of the contact of the links, kinematic pairs are divided into lower and higher. A pair is considered inferior if its links touch each other on one or more surfaces. These are all the pairs of lever mechanisms presented in Figure 1. Let us note in passing that a necessary feature of a lever mechanism is the presence of only lower pairs in it.

If the contact of the links occurs along lines or points (not along surfaces), it is called highest.

The highest are the cam and gear pairs (Fig. 2, a) and b)). The links of these pairs touch each other in a straight line.

A movable connection of more than two links is called kinematic chain. A chain, each link of which forms no more than two pairs with neighboring links, is called simple (Fig. 3, a). If the kinematic chain includes a link containing more than 2 kinematic pairs, then such a chain is called complex (Fig. 3, b).

all other links (slave) perform uniquely defined movements.

Mechanisms can be formed by both closed and open kinematic chains. A mechanism in which the output link (gripper) does not form a kinematic pair with the stand is called a mechanism with an open kinematic chain. An example is the mechanism of an elementary manipulator (Fig. 4,a). Most mechanisms are formed by closed kinematic chains, in which the output link is connected by a kinematic pair to a stand (Fig. 4b).

Figure 4 When considering theory, you have to analyze the movement not

only real, but also imaginary points of the mechanism. Let's assume that some place on the diagram or to the side of the diagram is designated by the letter K (Fig. 2, b). Then K 0 is a point K belonging to link 0, K 1 is a point K belonging to link 1, etc. – how many links, how many points K there can be in a mechanism.

The movement of the links, considered relative to the rack, is taken as absolute in the TMM. When indicating absolute and relative velocities, we will adhere to the following notation:

v K 2 - absolute speed of point K 2;

v K 2 1 - speed of point K 2 relative to link 1;

ω 2 - absolute angular velocity of link 2; ω 21 - angular velocity of link 2 relative to link 1.

Linear and angular accelerations are designated similarly - a and ε. Some problems related to the theory of gear and cam

mechanisms are solved more easily if higher pairs are replaced by lower ones. Let's look at the replacement rules. Let's do this using two-dimensional models as an example.

And the dynamics of mechanisms and machines during their analysis and synthesis.

Due to the brevity of our course, we will focus only on the structural and kinematic study of mechanisms. The purpose of these studies is to study the structure of mechanisms and analyze the movement of their links, regardless of the forces causing this movement.

In TMM, ideal mechanisms are studied: absolutely non-deformable; having no gaps in moving joints.

The basic provisions of TMM are common to mechanisms for various purposes. They are used at the first stage of design, that is, when developing a mechanism diagram and calculating its kinematic and dynamic parameters. After completing this design stage, you see the “skeleton” of your future product, the ideas embedded in it. In the future, implement your ideas in the form of design documentation and in the form of real products.

Structural analysis of mechanisms

Basic concepts and definitions

Detail- a separate, indivisible part of the mechanism (the part cannot be disassembled into parts).

Link- a part or several parts fixedly connected to each other.

Kinematic pair (KP)- a movable connection of two links. KP not a material quantity, it characterizes the connection of two links that are in direct contact.

KP element- the point, line or surface along which one link comes into contact with another. If the element KP is a point or a line - is it highest CP, if the surface is lowest CP.

According to the nature of the movement of the links KP there are: rotational, translational, with screw motion. Based on the type of contacting surfaces of the gearboxes, there are: planar, cylindrical, spherical, etc.

Gearbox class determined by the number of movement restrictions or the number of imposed connections S.

Total 6 degrees of freedom. Let us denote N as the number of degrees of freedom. You can write down

N + S = 6 or N = 6 - S, or S = 6 - N

It is often easier to determine how many degrees of freedom a link has left than how many connections have been applied. For example, how many degrees of freedom does a door or window have? one. What is an element of the CP - surface(no gaps). What is the nature of the movement? rotation. Therefore this is lower, rotational gearbox of the 5th class.

Quite often one has to deal with higher gearboxes, for example: contact of gear wheels; the cylinder rolls along a plane; cylinder by cylinder; cam pusher, etc. Such a connection is shown in Fig. 3.1.

