Introduction

1. Problems of dynamic research of mechanisms

2. Forces in mechanisms

3. Inertial forces

4. Kinetostatic calculation of mechanisms

5. Theorem N.E. Zhukovsky

Literature

mechanism resistance inertia kinetostatic

Introduction

Subject test work“Dynamic analysis of mechanisms” in the discipline “Theory of mechanisms and machines”.

Goal: formation of knowledge of dynamic analysis of mechanisms.

Objectives: become familiar with the methods of dynamic analysis of mechanisms.

The work addresses the following topics:

Problems of dynamic research of mechanisms;

Forces in mechanisms;

Inertia forces;

Kinetostatic calculation of mechanisms;

N.E. Zhukovsky’s theorem on a rigid lever.

1. Problems of dynamic research of mechanisms

The main objectives of mechanism dynamics are:

1) determination of the forces acting in the kinematic pairs of the mechanism;

2) determination of friction forces and their influence on the operation of the mechanism;

3) determination of the law of motion of a mechanism under the influence of certain forces;

4) identifying the conditions that ensure the given law of movement of the mechanism;

5) balancing mechanisms.

To solve the first problem, a force study of the mechanism is carried out.

2. Forces in mechanisms

The main forces that determine the nature of the movement of the mechanism are the driving forces that perform positive work, and the forces of useful (production) resistance that arise during the execution of the mechanism. useful work and doing negative work. The driving forces include: the pressure force of the working mixture on the piston of the engine cylinder, the torque developed by the electric motor on the drive shaft of the pump or compressor, etc.

Useful resistance forces are those forces that the mechanism is designed to overcome. These forces are: cutting resistance forces in lathe etc. In addition to these forces, it is also necessary to take into account the resistance forces of the environment in which the mechanism moves, and the gravity forces of the links, which produce positive or negative work depending on the direction of movement of the center of gravity of the links - down or up.

When calculating a mechanism, all driving forces of useful resistance must be specified - the so-called specified forces. These forces are usually specified in the form of mechanical characteristics.

The mechanical characteristic of an engine or working machine is the dependence of the torque applied to the driven shaft of the engine or the drive shaft of the working machine on one or more kinematic parameters. Mechanical characteristics are determined experimentally or using various mathematical relationships.

During operation of the mechanism, as a result of the action of all the indicated forces applied to its links, reactions occur in kinematic pairs that do not directly affect the nature of the movement of the mechanism, but cause friction forces on the surfaces of the elements of kinematic pairs. These forces are forces of harmful resistance.

Reactions in kinematic pairs arise not only due to the influence of external specified forces on the links of the mechanism, but also due to the movement of individual masses of the mechanism with acceleration, which can cause additional dynamic loads in kinematic pairs.

Therefore, the task of kinematic calculation is to determine the reactions in kinematic pairs of mechanisms or, in other words, the pressures that arise at the points of contact of the elements of kinematic pairs, as well as to determine the balancing moments or balancing forces.

By balancing forces or moments we mean those unknown and subject to determination forces or moments applied to the leading links that balance the system of all external forces and pairs of forces and all inertial forces and pairs of inertial forces.

If in a machine, during operation, the acceleration of the links reaches an insignificant value, then the reactions in kinematic pairs are determined from the condition of uniform motion of all links of the mechanism according to the conditions of static equilibrium:

∑ Fi=0; ∑ M (Fi)=0.

If the acceleration of the links in the machine reaches a significant value, then the links are subject to dynamic loads that can no longer be neglected. For force calculations in this case, it would be necessary to draw up a dynamic equation of motion, which is very difficult.

The problem posed can be solved using d'Alembert's principle, according to which, if inertial forces are also applied to the links of the mechanism along with all forces, then the mechanism can be considered to be in static equilibrium, and the dynamics equation can be replaced with static equations:

∑ M (Fi) + ∑ M (Fu) + Mu=0

3. Inertial forces

In the general case of plane-parallel motion of a link, the accelerations of its various material points are different (in magnitude and direction). Therefore, the elementary forces of inertia are also different

, conditionally applied at these points. This system of elementary forces is reduced to one inertia force Fu and one pair of inertia forces with a moment Mu, which are equal:

where: m – link mass;

WS - acceleration of the center of gravity of the link;

ε – angular acceleration of the link;

IS is the moment of inertia of the link relative to the axis passing through the center of gravity.

