The course examines: kinematics of a point and a rigid body (and from different points of view it is proposed to consider the problem of orientation of a rigid body), classical problems of the dynamics of mechanical systems and dynamics of a rigid body, elements of celestial mechanics, motion of systems of variable composition, the theory of impact, differential equations of analytical dynamics.

The course presents all the traditional sections of theoretical mechanics, but special attention is paid to the consideration of the most meaningful and valuable for the theory and applications of the sections of dynamics and methods of analytical mechanics; statics is studied as a section of dynamics, and in the section of kinematics, the concepts and mathematical apparatus necessary for the section of dynamics are introduced in detail.

Informational resources

Gantmakher F.R. Lectures on Analytical Mechanics. - 3rd ed. - M .: Fizmatlit, 2001.
Zhuravlev V.F. Foundations of theoretical mechanics. - 2nd ed. - M .: Fizmatlit, 2001; 3rd ed. - M .: Fizmatlit, 2008.
A.P. Markeev Theoretical mechanics. - Moscow - Izhevsk: Research Center "Regular and Chaotic Dynamics", 2007.

Requirements

The course is designed for students who own the apparatus of analytical geometry and linear algebra within the scope of the first year program of a technical university.

Course program

1. Kinematics of a point
1.1. Kinematic problems. Cartesian coordinate system. Decomposition of a vector in an orthonormal basis. The radius vector and coordinates of the point. Point speed and acceleration. Trajectory of movement.
1.2. Natural trihedron. Expansion of velocity and acceleration in the axes of a natural trihedron (Huygens' theorem).
1.3. Curvilinear coordinates of a point, examples: polar, cylindrical and spherical coordinate systems. Velocity components and acceleration projections on the axis of the curvilinear coordinate system.

2. Methods for setting the orientation of a rigid body
2.1. Solid. Fixed coordinate system associated with the body.
2.2. Orthogonal rotation matrices and their properties. Euler's finite turn theorem.
2.3. An active and passive point of view on orthogonal transformation. Adding turns.
2.4. Final rotation angles: Euler angles and airplane angles. Expression of an orthogonal matrix in terms of angles of final rotation.

3. Spatial movement of a rigid body
3.1. Translational and rotational motion of a rigid body. Angular velocity and angular acceleration.
3.2. Distribution of velocities (Euler's formula) and accelerations (Rivals's formula) points of a rigid body.
3.3. Kinematic invariants. Kinematic screw. Instantaneous helical axis.

4. Plane-parallel movement
4.1. The concept of plane-parallel body movement. Angular velocity and angular acceleration in the case of plane-parallel motion. Instant center of speeds.

5. Complex motion of a point and a rigid body
5.1. Stationary and moving coordinate systems. Absolute, relative and figurative movement of a point.
5.2. The theorem on the addition of velocities in a complex motion of a point, the relative and portable velocities of a point. Coriolis theorem on the addition of accelerations during complex motion of a point, relative, translational and Coriolis accelerations of a point.
5.3. Absolute, relative and translational angular velocity and angular acceleration of the body.

6. Motion of a rigid body with a fixed point (quaternionic presentation)
6.1. The concept of complex and hypercomplex numbers. Algebra of quaternions. Quaternion product. Conjugate and inverse quaternion, norm and modulus.
6.2. Trigonometric representation of the unit quaternion. Quaternionic way of specifying body rotation. Euler's finite turn theorem.
6.3. The relationship between the components of the quaternion in different bases. Adding turns. Rodrigues-Hamilton parameters.

7. Exam paper

8. Basic concepts of dynamics.
8.1 Impulse, angular momentum (angular momentum), kinetic energy.
8.2 Power of forces, work of forces, potential and total energy.
8.3 Center of mass (center of mass) of the system. The moment of inertia of the system about the axis.
8.4 Moments of inertia about parallel axes; Huygens – Steiner theorem.
8.5 Tensor and ellipsoid of inertia. Principal axes of inertia. Properties of axial moments of inertia.
8.6 Calculation of the angular momentum and kinetic energy of a body using the tensor of inertia.

9. Basic theorems of dynamics in inertial and non-inertial frames of reference.
9.1 The theorem about the change in the momentum of the system in the inertial frame of reference. The theorem on the motion of the center of mass.
9.2 The theorem about the change in the angular momentum of the system in the inertial frame of reference.
9.3 The theorem about the change in the kinetic energy of the system in the inertial frame of reference.
9.4 Potential, gyroscopic and dissipative forces.
9.5 Basic theorems of dynamics in non-inertial frames of reference.

10. The motion of a rigid body with a fixed point by inertia.
10.1 Dynamical Euler equations.
10.2 Euler's case, first integrals of dynamic equations; permanent rotation.
10.3 Interpretations by Poinsot and McCoolug.
10.4 Regular precession in the case of dynamic symmetry of the body.

