Ministry of Education and Science of the Russian Federation

Federal State Budgetary Educational Institution of Higher Professional Education

"Kuban State Technological University"

Theoretical mechanics

Part 2 dynamics

Approved by the Editorial and Publishing

university council as

study guide

Krasnodar

UDC 531.1/3 (075)

Theoretical mechanics. Part 2. Dynamics: Textbook / L.I.Draiko; Kuban. state technol.un-t. Krasnodar, 2011. 123 p.

ISBN 5-230-06865-5

The theoretical material is presented in a brief form, examples of problem solving are given, most of which reflect real technical issues, attention is paid to the choice of a rational solution method.

Designed for bachelors of correspondence and distance learning in construction, transport and engineering areas.

Tab. 1 Fig. 68 Bibliography. 20 titles

Scientific editor Candidate of Technical Sciences, Assoc. V.F. Melnikov

Reviewers: Head of the Department of Theoretical Mechanics and Theory of Mechanisms and Machines of the Kuban Agrarian University prof. F.M. Kanarev; Associate Professor of the Department of Theoretical Mechanics of the Kuban State Technological University M.E. Multykh

Published by decision of the Editorial and Publishing Council of the Kuban State Technological University.

Reissue

ISBN 5-230-06865-5 KubGTU 1998

Foreword

This study guide is intended for students correspondence forms s teaching construction, transport and engineering specialties, but can be used when studying the section "Dynamics" of the course of theoretical mechanics by part-time students of other specialties, as well as students daily form learning while working independently.

The manual is compiled in accordance with the current program of the course of theoretical mechanics, covers all issues of the main part of the course. Each section contains a brief theoretical material, provided with illustrations and guidelines for its use in solving problems. The manual analyzes the solution of 30 tasks that reflect the real issues of technology and the corresponding control tasks for independent decision. For each task, a calculation scheme is presented that clearly illustrates the solution. The design of the solution complies with the requirements for the design of examinations of part-time students.

The author expresses his deep gratitude to the teachers of the Department of Theoretical Mechanics and Theory of Mechanisms and Machines of the Kuban Agrarian University for great work for reviewing the textbook, as well as teachers of the Department of Theoretical Mechanics of the Kuban State Technological University for valuable comments and advice on preparing the textbook for publication.

All critical comments and wishes will be accepted by the author with gratitude in the future.

Introduction

Dynamics is the most important branch of theoretical mechanics. Most of the specific tasks that occur in engineering practice relate to dynamics. Using the conclusions of statics and kinematics, dynamics establishes the general laws of motion of material bodies under the action of applied forces.

The simplest material object is a material point. For a material point, one can take a material body of any shape, the dimensions of which in the problem under consideration can be neglected. A body of finite dimensions can be taken as a material point if the difference in the motion of its points is not significant for a given problem. This happens when the dimensions of the body are small compared to the distances that the points of the body pass. Every particle solid body it could be considered material point.

The forces applied to a point or a material body are evaluated in dynamics by their dynamic impact, i.e., by how they change the characteristics of the movement of material objects.

The movement of material objects over time takes place in space relative to a certain frame of reference. In classical mechanics, based on Newton's axioms, space is considered three-dimensional, its properties do not depend on material objects moving in it. The position of a point in such space is determined by three coordinates. Time is not connected with space and movement of material objects. It is considered the same for all reference systems.

The laws of dynamics describe the movement of material objects in relation to the absolute coordinate axes, conventionally taken as immovable. The origin of the absolute coordinate system is taken at the center of the Sun, and the axes are directed to distant, conditionally stationary stars. When solving many technical problems, conditionally stationary can be considered coordinate axes associated with the earth.

Options mechanical movement material objects in dynamics are established by mathematical deductions from the basic laws of classical mechanics.

First law (law of inertia):

The material point maintains a state of rest or uniform and rectilinear motion until the action of any forces will bring her out of this state.

Uniform and rectilinear motion of a point is called inertia motion. Rest is a special case of motion by inertia, when the speed of a point is zero.

Any material point has inertia, i.e., it tends to maintain a state of rest or uniform rectilinear motion. The frame of reference, in relation to which the law of inertia is fulfilled, is called inertial, and the motion observed in relation to this frame is called absolute. Any frame of reference that performs translational rectilinear and uniform motion relative to the inertial frame will also be an inertial frame.

The second law (basic law of dynamics):

The acceleration of a material point relative to the inertial frame of reference is proportional to the force applied to the point and coincides with the force in the direction:
.

It follows from the basic law of dynamics that with a force
acceleration
. The mass of a point characterizes the degree of resistance of a point to a change in its speed, that is, it is a measure of the inertia of a material point.

Third law (law of action and reaction):

The forces with which two bodies act on each other are equal in magnitude and directed along one straight line in opposite directions.