The moment of inertia of a link is a measure of the inertia of the link in rotational motion. Its magnitude depends only on the body itself: on its mass and mass distribution. The moment of inertia is generally determined by the formula:

where: ρ is the distance of each elementary mass from the axis passing through the center of gravity.

The inertial force Fu is applied at the center of gravity of the link S and is directed opposite to the acceleration vector of the center of gravity WS.

The moment of the pair of inertial forces is directed opposite to the angular acceleration of the link ε.

Let us consider what inertia forces come down to in various cases of link motion.

1. Translational movement of the link (Fig. 1).

The accelerations of all points are the same, therefore:

An inertial force is applied at the center of gravity. The moment of inertia of the link Mu=0, because when the link moves forward, it has no angular acceleration (ε=0).

2. The link rotates unevenly (ε≠0) around an axis passing through the center of gravity (Fig. 2).

Fig.2

The inertia force in this case is equal to Fu=0, because acceleration of the center of gravity WS=0.

The moment of inertia force is equal to: Mu=-IS·ε and is directed opposite to the angular acceleration ε.

3. The link rotates uniformly (ε=0) around an axis that does not pass through the center of gravity (Fig. 3).

In this case:

Where: .

The moment of inertia forces Mu=0, since the angular acceleration ε=0.

4. The link rotates uniformly (ε=0) around an axis passing through the center of gravity (Fig. 4).

In this case, the inertial force Fu=0, because аS=0 and moment of inertia µu=0 (since ε=0).

Such a link is called balanced.

5. The link rotates unevenly around an axis that does not pass through the center of gravity.

In this case, both the force of inertia and the moment of inertia arise:

; in size

The inertial force is applied at the center of gravity and is directed opposite to the acceleration of the center of gravity WS. The moment of the pair of inertial forces Mu is directed opposite to the angular acceleration.

It is often convenient to reduce the inertial force Fu and the moment of inertia Mu to one resultant force Fu (Fig. 6). To do this, we replace the moment Mu with a pair Fu and -Fu, the moment of which is equal to: Fu·h=Mu.

We apply the force -Fu of this pair at the center of gravity S. Then another force will be applied at some point “K” of the link. The forces Fu and -Fu applied at the center of gravity are mutually balanced, and thus there remains only one force applied at the point “K” of the link. This point is called the swing point.

The position of the swing point is determined from the equation.

V.B. Pokrovsky

THEORY OF MECHANISMS AND MACHINES. DYNAMIC ANALYSIS. GEARS

Lecture notes

Scientific editor Prof., Dr. Tech. Sciences V.V. Karzhavin

Ekaterinburg

UDC 621.01 (075.8) BBK 34.41.ya 73 P48

Reviewers: Department of Lifting and Transport Equipment of the Russian State Vocational Pedagogical University; Associate Professor of the Department of “Theoretical Mechanics” USTU-UPI, Ph.D. tech. Sciences B.V. Trukhin

P48 Theory of mechanisms and machines: dynamic analysis, gears: lecture notes / V.B. Pokrovsky. Ekaterinburg:

LLC Publishing House UMC UPI, 2004. 49 p.

Lecture notes are intended for distance learning students of mechanical engineering specialties, as well as other forms of education studying the theory of mechanisms and machines at a technical university.

Prepared by the Department of Machine Parts UDC 621.01 (075.8)

BBK 34.41.ya 73

© LLC Publishing House UMC UPI, 2004

	Dynamic analysis and synthesis of a machine unit....................................................
	Rigid dynamic models. Roughness assessment
	drive link in steady state operation....................................
	Determination of the reduced moments of inertia of machine parts
	unit………………………………………………………….................................. ................9
	Determination of the increment of kinetic energy of a machine unit....11
	Determination of the reduced moment of inertia of the drive mechanism.... 13
	Types of gear mechanisms. ........................... ................... .......................
	Kinematics of gears. ........................... ................... ...................
	Parameters of cylindrical gears and wheels....................................
	Basic law of gearing................................................................... ............................
	Construction of an involute. Properties of involute. ...........................................
	Equation of involute. Methods for manufacturing gears.
	Initial tool outline. Tool displacement during cutting
	gear wheels. ........................... ................... ............................................
	Elimination of undercut. Calculation of the minimum number of teeth. ........................
	Determination of the minimum displacement coefficient that excludes
	undercut when cutting a number of teeth less than the minimum………….... 36
	Geometric parameters gear transmission. ..........................................
	Center distance. The radii of the circles of the peaks and valleys of the teeth.
	The thickness of the teeth along the arc of the pitch circle. ....................................
Lecture 10.	Qualitative characteristics gear transmission. ....................................
	Overlap coefficient. Teeth sliding speed. Coefficient
	specific slip. ........................... ................... ................................