11. The movement of a heavy rigid body with a fixed point.
11.1 General formulation of the problem of the motion of a heavy rigid body around.
fixed point. Dynamical Euler equations and their first integrals.
11.2 Qualitative analysis of the motion of a rigid body in the Lagrange case.
11.3 Forced regular precession of a dynamically symmetric rigid body.
11.4 The basic formula of gyroscopy.
11.5 The concept of the elementary theory of gyroscopes.

12. Dynamics of a point in the central field.
12.1 Binet's equation.
12.2 Equation of the orbit. Kepler's laws.
12.3 The scattering problem.
12.4 Two-body problem. Equations of motion. Integral of areas, integral of energy, Laplace integral.

13. Dynamics of systems of variable composition.
13.1 Basic concepts and theorems about the change in basic dynamic quantities in systems of variable composition.
13.2 Movement of a material point of variable mass.
13.3 Equations of motion of a body of variable composition.

14. The theory of impulsive movements.
14.1 Basic concepts and axioms of the theory of impulsive movements.
14.2 Theorems about the change in basic dynamic quantities during impulsive movement.
14.3 Impulsive movement of a rigid body.
14.4 Collision of two rigid bodies.
14.5 Karnot's theorems.

15. Test work

Learning outcomes

As a result of mastering the discipline, the student must:

• Know:
  • the basic concepts and theorems of mechanics and the methods resulting from them for studying the motion of mechanical systems;
• Be able to:
  • correctly formulate problems in terms of theoretical mechanics;
  • develop mechanical and mathematical models that adequately reflect the basic properties of the phenomena under consideration;
  • apply the knowledge gained to solve relevant specific problems;
• Own:
  • skills in solving classical problems of theoretical mechanics and mathematics;
  • skills in the study of problems in mechanics and the construction of mechanical and mathematical models that adequately describe a variety of mechanical phenomena;
  • the skills of the practical use of the methods and principles of theoretical mechanics in solving problems: force calculation, determination of the kinematic characteristics of bodies with various methods of setting motion, determining the law of motion of material bodies and mechanical systems under the action of forces;
  • the skills to independently master new information in the process of industrial and scientific activities, using modern educational and information technologies;

