What is s in theoretical mechanics. Theoretical and analytical mechanics

1. Basic concepts of theoretical mechanics.

2. The structure of the theoretical mechanics course.

1. Mechanics (in a broad sense) is the science of the movement of material bodies in space and time. It unites a number of disciplines, the objects of study of which are solid, liquid and gaseous bodies. Theoretical mechanics , Elasticity Theory, Strength of Materials, Fluid Mechanics, Gas Dynamics and Aerodynamics- this is not a complete list of various sections of mechanics.

As can be seen from their names, they differ from each other primarily in the objects of study. Theoretical mechanics studies the motion of the simplest of them - rigid bodies. The simplicity of the objects studied in theoretical mechanics makes it possible to identify the most general laws of motion that are valid for all material bodies, regardless of their specific physical properties. Therefore, theoretical mechanics can be considered as the basis of general mechanics.

2. The course of theoretical mechanics consists of three sections: statics, kinematicsAndspeakers .

IN In statics, the general doctrine of forces is considered and equilibrium conditions for solid bodies are derived.

In kinematics mathematical methods for specifying the movement of bodies are outlined and formulas are derived that determine the main characteristics of this movement (speed, acceleration, etc.).

In dynamics by a given movement, they determine the forces that cause this movement and, conversely, by given forces they determine how the body moves.

Material point called a geometric point with mass.

System of material points a set of them is called in which the position and movement of each point depends on the position and movement of all other points of the given system. The system of material points is often called mechanical system . A special case of a mechanical system is an absolutely rigid body.

Absolutely solid is a body in which the distance between any two points always remains unchanged (i.e. it is an absolutely strong and non-deformable body).

Free called a rigid body whose movement is not limited by other bodies.

Unfree call a body whose movement is, one way or another, limited by other bodies. The latter in mechanics are called connections .

By force is a measure of the mechanical action of one body on another. Since the interaction of bodies is determined not only by its intensity, but also by its direction, force is a vector quantity and is depicted in drawings by a directed segment (vector). Per unit of force in the system SI accepted newton (N) . Forces are designated in capital letters of the Latin alphabet (A, Y, Z, J...). We will denote numerical values (or modules of vector quantities) with the same letters, but without the upper arrows (F, S, P, Q...).

Line of action of force is called a straight line along which the force vector is directed.

System of forces is any finite set of forces acting on a mechanical system. It is customary to divide systems of forces into flat (all forces act in one plane) and spatial . Each of them, in turn, can be either arbitrary or parallel (the lines of action of all forces are parallel) or system of converging forces (the lines of action of all forces intersect at one point).

The two systems of forces are called equivalent , if their actions on the mechanical system are the same (i.e., replacing one system of forces with another does not change the nature of the movement of the mechanical system).

If a certain system of forces is equivalent to one force, then this force is called resultant of this system of forces. Let us note that not every system of forces has a resultant force. A force equal to the resultant in magnitude, opposite in direction and acting along the same straight line is called balancing by force.

A system of forces under the influence of which a free rigid body is at rest or moves uniformly and rectilinearly is called balanced or equivalent to zero.

By internal forces are called the forces of interaction between material points of one mechanical system.

External forces- these are the forces of interaction between the points of a given mechanical system and the material points of another system.

The force applied to a body at any one point is called concentrated .

Forces acting on all points of a given volume or a given part of the surface of a body are called distributed (by volume and surface, respectively).

The above list of basic concepts is not exhaustive. Other, no less important concepts will be introduced and clarified in the process of presenting the course material.

Theoretical mechanics

Theoretical mechanics- the science of the general laws of mechanical motion and interaction of material bodies. Being essentially one of the branches of physics, theoretical mechanics, having absorbed a fundamental basis in the form of axiomatics, became an independent science and was widely developed due to its extensive and important applications in natural science and technology, of which it is one of the foundations.

In physics

In physics, theoretical mechanics refers to the part of theoretical physics that studies mathematical methods of classical mechanics that are alternative to the direct application of Newton's laws (so-called analytical mechanics). This includes, in particular, methods based on Lagrange equations, principles of least action, Hamilton-Jacobi equation, etc.

It should be emphasized that analytical mechanics can be either non-relativistic - then it intersects with classical mechanics, or relativistic. The principles of analytical mechanics are so general that its relativization does not lead to fundamental difficulties.

In technical sciences

In technical sciences, theoretical mechanics means a set of physical and mathematical methods that facilitate the calculations of mechanisms, structures, aircraft, etc. (the so-called applied mechanics or engineering mechanics). Almost always, these methods are derived from the laws of classical mechanics - mainly from Newton's laws, although in some technical problems some of the methods of analytical mechanics are useful.

Theoretical mechanics is based on a certain number of laws established in experimental mechanics, accepted as truths that do not require proof - axioms. These axioms replace the inductive truths of experimental mechanics. Theoretical mechanics is deductive in nature. Relying on axioms as a foundation known and tested by practice and experiment, theoretical mechanics erects its edifice with the help of strict mathematical deductions.

Theoretical mechanics, as a part of natural science that uses mathematical methods, deals not with real material objects themselves, but with their models. Such models studied in theoretical mechanics are

material points and systems of material points,
absolutely rigid bodies and systems of rigid bodies,
deformable continuous media.