Forces called action and reaction are applied to different bodies and therefore do not form a balanced system.

The fourth law (the law of the independence of the action of forces):

With the simultaneous action of several forces, the acceleration of a material point is equal to the geometric sum of the accelerations that the point would have under the action of each force separately:

, where
,
,…,
.

MINISTRY OF AGRICULTURE AND FOOD OF THE REPUBLIC OF BELARUS

Educational Institution "BELARUSIAN STATE AGRARIAN

TECHNICAL UNIVERSITY"

Department of Theoretical Mechanics and Theory of Mechanisms and Machines

THEORETICAL MECHANICS

methodological complex for students of the group of specialties

74 06 Agricultural engineering

In 2 parts Part 1

UDC 531.3(07) LBC 22.213ya7 T 33

Compiled by:

Candidate of Physical and Mathematical Sciences, Associate Professor Yu. S. Biza, candidate technical sciences, Associate Professor N. L. Rakova, Senior LecturerI. A. Tarasevich

Reviewers:

Department of Theoretical Mechanics of the Educational Establishment "Belarusian National Technical University» (head

Department of Theoretical Mechanics BNTU Doctor of Physical and Mathematical Sciences, Professor A. V. Chigarev);

Leading Researcher of the Laboratory "Vibroprotection of Mechanical Systems" State Scientific Institution "Joint Institute of Mechanical Engineering

National Academy of Sciences of Belarus”, Candidate of Technical Sciences, Associate Professor A. M. Goman

Theoretical mechanics. Section "Dynamics": educational

T33 method. complex. In 2 parts. Part 1 / comp.: Yu. S. Biza, N. L. Rakova, I. A. Tarasevich. - Minsk: BGATU, 2013. - 120 p.

ISBN 978-985-519-616-8.

AT educational and methodical complex presents materials on the study of the section "Dynamics", part 1, which is part of the discipline "Theoretical Mechanics". Includes a course of lectures, basic materials for the implementation practical exercises, tasks and samples of tasks for independent work and control learning activities full-time and part-time students.

UDC 531.3(07) LBC 22.213ya7

INTRODUCTION .................................................. .........................................
1. SCIENTIFIC AND THEORETICAL CONTENT OF EDUCATIONAL
OF THE METHODOLOGICAL COMPLEX .............................................. ..
1.1. Glossary................................................. ................................
1.2. Topics of lectures and their content .............................................. ..
Chapter 1. Introduction to dynamics. Basic concepts
classical mechanics .................................................................. ....................
Topic 1. Dynamics of a material point..............................................
1.1. Laws of material point dynamics
(laws of Galileo - Newton) .............................................. ..........
1.2. Differential equations of motion
1.3. Two main tasks of dynamics ..........................................................
Topic 2. Dynamics of relative motion
material point ................................................................ .........................
Review questions .................................................................. .............
Topic 3. Dynamics mechanical system.....................................
3.1. Mass geometry. Center of mass of a mechanical system......
3.2. Internal Forces .................................................................. .................
Review questions .................................................................. .............
Topic 4. Moments of inertia of a rigid body .......................................
4.1. Moments of inertia of a rigid body
relative to the axis and pole .................................................................. .....
4.2. Theorem on the moments of inertia of a rigid body
about parallel axes
(Huygens-Steiner theorem) .............................................. ....
4.3. Centrifugal moments of inertia ...............................................
Review questions .................................................................. ............
Chapter 2 General theorems material point dynamics

Topic 5. The theorem on the motion of the center of mass of the system ...............................
Review questions .................................................................. .............
Tasks for self-study ....................................................
Topic 6. The amount of movement of a material point
and mechanical system ............................................................... ...................
6.1. Quantity of movement of a material point 43
6.2. Impulse of force .................................................. .......................
6.3. Theorem on the change in momentum
material point ................................................................ ....................
6.4. Principal vector change theorem
momentum of a mechanical system ..........................................
Review questions .................................................................. .............
Tasks for self-study ....................................................
Topic 7. Moment of momentum of a material point
and mechanical system relative to the center and axis ..................................
7.1. Moment of momentum of a material point
relative to the center and axis .............................................................. ...........
7.2. Theorem on the change in angular momentum
material point relative to the center and axis .......................
7.3. Theorem on the change of the kinetic moment
mechanical system relative to the center and axis ..................................
Review questions .................................................................. .............
Tasks for self-study ....................................................
Topic 8. Work and power of forces ....................................... ............
Review questions .................................................................. .............
Tasks for self-study ....................................................
Topic 9. Kinetic energy of a material point
and mechanical system ............................................................... ...................
9.1. Kinetic energy of a material point
and mechanical system. Koenig's theorem...............................
9.2. Kinetic energy of a rigid body
with different movements .................................................................. .............
9.3. Change theorem kinetic energy
material point ................................................................ ....................

