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 main point external forces about 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. Its velocity in a fixed system of coordinates will be denoted by Theorem on the change in angular momentum material system with respect 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 of the angular momentum of the system relative to a fixed point: the time derivative of the angular momentum of the system, calculated relative to some fixed point, is equal to the main moment of all external forces relative 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 i-axis. 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.

The use of OZMS in solving problems is associated with certain difficulties. Therefore, additional relationships are usually established between the characteristics of motion and forces, which are more convenient for practical application. These ratios are general theorems of dynamics. They, being consequences of the OZMS, establish dependencies between the speed of change of some specially introduced measures of movement and the characteristics of external forces.

Theorem on the change in momentum. Let's introduce the concept of the momentum vector (R. Descartes) of a material point (Fig. 3.4):

i i = t v G (3.9)

Rice. 3.4.

For the system, we introduce the concept principal momentum vector of the system as a geometric sum:

Q \u003d Y, m "V r

In accordance with the OZMS: Xu, - ^ \u003d i), or X

R(E) .

Taking into account that /w, = const we get: -Ym,!" = R(E),

or in final form

do / di \u003d A (E (3.11)

those. the first time derivative of the main momentum vector of the system is equal to the main vector of external forces.

The theorem on the motion of the center of mass. Center of gravity of the system called geometric point, whose position depends on t, etc. on the mass distribution /r/, in the system and is determined by the expression of the radius vector of the center of mass (Fig. 3.5):

where g s - radius vector of the center of mass.

Rice. 3.5.

Let's call = t with the mass of the system. After multiplying the expression

(3.12) on the denominator and differentiating both parts of the semi-

valuable equality we will have: g s t s = ^t.U. = 0, or 0 = t s U s.

Thus, the main momentum vector of the system is equal to the product the mass of the system and the velocity of the center of mass. Using the momentum change theorem (3.11), we obtain:

t with dU s / dі \u003d A (E), or

Formula (3.13) expresses the theorem on the motion of the center of mass: the center of mass of the system moves as a material point with the mass of the system, which is affected by the main vector of external forces.

Theorem on the change in moment of momentum. Let us introduce the concept of moment of momentum of a material point as a vector product of its radius-vector and momentum:

k o o = bl X that, (3.14)

where to OI - angular momentum of a material point relative to a fixed point O(Fig. 3.6).

Now we define the moment of momentum mechanical system as a geometric sum:

K () \u003d X ko, \u003d ShchU,? O-15>

Differentiating (3.15), we get:

Ґ сік--- X t i w. + g yu X t i

Given that = U G U i X t i u i= 0, and formula (3.2), we obtain:

сіК a /с1ї - ї 0 .

Based on the second expression in (3.6), we will finally have a theorem on the change in the angular momentum of the system:

The first time derivative of the angular momentum of the mechanical system relative to the fixed center O is equal to the main moment of the external forces acting on this system relative to the same center.

When deriving relation (3.16), it was assumed that O- fixed point. However, it can be shown that in a number of other cases the form of relation (3.16) will not change, in particular, if, in the case of plane motion, the moment point is chosen at the center of mass, the instantaneous center of velocities or accelerations. In addition, if the point O coincides with a moving material point, equality (3.16), written for this point, will turn into the identity 0 = 0.

Theorem on the change in kinetic energy. When a mechanical system moves, both the “external” and internal energy systems. If the characteristics internal forces, the main vector and the main moment, do not affect the change in the main vector and the main moment of the number of accelerations, then internal forces can be included in process estimates energy state systems. Therefore, when considering changes in the energy of the system, one has to consider the movements of individual points, to which internal forces are also applied.

The kinetic energy of a material point is defined as the quantity

T^myTsg. (3.17)

The kinetic energy of a mechanical system is equal to the sum of the kinetic energies of the material points of the system:

notice, that T > 0.

We define the force power as the scalar product of the force vector by the velocity vector:

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 correspondence forms learning.

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 of a 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. Kinetic energy change theorem
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 - the science of general laws mechanical movement, equilibrium 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 section "Dynamics", 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 total classroom hours - 136, including 36 hours for the study of the "Dynamics" section.

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. 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 the work of 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 of translational motion of a rigid 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. Earth is a material point; b - forward movement solid 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 of 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 natural differential equations movement 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.

General theorems of the dynamics of a system of bodies. Theorems on the motion of the center of mass, on the change in the momentum, on the change in the main moment of the momentum, on the change in kinetic energy. Principles of d'Alembert, and possible displacements. General equation of dynamics. Lagrange's equations.