Usually in theoretical mechanics there are such sections as

Methods are widely used in theoretical mechanics

vector calculus and differential geometry,

Theoretical mechanics was the basis for the creation of many applied areas that have received great development. These are fluid and gas mechanics, mechanics of deformable solids, theory of oscillations, dynamics and strength of machines, gyroscopy, control theory, theory of flight, navigation, etc.

In higher education

Theoretical mechanics is one of the fundamental mechanical disciplines in the mechanics and mathematics faculties of Russian universities. In this discipline, annual All-Russian, national and regional student Olympiads, as well as the International Olympiad, are held.

Notes

Literature

Theoretical and analytical mechanics

Aizenberg T.B., Voronkov I.M., Ossetsky V.M.. Guide to solving problems in theoretical mechanics (6th edition). M.: Higher School, 1968 (djvu)
Yzerman M.A. Classical mechanics (2nd ed.). M.: Nauka, 1980 (djvu)
Aleshkevich V.A., Dedenko L.G., Karavaev V.A. Mechanics of solids. Lectures. M.: Physics Department of Moscow State University, 1997 (djvu)
Amelkin N.I. Kinematics and dynamics of a rigid body, MIPT, 2000 (pdf)
Appel P. Theoretical mechanics. Volume 1. Statistics. Dynamics of a point. M.: Fizmatlit, 1960 (djvu)
Appel P. Theoretical mechanics. Volume 2. System dynamics. Analytical mechanics. M.: Fizmatlit, 1960 (djvu)
Arnold V.I. Small denominators and problems of motion stability in classical and celestial mechanics. Advances in Mathematical Sciences vol. XVIII, no. 6 (114), pp.91-192, 1963 (djvu)
Arnold V.I., Kozlov V.V., Neishtadt A.I. Mathematical aspects of classical and celestial mechanics. M.: VINITI, 1985 (djvu)
Barinova M.F., Golubeva O.V. Problems and exercises in classical mechanics. M.: Higher. school, 1980 (djvu)
Bat M.I., Dzhanelidze G.Yu., Kelzon A.S. Theoretical mechanics in examples and problems. Volume 1: Statics and Kinematics (5th edition). M.: Nauka, 1967 (djvu)
Bat M.I., Dzhanelidze G.Yu., Kelzon A.S. Theoretical mechanics in examples and problems. Volume 2: Dynamics (3rd edition). M.: Nauka, 1966 (djvu)
Bat M.I., Dzhanelidze G.Yu., Kelzon A.S. Theoretical mechanics in examples and problems. Volume 3: Special chapters of mechanics. M.: Nauka, 1973 (djvu)
Bekshaev S.Ya., Fomin V.M. Fundamentals of the theory of oscillations. Odessa: OGASA, 2013 (pdf)
Belenky I.M. Introduction to Analytical Mechanics. M.: Higher. school, 1964 (djvu)
Berezkin E.N. Course of theoretical mechanics (2nd ed.). M.: Publishing house. Moscow State University, 1974 (djvu)
Berezkin E.N. Theoretical mechanics. Guidelines (3rd ed.). M.: Publishing house. Moscow State University, 1970 (djvu)
Berezkin E.N. Solving problems in theoretical mechanics, part 1. M.: Publishing house. Moscow State University, 1973 (djvu)
Berezkin E.N. Solving problems in theoretical mechanics, part 2. M.: Publishing house. Moscow State University, 1974 (djvu)
Berezova O.A., Drushlyak G.E., Solodovnikov R.V. Theoretical mechanics. Collection of problems. Kyiv: Vishcha School, 1980 (djvu)
Biderman V.L. Theory of mechanical vibrations. M.: Higher. school, 1980 (djvu)
Bogolyubov N.N., Mitropolsky Yu.A., Samoilenko A.M. Method of accelerated convergence in nonlinear mechanics. Kyiv: Nauk. Dumka, 1969 (djvu)
Brazhnichenko N.A., Kan V.L. and others. Collection of problems in theoretical mechanics (2nd edition). M.: Higher School, 1967 (djvu)
Butenin N.V. Introduction to Analytical Mechanics. M.: Nauka, 1971 (djvu)
Butenin N.V., Lunts Ya.L., Merkin D.R. Course of theoretical mechanics. Volume 1. Statics and kinematics (3rd edition). M.: Nauka, 1979 (djvu)
Butenin N.V., Lunts Ya.L., Merkin D.R. Course of theoretical mechanics. Volume 2. Dynamics (2nd edition). M.: Nauka, 1979 (djvu)
Buchgolts N.N. Basic course in theoretical mechanics. Volume 1: Kinematics, statics, dynamics of a material point (6th edition). M.: Nauka, 1965 (djvu)
Buchgolts N.N. Basic course in theoretical mechanics. Volume 2: Dynamics of a system of material points (4th edition). M.: Nauka, 1966 (djvu)
Buchgolts N.N., Voronkov I.M., Minakov A.P. Collection of problems on theoretical mechanics (3rd edition). M.-L.: GITTL, 1949 (djvu)
Vallee-Poussin C.-J. Lectures on theoretical mechanics, volume 1. M.: GIIL, 1948 (djvu)
Vallee-Poussin C.-J. Lectures on theoretical mechanics, volume 2. M.: GIIL, 1949 (djvu)
Webster A.G. Mechanics of material points of solid, elastic and liquid bodies (lectures on mathematical physics). L.-M.: GTTI, 1933 (djvu)
Veretennikov V.G., Sinitsyn V.A. Variable action method (2nd edition). M.: Fizmatlit, 2005 (djvu)
Veselovsky I.N. Dynamics. M.-L.: GITTL, 1941 (djvu)
Veselovsky I.N. Collection of problems on theoretical mechanics. M.: GITTL, 1955 (djvu)
