Solving problems in theoretical mechanics. Basic Mechanics for Dummies

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

Theoretical mechanics is a science in which the movements of bodies over time (mechanical movements) are studied. It serves as the basis for other sections of mechanics (the theory of elasticity, resistance of materials, the theory of plasticity, the theory of mechanisms and machines, hydroaerodynamics) 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 the mechanical movement changes or the relative position of body parts changes.

Rigid body statics

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

Basic concepts and laws of statics

Absolutely rigid body(solid body, 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.
loose body is a body, on the movement of which no restrictions are imposed.
Non-free (bound) body is a body whose movement is restricted.
Connections- these are bodies that prevent the movement of the object under consideration (a body or a system of bodies).
Communication reaction is a force that characterizes the action 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 reaction of the bond is a counteraction. In this case, the force - action is applied to the connection, and the reaction of the connection is applied to the solid body.
mechanical system is a set of interconnected bodies or material points.
Solid can be considered as a mechanical system, the positions and distance between the points of which do not change.
Force is a vector quantity characterizing the mechanical action of one material body on another.
Force as a vector is characterized by the point of application, the direction of action and the absolute value. The unit of measure for the modulus of force is Newton.
line of force is the straight line along which the force vector is directed.
Concentrated Power is the force applied at one point.
Distributed forces (distributed load)- these are forces acting on all points of the volume, surface or length of the body.
The distributed load is given by the force acting per unit 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 the totality of forces acting on a mechanical system.
Flat system of forces is a system of forces whose lines of action lie in the same plane.
Spatial system of forces is a system of forces whose lines of action do not lie in the same plane.
Converging force system is a system of forces whose lines of action intersect at one point.
Arbitrary system of forces is a system of forces whose lines of action do not intersect at one point.
Equivalent systems of forces- these are systems of forces, the replacement of which one for another does not change the mechanical state of the body.
Accepted designation: .
Equilibrium A state in which a body remains stationary or moves uniformly in a straight line under the action of forces.
Balanced system of forces- this is a system of forces that, when applied to a free solid body, does not change its mechanical state (does not unbalance it).
.
resultant force is a force whose action on a body is equivalent to the action of a system of forces.
.
Moment of power is a value that characterizes the rotational ability of the force.
Power couple is a system of two parallel equal in absolute value oppositely directed forces.
Accepted designation: .
Under the action of a couple of forces, the body will perform a rotational motion.
Projection of Force on the Axis- this 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 segment coincides with the positive direction of the axis.
Projection of Force on a Plane is a vector on a plane enclosed between the 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 uniformly and rectilinearly.
The uniform and rectilinear motion of a material point is a motion by inertia. The state of equilibrium of a material point and a rigid body is understood not only as a state of rest, but also as a movement by inertia. For a rigid body, there are various types of inertia motion, for example, uniform rotation of a rigid body around a fixed axis.
Law 2. A rigid body is in equilibrium under the action of two forces only if these forces are equal in magnitude and directed in opposite directions along a common line of action.
These two forces are called balanced.
In general, forces are said to be balanced if the rigid body to which these forces are applied is at rest.
Law 3. Without violating the state (the word "state" here means the state of motion or rest) of a rigid body, one can add and discard balancing forces.
Consequence. Without disturbing the state of a rigid body, the force can be transferred along its line of action to any point of the body.
Two systems of forces are called equivalent if one of them can be replaced by another without disturbing the state of the rigid body.
Law 4. The resultant of two forces applied at one point is applied at the same point, is equal in absolute value to the diagonal of the parallelogram built on these forces, and is directed along this
diagonals.
The modulus of the resultant is:
Law 5 (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 opposition- force applied to the body BUT, are not balanced, since they are attached to different bodies.
Law 6 (the law of hardening). The equilibrium of a non-solid body is not disturbed when it solidifies.
It should not be forgotten that the equilibrium conditions, which are necessary and sufficient for a rigid body, are necessary but insufficient for the corresponding non-rigid body.
Law 7 (the law of release from bonds). A non-free solid 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 restricts movement along the normal to the support surface. The reaction is directed perpendicular to the surface.
Articulated movable support limits 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 rod. The reaction will be directed along the line of the rod.
Blind termination counteracts any movement and rotation in the plane. Its action can be replaced by a force presented in the form of two components and a pair of forces with a moment.