• Aizenberg T.B., Voronkov I.M., Osetskiy V.M .. Guide to solving problems in theoretical mechanics (6th edition). M .: Higher school, 1968 (djvu)
• Aizerman M.A. Classical Mechanics (2nd ed.). Moscow: Nauka, 1980 (djvu)
• Aleshkevich V.A., Dedenko L.G., Karavaev V.A. Rigid body mechanics. Lectures. M .: Fizfak MGU, 1997 (djvu)
• Amelkin N.I. Rigid Body Kinematics and Dynamics, MIPT, 2000 (pdf)
• Appel P. Theoretical Mechanics. Volume 1. Statistics. Point dynamics. Moscow: Fizmatlit, 1960 (djvu)
• Appel P. Theoretical Mechanics. Volume 2. Dynamics of the system. Analytical mechanics. Moscow: 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), s91-192, 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., Janelidze G.Yu., Kelzon A.S. Theoretical mechanics in examples and problems. Volume 1: Statics and Kinematics (5th Edition). Moscow: Nauka, 1967 (djvu)
• Bat M.I., Janelidze G.Yu., Kelzon A.S. Theoretical mechanics in examples and problems. Volume 2: Dynamics (3rd edition). Moscow: Nauka, 1966 (djvu)
• Bat M.I., Janelidze G.Yu., Kelzon A.S. Theoretical mechanics in examples and problems. Volume 3: Special Mechanics Chapters. Moscow: Nauka, 1973 (djvu)
• Bekshaev S.Ya., Fomin V.M. Foundations of the theory of vibrations. Odessa: OGASA, 2013 (pdf)
• Belenkiy I.M. Introduction to Analytical Mechanics. M .: Higher. school, 1964 (djvu)
• Berezkin E.N. Theoretical Mechanics Course (2nd ed.). Moscow: Ed. Moscow State University, 1974 (djvu)
• Berezkin E.N. Theoretical mechanics. Guidelines (3rd ed.). Moscow: Ed. Moscow State University, 1970 (djvu)
• Berezkin E.N. Solving problems in theoretical mechanics, part 1. M .: Izd. Moscow State University, 1973 (djvu)
• Berezkin E.N. Solving problems in theoretical mechanics, part 2. M .: Izd. Moscow State University, 1974 (djvu)
• Berezova O.A., Drushlyak G.E., Solodovnikov R.V. Theoretical mechanics. Collection of tasks. Kiev: Vishcha school, 1980 (djvu)
• Biderman V.L. The theory of mechanical vibrations. M .: Higher. school, 1980 (djvu)
• Bogolyubov N.N., Mitropol'skiy Yu.A., Samoilenko A.M. Accelerated convergence method in nonlinear mechanics. Kiev: 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)
• N.V. Butenin Introduction to Analytical Mechanics. Moscow: Nauka, 1971 (djvu)
• Butenin N.V., Lunts Ya.L., Merkin D.R. Course of theoretical mechanics. Volume 1. Statics and kinematics (3rd edition). Moscow: Nauka, 1979 (djvu)
• Butenin N.V., Lunts Ya.L., Merkin D.R. Course of theoretical mechanics. Volume 2. Dynamics (2nd edition). Moscow: Nauka, 1979 (djvu)
• Bukhgolts N.N. Basic course of theoretical mechanics. Volume 1: Kinematics, statics, dynamics of a material point (6th edition). Moscow: Nauka, 1965 (djvu)
• Bukhgolts N.N. Basic course of theoretical mechanics. Volume 2: Dynamics of the system of material points (4th edition). Moscow: Nauka, 1966 (djvu)
• Bukhgolts N.N., Voronkov I.M., Minakov A.P. Collection of problems in 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. Alternating action method (2nd edition). Moscow: Fizmatlit, 2005 (djvu)
• Veselovsky I.N. Dynamics. M.-L .: GITTL, 1941 (djvu)
• Veselovsky I.N. Collection of problems in theoretical mechanics. M .: GITTL, 1955 (djvu)
• Wittenburg J. Dynamics of systems of rigid bodies. M .: Mir, 1980 (djvu)
• Voronkov I.M. Theoretical Mechanics Course (11th edition). Moscow: Nauka, 1964 (djvu)
• Ganiev R.F., Kononenko V.O. Vibrations of solids. Moscow: Nauka, 1976 (djvu)
• Gantmakher F.R. Lectures on Analytical Mechanics. Moscow: 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 main provisions). Moscow: Nauka, 1973 (djvu)
• Hertz G. Principles of mechanics set out in a new connection. M .: AN SSSR, 1959 (djvu)
• Goldstein G. Classical Mechanics. M .: Gostekhizdat, 1957 (djvu)
• O.V. Golubeva Theoretical mechanics. M .: Higher. school, 1968 (djvu)
• Dimentberg F.M. Screw calculus and its applications in mechanics. Moscow: Nauka, 1965 (djvu)
• Dobronravov V.V. Fundamentals of Analytical Mechanics. M .: Higher school, 1976 (djvu)
• Zhirnov N.I. Classic 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. Moscow: Institute for Problems in Mechanics RAS (preprint N 251), 1985 (djvu)
• Zhuravlev V.F. Foundations of Theoretical Mechanics (2nd Edition). M .: Fizmatlit, 2001 (djvu)
• Zhuravlev V.F., Klimov D.M. Applied methods in the theory of oscillations. Moscow: Nauka, 1988 (djvu)
• Zubov V.I., Ermolin V.S. and other Dynamics of a free rigid body and determination of its orientation in space. L .: Leningrad State University, 1968 (djvu)
• V.G. Zubov Mechanics. Series "Beginnings of Physics". Moscow: Nauka, 1978 (djvu)
• History of the mechanics of gyroscopic systems. Moscow: Nauka, 1975 (djvu)
• Ishlinsky A.Yu. (ed.). Theoretical mechanics. Letter designations of quantities. Issue 96. M: Science, 1980 (djvu)
• Ishlinsky A.Yu., Borzov V.I., Stepanenko N.P. Collection of problems and exercises on the theory of gyroscopes. M .: Publishing house of Moscow State University, 1979 (djvu)