Wittenburg J. Dynamics of rigid body systems. M.: Mir, 1980 (djvu)
Voronkov I.M. Course in Theoretical Mechanics (11th edition). M.: Nauka, 1964 (djvu)
Ganiev R.F., Kononenko V.O. Vibrations of solid bodies. M.: Nauka, 1976 (djvu)
Gantmakher F.R. Lectures on analytical mechanics. M.: Nauka, 1966 (2nd edition) (djvu)
Gernet M.M. Course of theoretical mechanics. M.: Higher school (3rd edition), 1973 (djvu)
Geronimus Ya.L. Theoretical mechanics (essays on the basic principles). M.: Nauka, 1973 (djvu)
Hertz G. Principles of mechanics set out in a new connection. M.: USSR Academy of Sciences, 1959 (djvu)
Goldstein G. Classical mechanics. M.: Gostekhizdat, 1957 (djvu)
Golubeva O.V. Theoretical mechanics. M.: Higher. school, 1968 (djvu)
Dimentberg F.M. Helical calculus and its applications in mechanics. M.: Nauka, 1965 (djvu)
Dobronravov V.V. Fundamentals of analytical mechanics. M.: Higher School, 1976 (djvu)
Zhirnov N.I. Classical mechanics. M.: Education, 1980 (djvu)
Zhukovsky N.E. Theoretical mechanics (2nd edition). M.-L.: GITTL, 1952 (djvu)
Zhuravlev V.F. Foundations of mechanics. Methodological aspects. M.: Institute of Problems of Mechanics RAS (preprint N 251), 1985 (djvu)
Zhuravlev V.F. Fundamentals of Theoretical Mechanics (2nd edition). M.: Fizmatlit, 2001 (djvu)
Zhuravlev V.F., Klimov D.M. Applied methods in the theory of vibrations. M.: Nauka, 1988 (djvu)
Zubov V.I., Ermolin V.S. and others. Dynamics of a free rigid body and determination of its orientation in space. L.: Leningrad State University, 1968 (djvu)
Zubov V.G. Mechanics. Series "Principles of Physics". M.: Nauka, 1978 (djvu)
History of the mechanics of gyroscopic systems. M.: Nauka, 1975 (djvu)
Ishlinsky A.Yu. (ed.). Theoretical mechanics. Letter designations of quantities. Vol. 96. M: Nauka, 1980 (djvu)
Ishlinsky A.Yu., Borzov V.I., Stepanenko N.P. Collection of problems and exercises on the theory of gyroscopes. M.: Moscow State University Publishing House, 1979 (djvu)
Kabalsky M.M., Krivoshey V.D., Savitsky N.I., Tchaikovsky G.N. Typical problems in theoretical mechanics and methods for solving them. Kyiv: GITL Ukrainian SSR, 1956 (djvu)
Kilchevsky N.A. Course of theoretical mechanics, vol. 1: kinematics, statics, dynamics of a point, (2nd ed.), M.: Nauka, 1977 (djvu)
Kilchevsky N.A. Course of theoretical mechanics, vol. 2: system dynamics, analytical mechanics, elements of potential theory, continuum mechanics, special and general theory of relativity, M.: Nauka, 1977 (djvu)
Kirpichev V.L. Conversations about mechanics. M.-L.: GITTL, 1950 (djvu)
Klimov D.M. (ed.). Mechanical problems: Sat. articles. To the 90th anniversary of the birth of A. Yu. Ishlinsky. M.: Fizmatlit, 2003 (djvu)
Kozlov V.V. Methods of qualitative analysis in rigid body dynamics (2nd ed.). Izhevsk: Research Center "Regular and Chaotic Dynamics", 2000 (djvu)
Kozlov V.V. Symmetries, topology and resonances in Hamiltonian mechanics. Izhevsk: Udmurt State Publishing House. University, 1995 (djvu)
Kosmodemyansky A.A. Course of theoretical mechanics. Part I. M.: Enlightenment, 1965 (djvu)
Kosmodemyansky A.A. Course of theoretical mechanics. Part II. M.: Education, 1966 (djvu)
Kotkin G.L., Serbo V.G. Collection of problems in classical mechanics (2nd ed.). M.: Nauka, 1977 (djvu)
Kragelsky I.V., Shchedrov V.S. Development of the science of friction. Dry friction. M.: USSR Academy of Sciences, 1956 (djvu)
Lagrange J. Analytical mechanics, volume 1. M.-L.: GITTL, 1950 (djvu)
Lagrange J. Analytical mechanics, volume 2. M.-L.: GITTL, 1950 (djvu)
Lamb G. Theoretical mechanics. Volume 2. Dynamics. M.-L.: GTTI, 1935 (djvu)
Lamb G. Theoretical mechanics. Volume 3. More complex issues. M.-L.: ONTI, 1936 (djvu)
Levi-Civita T., Amaldi U. Course in theoretical mechanics. Volume 1, part 1: Kinematics, principles of mechanics. M.-L.: NKTL USSR, 1935 (djvu)
Levi-Civita T., Amaldi U. Course in theoretical mechanics. Volume 1, part 2: Kinematics, principles of mechanics, statics. M.: From foreign. literature, 1952 (djvu)
Levi-Civita T., Amaldi U. Course in theoretical mechanics. Volume 2, part 1: Dynamics of systems with a finite number of degrees of freedom. M.: From foreign. literature, 1951 (djvu)
Levi-Civita T., Amaldi U. Course in theoretical mechanics. Volume 2, part 2: Dynamics of systems with a finite number of degrees of freedom. M.: From foreign. literature, 1951 (djvu)
Leach J.W. Classical mechanics. M.: Foreign. literature, 1961 (djvu)
Lunts Ya.L. Introduction to the theory of gyroscopes. M.: Nauka, 1972 (djvu)
Lurie A.I. Analytical mechanics. M.: GIFML, 1961 (djvu)
Lyapunov A.M. General problem of motion stability. M.-L.: GITTL, 1950 (djvu)
Markeev A.P. Dynamics of a body in contact with a solid surface. M.: Nauka, 1992 (djvu)