The connection contains two components of relative motion, that is, two degrees of freedom. The CP element is a line. Therefore this is Higher CP 4th class.

Kinematic chain- a system of links connected by kinematic pairs.

Mechanism- a kinematic chain in which, for a given movement of one or more leading links relative to the stationary

Fig.3.1 link ( racks), all other links ( slaves) make a certain movement. Slave the link that makes the movement for which the mechanism is created is called working level.

When drawing up diagrams of mechanisms and other kinematic chains, conventional images are used in accordance with GOST 2.770-68. In this case, kinematic pairs are designated by capital letters, and links by numbers. The leading link is indicated by an arrow. Fixed link ( rack) indicated by shading near kinematic pairs.

There are concepts structural scheme And kinematic diagram mechanism. Kinematic diagrams of mechanisms differ from structural ones in that they must be carried out strictly to scale and at a given position of the leading link. In reality, few people comply with this requirement. Take the passport of any machine or household appliance. Written - Kinematic diagram- , but there is no talk of any scale. In order not to violate GOST 2.770-68, we will simply call it - mechanism diagram.

IN hinged lever mechanisms the links have their own names:

Rotating link - crank;

Swinging link - rocker;

Performing plane-parallel motion - connecting rod;

Forward movement - slider;

Links forming a translational pair with sliders - guides;

Movable guides - backstage.

Rollers are the parts of the rotating links that transmit torque. Axis- a cylindrical part that is covered by elements of other links and forms rotational pairs with them - hinges. The axles do not transmit torque.

Degree of movement of the mechanism

The degree of mobility of a mechanism is the number of degrees of freedom of the mechanism relative to the fixed link ( racks).

The degree of mobility of a flat mechanism (all links move in parallel planes) is determined by the formula P.L. Chebysheva

W = 3n - 2P 5 - P 4,

where n is the number of moving parts; P 5 - number KP 5th grade; P 4 - number KP 4th grade.

Rice. 3.2 Mechanism diagrams

Figure 3.2 shows several diagrams of mechanisms. Let's write down the names of the links, characterize the kinematic pairs and determine the degree of mobility of each mechanism.

Scheme 1: 1 - stand; 1 1 - guide; 2 - crank; 3 - connecting rod; 4 - slider; A, B, C - lower rotational gearboxes of the 5th class; D - lower progressive CP of the 5th class.

Scheme 2: 1 - stand; 2 - crank; 3 - backstage; 4 - rocker arm; A, C, D - lower rotational gearboxes of the 5th class; B - lower progressive CP of the 5th class.

W = 3n - 2P 5 - P 4 = 3*3 - 2*4 = 1.

Scheme 3: 1 - guide; 2, 4 - sliders (pushers); 3 - rocker arm; A, E - lower progressive CP of the 5th class; C - lower rotational gearbox of the 5th class; B, D - higher CP of the 4th class.

W = 3n - 2P 5 - P 4 = 3*3 - 2*3 - 2 = 1.

Scheme 4: 1 - stand; 1 1 guide; 2 - cam; 3 - roller; 4 - slider (pusher); A, C - lower rotational gearboxes of the 5th class; D - lower progressive CP of the 5th class; B - higher CP 4th class.

W = 3n - 2P 5 - P 4 = 3*3 - 2*3 - 1 = 2.

Scheme 5: 1 - stand; 1 1 guide; 2 - cam; 3 - slider (pusher); A - lower rotational gearbox of the 5th class; C - lower progressive CP of the 5th class; B - higher CP 4th class.

W = 3n - 2P 5 - P 4 = 3*2 - 2*2 - 1 = 1.

Diagrams 4 and 5 show cam mechanisms that have 2 and 1 degrees of freedom, respectively, although it is obvious that the pushers of these mechanisms have one degree of freedom. The excess degree of mobility of the mechanism (diagram 4) is caused by the presence of link 3 (roller), which does not affect the law of motion working level(pusher). During structural and kinematic analyzes of mechanisms, such links are removed from the mechanism diagram.