• Kabalsky M.M., Krivoshey V.D., Savitsky N.I., Tchaikovsky G.N. Typical problems in theoretical mechanics and methods for their solution. Kiev: GITL Ukrainian SSR, 1956 (djvu)
• Kilchevsky N.A. Course of theoretical mechanics, vol. 1: kinematics, statics, dynamics of a point, (2nd ed.), Moscow: Nauka, 1977 (djvu)
• Kilchevsky N.A. Course of theoretical mechanics, v. 2: system dynamics, analytical mechanics, elements of potential theory, mechanics of a continuous medium, special and general theory of relativity, Moscow: Nauka, 1977 (djvu)
• Kirpichev V.L. Conversations about mechanics. M.-L .: GITTL, 1950 (djvu)
• Klimov D.M. (ed.). Problems of mechanics: Sat. articles. On the occasion of 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: Publishing house of the Udmurt state. University, 1995 (djvu)
• A.A. Kosmodemyanskiy Course of theoretical mechanics. Part I. M .: Education, 1965 (djvu)
• A.A. Kosmodemyanskiy Course of theoretical mechanics. Part II. M .: Education, 1966 (djvu)
• G.L. Kotkin, V.G. Serbo Collection of problems in classical mechanics (2nd ed.). Moscow: Nauka, 1977 (djvu)
• Kragelsky I.V., Shchedrov V.S. Development of the science of friction. Dry friction. M .: AN SSSR, 1956 (djvu)
• Lagrange J. Analytical mechanics, volume 1. Moscow-Leningrad: GITTL, 1950 (djvu)
• Lagrange J. Analytical mechanics, volume 2. Moscow-Leningrad: GITTL, 1950 (djvu)
• Lamb G. Theoretical Mechanics. Volume 2. Dynamics. M.-L .: GTTI, 1935 (djvu)
• Lamb G. Theoretical Mechanics. Volume 3. More difficult questions. M.-L .: ONTI, 1936 (djvu)
• Levi-Civita T., Amaldi W. Course in Theoretical Mechanics. Volume 1, part 1: Kinematics, principles of mechanics. M.-L .: NKTL USSR, 1935 (djvu)
• Levi-Civita T., Amaldi W. Course in Theoretical Mechanics. Volume 1, part 2: Kinematics, principles of mechanics, statics. M .: From-in. literature, 1952 (djvu)
• Levi-Civita T., Amaldi W. Course in Theoretical Mechanics. Volume 2, part 1: Dynamics of systems with a finite number of degrees of freedom. M .: From-in. literature, 1951 (djvu)
• Levi-Civita T., Amaldi W. Course in Theoretical Mechanics. Volume 2, part 2: Dynamics of systems with a finite number of degrees of freedom. M .: From-in. literature, 1951 (djvu)
• Leach J.W. Classic mechanics. M .: Inostr. literature, 1961 (djvu)
• Lunts Ya.L. Introduction to the theory of gyroscopes. Moscow: 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)
• A.P. Markeev Dynamics of a body in contact with a solid surface. Moscow: Nauka, 1992 (djvu)
• A.P. Markeev Theoretical Mechanics, 2nd Edition. Izhevsk: RHD, 1999 (djvu)
• Martynyuk A.A. Stability of movement of complex systems. Kiev: Nauk. dumka, 1975 (djvu)
• Merkin D.R. Introduction to the mechanics of flexible thread. Moscow: Nauka, 1980 (djvu)
• Mechanics in the USSR for 50 years. Volume 1. General and Applied Mechanics. Moscow: Nauka, 1968 (djvu)
• I. I. Metelitsyn Gyroscope theory. Stability theory. Selected Works. Moscow: Nauka, 1977 (djvu)
• I. V. Meshchersky Collection of problems in theoretical mechanics (34th edition). Moscow: Nauka, 1975 (djvu)
• Misurev M.A. Methodology for solving problems in theoretical mechanics. M .: Higher school, 1963 (djvu)
• Moiseev N.N. Asymptotic methods of nonlinear mechanics. Moscow: Nauka, 1969 (djvu)
• Yu.I. Neimark, N.A. Fufaev Dynamics of nonholonomic systems. Moscow: Nauka, 1967 (djvu)
• A.I. Nekrasov Course of theoretical mechanics. Volume 1. Statics and kinematics (6th ed.) M .: GITTL, 1956 (djvu)
• A.I. Nekrasov Course of theoretical mechanics. Volume 2. Dynamics (2nd ed.) M .: GITTL, 1953 (djvu)
• Nikolai E.L. Gyroscope and some of its technical applications in the public domain. M.-L .: GITTL, 1947 (djvu)
• Nikolai E.L. The theory of gyroscopes. L.-M .: GITTL, 1948 (djvu)
• Nikolai E.L. Theoretical mechanics. Part I. Statics. Kinematics (20th 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 .: Publishing house of Leningrad State University, 1966 (djvu)
• Olkhovsky I.I. A course in theoretical mechanics for physicists. M .: MGU, 1978 (djvu)
• Olkhovsky I.I., Pavlenko Yu.G., Kuzmenkov L.S. Tasks in theoretical mechanics for physicists. Moscow: Moscow State University, 1977 (djvu)
• Pars L.A. Analytical dynamics. Moscow: 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 of the classics of science. Moscow: Fizmatgiz, 1959 (djvu)
• Poincaré A. Lectures on celestial mechanics. Moscow: Nauka, 1965 (djvu)
• Poincare A. New Mechanics. Evolution of laws. M .: Modern problems: 1913 (djvu)
• Roze N.V. (ed.) Theoretical mechanics. Part 1. Mechanics of a material point. L.-M .: GTTI, 1932 (djvu)
• Roze N.V. (ed.) Theoretical mechanics. Part 2. Mechanics of a material system and a rigid body. L.-M .: GTTI, 1933 (djvu)
• Rosenblat G.M. Dry friction in problems and solutions. M.-Izhevsk: RKhD, 2009 (pdf)
• Rubanovsky V.N., Samsonov V.A. Stability of stationary movements in examples and problems. M.-Izhevsk: RKhD, 2003 (pdf)
• Samsonov V.A. Lecture notes on mechanics. M .: MGU, 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 .: Higher. school, 1986