9.4. Kinetic energy change theorem
mechanical system .................................................................. ................
Review questions .................................................................. .............
Tasks for self-study ....................................................
Topic 10. Potential force field
and potential energy ............................................................... .................
Review questions .................................................................. .............
Topic 11. Dynamics of a rigid body.................................................... .......
Review questions .................................................................. .............
2. MATERIALS FOR CONTROL
BY MODUL................................................... ...................................
INDEPENDENT WORK OF STUDENTS ..............................
4. REQUIREMENTS FOR THE DESIGN OF CONTROL
WORKS FOR FULL-TIME AND CORRESPONDENCE STUDENTS
FORMS OF TRAINING ................................................................ .........................
5. LIST OF PREPARATION QUESTIONS
TO THE EXAM (STUDY) OF STUDENTS
FULL-TIME AND CORRESPONDENCE EDUCATION..................................................
6. LIST OF REFERENCES ............................................... ............

INTRODUCTION

Theoretical mechanics is the science of the general laws of mechanical motion, balance and interaction of material bodies.

This is one of the fundamental general scientific physical and mathematical disciplines. It is the theoretical basis of modern technology.

The study of theoretical mechanics, along with other physical and mathematical disciplines, contributes to the expansion of scientific horizons, forms the ability to concrete and abstract thinking and contributes to the improvement of the general technical culture of the future specialist.

Theoretical mechanics, being the scientific basis of all technical disciplines, contributes to the development of skills rational decisions engineering tasks associated with the operation, repair and design of agricultural and reclamation machinery and equipment.

According to the nature of the tasks under consideration, mechanics is divided into statics, kinematics and dynamics. Dynamics is a section of theoretical mechanics that studies the motion of material bodies under the action of applied forces.

AT educational and methodical complex (UMK) presents materials on the study of the "Dynamics" section, which includes a course of lectures, basic materials for conducting practical work, tasks and samples of execution for independent work and control of educational activities of full-time part-time students.

AT as a result of studying the "Dynamics" section, the student must learn theoretical basis dynamics and master the basic methods for solving problems of dynamics:

Know methods for solving problems of dynamics, general theorems of dynamics, principles of mechanics;

To be able to determine the laws of motion of a body depending on the forces acting on it; apply the laws and theorems of mechanics to solve problems; determine the static and dynamic reactions of the bonds that limit the movement of bodies.

The curriculum of the discipline "Theoretical Mechanics" provides for a total number of classroom hours - 136, including 36 hours for studying the section "Dynamics".

1. SCIENTIFIC AND THEORETICAL CONTENT OF THE EDUCATIONAL AND METHODOLOGICAL COMPLEX

1.1. Glossary

Statics is a section of mechanics that outlines the general doctrine of forces, the reduction is studied complex systems forces to the simplest form and equilibrium conditions are established various systems forces.

Kinematics is a section of theoretical mechanics in which the movement of material objects is studied, regardless of the causes that cause this movement, i.e., regardless of the forces acting on these objects.

Dynamics is a section of theoretical mechanics that studies the motion of material bodies (points) under the action of applied forces.

Material point- a material body, the difference in the movement of points of which is insignificant.

The mass of a body is a scalar positive value that depends on the amount of matter contained in a given body and determines its measure of inertia during translational motion.

Reference system - a coordinate system associated with the body, in relation to which the motion of another body is being studied.

inertial system- a system in which the first and second laws of dynamics are fulfilled.

The momentum of a force is a vector measure of the action of a force over some time.

Quantity of movement of a material point is the vector measure of its motion, equal to the product the mass of a point by its velocity vector.

Kinetic energy is a scalar measure of mechanical motion.

Elementary work of force is an infinitesimal scalar equal to dot product vector of force to the vector of infinitesimal displacement of the point of application of the force.

Kinetic energy is a scalar measure of mechanical motion.

The kinetic energy of a material point is a scalar

a positive value equal to half the product of the mass of a point and the square of its speed.

The kinetic energy of a mechanical system is an arithme-

the kinetic sum of the kinetic energies of all material points of this system.

Force is a measure of the mechanical interaction of bodies, characterizing its intensity and direction.

1.2. Lecture topics and their content

Section 1. Introduction to dynamics. Basic concepts

classical mechanics

Topic 1. Dynamics of a material point

The laws of dynamics of a material point (the laws of Galileo - Newton). Differential equations of motion of a material point. Two main tasks of dynamics for a material point. Solution of the second problem of dynamics; integration constants and their determination from initial conditions.

References:, pp. 180-196, , pp. 12-26.

Topic 2. Dynamics of the relative motion of the material

Relative motion of a material point. Differential equations of relative motion of a point; portable and Coriolis forces of inertia. The principle of relativity in classical mechanics. A case of relative rest.