Kinematics

Kinematics- a section of theoretical mechanics, which considers the general geometric properties of mechanical motion, as a process occurring 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 locus of the positions 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 the kinematic characteristics of a point

Point trajectory
In the vector reference system, the trajectory is described by the expression: .
In the coordinate reference system, the trajectory is determined according to the law of point motion and is described by the expressions z = f(x,y) in space, or y = f(x)- in the plane.
In a natural reference system, the trajectory is predetermined.
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 movement to the time interval is called the average value of the speed in this time interval: .
Taking the time interval as an infinitesimal value, we obtain the speed value at a given time (instantaneous speed value): .
The average velocity vector is directed along the vector in the direction of the point movement, the instantaneous velocity vector is directed tangentially to the trajectory in the direction of the point movement.
Conclusion: 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 time derivative of any value determines the rate of change of this value.
Determining the speed of a point in a coordinate reference system
Rate of change of point coordinates:
.
The module 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 steering angles:
,
where are the angles between the velocity vector and the coordinate axes.
Determining the speed of a point in a natural reference system
The speed of a point in a natural reference system is defined 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 the point movement and in the axes is determined by only one projection .

Rigid Body Kinematics

In the kinematics of rigid bodies, two main problems are solved:
1) task of movement and determination of the kinematic characteristics of the body as a whole;
2) determination of the kinematic characteristics of the points of the body.
Translational motion of a rigid body
Translational motion is a motion in which a straight line drawn through two points of the body remains parallel to its original position.
Theorem: in translational motion, all points of the body move along the same trajectories and at each moment of time have the same speed and acceleration in absolute value and direction.
Conclusion: the translational motion of a rigid body is determined by the motion of any of its points, and therefore, the task and study of its motion is reduced to the kinematics of a point.
Rotational motion of a rigid body around a fixed axis
The rotational motion of a rigid body around a fixed axis is the motion 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 unit of measurement for an angle is radians. (A radian is the central angle of a circle whose arc length is equal to the radius, the full angle of the circle contains 2π radian.)
The law of rotational motion of a body around a fixed axis.
The angular velocity and angular acceleration of the body will be determined by the differentiation method:
— angular velocity, rad/s;
— angular acceleration, rad/s².
If we cut the body by a plane perpendicular to the axis, choose a point on the axis of rotation With and an arbitrary point M, then the point M will describe around the point With radius circle R. During dt there is an elementary rotation through the angle , while the point M will move along the trajectory for a distance .
Linear speed module:
.
point acceleration M with a known trajectory is determined by its components:
,
where .
As a result, we get formulas
tangential acceleration: ;
normal acceleration: .

Dynamics

Dynamics- This is a branch of theoretical mechanics, which studies the mechanical movements of material bodies, depending on the causes 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 the inertia of a body. The unit of mass is kilogram (kg).
Material point is a body with a mass, the dimensions of which are neglected in solving this problem.
Center of mass of a mechanical system is a geometric point whose coordinates are determined by the formulas:

where m k , x k , y k , z k- mass and coordinates k- that point of the mechanical system, m is the mass of the system.
In a uniform field of gravity, 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 mass of the point and the square of the distance of the point 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 absolute value to the product of the mass of a point and the module of acceleration and directed opposite to the acceleration vector:
Force of inertia of a material body is a vector quantity equal in absolute value to the product of the body mass and the module of acceleration of the center of mass of the body and directed opposite to the acceleration vector of the center of mass: ,
where is the acceleration of the center of mass of the body.
Elemental Force Impulse is a vector quantity equal to the product of the force vector by an infinitesimal time interval dt:
.
The total impulse of force for Δt is equal to the integral of elementary impulses:
.
Elementary work of force is a scalar dA, equal to the scalar

List of exam questions

Technical mechanics, its definition. Mechanical motion and mechanical interaction. Material point, mechanical system, absolutely rigid body.