Replacement of higher kinematic pairs with lower ones

In structural, kinematic and power studies of mechanisms, in some cases it is advisable to replace a mechanism with higher pairs of the 4th class with an equivalent mechanism with lower pairs of the 5th class. In this case, the number of degrees of freedom and the instantaneous movement of the links equivalent replacement mechanism should be the same as replacement mechanism.

Figure 3.3, a) shows the replacement of the cam mechanism, consisting of links 1, 2, 3, with a four-link hinge, composed of links 1, 4, 5, 6. Higher kinematic pair IN replaced by lower pairs D, E. In Fig. 3.3, b) cam mechanism 1, 2, 3 is replaced

Rice. 3.3 crank mechanism 1, 4, 5, 3. Highest pair IN replaced by lower pairs D, E.

The algorithm for replacing higher kinematic pairs with lower ones is as follows:

1) a normal is drawn through the point of contact of the links in the highest gearbox;

2) on the normal at distances of the radii of curvature (R1 and R2, Fig. 3.3, a) lower CPs are placed;

3) the resulting CPs are connected by links to lower CPs that were already in the mechanism.

Structural synthesis and analysis of mechanisms

Structural synthesis of mechanisms is the initial stage of drawing up a diagram of a mechanism that satisfies given conditions. The initial data are usually the types of movement of the driving and working links of the mechanism. If an elementary three- or four-bar mechanism does not solve the problem of the required motion transformation, the mechanism diagram is drawn up by connecting several elementary mechanisms in series.

The basic principles of structural synthesis and analysis of mechanisms with class 5 CPs and the classification of such mechanisms were first proposed by the Russian scientist L.V. Assur in 1914, and developed the ideas of L.V. Assura Academician I.I. Artobolevsky. According to the proposed classification, mechanisms are combined into classes from the first and higher according to structural characteristics. The first class mechanism consists of a drive link and a rack connected by a 5th class kinematic pair.

Mechanisms of higher classes are formed by sequentially attaching to the mechanism of the first class kinematic chains that do not change the degree of mobility of the original mechanism, that is, having a degree of mobility equal to zero. Such a kinematic chain is called structural group. Since the structural group includes only class 5 CPs, and the degree of mobility of the group is zero, we can write

W = 3n - 2P 5 = 0, whence P 5 = 3/2 n.

Therefore, a structural group can only contain an even number of units, since P 5 can only be an integer.

Structural groups are distinguished by class And in order. A group of 2nd class and 2nd order consists of two links and three command posts. Group class(above the 2nd) is determined by the number of internal gearboxes that form a moving closed loop from the largest number of links in the group.

Group order is determined by the number of free elements of the links with which the group is attached to the mechanism.

Figure 3.4 shows the mechanism of the 1st class, as well as the structural groups of the 2nd and 3rd classes. As a result of structural synthesis (attachment of structural groups to a mechanism of the 1st class), four-link mechanisms of the 2nd class and a six-link mechanism of the 3rd class were obtained (Fig. 3.4).

Structural analysis determines the degree of mobility of the mechanism and the decomposition of its kinematic chain into structural groups and leading links. In this case, excess degrees of freedom (if any) and passive connections (if any) are removed.

Kinematic analysis of mechanisms

The purpose of kinematic analysis is the study of the movement of mechanism links regardless of the forces acting on them. In this case, the following assumptions are made: the links are absolutely rigid and there are no gaps in the kinematic pairs.

The following are solved main goals: a) determining the positions of links and constructing trajectories of movement of individual points or links as a whole; b) finding the linear velocities of the points of the mechanism and the angular velocities of the links; c) determination of linear accelerations of points of the mechanism and angular accelerations of links.

Initial data are: kinematic diagram of the mechanism; dimensions of all links; laws of motion of leading links.

In the kinematic analysis of mechanisms, analytical, graphic-analytical and graphical methods are used. Usually the full cycle of movement of the mechanism is considered.