References: , pp. 180-196, , pp. 127-155.

Topic 3. Geometry of masses. Center of mass of a mechanical system

Mass of the system. The center of mass of the system and its coordinates.

Literature:, pp. 86-93, pp. 264-265

Topic 4. Moments of inertia of a rigid body

Moments of inertia of a rigid body about the axis and pole. Radius of inertia. Theorem about moments of inertia about parallel axes. Axial moments of inertia of some bodies.

Centrifugal moments of inertia as a characteristic of body asymmetry.

References: , pp. 265-271, , pp. 155-173.

Section 2. General theorems of the dynamics of a material point

and mechanical system

Topic 5. The theorem on the motion of the center of mass of the system

The theorem on the motion of the center of mass of the system. Consequences from the theorem on the motion of the center of mass of the system.

References: , pp. 274-277, , pp. 175-192.

Topic 6. The amount of movement of a material point

and mechanical system

Quantity of motion of a material point and a mechanical system. Elementary impulse and momentum of force for end interval time. The theorem on the change in the momentum of a point and a system in differential and integral forms. Law of conservation of momentum.

Literature: , pp. 280-284, , pp. 192-207.

Topic 7. Moment of momentum of a material point

and mechanical system relative to the center and axis

The moment of momentum of a point about the center and axis. The theorem on the change in the angular momentum of a point. Kinetic moment of a mechanical system about the center and axis.

The angular momentum of a rotating rigid body about the axis of rotation. Theorem on the change in the kinetic moment of the system. Law of conservation of momentum.

References: , pp. 292-298, , pp. 207-258.

Topic 8. Work and power of forces

Elementary work of force, its analytical expression. The work of the force on final path. The work of gravity, elastic force. Equality to zero of the sum of works internal forces acting in a solid. The work of forces applied to a rigid body rotating around a fixed axis. Power. Efficiency.

References: , pp. 208-213, , pp. 280-290.

Topic 9. Kinetic energy of a material point

and mechanical system

Kinetic energy of a material point and a mechanical system. Calculation of the kinetic energy of a rigid body in various cases of its motion. Koenig's theorem. Theorem on the change in the kinetic energy of a point in differential and integral forms. Theorem on the change in the kinetic energy of a mechanical system in differential and integral forms.

References: , pp. 301-310, , pp. 290-344.

Topic 10. Potential force field and potential

The concept of a force field. Potential force field and force function. The work of a force on the final displacement of a point in a potential force field. Potential energy.

References: , pp. 317-320, , pp. 344-347.

Topic 11. Rigid body dynamics

Differential Equations forward movement solid body. Differential equation rotary motion rigid body around a fixed axis. physical pendulum. Differential equations of plane motion of a rigid body.

References: , pp. 323-334, , pp. 157-173.

Section 1. Introduction to dynamics. Basic concepts

classical mechanics

Dynamics is a section of theoretical mechanics that studies the motion of material bodies (points) under the action of applied forces.

material body- a body that has mass.

Material point- a material body, the difference in the movement of points of which is insignificant. This can be either a body, the dimensions of which can be neglected during its movement, or a body of finite dimensions, if it moves forward.

Particles are also called material points, into which a solid body is mentally divided when determining some of its dynamic characteristics. Examples of material points (Fig. 1): a - the movement of the Earth around the Sun. The earth is a material point; b is the translational motion of a rigid body. The solid body is mother-

al point, since V B \u003d V A; a B = a A ; c - rotation of the body around the axis.

A body particle is a material point.

Inertia is the property of material bodies to change the speed of their movement faster or slower under the action of applied forces.

The mass of a body is a scalar positive value that depends on the amount of matter contained in a given body and determines its measure of inertia during translational motion. In classical mechanics, mass is a constant.

Force - quantitative measure mechanical interaction between bodies or between a body (point) and a field (electric, magnetic, etc.).

Force is a vector quantity characterized by magnitude, point of application and direction (line of action) (Fig. 2: A - point of application; AB - line of action of the force).

Rice. 2

In dynamics, along with constant forces, there are also variable forces that can depend on time t, speed ϑ, distance r, or on a combination of these quantities, i.e.

F = const;

F = F(t);

F = F(ϑ ) ;

F = F(r) ;

F = F(t, r, ϑ ) .

	Examples of such forces are shown in Figs. 3: a										- body weight;
			(ϑ) – air resistance force;b ​​−		T =						- traction force

electric locomotive; c − F = F (r) is the force of repulsion from the center O or attraction to it.

Reference system - a coordinate system associated with the body, in relation to which the motion of another body is being studied.

An inertial system is a system in which the first and second laws of dynamics are fulfilled. This is a fixed coordinate system or a system moving uniformly and rectilinearly.

Movement in mechanics is a change in the position of a body in space and time in relation to other bodies.