Markeev A.P. Theoretical Mechanics, 2nd edition. Izhevsk: RHD, 1999 (djvu)
Martynyuk A.A. Stability of motion of complex systems. Kyiv: Nauk. Dumka, 1975 (djvu)
Merkin D.R. Introduction to the mechanics of flexible filament. M.: Nauka, 1980 (djvu)
Mechanics in the USSR for 50 years. Volume 1. General and applied mechanics. M.: Nauka, 1968 (djvu)
Metelitsyn I.I. Gyroscope theory. Theory of stability. Selected works. M.: Nauka, 1977 (djvu)
Meshchersky I.V. Collection of problems on theoretical mechanics (34th edition). M.: Nauka, 1975 (djvu)
Misyurev M.A. Methods for solving problems in theoretical mechanics. M.: Higher School, 1963 (djvu)
Moiseev N.N. Asymptotic methods of nonlinear mechanics. M.: Nauka, 1969 (djvu)
Neimark Yu.I., Fufaev N.A. Dynamics of nonholonomic systems. M.: Nauka, 1967 (djvu)
Nekrasov A.I. Course of theoretical mechanics. Volume 1. Statics and kinematics (6th ed.) M.: GITTL, 1956 (djvu)
Nekrasov A.I. Course of theoretical mechanics. Volume 2. Dynamics (2nd ed.) M.: GITTL, 1953 (djvu)
Nikolai E.L. The gyroscope and some of its technical applications in a publicly accessible presentation. M.-L.: GITTL, 1947 (djvu)
Nikolai E.L. Theory of gyroscopes. L.-M.: GITTL, 1948 (djvu)
Nikolai E.L. Theoretical mechanics. Part I. Statics. Kinematics (twentieth edition). M.: GIFML, 1962 (djvu)
Nikolai E.L. Theoretical mechanics. Part II. Dynamics (thirteenth edition). M.: GIFML, 1958 (djvu)
Novoselov V.S. Variational methods in mechanics. L.: Leningrad State University Publishing House, 1966 (djvu)
Olkhovsky I.I. Course of theoretical mechanics for physicists. M.: MSU, 1978 (djvu)
Olkhovsky I.I., Pavlenko Yu.G., Kuzmenkov L.S. Problems in theoretical mechanics for physicists. M.: MSU, 1977 (djvu)
Pars L.A. Analytical dynamics. M.: Nauka, 1971 (djvu)
Perelman Ya.I. Entertaining mechanics (4th edition). M.-L.: ONTI, 1937 (djvu)
Planck M. Introduction to Theoretical Physics. Part one. General mechanics (2nd edition). M.-L.: GTTI, 1932 (djvu)
Polak L.S. (ed.) Variational principles of mechanics. Collection of articles by classics of science. M.: Fizmatgiz, 1959 (djvu)
Poincare A. Lectures on celestial mechanics. M.: Nauka, 1965 (djvu)
Poincare A. New mechanics. Evolution of laws. M.: Modern problems: 1913 (djvu)
Rose N.V. (ed.) Theoretical mechanics. Part 1. Mechanics of a material point. L.-M.: GTTI, 1932 (djvu)
Rose N.V. (ed.) Theoretical mechanics. Part 2. Mechanics of material systems and solids. L.-M.: GTTI, 1933 (djvu)
Rosenblat G.M. Dry friction in problems and solutions. M.-Izhevsk: RHD, 2009 (pdf)
Rubanovsky V.N., Samsonov V.A. Stability of stationary motions in examples and problems. M.-Izhevsk: RHD, 2003 (pdf)
Samsonov V.A. Lecture notes on mechanics. M.: MSU, 2015 (pdf)
Sugar N.F. Course of theoretical mechanics. M.: Higher. school, 1964 (djvu)
Collection of scientific and methodological articles on theoretical mechanics. Issue 1. M.: Higher. school, 1968 (djvu)
Collection of scientific and methodological articles on theoretical mechanics. Issue 2. M.: Higher. school, 1971 (djvu)
Collection of scientific and methodological articles on theoretical mechanics. Issue 3. M.: Higher. school, 1972 (djvu)
Collection of scientific and methodological articles on theoretical mechanics. Issue 4. M.: Higher. school, 1974 (djvu)
Collection of scientific and methodological articles on theoretical mechanics. Issue 5. M.: Higher. school, 1975 (djvu)
Collection of scientific and methodological articles on theoretical mechanics. Issue 6. M.: Higher. school, 1976 (djvu)
Collection of scientific and methodological articles on theoretical mechanics. Issue 7. M.: Higher. school, 1976 (djvu)
Collection of scientific and methodological articles on theoretical mechanics. Issue 8. M.: Higher. school, 1977 (djvu)
Collection of scientific and methodological articles on theoretical mechanics. Issue 9. M.: Higher. school, 1979 (djvu)
Collection of scientific and methodological articles on theoretical mechanics. Issue 10. M.: Higher. school, 1980 (djvu)
Collection of scientific and methodological articles on theoretical mechanics. Issue 11. M.: Higher. school, 1981 (djvu)
Collection of scientific and methodological articles on theoretical mechanics. Issue 12. M.: Higher. school, 1982 (djvu)
Collection of scientific and methodological articles on theoretical mechanics. Issue 13. M.: Higher. school, 1983 (djvu)
Collection of scientific and methodological articles on theoretical mechanics. Issue 14. M.: Higher. school, 1983 (djvu)
Collection of scientific and methodological articles on theoretical mechanics. Issue 15. M.: Higher. school, 1984 (djvu)
Collection of scientific and methodological articles on theoretical mechanics. Issue 16. M.: Vyssh. school, 1986