Dynamic analysis and synthesis of a machine unit.

In dynamic analysis and synthesis, dynamic models or equivalent circuits of a real machine unit are considered.

There are two forms of dynamic models:

1. Model with rigid links (Fig. 1, 2). When forming such a model, the assumption is made that all links are absolutely rigid bodies, and kinematic pairs have no gaps between the elements.

2. Model with elastic links (Fig. 3). In such a model, the deformation of the links is taken into account and the forces and moments of elastic forces are determined based on the solution of the Lagrange equation of the 2nd kind.

F c pr		F dvpr

m pr

M c pr

M dvpr

I pr

Execute mechanism

The positions in Fig. 3 indicate: 1 - electric motor 2.4 - couplings 3 - gearbox 5 - flywheel

6 – drive mechanism

7 – machine unit

Rigid dynamic models.

When forming a model, a reduction link is selected. This can be any part of the machine, but, as a rule, the driving link of the drive mechanism or the driving link of the actuator is selected.

If the drive link makes translational motion, then the dynamic model has the form (see Fig. 1).

When the drive link rotates, the shape of the model is shown in Fig. 2.

V 1, ω 1 – linear or angular velocity of the actuation link;

F c pr , M c pr – reduced force or moment of resistance;

F dv pr, M dv pr - reduced driving force or moment; m pr . , I ave. – reduced mass or moment of inertia.

The movement of the links occurs under the influence of forces applied to them, which perform work.

For a rigid dynamic model, the motion is described by the work equation

A Σ = A = A dv. − A c =T i −T 0 ,

where A dv. – work of driving forces;

A c is the work of resistance forces;

T i , T 0 – the sum of the kinetic energies of all links in the i-th position and

zero (at the beginning of the countdown).

Three periods of machine movement are considered (Fig. 4).

During the start-up period A motor. > And with .

IN period of steady motion A door = A s at the beginning and end

During the run-down period A motor.< А с .

Cycle is the time after which the drive link returns to its original position, having the original speed value.

			Steady motion Coasting

Assessment of the uneven movement of the drive link under steady-state operating conditions.

In differential form, the work equation can be represented as

DA Σ

(A − Ŕ );

Ŕ = ∫

d ϕ ;

Ŕ = ∫

d ϕ ,

where ϕ 1 – angle of rotation of the reduction link

dAΣ

ďđ − Ě

ďđ.

dϕ 1

Thus

Ě ďđ

ω 2

ďđ,

where d ω 1

– analogue of the angular acceleration of the reduction link

dω 1

D ω 1

Ě Ĥâďđ = I ďđ ε 1 + ω 2 12 dI d ϕ ďđ + M cďđ .

The first two terms in the differential equation of motion take into account the inertial loads that arise when the links move unevenly.

cast is a variable.

The criterion for uneven rotation of the drive link during steady motion is the coefficient of unevenness of steady motion:

δ= ω 1 max −ω 1 min ,

ω 1ńđ

where ω 1max – maximum angular velocity;

Determination of the reduced moments of inertia of the masses of the machine unit links.

The machine unit has a structure that is shown in Fig. 3. One of the tasks of dynamic synthesis is to determine the moment of inertia of the flywheel, providing a given coefficient

unevenness of steady motion δ.

The moment of inertia of the flywheel installed on the shaft of the drive link of the actuator, which is the drive link, is determined by the formula

I m = I prpr − 1.1 I dvpr − I redpr ,

where I dv pr – reduced moment of inertia of the rotor (armature)

electric motor, (1.1 – coefficient taking into account the reduced masses of the couplings);

I ed pr – reduced moment of inertia of gears and shafts

gearbox;

I pr pr – reduced moment of inertia of the drive mechanism;

Ipr pr = f (Tpr ) ,

where T pr – increment of kinetic energy of the drive.