The space in classical mechanics is three-dimensional, obeying Euclidean geometry.

Time is a scalar quantity that flows in the same way in any reference systems.

A system of units is a set of units of measurement physical quantities. To measure all mechanical quantities, three basic units are sufficient: units of length, time, mass or force.

Mechanical	Dimension		Notation	Dimension	Notation
magnitude
				centimeter
	kilogram-

All other units of measurement of mechanical quantities are derivatives of these. Two types of systems of units are used: international system SI units (or smaller - CGS) and the technical system of units - MKGSS.

Topic1. Material point dynamics

1.1. The laws of dynamics of a material point (the laws of Galileo - Newton)

The first law (of inertia).

isolated from external influences a material point maintains its state of rest or moves uniformly and rectilinearly until the applied forces force it to change this state.

The movement made by a point in the absence of forces or under the action of a balanced system of forces is called inertia motion.

For example, the movement of a body along a smooth (friction force is zero) go-

horizontal surface (Fig. 4: G - body weight; N - normal reaction planes).

Since G = − N , then G + N = 0.

When ϑ 0 ≠ 0 the body moves at the same speed; at ϑ 0 = 0 the body is at rest (ϑ 0 is the initial velocity).

The second law (basic law of dynamics).

The product of the mass of a point and the acceleration that it receives under the action of a given force is equal in absolute value to this force, and its direction coincides with the direction of acceleration.

a b

Mathematically, this law is expressed by the vector equality

For F = const,

a = const - the motion of the point is uniform. EU-

whether a ≠ const, α

- slow motion (Fig. 5, but);

a ≠ const,

a -

– accelerated motion (Fig. 5, b); m – point mass;

acceleration vector;

– vector force; ϑ 0 is the velocity vector).

At F = 0,a 0 = 0 = ϑ 0 = const - the point moves uniformly and rectilinearly, or at ϑ 0 = 0 - it is at rest (the law of inertia). Second

the law allows you to establish a relationship between the mass m of a body located near earth's surface, and its weight G .G = mg , where g is

acceleration of gravity.

The third law (the law of equality of action and reaction). Two material points act on each other with forces equal in magnitude and directed along the straight line connecting

these points, in opposite directions.

Since the forces F 1 = − F 2 are applied to different points, then the system of forces (F 1 , F 2 ) is not balanced, i.e. (F 1 , F 2 )≈ 0 (Fig. 6).

In its turn

m a = m a

- attitude

the masses of the interacting points are inversely proportional to their accelerations.

The fourth law (the law of the independence of the action of forces). The acceleration received by a point under the action of a simultaneous

but several forces geometric sum those accelerations that a point would receive under the action of each force separately on it.

Explanation (Fig. 7).

t a n

a 1 a kF n

The resultant R forces (F 1 ,...F k ,...F n ) .

Since ma = R ,F 1 = ma 1 , ...,F k = ma k , ...,F n = ma n , then

a = a 1 + ...+ a k + ...+ a n = ∑ a k , i.e. the fourth law is equivalent to

k = 1

the rule of addition of forces.

1.2. Differential equations of motion of a material point

Let several forces act simultaneously on a material point, among which there are both constants and variables.

We write the second law of dynamics in the form

= ∑

(t ,

k = 1

, ϑ=

r is the radius vector of the moving

points, then (1.2) contains derivatives of r and is a differential equation of motion of a material point in vector form or the basic equation of the dynamics of a material point.

Projections of vector equality (1.2): - on the axis of Cartesian coordinates (Fig. 8, but)

max=md
					= ∑Fkx;
					k = 1
may=md
					= ∑Fky;	(1.3)
					k = 1
maz=m					= ∑Fkz;
					k = 1

On the natural axis (Fig. 8, b)

mat				= ∑ Fk τ ,
				k = 1
				= ∑ F k n ;
				k = 1

mab = m0 = ∑ Fk b

k = 1

M t oM oa

b on o

Equations (1.3) and (1.4) are differential equations of motion of a material point in the Cartesian coordinate axes and natural axes, respectively, i.e., natural differential equations that are usually used for curvilinear motion of a point if the trajectory of the point and its radius of curvature are known.

1.3. Two main problems of dynamics for a material point and their solution

The first (direct) task.

Knowing the law of motion and the mass of the point, determine the force acting on the point.

To solve this problem, you need to know the acceleration of the point. In problems of this type, it can be given directly, or the law of motion of a point is given, in accordance with which it can be determined.