Theoretical mechanics- this is a section of mechanics, which sets out the basic laws of mechanical motion and mechanical interaction of material bodies.

Theoretical mechanics is the science in which the movements of bodies over time (mechanical movements) are studied. It serves as the basis for other branches of mechanics (theory of elasticity, resistance of materials, theory of plasticity, theory of mechanisms and machines, hydro-aerodynamics) and many technical disciplines.

Mechanical movement- this is a change over time in the relative position in space of material bodies.

Mechanical interaction- this is such an interaction as a result of which mechanical movement changes or the relative position of body parts changes.

Rigid body statics

Statics- this is a section of theoretical mechanics, which deals with the problems of equilibrium of rigid bodies and the transformation of one system of forces into another, equivalent to it.

Basic concepts and laws of statics
• Absolutely solid(solid, body) is a material body, the distance between any points in which does not change.
• Material point Is a body whose dimensions, according to the conditions of the problem, can be neglected.
• Free body Is a body, the movement of which is not subject to any restrictions.
• Unfree (bound) body Is a body with restrictions imposed on its movement.
• Connections- these are bodies that prevent the movement of the object under consideration (body or system of bodies).
• Communication reaction Is a force that characterizes the effect of a bond on a rigid body. If we consider the force with which a rigid body acts on a bond as an action, then the bond reaction is a reaction. In this case, the force - the action is applied to the bond, and the bond reaction is applied to the solid.
• Mechanical system Is a set of interconnected bodies or material points.
• Solid can be considered as a mechanical system, the position and distance between the points of which do not change.
• Force Is a vector quantity that characterizes the mechanical action of one material body on another.
  Force as a vector is characterized by the point of application, direction of action and absolute value. The unit of measure for the modulus of force is Newton.
• Force action line Is a straight line along which the force vector is directed.
• Concentrated power- force applied at one point.
• Distributed forces (distributed load)- these are the forces acting on all points of the volume, surface or length of the body.
  The distributed load is set by the force acting on a unit of volume (surface, length).
  The dimension of the distributed load is N / m 3 (N / m 2, N / m).
• External force Is a force acting from a body that does not belong to the considered mechanical system.
• Inner strength Is a force acting on a material point of a mechanical system from another material point belonging to the system under consideration.
• Force system Is a set of forces acting on a mechanical system.
• Flat system of forces Is a system of forces, the lines of action of which lie in the same plane.
• Spatial system of forces Is a system of forces, the lines of action of which do not lie in the same plane.
• System of converging forces Is a system of forces whose lines of action intersect at one point.
• Arbitrary system of forces Is a system of forces, the lines of action of which do not intersect at one point.
• Equivalent systems of forces- these are systems of forces, the replacement of which one with another does not change the mechanical state of the body.
  Accepted designation:.
• Equilibrium- this is a state in which the body under the action of forces remains stationary or moves uniformly in a straight line.
• Balanced system of forces Is a system of forces that, when applied to a free solid, does not change its mechanical state (does not unbalance).
  .
• Resultant force Is a force, the action of which on the body is equivalent to the action of the system of forces.
  .
• Moment of power Is a value that characterizes the rotational ability of a force.
• A couple of forces Is a system of two parallel, equal in magnitude, oppositely directed forces.
  Accepted designation:.
  Under the action of a pair of forces, the body will rotate.
• Axis force projection Is a segment enclosed between perpendiculars drawn from the beginning and end of the force vector to this axis.
  The projection is positive if the direction of the line segment coincides with the positive direction of the axis.
• Force projection onto plane Is a vector on a plane, enclosed between perpendiculars drawn from the beginning and end of the force vector to this plane.
• Law 1 (law of inertia). An isolated material point is at rest or moves evenly and rectilinearly.
  The uniform and rectilinear motion of a material point is motion by inertia. The state of equilibrium between a material point and a rigid body is understood not only as a state of rest, but also as motion by inertia. For a rigid body, there are various types of inertial motion, for example, uniform rotation of a rigid body around a fixed axis.
• Law 2. A solid body is in equilibrium under the action of two forces only if these forces are equal in magnitude and directed in opposite directions along the common line of action.
  These two forces are called balancing forces.
  In general, forces are called balancing if the rigid body to which these forces are applied is at rest.
• Law 3. Without disturbing the state (the word "state" here means a state of motion or rest) of a rigid body, one can add and drop counterbalancing forces.
  Consequence. Without violating the state of a rigid body, force can be transferred along its line of action to any point in the body.
  Two systems of forces are called equivalent if one of them can be replaced by another without violating the state of a rigid body.
• Law 4. The resultant of two forces applied at one point, applied at the same point, is equal in magnitude to the diagonal of the parallelogram built on these forces, and is directed along this
  diagonals.
  The modulus of the resultant is equal to:
  