Examples of solving problems in theoretical mechanics

Statics

Problem conditions

Kinematics

Kinematics of a material point

Problem condition

Determining the speed and acceleration of a point using the given equations of its motion.
Using the given equations of motion of a point, establish the type of its trajectory and for the moment of time t = 1 s find the position of the point on the trajectory, its speed, total, tangential and normal acceleration, as well as the radius of curvature of the trajectory.
Equations of motion of a point:
x = 12 sin(πt/6), cm;
y= 6 cos 2 (πt/6), cm.

Kinematic analysis of a flat mechanism

Problem condition

The flat mechanism consists of rods 1, 2, 3, 4 and a slider E. The rods are connected to each other, to the sliders and fixed supports using cylindrical hinges. Point D is located in the middle of rod AB. The lengths of the rods are equal, respectively
l 1 = 0.4 m; l 2 = 1.2 m; l 3 = 1.6 m; l 4 = 0.6 m.

The relative arrangement of the mechanism elements in a specific version of the problem is determined by the angles α, β, γ, φ, ϑ. Rod 1 (rod O 1 A) rotates around a fixed point O 1 counterclockwise with a constant angular velocity ω 1.

For a given position of the mechanism it is necessary to determine:

linear velocities V A, V B, V D and V E of points A, B, D, E;
angular velocities ω 2, ω 3 and ω 4 of links 2, 3 and 4;
linear acceleration a B of point B;
angular acceleration ε AB of link AB;
positions of instantaneous speed centers C 2 and C 3 of links 2 and 3 of the mechanism.

Determination of absolute speed and absolute acceleration of a point

Problem condition

The diagram below considers the motion of point M in the trough of a rotating body. Using the given equations of portable motion φ = φ(t) and relative motion OM = OM(t), determine the absolute speed and absolute acceleration of a point at a given point in time.

Download the solution to the problem >>>

Dynamics

Integration of differential equations of motion of a material point under the influence of variable forces

Problem condition

A load D of mass m, having received an initial speed V 0 at point A, moves in a curved pipe ABC located in a vertical plane. In a section AB, the length of which is l, the load is acted upon by a constant force T (its direction is shown in the figure) and a force R of the medium resistance (the modulus of this force R = μV 2, the vector R is directed opposite to the speed V of the load).

The load, having finished moving in section AB, at point B of the pipe, without changing the value of its speed module, moves to section BC. In section BC, the load is acted upon by a variable force F, the projection F x of which on the x axis is given.

Considering the load to be a material point, find the law of its motion in section BC, i.e. x = f(t), where x = BD. Neglect the friction of the load on the pipe.

Download the solution to the problem >>>

Theorem on the change in kinetic energy of a mechanical system

Problem condition

The mechanical system consists of weights 1 and 2, a cylindrical roller 3, two-stage pulleys 4 and 5. The bodies of the system are connected by threads wound on the pulleys; sections of threads are parallel to the corresponding planes. The roller (a solid homogeneous cylinder) rolls along the supporting plane without sliding. The radii of the stages of pulleys 4 and 5 are respectively equal to R 4 = 0.3 m, r 4 = 0.1 m, R 5 = 0.2 m, r 5 = 0.1 m. The mass of each pulley is considered to be uniformly distributed along its outer rim . The supporting planes of loads 1 and 2 are rough, the sliding friction coefficient for each load is f = 0.1.

Under the action of a force F, the modulus of which changes according to the law F = F(s), where s is the displacement of the point of its application, the system begins to move from a state of rest. When the system moves, pulley 5 is acted upon by resistance forces, the moment of which relative to the axis of rotation is constant and equal to M 5 .

Determine the value of the angular velocity of pulley 4 at the moment in time when the displacement s of the point of application of force F becomes equal to s 1 = 1.2 m.

Download the solution to the problem >>>

Application of the general equation of dynamics to the study of the motion of a mechanical system

Problem condition

For a mechanical system, determine the linear acceleration a 1 . Assume that the masses of blocks and rollers are distributed along the outer radius. Cables and belts should be considered weightless and inextensible; there is no slippage. Neglect rolling and sliding friction.

Download the solution to the problem >>>

Application of d'Alembert's principle to determining the reactions of the supports of a rotating body

Problem condition

The vertical shaft AK, rotating uniformly with an angular velocity ω = 10 s -1, is fixed by a thrust bearing at point A and a cylindrical bearing at point D.