Technical mechanics - the science of mechanical motion and interaction of material bodies.

Mechanics is one of the most ancient sciences. The term "Mechanics" was introduced by the outstanding philosopher of antiquity Aristotle.

The achievements of scientists in the field of mechanics make it possible to solve complex practical problems in the field of technology, and in essence, not a single phenomenon of nature can be understood without understanding it from the mechanical side. And not a single creation of technology can be created without taking into account certain mechanical laws.

mechanical movement - this is a change over time in the relative position in space of material bodies or the relative position of parts of a given body.

Mechanical interaction - these are the actions of material bodies on each other, as a result of which there is a change in the movement of these bodies or a change in their shape (deformation).

Basic concepts:

Material point is a body whose dimensions under given conditions can be neglected. It has mass and the ability to interact with other bodies.

mechanical system is a set of material points, the position and movement of each of which depend on the position and movement of other points in the system.

Absolutely rigid body (ATT) is a body, the distance between any two points of which always remains unchanged.

Theoretical mechanics and its sections. Problems of theoretical mechanics.

Theoretical mechanics is a branch of mechanics that studies the laws of motion of bodies and the general properties of these motions.

Theoretical mechanics consists of three sections: statics, kinematics and dynamics.

Statics considers the equilibrium of bodies and their systems under the action of forces.

Kinematics considers the general geometric properties of the motion of bodies.

Dynamics studies the motion of bodies under the action of forces.

Static tasks:

1. Transformation of systems of forces acting on ATT into systems equivalent to them, i.e. reduction of this system of forces to the simplest form.

2. Determination of the equilibrium conditions for the system of forces acting on the ATT.

To solve these problems, two methods are used: graphical and analytical.

Equilibrium. Force, system of forces. Resultant force, concentrated force and distributed forces.

Equilibrium is the state of rest of a body in relation to other bodies.

Force - this is the main measure of the mechanical interaction of material bodies. Is a vector quantity, i.e. Strength is characterized by three elements:

application point;

Line of action (direction);

Module (numerical value).

Force system is the totality of all forces acting on the considered absolutely rigid body (ATT)

The force system is called converging if the lines of action of all forces intersect at one point.

The system is called flat , if the lines of action of all forces lie in the same plane, otherwise spatial.

The force system is called parallel if the lines of action of all forces are parallel to each other.

The two systems of forces are called equivalent , if one system of forces acting on an absolutely rigid body can be replaced by another system of forces without changing the state of rest or motion of the body.

Balanced or equivalent to zero called a system of forces under the action of which a free ATT can be at rest.

resultant force is a force whose action on a body or material point is equivalent to the action of a system of forces on the same body.

Outside forces

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

Forces acting on all points of a certain volume or surface are called distributed .

A body that is not prevented from moving in any direction by any other body is called a free body.

External and internal forces. Free and non-free body. The principle of release from bonds.

Outside forces called the forces with which the parts of a given body act on each other.

When solving most problems of statics, it is required to represent a non-free body as a free one, which is done using the principle of freeing the body, which is formulated as follows:

any non-free body can be considered as free, if we discard the connections, replacing them with reactions.

As a result of applying this principle, a body is obtained that is free from bonds and is under the action of a certain system of active and reactive forces.

Axioms of statics.

Conditions under which a body can be in equal Vesii, are derived from several basic provisions, accepted without evidence, but confirmed by experiments , and called axioms of statics. The basic axioms of statics were formulated by the English scientist Newton (1642-1727), and therefore they are named after him.