1. So, if the movement of a point is given in Cartesian coordinates

x \u003d f 1 (t) , y \u003d f 2 (t) and z \u003d f 3 (t) then the projections of the acceleration are determined

on the coordinate axis x =		d2x			d2y			d2z		And then - project-
F x ,F y and F z forces on these axes:
				,k ) = F F z . (1.6) 2. If the point commits curvilinear motion and the law of motion is known s = f (t), the trajectory of the point and its radius of curvature ρ, then it is convenient to use natural axes, and the acceleration projections on these axes are determined by the well-known formulas: Tangential axis a τ = d ϑ = d 2 2 s – tangential acceleration;dt dt HomeNormal ds 2 a n = ϑ 2 = dt is normal acceleration. The projection of the acceleration onto the binormal is zero. Then the projections of the force on the natural axes F=m F=m The modulus and direction of the force are determined by the formulas: F \u003d F τ 2 + F n 2; cos ( ; cos( The second (inverse) task. Knowing the forces acting on the point, its mass and initial conditions movement, determine the law of motion of a point or any other of its kinematic characteristics. The initial conditions for the movement of a point in the Cartesian axes are the coordinates of the point x 0, y 0, z 0 and the projection of the initial velocity ϑ 0 onto these axes ϑ 0 x \u003d x 0, ϑ 0 y \u003d y 0 and ϑ 0 z \u003d z 0 at the time corresponding to giving the beginning of the point motion and taken equal to zero. Solving problems of this type is reduced to compiling a differential differential equations (or one equation) of motion of a material point and their subsequent solution by direct integration or using the theory differential equations. Review questions 1. What does dynamics study? 2. What kind of motion is called inertial motion? 3. Under what condition will a material point be at rest or move uniformly and rectilinearly? 4. What is the essence of the first main problem of the dynamics of a material point? Second task? 5. Write down the natural differential equations of motion of a material point. Tasks for self-study 1. A point of mass m = 4 kg moves along a horizontal straight line with an acceleration a = 0.3 t. Determine the module of the force acting on the point in the direction of its movement at the time t = 3 s. 2. A part of mass m = 0.5 kg slides down the tray. At what angle to horizontal plane a tray must be located so that the part moves with acceleration a = 2 m / s 2? Angle express in degrees. 3. A point with a mass m = 14 kg moves along the Ox axis with an acceleration a x = 2 t . Determine the modulus of the force acting on the point in the direction of motion at time t = 5 s.

(MECHANICAL SYSTEMS) - IV option

1. The basic equation of the dynamics of a material point, as is known, is expressed by the equation . Differential equations of motion of arbitrary points of a non-free mechanical system, according to two methods of dividing forces, can be written in two forms:

(1) , where k=1, 2, 3, … , n is the number of points of the material system.

(2)

where is the mass of the k-th point; - radius vector of the k-th point, - given (active) force acting on the k-th point or the resultant of all active forces acting on the k-th point. - the resultant of the reaction forces of the bonds, acting on the k-th point; - resultant of internal forces acting on the k-th point; - resultant external forces acting on the kth point.

Equations (1) and (2) can be used to solve both the first and second problems of dynamics. However, the solution of the second problem of dynamics for the system becomes very complicated not only with mathematical point vision, but also because we face fundamental difficulties. They lie in the fact that both for system (1) and for system (2) the number of equations is significantly less than number unknown.

So, if we use (1), then the known for the second (inverse) problem of dynamics will be and , and the unknowns will be and . Vector equations will be " n", and unknown - "2n".

If we proceed from the system of equations (2), then the known and part of the external forces . Why a part? The fact is that the number of external forces includes external reactions connections that are unknown. In addition, there will also be unknowns.

Thus, both system (1) and system (2) are OPEN. We need to add equations, taking into account the equations of relations, and perhaps we still need to impose some restrictions on the relations themselves. What to do?

If we proceed from (1), then we can follow the path of compiling the Lagrange equations of the first kind. But this way is not rational because easier task(less degrees of freedom), the more difficult it is to solve it from the point of view of mathematics.

Then let's pay attention to the system (2), where - are always unknown. The first step in solving the system is to eliminate these unknowns. It should be borne in mind that, as a rule, we are not interested in internal forces during the movement of the system, that is, when the system moves, it is not necessary to know how each point of the system moves, but it is enough to know how the system as a whole moves.

Thus, if different ways exclude from the system (2) unknown forces, then we obtain some relations, i.e., some General characteristics for the system, the knowledge of which makes it possible to judge how the system moves in general. These characteristics are introduced using the so-called general theorems of dynamics. There are four such theorems:

1. Theorem about movement of the center of mass of the mechanical system;

2. Theorem about change in the momentum of a mechanical system;

3. Theorem about change in the angular momentum of a mechanical system;

4. Theorem about change in the kinetic energy of a mechanical system.

The theorem on the motion of the center of mass. Differential equations of motion of a mechanical system. The theorem on the motion of the center of mass of a mechanical system. Law of conservation of motion of the center of mass.