The results of kinematic analysis allow, if necessary, to adjust the mechanism design; in addition, they are necessary for solving problems of mechanism dynamics.

Determination of positions and movements of mechanism links

We will solve the problem using graphical and analytical methods. As an example, let's take a crank-slider mechanism.

Given: crank length r = 150 mm; connecting rod length l = 450 mm; driving crank (ω = const.)

The position of the crank is determined by the angle φ. The movement cycle of such a mechanism is carried out in one full revolution of the crank - cycle period T = 60/n = 2π/ω, s. Where n is the number of revolutions per minute; ω - angular velocity, s -1. In this case φ = 2π, rad.

We draw the kinematic diagram of the mechanism on the selected scale (Fig. 3.5). In Fig. 3.5, the scale is 1:10. We build a diagram of the mechanism in eight crank positions (the more positions of the mechanism, the higher the accuracy of the results obtained). Mark the position of the slider ( working link). Based on the data obtained, we construct a graph of the dependence of the movement of point B of the slider on the angle of rotation of the crank φ (S B = f(φ)). This graph is called the kinematic diagram of the displacements of point B.

Analytical method

The movement of the slider is counted from the extreme right position (Fig. 3.5). Analyzing the figure, we can write the equations

S = (r + l) - (r * cosφ + l * cosβ) (3.1)

r * sin φ = l * sin β

Denoting r/ l = λ, we can write

β = arcsin(λ * sinφ).

Therefore, for each angle φ it is not difficult to determine the corresponding angle β and then solve the first equation of system (3.1). In this case, the accuracy of the results will be determined only by the specified accuracy of the calculations.

An approximate formula for determining the movements of the slider is given

S = r*(1 - cos φ + sin 2 φ* λ /2) (3.2)

Determination of speeds and accelerations of points and links of the mechanism

The speeds and accelerations of the driven links of the mechanism can be determined by the methods of plans, kinematic diagrams and analytical ones. In all cases, the following must be known as the initial ones: the diagram of the mechanism at a certain position of the driving link, its speed and acceleration.

Let us consider the application of these methods using the example of a crank-slider mechanism (Fig. 3.5) with φ = 45 o And n = 1200 rpm, respectively ω = π*n/30 = 125.7 s -1.

Plan of speeds (accelerations) of the mechanism.

The speed (acceleration) plan of a mechanism is the figure formed by the speed (acceleration) vectors of the points of the links at a given position of the mechanism.

Building a speed plan

Known

By size V AO = ω* r= 125.7*0.15 = 18.9 m/s.

Select the construction scale, for example, 1m/(s*mm).

Mark some point as a pole R when constructing a speed plan (Fig. 3.6).

We lay off the vector from the pole,

Rice. 3.6 perpendicular JSC. Point velocity vector IN we find by graphically solving the equation The direction of the vectors is known. The vector lies on a horizontal line, and the vector is perpendicular VA. From the pole and end of the vector we draw the corresponding straight lines and close the vector equation. Measuring the distance Pb And ba and, taking into account the scale, we find

V V= 16.6 m/s, V VA= 13.8 m/s.

Building an acceleration plan(Fig. 3.7)

Point acceleration A equals since = 0. . Magnitude of normal acceleration a n AO = ω 2 * r =

= 125.7 2 *0.15 = 2370 m/s 2.

Tangential acceleration a t AO = ε* r = 0, since angular acceleration ε = 0, because the ω = const.

Select the construction scale, for example, 100m/(s 2 *mm). Set aside from the pole r a vector, parallel JSC from A To ABOUT. Point acceleration vector IN we find by graphically solving the equation. The vector is directed parallel VA from IN To A, its value is equal a n VA = V VA 2 / l = 13.8 2 /0.45 = 423 m/s 2 .

a B = 1740 m/s 2 ; a t VA = 1650 m/s 2.

Kinematic diagram method (Fig. 3.8)

The kinematic diagram method is a graphical method. It includes graphical differentiation of first the displacement graph and then the speed graph. In this case, the displacement and velocity curves are replaced by a broken line. The value of the average speed on an elementary section of the track can be expressed as

µ S - displacement scale.