General theorems of rigid body dynamics and systems of bodies

General theorems of dynamics- this is a theorem on the movement of the center of mass of a mechanical system, a theorem on a change in the momentum, a theorem on a change in the main moment of the momentum (kinetic moment) and a theorem on a change in the kinetic energy of a mechanical system.

Theorem on the motion of the center of mass of a mechanical system

The theorem on the motion of the center of mass.
The product of the mass of the system and the acceleration of its center of mass is equal to the vector sum of all external forces acting on the system:
.

Here M is the mass of the system:
;
a C - acceleration of the center of mass of the system:
;
v C - speed of the center of mass of the system:
;
r C - radius vector (coordinates) of the center of mass of the system:
;
- coordinates (with respect to the fixed center) and masses of points that make up the system.

Theorem on the change in momentum (momentum)

The amount of motion (momentum) of the system is equal to the product of the mass of the entire system and the speed of its center of mass or the sum of the momentum (sum of impulses) of individual points or parts that make up the system:
.

Theorem on the change in momentum in differential form.
The time derivative of the amount of motion (momentum) of the system is equal to the vector sum of all external forces acting on the system:
.

Theorem on the change in momentum in integral form.
The change in the amount of motion (momentum) of the system for a certain period of time is equal to the sum of the impulses of external forces for the same period of time:
.

The law of conservation of momentum (momentum).
If the sum of all external forces acting on the system is zero, then the momentum vector of the system will be constant. That is, all its projections on the coordinate axes will retain constant values.

If the sum of the projections of external forces on any axis is equal to zero, then the projection of the momentum of the system on this axis will be constant.

Theorem on the change in the main moment of momentum (theorem of moments)

The main moment of the amount of motion of the system relative to a given center O is the value equal to the vector sum of the moments of the quantities of motion of all points of the system relative to this center:
.
Here square brackets denote the vector product.

Fixed systems

The following theorem refers to the case when the mechanical system has fixed point or an axis that is fixed relative to an inertial frame of reference. For example, a body fixed with a spherical bearing. Or a system of bodies moving around a fixed center. It can also be a fixed axis around which a body or system of bodies rotates. In this case, moments should be understood as moments of momentum and forces relative to fixed axle.

Theorem on the change in the main moment of momentum (theorem of moments)
The time derivative of the principal angular momentum of the system with respect to some fixed center O is equal to the sum of the moments of all external forces of the system with respect to the same center.

The law of conservation of the main moment of momentum (moment of momentum).
If the sum of the moments of all external forces applied to the system relative to a given fixed center O is equal to zero, then the main moment of the system's momentum relative to this center will be constant. That is, all its projections on the coordinate axes will retain constant values.

If the sum of the moments of external forces about some fixed axis is equal to zero, then the moment of momentum of the system about this axis will be constant.

Arbitrary systems

The following theorem has a universal character. It is applicable to both fixed systems and freely moving ones. In the case of fixed systems, it is necessary to take into account the reactions of the bonds at the fixed points. It differs from the previous theorem in that the center of mass C of the system should be taken instead of the fixed point O.

Theorem of moments about the center of mass
The time derivative of the main angular momentum of the system about the center of mass C is equal to the sum of the moments of all external forces of the system about the same center.

Law of conservation of angular momentum.
If the sum of the moments of all external forces applied to the system relative to the center of mass C is equal to zero, then the main moment of the system's momentum relative to this center will be constant. That is, all its projections on the coordinate axes will retain constant values.

moment of inertia of the body

If the body rotates around the z axis with angular velocityω z , then its angular momentum (kinetic moment) relative to the z-axis is determined by the formula:
L z = J z ω z ,
where J z is the moment of inertia of the body about the z axis.

Moment of inertia of the body about the z-axis is determined by the formula:
,
where h k is the distance from a point of mass m k to the z axis.
For a thin ring of mass M and radius R or a cylinder whose mass is distributed along its rim,
J z = M R 2 .
For a solid homogeneous ring or cylinder,
.

The Steiner-Huygens theorem.
Let Cz be the axis passing through the center of mass of the body, Oz be the axis parallel to it. Then the moments of inertia of the body about these axes are related by the relation:
J Oz = J Cz + M a 2 ,
where M is the body weight; a - distance between axles.

In more general case :
,
where is the inertia tensor of the body.
Here is a vector drawn from the center of mass of the body to a point with mass m k .

Kinetic energy change theorem

Let a body of mass M perform translational and rotational motion with an angular velocity ω around some axis z. Then kinetic energy body is determined by the formula:
,
where v C is the speed of movement of the center of mass of the body;
J Cz - moment of inertia of the body about the axis passing through the center of mass of the body parallel to the axis of rotation. The direction of the axis of rotation can change over time. Specified formula gives the instantaneous value of kinetic energy.