• Law 5 (the law of equality of action and reaction)... The forces with which two bodies act on each other are equal in magnitude and directed in opposite directions along one straight line.
  It should be borne in mind that action- force applied to the body B, and counteraction- force applied to the body A are not balanced, since they are attached to different bodies.
• Law 6 (law of hardening)... The equilibrium of a non-solid body is not disturbed when it solidifies.
  It should not be forgotten that the conditions of equilibrium, which are necessary and sufficient for a solid, are necessary, but not sufficient for the corresponding non-solid.
• Law 7 (the law of release from ties). A non-free rigid body can be considered as free if it is mentally freed from bonds, replacing the action of bonds with the corresponding reactions of bonds.

Connections and their reactions
• Smooth surface constrains movement along the normal to the support surface. The reaction is directed perpendicular to the surface.
• Articulated movable support constrains the movement of the body along the normal to the reference plane. The reaction is directed along the normal to the support surface.
• Articulated fixed support counteracts any movement in a plane perpendicular to the axis of rotation.
• Articulated weightless rod counteracts the movement of the body along the line of the bar. The reaction will be directed along the line of the bar.
• Blind termination counteracts any movement and rotation in the plane. Its action can be replaced by a force represented in the form of two components and a pair of forces with a moment.

Kinematics

Kinematics- a section of theoretical mechanics, which examines the general geometric properties of mechanical motion, as a process that occurs in space and time. Moving objects are considered as geometric points or geometric bodies.

Basic concepts of kinematics
• The law of motion of a point (body) Is the dependence of the position of a point (body) in space on time.
• Point trajectory Is the geometrical position of a point in space during its movement.
• Point (body) speed- This is a characteristic of the change in time of the position of a point (body) in space.
• Point (body) acceleration- This is a characteristic of the change in time of the speed of a point (body).

Determination of kinematic characteristics of a point
• Point trajectory
  In the vector frame of reference, the trajectory is described by the expression:.
  In the coordinate system of reference, the trajectory is determined according to the law of motion of a point and is described by the expressions z = f (x, y)- in space, or y = f (x)- in the plane.
  In the natural frame of reference, the trajectory is set in advance.
• Determining the speed of a point in a vector coordinate system
  When specifying the movement of a point in a vector coordinate system, the ratio of the movement to the time interval is called the average value of the speed in this time interval:.
  Taking the time interval as an infinitely small value, the speed value is obtained at a given time (instantaneous speed value): .
  The average velocity vector is directed along the vector in the direction of the point's movement, the instantaneous velocity vector is directed tangentially to the trajectory in the direction of the point's movement.
  Output: the speed of a point is a vector quantity equal to the derivative of the law of motion with respect to time.
  Derivative property: the derivative of any quantity with respect to time determines the rate of change of this quantity.
• Determining the speed of a point in a coordinate system
  Point coordinates change rates:
  .
  The modulus of the full speed of a point with a rectangular coordinate system will be equal to:
  .
  The direction of the velocity vector is determined by the cosines of the direction angles:
  ,
  where are the angles between the velocity vector and the coordinate axes.
• Determining the speed of a point in the natural frame of reference
  The speed of a point in the natural frame of reference is determined as a derivative of the law of motion of a point:.
  According to the previous conclusions, the velocity vector is directed tangentially to the trajectory in the direction of movement of the point and in the axes is determined by only one projection.

Rigid body kinematics
• In the kinematics of solids, two main tasks are solved:
  1) the task of movement and the determination of the kinematic characteristics of the body as a whole;
  2) determination of the kinematic characteristics of the points of the body.
• The translational motion of a rigid body
  Translational movement is a movement in which a straight line drawn through two points of the body remains parallel to its original position.
  Theorem: during translational motion, all points of the body move along the same trajectories and at each moment of time have the same velocity and acceleration in magnitude and direction.
  Output: the translational movement of a rigid body is determined by the movement of any of its points, and therefore, the task and study of its movement is reduced to the kinematics of the point.
• Rotational movement of a rigid body around a fixed axis
  The rotational movement of a rigid body around a fixed axis is the movement of a rigid body in which two points belonging to the body remain motionless during the entire time of movement.
  The position of the body is determined by the angle of rotation. The angle unit is radians. (Radian is the central angle of a circle whose arc length is equal to the radius, the total angle of the circle contains 2π radians.)
  The law of rotational motion of a body around a fixed axis.
  The angular velocity and angular acceleration of the body is determined by the differentiation method:
  - angular velocity, rad / s;
  - angular acceleration, rad / s².
  If you cut the body with a plane perpendicular to the axis, select the point on the axis of rotation WITH and an arbitrary point M then point M will describe around the point WITH circle radius R... During dt an elementary rotation through an angle occurs, while the point M will move along the trajectory at a distance .
  Linear speed module:
  .
  Point acceleration M with a known trajectory, it is determined by its components:
  ,
  where .
  As a result, we get the formulas
  tangential acceleration: ;
  normal acceleration: .