µ t - time scale.

In our case

µ S = 0.01 m/mm;

µ t = 0.000625 s/mm.

The speed scale is:

µ V = µ S /(µ t *H V) =

0,01/(0,000625*30) =

0.533 m/(s*mm).

The acceleration scale is:

µ a = µ V /(µ t * H a) =

0,533/(0,000625*30) =

28.44 m/(s 2 *mm).

The procedure for constructing a speed diagram.

At a distance H V (20-40 mm) point O is placed - the construction pole. Straight lines are drawn from the pole, parallel to the segments of the broken line of the displacement graph, until they intersect the ordinate axis. The ordinates are transferred to the velocity graph in the middle of the corresponding sections. A curve is drawn from the obtained points - this is the speed diagram.

The acceleration diagram is constructed in a similar way, only the speed diagram becomes the original graph, replaced by a broken line.

To indicate the numerical values ​​of velocity and acceleration, the plotting scale is calculated as shown above.

The speeds and accelerations of the slider can also be determined analytically, by sequentially differentiating the approximate equation (3.2).

Knowledge of the speeds and accelerations of the mechanism links is necessary for the dynamic analysis of the mechanism, in particular, for determining the inertia forces that can occur at high accelerations.(as in our case) exceed static loads many times over, for example, the weight of a link.

Due to the brevity of our course, we do not conduct a force study of mechanisms, but you can familiarize yourself with it from the literature, in particular, those recommended in this section.

The theory of mechanisms and machines deals with issues of gear geometry, as well as issues of friction in kinematic pairs. We will also consider these issues, but in the “machine parts” section, in relation to specific cases and tasks.

Literature

1. Pervitsky Yu.D. Calculation and design of precise mechanisms. - L.: Mechanical engineering,

2. Zablonsky K.I. Applied mechanics. - Kyiv: Vishcha School, 1984. - 280 p.

3. Korolev P.V. Theory of mechanisms and machines. Lecture notes. - Irkutsk: Publishing house

The development of mankind is accompanied by the continuous creation of machines, mechanisms and gears that facilitate the work of humans and animals and increase their productivity. The creation of new machines, mechanisms, various devices and installations that meet modern requirements is based on the achievements of fundamental and applied sciences.

Theory of mechanisms and machines– a science that studies general methods for studying the properties of mechanisms and machines and their design. The methods outlined in the theory of mechanisms and machines are suitable for the design of any mechanism and do not depend on its technical purpose, as well as the physical nature of the machine’s working process.

Car– a device that performs mechanical movements to transform energy, materials and information in order to replace or facilitate human physical and mental labor. Materials are understood as processed objects, transported loads and other objects of labor.

The machine carries out its work process by performing regular mechanical movements. The carrier of these movements is the mechanism. Hence, mechanism- a system of solid bodies, movably connected by contact and moving in a certain, required way relative to one of them, taken as stationary. Many mechanisms perform the function of transforming the mechanical motion of solid bodies.

The simplest mechanisms (lever, gear, etc.) have been known since ancient times; the process of their research, improvement and implementation into practice gradually took place in order to facilitate human labor and increase labor productivity.

Thus, it is known that the outstanding cultural figure of the Renaissance and scientist Leonardo da Vinci (1452–1519) developed designs for the mechanisms of weaving machines, printing and woodworking machines, and he attempted to experimentally determine the coefficient of friction. The Italian physician and mathematician D. Cardan (1501–1576) studied the movement of clock and mill mechanisms. French scientists G. Amonton (1663–1705) and C. Coulomb (1736–1806) were the first to propose formulas for determining the force of static and sliding friction.

The outstanding mathematician and mechanic L. Euler (1707–1783), Swiss by birth, lived and worked in Russia for thirty years, professor, and then full member of the St. Petersburg Academy of Sciences, author of 850 scientific papers, solved a number of problems in the kinematics and dynamics of a rigid body, studied the vibrations and stability of elastic bodies, dealt with issues of practical mechanics, studied, in particular, various profiles of gear teeth and came to the conclusion that the most promising profile is involute.