Ň ďđ = T ě. ŕ − Ň č. ě,

where T ě . ŕ – increment of kinetic energy of the machine unit;

Tč. ě – increment of kinetic energy of executive units

mechanism.

According to the work equation

Ň ě. ŕ = Ŕ = Ŕ Σ = Ŕ äâ − Ŕ ń.

The increment in the kinetic energy of the actuator links is determined by the formula

Ň č. ěi = Ň č. ě i − Ň č. ě 0 ,

where Ň č . ě i – kinetic energy of links in the i-th position.

Ň č . ě i – kinetic energy in the initial position (minimum value).

			I ďđ					I ďđ
			č. ěi					č. ě
č. ě i						č. ě
where I		ďđ ,		I ďđ		– given				moment of inertia of links
	č. ěi			č. ě

actuator in the i-th and initial position, kg m2; ω1 – angular velocity of the reduction link, 1/s.

Bringing the moments of inertia of the actuator links.

According to the law of conservation of energy, the kinetic energy of the reduced mass (moment of inertia) is equal to the sum of the kinetic energies of the reduced masses and moments of inertia.

I ďđ
č. ěi			∑n			k i +			ski.
			k = 1

Reduced moment of inertia of the actuator links in the i-th position

I ďđ =
č. ěi				k = 1
where I k		moment of inertia of the k-th link, kg m2;
			mass of the k-th link, kg;

ω ki – angular velocity of the k-th link in the i-th position, 1/s;

Issues discussed in the lecture. Forces acting on the links of mechanisms. Determination of link inertia forces. Kinetostatic analysis of mechanisms.

Some basic concepts.

Driving forces- These are the forces applied to the links of the mechanism that tend to accelerate the movement of the leading link; their elementary work is positive.

Resistance forces - These are the forces applied to the links of the mechanism that tend to slow down the movement of the leading link; their elementary work is negative. Distinguish between forces useful and harmful resistance.

Under the influence of forces applied to the machine, the angular velocity of the main shaft of the machine changes during the period of steady motion of the machine, oscillating around a certain average value.

The magnitude of the difference between the largest and smallest values ​​of angular velocity depends, for given forces, on the magnitude of the moment of inertia of the machine reduced to the main shaft. The greater the reduced moment, the smaller this difference. Thus, by increasing the reduced moment of inertia of the machine, the magnitude of the difference can be reduced.

The magnitude of this difference is taken into account by the coefficient of uneven running of the machine

.

Established by practice upper limits coefficient values d for various types of machines, these values ​​are tabulated and given in the literature on TMM.

To increase the reduced moment of inertia of the machine, most often a solid body in the form of a disk or rim with spokes, called a flywheel, is installed on the main shaft of the machine. flywheel.

The task is to determine the moment of inertia of the flywheel relative to the axis of rotation of the main shaft, at which the limits of fluctuation of the angular velocity of the main shaft during steady motion, specified by the unevenness coefficient, would be ensured d.

When solving the problem, they use the well-known technique of machine dynamics, according to which the study of the movement of the entire machine is replaced by the study of the movement of one link (the drive link). The main shaft of the machine is often taken as the drive link.

To determine the reduced moment of the flywheel, it is recommended to use the Wittenbauer method, which is the most successful methodologically compared to others. The method consists in determining the moment of inertia of the flywheel by plotting energy mass diagrams , which is constructed by excluding the parameter j from diagrams of changes in the kinetic energy of the mechanism and the reduced moment of inertia, for which diagrams of the reduced moments of driving forces and resistance forces, the work of driving forces and resistance forces must first be constructed.

When determining the law of motion of a mechanism, the masses of all moving links are replaced by the mass of the drive link. If the reduction link performs a rotational movement, then the concept is used reduced moment of inertia .

where is the linear speed of the center of gravity of the i-th link;

Mass of the i-th link;

Angular velocity of the i-th link;

Central moment of inertia of the i-th link.

Zhgurova I. A.

Dynamic analysis of mechanisms

Dynamic analysis mechanism is the determination of the movement of the mechanism under the action of applied forces or the determination of forces by a given movement of the links. Depending on the sign of the elementary work, all forces acting on the links of the mechanism are divided into driving forces and resistance forces. Driving force is called a force whose elementary work is positive, and resistance force– a force whose elementary work is negative. Elementary work of force is defined as the scalar product of force and the elementary displacement of the point of its application. Driving and resisting forces are usually functions of the displacement and velocities of the points of application of forces, and sometimes functions of time.