Theorem on the change in momentum. The amount of movement of a material point. Elemental impulse of force. Impulse of force over a finite period of time and its projections on the coordinate axes. Theorem on the change in momentum of a material point in differential and finite forms.

The amount of movement of the mechanical system; its expression in terms of the mass of the system and the velocity of its center of mass. The theorem on the change in the momentum of a mechanical system in differential and finite forms. Law of conservation of mechanical momentum

(The concept of a body and a point of variable mass. Meshchersky's equation. Tsiolkovsky's formula.)

Theorem on the change in moment of momentum. The moment of momentum of a material point relative to the center and relative to the axis. The theorem on the change in the angular momentum of a material point. Central force. Conservation of angular momentum of a material point in the case of a central force. (The concept of sector speed. The law of areas.)

The main moment of momentum or the kinetic moment of a mechanical system about the center and about the axis. The angular momentum of a rotating rigid body about the axis of rotation. Theorem on the change in the kinetic moment of a mechanical system. The law of conservation of the kinetic moment of a mechanical system. (Theorem about the change in the kinetic moment of a mechanical system in relative motion with respect to the center of mass.)

Theorem on the change in kinetic energy. Kinetic energy of a material point. Elementary work of force; analytical expression for elementary work. The work of a force on the final displacement of the point of its application. The work of the force of gravity, the force of elasticity and the force of gravity. Theorem on the change in the kinetic energy of a material point in differential and finite forms.

Kinetic energy of a mechanical system. Formulas for calculating the kinetic energy of a rigid body during translational motion, during rotation around a fixed axis and in general case movement (in particular, with plane-parallel movement). Theorem on the change in the kinetic energy of a mechanical system in differential and finite forms. Equality to zero of the sum of the work of internal forces in a solid. Work and power of forces applied to a rigid body rotating around a fixed axis.

The concept of a force field. Potential force field and force function. Expression of force projections in terms of force function. Surfaces of equal potential. The work of a force on the final displacement of a point in a potential force field. Potential energy. Examples of potential force fields: a uniform gravitational field and a gravitational field. The law of conservation of mechanical energy.

Rigid Body Dynamics. Differential equations of translational motion of a rigid body. Differential equation of rotation of a rigid body around a fixed axis. physical pendulum. Differential equations of plane motion of a rigid body.

d'Alembert principle. d'Alembert's principle for a material point; force of inertia. d'Alembert's principle for a mechanical system. Bringing the forces of inertia of the points of a rigid body to the center; main vector and main point forces of inertia.

(Determination of dynamic reactions of bearings during rotation of a rigid body around a fixed axis. The case when the axis of rotation is the main central axis of inertia of the body.)

Principle possible movements and the general equation of dynamics. Relationships imposed on a mechanical system. Possible (or virtual) displacements of a material point and a mechanical system. The number of degrees of freedom of the system. Ideal connections. The principle of possible movements. General Equation dynamics.

Equations of system motion in generalized coordinates (Lagrange equations). Generalized system coordinates; generalized speeds. Expression of elementary work in generalized coordinates. Generalized forces and their calculation; the case of forces with potential. Equilibrium conditions for the system in generalized coordinates. Differential equations of system motion in generalized coordinates or Lagrange equations of the 2nd kind. Lagrange equations in case of potential forces; Lagrange function (kinetic potential).

The concept of equilibrium stability. Small free vibrations mechanical system with one degree of freedom near the position of stable equilibrium of the system and their properties.

Elements of impact theory. Impact phenomenon. Impact force and impact impulse. Action striking force to a material point. Theorem on the change in the momentum of a mechanical system upon impact. Direct central impact of the body on a fixed surface; elastic and inelastic impacts. Impact recovery coefficient and its experimental determination. Direct central blow of two bodies. Carnot's theorem.

BIBLIOGRAPHY

Basic

Butenin N. V., Lunts Ya-L., Merkin D. R. Course of theoretical mechanics. Vol. 1, 2. M., 1985 and previous editions.

Dobronravov V. V., Nikitin N. N. Course of theoretical mechanics. M., 1983.

Starzhinsky V. M. Theoretical mechanics. M., 1980.

Targ S. M. A short course in theoretical mechanics. M., 1986 and previous editions.

Yablonsky A. A., Nikiforova V. M. Course of theoretical mechanics. Part 1. M., 1984 and previous editions.

Yablonsky A. A. Course of theoretical mechanics. Part 2. M., 1984 and previous editions.

Meshchersky I.V. Collection of problems in theoretical mechanics. M., 1986 and previous editions.

Collection of problems in theoretical mechanics / Ed. K. S. Kolesnikova. M., 1983.

Additional

Bat M. I., Dzhanelidze G. Yu., Kelzon A. S. Theoretical mechanics in examples and problems. Ch. 1, 2. M., 1984 and previous editions.