Theorem on the change in the kinetic energy of the system in differential form.
The differential (increment) of the kinetic energy of the system during some of its displacement is equal to the sum of the differentials of work on this displacement of all external and internal forces applied to the system:
.

Theorem on the change in the kinetic energy of the system in integral form.
The change in the kinetic energy of the system during some of its displacement is equal to the sum of the work on this displacement of all external and internal forces applied to the system:
.

The work done by the force, is equal to the scalar product of the force vectors and the infinitesimal displacement of the point of its application :
,
that is, the product of the modules of the vectors F and ds and the cosine of the angle between them.

The work done by the moment of force, is equal to the scalar product of the vectors of the moment and the infinitesimal angle of rotation :
.

d'Alembert principle

The essence of d'Alembert's principle is to reduce the problems of dynamics to the problems of statics. To do this, it is assumed (or it is known in advance) that the bodies of the system have certain (angular) accelerations. Next, the forces of inertia and (or) moments of inertia forces are introduced, which are equal in magnitude and reciprocal in direction to the forces and moments of forces, which, according to the laws of mechanics, would create given accelerations or angular accelerations

Consider an example. The body makes a translational motion and external forces act on it. Further, we assume that these forces create an acceleration of the center of mass of the system . According to the theorem on the movement of the center of mass, the center of mass of a body would have the same acceleration if a force acted on the body. Next, we introduce the force of inertia:
.
After that, the task of dynamics is:
.
;
.

For rotational movement proceed in a similar way. Let the body rotate around the z axis and external moments of forces M e zk act on it. We guess these moments create angular accelerationεz . Next, we introduce the moment of inertia forces M И = - J z ε z . After that, the task of dynamics is:
.
Turns into a static task:
;
.

The principle of possible movements

The principle of possible displacements is used to solve problems of statics. In some problems, it gives a shorter solution than writing equilibrium equations. This is especially true for systems with connections (for example, systems of bodies connected by threads and blocks), consisting of many bodies

The principle of possible movements.
For the equilibrium of a mechanical system with ideal constraints, it is necessary and sufficient that the sum of the elementary works of all active forces acting on it for any possible relocation system was zero.

Possible system relocation- this is a small displacement, at which the connections imposed on the system are not broken.

Perfect Connections- these are bonds that do not do work when the system is moved. More precisely, the sum of work performed by the links themselves when moving the system is zero.

General equation of dynamics (d'Alembert - Lagrange principle)

The d'Alembert-Lagrange principle is a combination of the d'Alembert principle with the principle of possible displacements. That is, when solving the problem of dynamics, we introduce the forces of inertia and reduce the problem to the problem of statics, which we solve using the principle of possible displacements.

d'Alembert-Lagrange principle.
When a mechanical system moves with ideal constraints at each moment of time, the sum of the elementary works of all applied active forces and all inertia forces on any possible displacement of the system is equal to zero:
.
This equation is called general equation speakers.

Lagrange equations

Generalized coordinates q 1 , q 2 , ..., q n is a set of n values ​​that uniquely determine the position of the system.

The number of generalized coordinates n coincides with the number of degrees of freedom of the system.

Generalized speeds are the derivatives of the generalized coordinates with respect to time t.

Generalized forces Q 1 , Q 2 , ..., Q n .
Consider a possible displacement of the system, in which the coordinate q k will receive a displacement δq k . The rest of the coordinates remain unchanged. Let δA k be the work done by external forces during such a displacement. Then
δA k = Q k δq k , or
.

If, with a possible displacement of the system, all coordinates change, then the work done by external forces during such a displacement has the form:
δA = Q 1 δq 1 + Q 2 δq 2 + ... + Q n δq n.
Then the generalized forces are partial derivatives of the displacement work:
.

For potential forces with potential Π,
.

Lagrange equations are the equations of motion of a mechanical system in generalized coordinates:

Here T is the kinetic energy. It is a function of generalized coordinates, velocities, and possibly time. Therefore, its partial derivative is also a function of generalized coordinates, velocities, and time. Next, you need to take into account that the coordinates and velocities are functions of time. Therefore, to find the total derivative with respect to time, one must apply the differentiation rule complex function:
.

References:
S. M. Targ, Short Course theoretical mechanics, graduate School", 2010.

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 textbook is intended for part-time students of construction, transport and engineering specialties, but can be used when studying the "Dynamics" section of the theoretical mechanics course 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. Each particle of a rigid body can be considered a 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, coordinate axes associated with the Earth can be considered conditionally immovable.

The parameters of the mechanical motion of 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 satisfied, 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
,
,…,
.