Axiom I (axiom of inertia or Newton's first law).

Any body retains its state of rest or rectilinear uniform motion, as long as some Forces will not bring him out of this state.

The ability of a body to maintain its state of rest or rectilinear uniform motion is called inertia. On the basis of this axiom, we consider the state of equilibrium to be such a state when the body is at rest or moves in a straight line and uniformly (i.e., the PO of inertia).

Axiom II (the axiom of interaction or Newton's third law).

If one body acts on the second with a certain force, then the second body simultaneously acts on the first with a force equal in magnitude to the opposite in direction.

The totality of forces applied to a given body (or system of bodies) is called force system. The force of action of a body on a given body and the force of reaction of a given body do not represent a system of forces, since they are applied to different bodies.

If some system of forces has such a property that, after being applied to a free body, it does not change its state of equilibrium, then such a system of forces is called balanced.

Axiom III (condition of balance of two forces).

For the equilibrium of a free rigid body under the action of two forces, it is necessary and sufficient that these forces be equal in absolute value and act in one straight line in opposite directions.

necessary to balance the two forces. This means that if the system of two forces is in equilibrium, then these forces must be equal in absolute value and act in one straight line in opposite directions.

The condition formulated in this axiom is sufficient to balance the two forces. This means that the reverse formulation of the axiom is true, namely: if two forces are equal in absolute value and act in the same straight line in opposite directions, then such a system of forces is necessarily in equilibrium.

In the following, we will get acquainted with the equilibrium condition, which will be necessary, but not sufficient for equilibrium.

Axiom IV.

The equilibrium of a rigid body will not be disturbed if a system of balanced forces is applied to it or removed.

Consequence from the axioms III and IV.

The equilibrium of a rigid body is not disturbed by the transfer of a force along its line of action.

Parallelogram axiom. This axiom is formulated as follows:

The resultant of two forces applied to body at one point, is equal in absolute value and coincides in direction with the diagonal of the parallelogram built on these forces, and is applied at the same point.

Connections, reactions of connections. Connection examples.

connections bodies that limit the movement of a given body in space are called. The force with which the body acts on the bond is called pressure; the force with which a bond acts on a body is called reaction. According to the axiom of interaction, the reaction and pressure modulo equal and act in the same straight line in opposite directions. Reaction and pressure are applied to different bodies. The external forces acting on the body are divided into active and reactive. Active forces tend to move the body to which they are applied, and reactive forces, through bonds, prevent this movement. The fundamental difference between active forces and reactive forces is that the magnitude of reactive forces, generally speaking, depends on the magnitude of active forces, but not vice versa. Active forces are often called

The direction of the reactions is determined by the direction in which this connection prevents the body from moving. The rule for determining the direction of reactions can be formulated as follows:

the direction of the reaction of the connection is opposite to the direction of the displacement destroyed by this connection.

1. Perfectly smooth plane

In this case, the reaction R directed perpendicular to the reference plane towards the body.

2. Ideally smooth surface (Fig. 16).

In this case, the reaction R is directed perpendicular to the tangent plane t - t, i.e., along the normal to the supporting surface towards the body.

3. Fixed point or corner edge (Fig. 17, edge B).

In this case, the reaction R in directed along the normal to the surface of an ideally smooth body towards the body.

4. Flexible connection (Fig. 17).

The reaction T of a flexible bond is directed along c to i s and. From fig. 17 it can be seen that the flexible connection, thrown over the block, changes the direction of the transmitted force.

5. Ideally smooth cylindrical hinge (Fig. 17, hinge BUT; rice. 18, bearing D).

In this case, it is only known in advance that the reaction R passes through the hinge axis and is perpendicular to this axis.

6. Perfectly smooth thrust bearing (Fig. 18, thrust bearing BUT).

The thrust bearing can be considered as a combination of a cylindrical hinge and a bearing plane. Therefore, we will

7. Perfectly smooth ball joint (Fig. 19).

In this case, it is only known in advance that the reaction R passes through the center of the hinge.