Rigidly attached to the shaft are a weightless rod 1 with a length of l 1 = 0.3 m, at the free end of which there is a load with a mass of m 1 = 4 kg, and a homogeneous rod 2 with a length of l 2 = 0.6 m, having a mass of m 2 = 8 kg. Both rods lie in the same vertical plane. The points of attachment of the rods to the shaft, as well as the angles α and β are indicated in the table. Dimensions AB=BD=DE=EK=b, where b = 0.4 m. Take the load as a material point.

Neglecting the mass of the shaft, determine the reactions of the thrust bearing and the bearing.

Introduction

Theoretical mechanics is one of the most important fundamental general scientific disciplines. It plays a significant role in the training of engineers of any specialization. General engineering disciplines are based on the results of theoretical mechanics: strength of materials, machine parts, theory of mechanisms and machines, and others.

The main task of theoretical mechanics is the study of the movement of material bodies under the influence of forces. An important particular task is the study of the equilibrium of bodies under the influence of forces.

Course of Lectures. Theoretical mechanics

The structure of theoretical mechanics. Basics of statics

Equilibrium conditions for an arbitrary system of forces.

Equilibrium equations for a rigid body.

Flat system of forces.

Special cases of rigid body equilibrium.

Balance problem for a beam.

Determination of internal forces in rod structures.

Fundamentals of point kinematics.

Natural coordinates.

Euler's formula.

Distribution of accelerations of points of a rigid body.

Translational and rotational movements.

Plane-parallel motion.

Complex point movement.

Basics of point dynamics.

Differential equations of motion of a point.

Particular types of force fields.

Fundamentals of the dynamics of a system of points.

General theorems on the dynamics of a system of points.

Dynamics of rotational motion of the body.

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

Butenin N.V., Lunts Ya.L., Merkin D.R. Course of theoretical mechanics, parts 1 and 2. M., Higher School, 1971.

Petkevich V.V. Theoretical mechanics. M., Nauka, 1981.

Collection of assignments for coursework in theoretical mechanics. Ed. A.A. Yablonsky. M., Higher School, 1985.

Lecture 1. The structure of theoretical mechanics. Basics of statics

In theoretical mechanics, the motion of bodies relative to other bodies, which are physical reference systems, is studied.

Mechanics allows not only to describe, but also to predict the movement of bodies, establishing causal relationships in a certain, very wide range of phenomena.

Basic abstract models of real bodies:

material point – has mass, but no size;

absolutely rigid body – a volume of finite dimensions, completely filled with a substance, and the distances between any two points of the medium filling the volume do not change during movement;

continuous deformable medium – fills a finite volume or unlimited space; the distances between points in such a medium can vary.

Of these, systems:

System of free material points;

Connected systems;

An absolutely solid body with a cavity filled with liquid, etc.

"Degenerates" models:

Infinitely thin rods;

Infinitely thin plates;

Weightless rods and threads connecting material points, etc.

From experience: mechanical phenomena occur differently in different places of the physical reference system. This property is the heterogeneity of space determined by the physical reference system. Here, heterogeneity is understood as the dependence of the nature of the occurrence of a phenomenon on the place in which we observe this phenomenon.

Another property is anisotropy (non-isotropy), the movement of a body relative to a physical reference system can be different depending on the direction. Examples: river flow along the meridian (from north to south - Volga); projectile flight, Foucault pendulum.

The properties of the reference system (inhomogeneity and anisotropy) make it difficult to observe the movement of a body.

Practically free from this - geocentric system: the center of the system is in the center of the Earth and the system does not rotate relative to the “fixed” stars). The geocentric system is convenient for calculating movements on Earth.

For celestial mechanics(for solar system bodies): a heliocentric reference frame that moves with the center of mass of the Solar System and does not rotate relative to the “fixed” stars. For this system not yet discovered heterogeneity and anisotropy of space

in relation to mechanical phenomena.

So, the abstract is introduced inertial frame of reference for which space is homogeneous and isotropic in relation to mechanical phenomena.

Inertial reference frame- one whose own motion cannot be detected by any mechanical experiment. Thought experiment: “a point alone in the whole world” (isolated) is either at rest or moving in a straight line and uniformly.

All reference systems moving relative to the original one rectilinearly and uniformly will be inertial. This allows the introduction of a unified Cartesian coordinate system. Such a space is called Euclidean.

Conventional agreement - take the right coordinate system (Fig. 1).

IN time– in classical (non-relativistic) mechanics absolutely, the same for all reference systems, that is, the initial moment is arbitrary. In contrast to relativistic mechanics, where the principle of relativity is applied.

The state of motion of the system at time t is determined by the coordinates and velocities of the points at this moment.

Real bodies interact and forces arise that change the state of motion of the system. This is the essence of theoretical mechanics.

How is theoretical mechanics studied?

The doctrine of the equilibrium of a set of bodies of a certain frame of reference - section statics.

Chapter kinematics: part of mechanics in which dependencies between quantities characterizing the state of motion of systems are studied, but the reasons causing a change in the state of motion are not considered.

After this, we will consider the influence of forces [MAIN PART].

Chapter dynamics: part of mechanics that deals with the influence of forces on the state of motion of systems of material objects.

Principles for constructing the main course - dynamics:

1) based on a system of axioms (based on experience, observations);

Constantly - ruthless control of practice. Sign of exact science – presence of internal logic (without it - a set of unrelated recipes)!