The famous Russian mechanic and inventor I.I. Polzunov (1728–1766) was the first to develop a design for the mechanism of a two-cylinder steam engine (which, unfortunately, he failed to implement), designed an automatic regulator for feeding the boiler with water, a device for supplying water and steam, and other mechanisms. Outstanding mechanic I.I. Kulibin (1735–1818) created the famous egg-shaped clock, which was the most complex automatic mechanism for those times.

In connection with the development of mechanical engineering as a branch of industry, there was a need to develop general scientific methods for studying and designing the mechanisms that make up the machines. These methods contributed to the creation of the most advanced machines for their time, performing the best defined, required functions. It is known that mechanical engineering as a branch of industry began to take shape in the 18th century, and in the 19th century. it began to develop rapidly, especially in England and the USA.

In Russia, the first machine-building factories appeared in the 18th century; in 1861 there were already over 100 of them, and in 1900 - approximately 1410. However, at the beginning of the 20th century. domestic mechanical engineering lagged behind both in terms of development and scale of production: half of all machines were imported from abroad. Only in the 30–50s did powerful mechanical engineering begin to develop in our country, successfully creating various machines and mechanisms that are not inferior to the best world models, and in some cases superior to them.

Highly developed domestic mechanical engineering was one of the factors that ensured victory in the Great Patriotic War.

As a science, the theory of mechanisms and machines under the name “Applied Mechanics” began to take shape at the beginning of the 19th century, and then mainly methods of structural, kinematic and dynamic analysis of mechanisms were developed. And only from the middle of the 19th century. In the theory of mechanisms and machines, general methods for the synthesis of mechanisms are being developed. Thus, the famous Russian scientist, mathematician and mechanic, academician P.L. Chebyshev (1821–1894) published 15 works on the structure and synthesis of lever mechanisms, while, based on the methods developed, he invented and built over 40 different new mechanisms that carry out a given trajectory, stop some links while others are moving, etc.; The structural formula of plane mechanisms is now called the Chebyshev formula.

The German scientist F. Grashof (1826–1893) gave a mathematical formulation of the condition for the rotation of a link in a flat lever mechanism, which is necessary in its synthesis. English mathematicians D. Sylvester (1814–1897) and S. Roberts (1827–1913) developed the theory of lever mechanisms for transforming curves (pantographs).

I.A. Vyshnegradsky (1831–1895), known as one of the founders of the theory of automatic control, designed a number of machines and mechanisms (automatic press, lifting machines, pump regulator) and, as a professor at the St. Petersburg Institute of Technology, created a scientific school of machine design.

Methods for the synthesis of gear mechanisms used in various machines are characterized by a certain complexity. Many scientists have worked in this area. The French geometer T. Olivier (1793–1858) substantiated the method of synthesizing conjugate surfaces in plane and spatial engagements using a generating surface. The English scientist R. Willis (1800–1875) proved the basic theorem of plane gearing and proposed an analytical method for studying planetary gear mechanisms. The German mechanical engineer F. Reuleaux (1829–1905) developed a graphical method for synthesizing conjugate profiles, currently known as the “method of normals.” Releaux is also the author of works on the structure (structure) and kinematics of mechanisms. Russian scientist H.I. Gokhman (1851–1916) was one of the first to publish work on the analytical theory of gearing.

A significant contribution to the dynamics of machines was made by the “father of Russian aviation” N.E. Zhukovsky (1847–1921). He was not only the founder of modern aerodynamics, but also the author of a number of works on applied mechanics and the theory of machine control.

The development of machine mechanics was facilitated by the work of N.P. Petrov (1836–1920), who laid the foundations of the hydrodynamic theory of lubrication; V.P. Goryachkin (1868–1935), who developed the theoretical foundations for the calculation and construction of agricultural machines, the whole complexity of the calculation of which lies in the fact that their actuators must reproduce the movements of the human hand.