Dynamics

Dynamics- This is a section of theoretical mechanics in which the mechanical movements of material bodies are studied, depending on the reasons that cause them.

Basic concepts of dynamics
• Inertia- this is the property of material bodies to maintain a state of rest or uniform rectilinear motion until external forces change this state.
• Weight Is a quantitative measure of body inertia. The unit of measure for mass is kilogram (kg).
• Material point Is a body with a mass, the dimensions of which are neglected when solving this problem.
• Center of gravity of the mechanical system- geometric point, the coordinates of which are determined by the formulas:
  
  where m k, x k, y k, z k- mass and coordinates k-th point of the mechanical system, m Is the mass of the system.
  In a homogeneous gravity field, the position of the center of mass coincides with the position of the center of gravity.
• Moment of inertia of a material body about the axis Is a quantitative measure of inertia during rotational motion.
  The moment of inertia of a material point about the axis is equal to the product of the point's mass by the square of the point's distance from the axis:
  .
  The moment of inertia of the system (body) about the axis is equal to the arithmetic sum of the moments of inertia of all points:
  
• The force of inertia of a material point Is a vector quantity equal in magnitude to the product of the point mass by the acceleration modulus and directed opposite to the acceleration vector:
• The force of inertia of a material body Is a vector quantity equal in modulus to the product of the body mass by the modulus of acceleration of the center of mass of the body and directed opposite to the vector of acceleration of the center of mass:,
  where is the acceleration of the center of mass of the body.
• Elementary Force Impulse Is a vector quantity equal to the product of the force vector by an infinitely small time interval dt:
  .
  The total impulse of force for Δt is equal to the integral of elementary impulses:
  .
• Elementary work of strength Is a scalar dA equal to the scalar proi

Kinematics

Material point kinematics

Determination of the speed and acceleration of a point according to the given equations of its motion

Given: Equations of motion of a point: x = 12 sin (πt / 6), cm; y = 6 cos 2 (πt / 6), cm.

Set the type of its trajectory and for the moment of time t = 1 s find the position of a point on the trajectory, its speed, total, tangential and normal accelerations, as well as the radius of curvature of the trajectory.

Translational and rotational motion of a rigid body

Given:
t = 2 s; r 1 = 2 cm, R 1 = 4 cm; r 2 = 6 cm, R 2 = 8 cm; r 3 = 12 cm, R 3 = 16 cm; s 5 = t 3 - 6t (cm).

Determine at time t = 2 the speeds of points A, C; angular acceleration of wheel 3; point B acceleration and staff acceleration 4.

Kinematic Analysis of a Plane Mechanism

Given:
R 1, R 2, L, AB, ω 1.
Find: ω 2.

The flat mechanism consists of rods 1, 2, 3, 4 and slide E. The rods are connected by means of cylindrical hinges. Point D is located in the middle of bar AB.
Given: ω 1, ε 1.
Find: speeds V A, V B, V D and V E; angular velocities ω 2, ω 3 and ω 4; acceleration a B; angular acceleration ε AB link AB; positions of instant centers of speeds P 2 and P 3 of links 2 and 3 of the mechanism.

Determination of the absolute speed and absolute acceleration of a point

The rectangular plate rotates around a fixed axis according to the law φ = 6 t 2 - 3 t 3... The positive direction of the angle φ is shown in the figures with an arc arrow. Rotation axis OO 1 lies in the plane of the plate (the plate rotates in space).

Point M moves along the line BD on the plate. The law of its relative motion is given, i.e., the dependence s = AM = 40 (t - 2 t 3) - 40(s - in centimeters, t - in seconds). Distance b = 20 cm... In the figure, point M is shown in a position at which s = AM > 0 (for s< 0 point M is on the other side of point A).

Find the absolute speed and absolute acceleration of point M at time t 1 = 1 s.

Dynamics

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

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

The load, having finished its movement on the section AB, at the point B of the pipe, without changing the value of the modulus of its speed, goes to the section BC. In section BC, a variable force F acts on the load, the projection F x of which on the x axis is given.