Gravity forces can be either driving forces or resistance forces, depending on the direction of elementary movements. The friction forces in kinematic pairs are functions of the forces of normal pressure on the surface, the relative speed of movement of the links, lubrication parameters, etc.

General methods It is advisable to apply dynamic analysis of mechanisms to mechanisms with one degree of freedom. In dynamic analysis, the task is to determine the movement of the initial link according to given forces. The solution to this problem is to find the law of motion of the initial link - the dependence of the generalized coordinate on time.

The law of motion of the initial link is the solution to the equation of motion of the mechanism. Most simple form the equation of motion is obtained based on the theorem on the change in kinetic energy of a mechanical system. The mass of the drive link is determined from the condition that its kinetic energy is equal to the sum of the kinetic energies of all links of the mechanism, and the power of the reduced force is equal to the sum of the powers of all driven forces. It is convenient to determine the reduced force using the lever method of N. E. Zhukovsky.

When considering the movement of a mechanism, three modes are distinguished: run-up, steady motion and run-down. Kinematic characteristics of steady motion:

coefficient of uneven movement of the mechanism, estimating the relative fluctuation in the speed of the drive link,

the efficiency of a mechanism, equal to the ratio of the work expended during the period of steady motion to overcome useful resistances to the work of the driving forces.

One of the tasks of the dynamic analysis of the mechanism is to carry out a kinetostatic calculation, in which reactions in kinematic pairs and the balancing moment applied to the initial link from the action of external forces and inertial forces are determined.

The force calculation of a plane and spatial mechanism is carried out using individual Assur structural groups, which are static definable kinematic chains. The presence of redundant connections leads to an excess of the number of unknown reactions over the number of kinetostatic conditions, i.e., to the static indetermination of the problem. Therefore, mechanisms without redundant connections are also called statically definable mechanisms.

The analytical determination of reactions in kinematic pairs of statically definable mechanisms is reduced to a sequential consideration of the equilibrium conditions of the links forming structural groups. Along with the analytical solution of force calculation problems, graphical determination of reactions is used by constructing force plans.

If we take into account the friction forces at power calculation mechanism, then it is possible to identify such relationships between the parameters of the mechanism in which, due to friction, the movement of the link in the required direction cannot begin, regardless of the magnitude of the driving force. This phenomenon is called self-braking of the mechanism, which in most cases is unacceptable, but is sometimes used to prevent the mechanism from moving in the opposite direction.

When designing a mechanism, the task is set to rationally select the masses of the mechanism links, ensuring the absorption of dynamic loads - the task of balancing the masses of the mechanism, or the task of balancing the inertial forces arising in the links of the mechanism.

She shares:

For the problem of balancing dynamic loads on the foundation,

On the problem of balancing dynamic loads in kinematic pairs.

When considering the case of balancing a rotating link consisting of a rotating shaft with rigidly connected given masses, it is possible to achieve complete balancing of all masses fixed on the shaft by installing two counterweights in arbitrarily selected planes, using the construction of a polygon of forces and a polygon of moments along closing vectors. All forces and moments of force pairs can be reduced to one link, called link of reduction.

Balancing called balancing of rotating or translationally moving masses of mechanisms in order to destroy the influence of inertial forces. Imbalance rotor (rotating in the supports of a body) is its state, characterized by such a distribution of masses, which during rotation causes variable loads on the supports. These loads cause shocks and vibrations, premature wear, and reduce efficiency. and machine performance. Static imbalance of a body is a state when its center of gravity does not lie on the axis of rotation. To balance a rotating body, it is necessary that its center of gravity lies on the axis of rotation. To balance the main vector of inertia forces of a flat mechanism, it is sufficient that the common center of mass of all links corresponds to the condition of constant coordinates.

Rotor imbalance is characterized by the magnitude of the imbalance. The product of an unbalanced mass and its eccentricity is called the imbalance value and is expressed in g-mm.

If static and moment imbalance exist simultaneously, then such imbalance is called dynamic. If there is significant imbalance, counterweights are installed.