Static is called that part of mechanics where the conditions are studied that the forces acting on a system of material points must satisfy in order for the system to be in equilibrium, and the conditions for the equivalence of systems of forces.

Equilibrium problems in elementary statics will be considered using exclusively geometric methods based on the properties of vectors. This approach is used in geometric statics(in contrast to analytical statics, which is not considered here).

The positions of various material bodies will be related to the coordinate system, which we will take as stationary.

Ideal models of material bodies:

1) material point – a geometric point with mass.

2) an absolutely rigid body is a collection of material points, the distances between which cannot be changed by any actions.

By forces we will call objective causes that are the result of the interaction of material objects, capable of causing the movement of bodies from a state of rest or changing the existing movement of the latter.

Since force is determined by the movement it causes, it also has a relative nature, depending on the choice of reference system.

The question of the nature of forces is considered in physics.

A system of material points is in equilibrium if, being at rest, it does not receive any movement from the forces acting on it.

From everyday experience: forces have a vector nature, that is, magnitude, direction, line of action, point of application. The condition for the equilibrium of forces acting on a rigid body is reduced to the properties of vector systems.

Summarizing the experience of studying the physical laws of nature, Galileo and Newton formulated the basic laws of mechanics, which can be considered as axioms of mechanics, since they have are based on experimental facts.

Axiom 1. The action of several forces on a point of a rigid body is equivalent to the action of one resultant force constructed according to the rule of vector addition (Fig. 2).

Consequence. The forces applied to a point on a rigid body add up according to the parallelogram rule.

Axiom 2. Two forces applied to a rigid body mutually balanced if and only if they are equal in size, directed in opposite directions and lie on the same straight line.

Axiom 3. The action of a system of forces on a rigid body will not change if add to this system or discard from it two forces of equal magnitude, directed in opposite directions and lying on the same straight line.

Consequence. The force acting on a point of a rigid body can be transferred along the line of action of the force without changing the equilibrium (that is, the force is a sliding vector, Fig. 3)

1) Active - create or are capable of creating the movement of a rigid body. For example, weight force.

2) Passive - do not create movement, but limit the movement of a solid body, preventing movement. For example, the tension force of an inextensible thread (Fig. 4).

Axiom 4. The action of one body on a second is equal and opposite to the action of this second body on the first ( action equals reaction).

We will call the geometric conditions limiting the movement of points connections.

Terms of communication: for example,

- rod of indirect length l.

- flexible non-stretchable thread of length l.

Forces caused by connections and preventing movement are called forces of reactions.

Axiom 5. The connections imposed on a system of material points can be replaced by reaction forces, the action of which is equivalent to the action of the connections.

When passive forces cannot balance the action of active forces, movement begins.

Two particular problems of statics

1. System of converging forces acting on a rigid body

A system of converging forces is called such a system of forces, the lines of action of which intersect at one point, which can always be taken as the origin of coordinates (Fig. 5).

Projections of the resultant:

;

If , then the force causes the motion of the rigid body.

Equilibrium condition for a converging system of forces:

2. Balance of three forces

If three forces act on a rigid body, and the lines of action of the two forces intersect at some point A, equilibrium is possible if and only if the line of action of the third force also passes through point A, and the force itself is equal in magnitude and opposite in direction to the sum (Fig. 6).

Examples:

Moment of force about point O let's define it as a vector, in size equal to twice the area of a triangle, the base of which is the force vector with the vertex at a given point O; direction– orthogonal to the plane of the triangle in question in the direction from where the rotation produced by the force around point O is visible counterclockwise. is the moment of the sliding vector and is free vector(Fig.9).

So: or

Where ;;.

Where F is the force modulus, h is the shoulder (the distance from the point to the direction of the force).

Moment of force about the axis is the algebraic value of the projection onto this axis of the vector of the moment of force relative to an arbitrary point O taken on the axis (Fig. 10).

This is a scalar independent of the choice of point. Indeed, let us expand :|| and in the plane.

About moments: let O 1 be the point of intersection with the plane. Then:

a) from - moment => projection = 0.

b) from - moment along => is a projection.

So, moment about an axis is the moment of the force component in a plane perpendicular to the axis relative to the point of intersection of the plane and the axis.

Varignon's theorem for a system of converging forces:

Moment of resultant force for a system of converging forces relative to an arbitrary point A is equal to the sum of the moments of all component forces relative to the same point A (Fig. 11).

Proof in the theory of convergent vectors.

Explanation: addition of forces according to the parallelogram rule => the resulting force gives a total moment.

Security questions:

1. Name the main models of real bodies in theoretical mechanics.

2. Formulate the axioms of statics.

3. What is called the moment of force about a point?

Lecture 2. Equilibrium conditions for an arbitrary system of forces

From the basic axioms of statics, elementary operations on forces follow:

1) force can be transferred along the line of action;

2) forces whose lines of action intersect can be added according to the parallelogram rule (according to the rule of vector addition);

3) to the system of forces acting on a rigid body, you can always add two forces, equal in magnitude, lying on the same straight line and directed in opposite directions.

Elementary operations do not change the mechanical state of the system.

Let's call two systems of forces equivalent, if one from the other can be obtained using elementary operations (as in the theory of sliding vectors).

A system of two parallel forces, equal in magnitude and directed in opposite directions, is called a couple of forces(Fig. 12).