Considering the load as a material point, find the law of its movement on the BC section, i.e. x = f (t), where x = BD. Disregard the friction of the load on the pipe.

Download problem solution

The theorem on the change in the kinetic energy of a mechanical system

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; the thread sections are parallel to the corresponding planes. The roller (solid homogeneous cylinder) rolls on the reference plane without sliding. The radii of the steps of the pulleys 4 and 5 are, respectively, 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 uniformly distributed along its outer rim ... The support planes of weights 1 and 2 are rough, the sliding friction coefficient for each load is f = 0.1.

Under the action of the 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 starts to move from a state of rest. When the system moves, resistance forces act on the pulley 5, the moment of which relative to the axis of rotation is constant and equal to M 5.

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

Download problem solution

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

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

Download problem solution

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

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.

A weightless rod 1 with a length of l 1 = 0.3 m is rigidly attached to the shaft, 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.

By neglecting the mass of the shaft, determine the reaction of the thrust bearing and bearing.

Point kinematics.

1. The subject of theoretical mechanics. Basic abstractions.

Theoretical mechanicsis a science in which the general laws of mechanical motion and mechanical interaction of material bodies are studied

Mechanical movementis called the movement of a body in relation to another body, which occurs in space and time.

Mechanical interaction such an interaction of material bodies is called, which changes the nature of their mechanical movement.

Statics - This is a branch of theoretical mechanics, in which methods of transforming systems of forces into equivalent systems are studied and conditions for the equilibrium of forces applied to a solid are established.

Kinematics - this is a branch of theoretical mechanics that studies the movement of material bodies in space from a geometric point of view, regardless of the forces acting on them.

Dynamics - this is a section of mechanics, which studies the movement of material bodies in space, depending on the forces acting on them.

Objects of study in theoretical mechanics:

material point,

system of material points,

Absolutely solid.

Absolute space and absolute time are independent of one another. Absolute space - three-dimensional, homogeneous, stationary Euclidean space. Absolute time - flows from the past to the future continuously, it is homogeneous, the same at all points in space and does not depend on the movement of matter.

2. The subject of kinematics.

Kinematics - this is a branch of mechanics in which the geometric properties of the motion of bodies are studied without taking into account their inertia (i.e. mass) and the forces acting on them

To determine the position of a moving body (or point) with the body, in relation to which the movement of the given body is being studied, some coordinate system is rigidly connected, which, together with the body, forms frame of reference.

The main task of kinematics is, knowing the law of motion of a given body (point), to determine all the kinematic quantities that characterize its motion (speed and acceleration).

3. Methods for specifying the movement of a point

· Natural way

It should be known:

Point motion trajectory;

Start and direction of counting;

The law of motion of a point along a given trajectory in the form (1.1)

· Coordinate way

Equations (1.2) are the equations of motion of the point M.

The equation of the trajectory of point M can be obtained by excluding the time parameter « t » from equations (1.2)

· Vector way

The relationship between coordinate and natural ways of specifying the movement of a point

Determine the trajectory of a point, excluding time from equations (1.2);

-- find the law of motion of a point along a trajectory (use the expression for the differential of the arc)

After integration, we obtain the law of motion of a point along a given trajectory:

The relationship between the coordinate and vector methods of specifying the motion of a point is determined by the equation (1.4)

4. Determination of the speed of a point in the vector method of specifying the movement.

Let at the moment of timetthe position of the point is determined by the radius vector, and at the moment of timet 1 - radius vector, then for a period of time the point will move.

Point speed at a given time

To get the speed of a point at a given moment in time, it is necessary to make the passage to the limit

(1.6)

(1.7)

The velocity vector of a point at a given time is equal to the first derivative of the radius vector in time and is directed tangentially to the trajectory at a given point.

(unit¾ m / s, km / h)

Average acceleration vector has the same direction as the vectorΔ v , that is, directed towards the concavity of the trajectory.

Acceleration vector of a point at a given time is equal to the first derivative of the velocity vector or the second derivative of the radius vector of the point with respect to time.

(unit of measure -)

How is the vector positioned in relation to the path of the point?

In straight-line motion, the vector is directed along the straight line along which the point is moving. If the trajectory of a point is a flat curve, then the acceleration vector, as well as the vector cp, lies in the plane of this curve and is directed towards its concavity. If the trajectory is not a plane curve, then the vector cp will be directed towards the concavity of the trajectory and will lie in the plane passing through the tangent to the trajectory at the pointM and a straight line parallel to the tangent at an adjacent pointM 1 . V limit when pointM 1 strives for M this plane occupies the position of the so-called contacting plane. Therefore, in the general case, the acceleration vector lies in the contacting plane and is directed towards the concavity of the curve.