Depending on the state of the surfaces of the rubbing bodies, types of sliding friction are distinguished: friction clean(on surfaces without adsorbed films or chemical compounds), friction dry(friction of non-lubricated surfaces), boundary friction (with a slight layer of lubrication) and friction liquid(friction of lubricated surfaces). Deformations of the protrusions can be elastic or inelastic. The force of resistance relative to the movement of surfaces creates the force of friction. If the protruding surface irregularities come into contact, dry friction occurs; if there is a layer of lubricant between the surfaces, liquid friction occurs. By friction slip the same areas of the contacting surfaces of one body come into contact with different areas of another body. By friction rolling various areas of the contacting surfaces of one body consistently coincide with the corresponding areas of another body.

The dependence of the moment applied to the driven shaft of a machine-engine or to the drive shaft of a working machine on the angular velocity of these machines is called mechanical characteristics of the machine. Engine machines are characterized by a decrease in torque with an increase in angular speed; in working machines, with an increase in angular speed, the torque increases.

The mechanism's run-up mode occurs when starting a machine or mechanism and when transferring the mechanism from a lower speed to a higher one. The period of change of forces during steady motion of the mechanism usually corresponds to one, two or several revolutions of the drive link and can be repeated an unlimited number of times if the operating conditions of the mechanism do not change. The run-down mode of the mechanism corresponds to the time during which the mechanism stops or is transferred from a higher speed to a lower one. For most machines, the main motion is steady-state motion, and the run-up and run-down only occur when the machine is started and stopped.

Problems of dynamics: Direct problem of dynamics - force analysis of a mechanism according to a given law of motion, determine the forces acting on its links, as well as the reactions in the kinematic pairs of the mechanism. Various forces are applied to the mechanism of the machine unit during its movement. These driving forces are resistance forces, sometimes called useful resistance forces, gravity, friction, and many other forces. By their action, the applied forces impart to the mechanism one or another law of motion.

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Lecture N6

Dynamics of mechanisms.

Dynamics tasks:

1. The direct task of dynamics (force analysis of a mechanism) is to determine, based on a given law of motion, the forces acting on its links, as well as the reactions in the kinematic pairs of the mechanism.
2. Inverse problem of dynamics using the given forces applied to the mechanism, determine the true law of motion of the mechanism.

The dynamic analysis of mechanisms can also include balancing and vibration protection problems.

Let's first start solving the inverse problem of dynamics, considering all links of the mechanisms to be rigid.

Various forces are applied to the mechanism of the machine unit during its movement. These are driving forces, drag forces (sometimes called useful drag forces), gravity forces, friction forces and many other forces. The nature of their action may be different:

A) some depend on the position of the mechanism links;

B) some from changes in their speed;

C) some are permanent.

By their action, the applied forces impart to the mechanism one or another law of motion.

Forces acting in machines and their characteristics

Forces and pairs of forces (moments) applied to the machine mechanism can be divided into the following groups.

1. Driving forces and moments, making a positivework during its duration or during one cycle if they change periodically. These forces and moments are applied to the links of the mechanism, which are called driving links.

2. Forces and moments of resistance, committing negativework during its action or in one cycle. These forces and moments are divided, firstly, into forces and moments of useful resistance, which perform the work required from the machine and are applied to the links called driven ones, and, secondly, into forces and moments of resistance of the medium (gas, liquid), in which the links of the mechanism move. The resistance forces of the medium are usually small compared to other forces, so in what follows they will not be taken into account, and the forces and moments of useful resistance will be simply called forces and moments of resistance.

3. Gravity moving links and elastic force of springs. In certain areas of the movement of the mechanism, these forces can perform both positive and negative work. However, for a full kinematic cycle, the work of these forces is zero, since their points of application move cyclically.

4. Forces and moments applied to the machine body(i.e. to the rack) from the outside. In addition to the gravity of the body, these include the reaction of the base (foundation) of the machine on its body and many other forces. All these forces and moments, since they are applied to a stationary body (stand), do not do any work.

5. Interaction forces between mechanism links, i.e. the forces acting in its kinematic pairs. These forces, according to Newton's 3rd law, are always reciprocal. Their normal parts of work are not commit and the tangential components, i.e. the friction forces, do the work, and the work of the friction force on the relative movement of the links of the kinematic pair negative

The forces and moments of the first three groups are classified as active. They are usually known or can be estimated. All these forces and moments are applied to the mechanism from the outside, and therefore are external External forces also include all forces and moments of the 4th group. However, not all of them are active.