Russian scientist L.V. Assur (1878–1920) discovered a general pattern in the structure of multi-link flat mechanisms, which is still used in their analysis and synthesis. He also developed the “singular point” method for the kinematic analysis of complex lever mechanisms; A.P. Malyshev (1879–1962) proposed the theory of structural analysis and synthesis as applied to complex plane and spatial mechanisms.

A significant contribution to the development of machine mechanics as an integral theory of mechanical engineering was made by I.I. Artobolevsky (1905–1977). He was the organizer of the national school of theory of mechanisms and machines; he wrote numerous works on the structure, kinematics and synthesis of mechanisms, machine dynamics and the theory of automatic machines, as well as textbooks that received universal recognition.

Disciples and followers of I.I. Artobolevsky - A.P. Bessonov, V. A. Zinoviev (1899–1975), N.I. Levitsky, N.V. Umnov, S.A. Cherkudinov and others - with their work in the field of machine dynamics (including acoustic and nonholonomic), optimization synthesis of mechanisms, the theory of automatic machines and in other areas of the theory of mechanisms and machines, contributed to their further development.

In the 30s and subsequent years, N.G. made a great contribution to the theory of mechanisms and machines with their research. Bruevich (1896–1987), one of the creators of the theory of accuracy of mechanisms, G.G. Baranov (1899–1968), author of works on the kinematics of spatial mechanisms, S.N. Kozhevnikov (1906–1988), who developed general methods for dynamic analysis of mechanisms with elastic links and mechanisms of heavily loaded machines.

It is worth noting the works of scientists: F.E. Orlova (1843–1892), D.S. Zernova (1860–1922) - expanded the theory of gears; N.I. Mertsalova (1866–1948) - supplemented the kinematic study of plane mechanisms with the theory of spatial mechanisms and developed a simple and reliable method for calculating the flywheel; L.P. Smirnova (1877–1954) - brought into a strict unified system graphic methods for studying the kinematics of mechanisms and the dynamics of machines; V.A. Gavrilenko (1899–1977) - developed the geometric theory of gears; L.N. Reshetova (1906–1998) - developed the theory of correction of gears, as well as planetary and cam mechanisms and laid the foundation for the theory of self-aligning mechanisms.

The most important concept of “machine” was given above. Let us add that machines not only replace or facilitate human labor, but also increase their productivity a thousandfold. The essential thing is that the transformation of energy, materials and information occurs thanks to mechanical movement. Keeping this in mind, let us explore the concept of “machine” in detail using specific examples.

The electric motor takes electricity from the network and converts it into mechanical energy, which it delivers to the consumer. It can be a compressor that converts the received mechanical energy into compressed air energy. The main thing is that energy conversion occurs due to the mechanical movement of the working parts: in an electric motor, this is the rotation of the rotor 1 (Fig. 1.1) in the compressor - piston movement 3 up and down (Fig. 1.2).

Rice. 1.1. Electric motor

Rice. 1.2. Compressor

The consumer of the mechanical energy of an electric motor can also be a machine tool, a press, or some other technological machine. In this case, mechanical energy is spent on performing work caused by the technological process. A machine or press also carries out transformation, but not of energy, but of the size and shape of the product being processed: the machine - by cutting, the press - by pressure. And in these examples it is shown that the transformation is carried out through mechanical movement: in a machine - a cutting tool or product, in a press - a stamp.

In a conveyor, mechanical energy is used to move the load. The transformation process inherent in the machine consists of transporting the load (changing its location) and is carried out, naturally, thanks to the mechanical movement of the conveyor belt on which the load lies.

A consumer of mechanical energy also includes a printing machine. In it, information is converted into repeatedly reproduced printed products through mechanical movement performed by the working parts of the machine.

The working process in a machine is carried out through mechanical movement, so it must have a carrier for this movement. Such a carrier is a mechanism. Consequently, the concept of “machine” is inextricably linked with the concept of “mechanism”. The mechanism, no matter how simple it may be, is necessarily part of the machine; it is its kinematic basis, and therefore the study of the mechanics of machines is inextricably linked with the study of the properties of their mechanisms.