Collection of problems in theoretical mechanics / 5raznichen / co N. A., Kan V. L., Mintsberg B. L. et al. M., 1987.

Novozhilov I. V., Zatsepin M. F. Standard calculations in theoretical mechanics based on a computer. M., 1986,

Collection of tasks for term papers on Theoretical Mechanics / Ed. A. A. Yablonsky. M., 1985 and previous editions (contains examples of problem solving).

Consider the motion of a certain system of material volumes relative to a fixed coordinate system. When the system is not free, then it can be considered as free, if we discard the constraints imposed on the system and replace their action with the corresponding reactions.

Let us divide all the forces applied to the system into external and internal ones; both may include reactions of discarded

connections. Denote by and the main vector and the main moment of external forces relative to point A.

1. Theorem on the change in momentum. If is the momentum of the system, then (see )

i.e., the theorem is valid: the time derivative of the momentum of the system is equal to the main vector of all external forces.

Replacing the vector through its expression where is the mass of the system, is the velocity of the center of mass, equation (4.1) can be given a different form:

This equality means that the center of mass of the system moves as a material point whose mass is equal to the mass of the system and to which a force is applied that is geometrically equal to the main vector of all external forces of the system. The last statement is called the theorem on the motion of the center of mass (center of inertia) of the system.

If then from (4.1) it follows that the momentum vector is constant in magnitude and direction. Projecting it on the coordinate axis, we obtain three scalar first integrals of the differential equations of the system's double chain:

These integrals are called momentum integrals. When the speed of the center of mass is constant, i.e., it moves uniformly and rectilinearly.

If the projection of the main vector of external forces on any one axis, for example, on the axis, is equal to zero, then we have one first integral, or if two projections of the main vector are equal to zero, then there are two integrals of the momentum.

2. Theorem on the change of the kinetic moment. Let A be some arbitrary point space (moving or stationary), which does not necessarily coincide with any particular material point of the system during the entire time of movement. We denote its velocity in a fixed system of coordinates as The theorem on the change in the angular momentum of a material system relative to point A has the form

If point A is fixed, then equality (4.3) takes a simpler form:

This equality expresses the theorem on the change in the angular momentum of the system with respect to fixed point: the time derivative of the angular momentum of the system, calculated with respect to some fixed point, is equal to the principal moment of all external forces with respect to this point.

If then, according to (4.4), the angular momentum vector is constant in magnitude and direction. Projecting it on the coordinate axis, we obtain the scalar first integrals of the differential equations of the motion of the system:

These integrals are called the integrals of the angular momentum or the integrals of the areas.

If point A coincides with the center of mass of the system, Then the first term on the right side of equality (4.3) vanishes and the theorem on the change in angular momentum has the same form (4.4) as in the case of a fixed point A. Note (see 4 § 3) that in the case under consideration the absolute angular momentum of the system on the left side of equality (4.4) can be replaced by the equal angular momentum of the system in its motion relative to the center of mass.

Let be some constant axis or an axis of constant direction passing through the center of mass of the system, and let be the angular momentum of the system relative to this axis. From (4.4) it follows that

where is the moment of external forces about the axis. If during the whole time of motion then we have the first integral

In the works of S. A. Chaplygin, several generalizations of the theorem on the change in angular momentum were obtained, which were then applied in solving a number of problems on the rolling of balls. Further generalizations of the theorem on the change of the kpnetological moment and their applications in problems of the dynamics of a rigid body are contained in the works. The main results of these works are related to the theorem on the change in the kinetic moment relative to the moving one, constantly passing through some moving point A. Let - unit vector directed along this axis. Multiplying scalarly by both sides of equality (4.3) and adding the term to both its parts, we obtain

When the kinematic condition is met

equation (4.5) follows from (4.7). And if condition (4.8) is satisfied during the whole time of motion, then the first integral (4.6) exists.

If the connections of the system are ideal and allow rotation of the system as a rigid body around the axis and in the number of virtual displacements, then the main moment of reactions about the axis and is equal to zero, and then the value on the right side of equation (4.5) is the main moment of all external active forces about the axis and . The equality to zero of this moment and the satisfiability of relation (4.8) will be in the case under consideration sufficient conditions for the existence of the integral (4.6).

If the direction of the axis and is unchanged, then condition (4.8) can be written as

This equality means that the projections of the velocity of the center of mass and the velocity of point A on the axis and on the plane perpendicular to this are parallel. In the work of S. A. Chaplygin, instead of (4.9), it is required that less than general condition where X is an arbitrary constant.

Note that condition (4.8) does not depend on the choice of a point on . Indeed, let P be an arbitrary point on the axis. Then

and hence

In conclusion, we note Resal's geometric interpretation of equations (4.1) and (4.4): the vectors absolute speeds the ends of the vectors and are equal respectively to the main vector and the main moment of all external forces relative to point A.