The forces of the 5th group, if we consider the mechanism as a whole, without singling out its individual parts, are internal. These forces are reactions to the action of active forces. The reaction will also be the force (or moment) with which the base (foundation) of the machine acts on its body (i.e., on the mechanism stand). Reactions are unknown in advance. They depend on active forces and moments and on the accelerations of the mechanism links.

The greatest influence on the law of motion of a mechanism is exerted by driving forces and moments, as well as forces and moments of resistance. Their physical nature, magnitude and nature of action are determined by the working process of the machine or device in which the mechanism in question is used. In most cases, these forces and moments do not remain constant, but change their magnitude when the position of the mechanism links or their speed changes. These functional dependencies, represented graphically, or by an array of numbers, or analytically, are calledmechanical characteristicsand when solving problems are considered known.

When depicting mechanical characteristics, we will adhere to the following sign rule: force and moment will be considered positive if on the considered section of the path (linear or angular) they produce positive work.

Characteristics of speed-dependent forces.In Fig. 6.1 shown mechanical characteristics asynchronous electric motor dependence of the driving torque on the angular velocity of the machine rotor. The working part of the characteristic is the section ab, at which the driving torque decreases sharply even with a slight increase in rotation speed.

Forces and moments also depend on speed, acting also in rotary machines such as electric generators, fans, blowers, centrifugal pumps(Fig. 6.2) and many others.

Figure 6.3

As the speed increases, the torque of the motors usually decreases, and the torque of consumer machines mechanical energy usually increases. This property is very useful, as it automatically contributes to the stable maintenance of the machine’s motion mode, and the more pronounced it is, the greater the stability. Let's call this property of machines self-regulation.

Characteristics of forces depending on displacement. In Fig. 6.3 shown kinematic diagram mechanism two stroke engine internal combustion (ICE) and its mechanical characteristics. Force, applied to the piston 3, always acts to the left. Therefore, when the piston moves to the left (the process of gas expansion), it does positive work and is shown with a plus sign (branch czd). When the piston moves to the right (gas compression process), the forcereceives a minus sign (branch dac). If the fuel supply to the internal combustion engine does not change, then at the next revolution of the initial link (link 1 ) the mechanical characteristic will repeat its shape. This means that strengthwill change periodically.

Work of force graphically represented by the area bounded by the curve(s c ). In Fig. 6.3, this area has two parts: positive and negative, with the first being larger than the second. Therefore, the work done by the force over the full period will be positive. Consequently, the force is driving, although it is alternating in sign. Let us note in passing that if a force, being alternating in sign, does negative work in one period, then it is a resistance force.

Forces that depend only on movement act in many other machines and devices (in piston compressors, forging machines, planing and slotting machines, various devices with both pneumatic drive and spring motors, etc.), and the action of forces 6 can be both periodic and non-periodic.

At the same time, it should be noted that the torque of rotor-type machines does not depend on movement, i.e., on the angle of rotation of the rotor; the characteristics of such machines are shown in Fig. 6.4, a, b . At the same time, for machines that are engines, and for machines that consume mechanical energy (i.e., working machines).

If you change the fuel supply to the internal combustion engine, its mechanical characteristics will take the form of a family of curves (Fig. 6.5, A ): the larger the fuel supply (parameter h family), the higher the characteristic is located. The family of curves also depicts the mechanical characteristics of the shunt electric motor (Fig. 6.5, b ): the greater the resistance of the motor field winding circuit (parameter h ), the further to the right the curve is placed. The characteristics of a hydrodynamic coupling also take the form of a family of curves (Fig. 6.5, c): the greater the filling of the coupling with liquid (parameter h ), the further to the right and higher the characteristics are located.

Thus, influencing the parameter h , you can control the operating mode of the drive thermal, electric or hydraulic, increasing its driving force or speed. However, the control parameter h is associated with the amount of energy flowing through the machine, i.e. it determines its load and productivity.

The mechanism of a machine unit is usually a multi-link system loaded with forces and moments applied to its various links. To better imagine it, consider the power pumping unit driven by an asynchronous electric motor.

The fluid resistance force is applied to the piston 3, and the driving moment is applied to the rotor 4 of the electric motor. If the pump is multi-cylinder, then a resistance force will act on each piston, so the loading pattern will become more complex.