Let us recall that the mechanism, being a system of movably connected and in contact with each other, converts the movement of some into the required movements of others.

Let us explore this definition in detail using specific examples.

The electric motor mechanism is a system of two solid bodies: a rotor 1, rotating inside a stationary stator, and the stator itself 2 (see Fig. 1.1); these solids are called links of the mechanism. The rotor rotates relative to the stator, which means that the links are movably connected to each other. This connection is structurally made using bearings and is carried out by contact. Indeed, let the electric motor have plain bearings; then the cylindrical surface of the rotor shaft comes into contact with the cylindrical surface of the stationary stator bearing liners. Such a connection of contacting links, which allows their relative movement, is called kinematic pair. In this case, the rotor 1 and stator 2 form a kinematic pair 1/2. Finally, we note that the rotational motion of the rotor is the motion that is required to transfer mechanical energy from the engine to its consumer (compressor, machine tool, forging machine, crane, printing machine, etc.). Consequently, the rotor-stator system has all the features that, by definition, are inherent in any mechanism and is, therefore, a mechanism.

The considered example clearly shows that the mechanism of an electric motor, consisting of only two links - the rotor and the stator, has a simple structure or, as they say, structure. The mechanisms of many machines have the same simple structure: steam, gas and hydraulic turbines, axial compressors, fans, blowers, centrifugal pumps, electric generators and other machines called rotary.

Note that many mechanisms have a more complex structure. The need for complication arises when, in order to carry out the required movements, the mechanism must perform the functions of transmitting and converting movement. To illustrate this, let's look at another example.

For a piston compressor, which is designed to produce compressed air, the mechanical energy required for this process is supplied to a rotating crankshaft 1 and through the connecting rod 2 transferred to the piston 3, reciprocating up and down inside the working cylinder C(see Fig. 1.2). When the piston moves downward, air is sucked in from the atmosphere; when it moves up, the air is first compressed and then pumped into a special reservoir. The required movements here are the continuous rotational movement of the shaft and the reciprocating movement of the piston. Therefore, to implement them, it is necessary to transform the movement of the shaft into the movement of the piston, which is performed by the compressor mechanism, called the crank-slider. Therefore, the compressor mechanism is much more complex than the electric motor mechanism, which does not convert motion. The crank-slider mechanism no longer consists of two, but of four links: three movable 1, 2, 3 and one fixed thing, which is the body 4 compressor (see Fig. 1.2).

The links of the crank-slider mechanism, interconnected, form pairs 1/4, 1/2, 2/3, 3/4. The links touch each other in bearings A, IN And WITH, and, in addition, the piston is in contact with the stationary surface of the working cylinder C. All these connections allow the links to move relative to each other: link 1 rotates relative to the link 4, link 2 rotates relative to the link 1, since the angle ABC changes during movement, etc. Thus, the system of rigid bodies (1 – 2 – 3 – 4) possesses all the features that, by definition, must be inherent in a mechanism, and therefore is a mechanism.

The considered crank-slider mechanism is widely used: it is used in stationary and marine internal combustion engines, piston expanders and hydraulic pumps, technological, transport (cars, tractors, diesel locomotives) and many other machines.

Thus, the concept of “mechanism” is broader than “the kinematic basis of the machine.” First of all, the mechanism is the kinematic basis not only of machines, but also of many instruments and apparatus (gyros, regulators, relays, contactors, electrical measuring instruments, automatic protection devices, etc.). In addition, many mechanisms exist independently, not relating to any machine specifically, not being an integral part of it. These include transmission mechanisms (reducers, variators, gears and other transmissions), connecting individual machines into entire units.

In conclusion, we give definitions of some terms in the theory of mechanisms and machines. Link– a rigid body participating in a given transformation of motion. A link can consist of either one part or several parts that do not have relative movement among themselves. Detail- a product that cannot be divided into smaller parts without preventing them from performing their functions. 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– a connection of two rigid bodies of a mechanism, allowing their specified relative motion.