Moment of a couple of forces- a vector equal in size to the area of the parallelogram constructed on the vectors of the pair, and directed orthogonally to the plane of the pair in the direction from where the rotation imparted by the vectors of the pair is seen to occur counterclockwise.

, that is, the moment of force relative to point B.

A pair of forces is completely characterized by its moment.

A pair of forces can be transferred by elementary operations to any plane parallel to the plane of the pair; change the magnitude of the forces of the pair in inverse proportion to the shoulders of the pair.

Pairs of forces can be added, and the moments of pairs of forces are added according to the rule of addition of (free) vectors.

Bringing a system of forces acting on a rigid body to an arbitrary point (center of reduction)- means replacing the current system with a simpler one: a system of three forces, one of which passes through a predetermined point, and the other two represent a pair.

It can be proven using elementary operations (Fig. 13).

A system of converging forces and a system of pairs of forces.

- resultant force.

Resulting pair.

That's what needed to be shown.

Two systems of forces will equivalent if and only if both systems are reduced to one resultant force and one resultant pair, that is, when the conditions are met:

General case of equilibrium of a system of forces acting on a rigid body

Let us reduce the system of forces to (Fig. 14):

Resultant force through the origin;

The resulting pair, moreover, through point O.

That is, they led to and - two forces, one of which passes through a given point O.

Equilibrium, if the two on the same straight line are equal and opposite in direction (axiom 2).

Then it passes through point O, that is.

So, general conditions for the equilibrium of a solid body:

These conditions are valid for an arbitrary point in space.

Security questions:

1. List the elementary operations on forces.

2. What systems of forces are called equivalent?

3. Write the general conditions for the equilibrium of a rigid body.

Lecture 3. Equilibrium equations for a rigid body

Let O be the origin of coordinates; – resultant force; – moment of the resultant pair. Let point O1 be the new center of reduction (Fig. 15).

New power system:

When the reduction point changes, => only changes (in one direction with one sign, in the other direction with another). That is, the point: the lines match

Analytically: (colinearity of vectors)

; coordinates of point O1.

This is the equation of a straight line, for all points of which the direction of the resulting vector coincides with the direction of the moment of the resulting pair - the straight line is called dynamo.

If the dynamism => on the axis, then the system is equivalent to one resultant force, which is called resultant force of the system. At the same time, always, that is.

Four cases of bringing forces:

1.) ;- dynamism.

2.) ;- resultant.

3.) ;- pair.

4.) ;- balance.

Two vector equilibrium equations: the main vector and the main moment are equal to zero,.

Or six scalar equations in projections onto Cartesian coordinate axes:

Here:

The complexity of the type of equations depends on the choice of the reduction point => the skill of the calculator.

Finding the equilibrium conditions for a system of solid bodies in interaction<=>the problem of the equilibrium of each body separately, and the body is acted upon by external forces and internal forces (the interaction of bodies at points of contact with equal and oppositely directed forces - axiom IV, Fig. 17).

Let us choose for all bodies of the system one adduction center. Then for each body with the equilibrium condition number:

, , (= 1, 2, …, k)

where , is the resulting force and moment of the resulting pair of all forces, except internal reactions.

The resulting force and moment of the resulting pair of forces of internal reactions.

Formally summing by and taking into account the IV axiom

we get necessary conditions for the equilibrium of a solid body:

Example.

Equilibrium: = ?

Security questions:

1. Name all cases of bringing a system of forces to one point.

2. What is dynamism?

3. Formulate the necessary conditions for equilibrium of a system of solid bodies.

Lecture 4. Flat force system

A special case of the general delivery of the problem.

Let all the acting forces lie in the same plane - for example, a sheet. Let us choose point O as the reduction center - in the same plane. We obtain the resulting force and the resulting steam in the same plane, that is (Fig. 19)

Comment.

The system can be reduced to one resultant force.

Equilibrium conditions:

or scalar:

Very common in applications such as strength of materials.

Example.

With the friction of the ball on the board and on the plane. Equilibrium condition: = ?

The problem of the equilibrium of a non-free rigid body.

A rigid body whose movement is constrained by bonds is called unfree. For example, other bodies, hinged fastenings.

When determining equilibrium conditions: a non-free body can be considered as free, replacing bonds with unknown reaction forces.

Example.

Security questions:

1. What is called a plane system of forces?

2. Write the equilibrium conditions for a plane system of forces.

3. Which solid body is called non-free?

Lecture 5. Special cases of rigid body equilibrium

Theorem. Three forces balance a rigid body only if they all lie in the same plane.

Proof.

Let us choose a point on the line of action of the third force as the reduction point. Then (Fig. 22)

That is, the planes S1 and S2 coincide, and for any point on the force axis, etc. (Simpler: in the plane only there for balancing).

What is s in theoretical mechanics. Theoretical and analytical mechanics

In physics

In technical sciences

In higher education

Notes

Literature

See also

See what “Theoretical mechanics” is in other dictionaries:

Theoretical and analytical mechanics

Examples of solving problems in theoretical mechanics

Statics

Kinematics

Kinematics of a material point

Kinematic analysis of a flat mechanism

Determination of absolute speed and absolute acceleration of a point

Dynamics

Integration of differential equations of motion of a material point under the influence of variable forces

Theorem on the change in kinetic energy of a mechanical system

Application of the general equation of dynamics to the study of the motion of a mechanical system

Application of d'Alembert's principle to determining the reactions of the supports of a rotating body