To determine the law of motion of a mechanism under the action of given external (active) forces, it is necessary to solve the equation of its motion. The basis for drawing up the equation of motion is the theorem about the change in the kinetic energy of the mechanism with W =1, which is formulated as follows:

The change in the kinetic energy of the mechanism occurs due to the work of all forces and moments applied to the mechanism

=
(6.1)

In a flat mechanism, the links perform rotational, translational and plane-parallel movements, then the kinematic energy of the mechanism

(6.2)

for all moving parts of the mechanism

=
(6.3)

Total work of all external forces and moments

(6.4)

After substitution we get

(
+
) - =(
)

The transition from many unknowns to one is carried out using methods of bringing forces and masses. To do this, we move from the real mechanism to the model, i.e. We replace the entire complex mechanism with one conditional link.

In the example under consideration, the mechanism has one degree of freedom ( W =1). This means that it is necessary to determine the law of motion of just one of its links, which will thereby be the initial one.

Dynamic model

Mechanism position with W =1 is completely determined by one coordinate, which is called the generalized coordinate. The angular coordinate of the link performing rotational motion is most often taken as a generalized coordinate. In this case, the dynamic model will be presented as:

Generalized angular coordinate of the model

Angular velocity of the model

Total reduced moment (generalized force - the equivalent of the entire given load applied to the mechanism)

The total reduced moment of inertia, which is equivalent to the inertia of the mechanism.

In the case of reduction, we replace the actual forces and moments with the total reduced moment applied to the dynamic model.

It should be emphasized that the replacement made should not violate the law of motion of the mechanism, determined by the action of the actually applied forces and moments.

The basis for bringing forces and moments should be the condition of equality of elementary works, i.e. the elementary work of each force on a possible displacement of the point of its application or moment on a possible angular displacement of the link on which it acts must be equal toelementary work of the reduced moment on the possible angular displacement of the dynamic model.

Let us consider, as an example, the reduction of forces and moments applied to the links of a machine unit (Fig. 6.6), assigning the angular coordinate as a generalized coordinate.

Let us define a substitute for the applied force
. According to the condition of equality of elementary works

Having solved for the desired value and divided the possible movements by time, we get

=

cos(
,
), where cos(

)= 1

=

=

= , where

for solving on a computer,

Using speeds.

Similarly, we will reduce the forces to the dynamic model (link 1)
,
, And
.

=
cos(
,
) = 0.0 t. To . cos(
,
) = 0.

=
=

Center of mass velocity projection
to the y axis.

We'll find it in the same way.

If we algebraically add up all the given moments applied to the initial link, we gettotal reduced moment, which replaces all forces and moments acting on the mechanism.

(6.5)

Bringing the masses.

Bringing masses is done for the same purpose as bringing forces:

modify and simplify dynamic scheme mechanism, i.e. to arrive at the corresponding dynamic model, and, consequently, to simplify the solution of the equation of motion.

If the initial link with a generalized coordinate is taken as a dynamic model, then the kinetic energy of the model must be equal to the sum of the kinetic energies of all links of the mechanism, i.e. the basis bringing the masses The initial link is subject to the condition of equality of kinetic energies.

The reduced moment of inertia is a parameter of the dynamic model, the kinetic energy of which is equal to the sum of the kinetic energies of the actually moving links.

Let us write down the condition for the equality of the kinetic energy of an individual link, the entire mechanism and the model for an individual link:

(6.6)

where for the model, for real parts of the mechanism

(6.7)

The transfer functions in brackets do not depend on, and therefore can be determined further if the law of motion of the model (initial link) is unknown. At
=

Where,

Let's define the given moments of inertia

All these moments of inertia do not depend on the angular position of the initial link. This group of links connected to the dynamic model by linear transmission ratios is called links of the first group, and their moments of inertia are called moments of inertia of the first group.

Let us determine the moments of inertia of the 2nd and 3rd links

The moments of inertia of the first and second groups of links and the total reduced moment of inertia of the installation under consideration are shown in Fig. 6.7

Test questions for the lecture N 6

1. Formulate the definition of direct and inverse dynamics problems.
2. What is meant by a dynamic model of a mechanism?
3. What is the purpose of bringing forces and moments into a mechanism? What condition is the basis for the reduction of forces and moments?
4. What condition underlies the replacement of masses and moments of inertia during reduction?
5. Write the formula for kinetic energy for a crank-slider mechanism.

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