8. A rod fixed at both ends in ideally smooth hinges and loaded only at the ends (Fig. 18, rod BC).

In this case, the reaction of the rod is directed along the rod, since, according to axiom III, the reactions of the hinges B and C in equilibrium, the rod can only be directed along the line sun, i.e. along the rod.

System of converging forces. Addition of forces applied at one point.

converging called forces whose lines of action intersect at one point.

This chapter deals with systems of converging forces whose lines of action lie in the same plane (flat systems).

Imagine that a flat system of five forces acts on the body, the lines of action of which intersect at the point O (Fig. 10, a). In § 2 it was established that the force- sliding vector. Therefore, all forces can be transferred from the points of their application to the point O of the intersection of the lines of their action (Fig. 10, b).

Thus, any system of converging forces applied to different points of the body can be replaced by an equivalent system of forces applied to one point. This system of forces is often called bundle of forces.

The course covers: 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 the dynamics of a rigid body, elements of celestial mechanics, motion of systems of variable composition, impact theory, differential equations of analytical dynamics.

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

Informational resources

Gantmakher F.R. Lectures on Analytical Mechanics. - 3rd ed. – M.: Fizmatlit, 2001.
Zhuravlev V.F. Fundamentals of theoretical mechanics. - 2nd ed. - M.: Fizmatlit, 2001; 3rd ed. – M.: Fizmatlit, 2008.
Markeev A.P. 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 in the scope of the first-year program of a technical university.

Course program

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

2. Methods for specifying the orientation of a rigid body
2.1. Solid. Fixed and body-bound coordinate systems.
2.2. Orthogonal rotation matrices and their properties. Euler's finite turn theorem.
2.3. Active and passive points of view on orthogonal transformation. Addition of turns.
2.4. Finite rotation angles: Euler angles and "airplane" angles. Expression of an orthogonal matrix in terms of finite rotation angles.

3. Spatial motion 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' formula) of points of a rigid body.
3.3. Kinematic invariants. Kinematic screw. Instant screw axle.

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

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

6. Motion of a rigid body with a fixed point (quaternion 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. Quaternion method of specifying body rotation. Euler's finite turn theorem.
6.3. Relationship between quaternion components in different bases. Addition of turns. Rodrigues-Hamilton parameters.

7. Exam work

8. Basic concepts of dynamics.
8.1 Momentum, angular momentum (kinetic moment), kinetic energy.
8.2 Power of forces, work of forces, potential and total energy.
8.3 Center of mass (center of inertia) of the system. The moment of inertia of the system about the axis.
8.4 Moments of inertia about parallel axes; the 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 the body using the inertia tensor.

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

10. Movement of a rigid body with a fixed point by inertia.
10.1 Euler dynamic equations.
10.2 Euler case, first integrals of dynamical equations; permanent rotations.
10.3 Interpretations of Poinsot and Macculag.
10.4 Regular precession in the case of dynamic symmetry of the body.

11. Motion 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. Euler dynamic equations and their first integrals.
11.2 Qualitative analysis of the motion of a rigid body in the case of Lagrange.
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 Orbit equation. Kepler's laws.
12.3 The scattering problem.
12.4 The problem of two bodies. Equations of motion. Area integral, energy integral, Laplace integral.

13. Dynamics of systems of variable composition.
13.1 Basic concepts and theorems on the change of 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. Theory of impulsive movements.
14.1 Basic concepts and axioms of the theory of impulsive movements.
14.2 Theorems about changing the basic dynamic quantities during impulsive motion.
14.3 Impulsive motion of a rigid body.
14.4 Collision of two rigid bodies.
14.5 Carnot's theorems.

15. Control work

Learning Outcomes

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

Know:
- basic concepts and theorems of mechanics and the methods of studying the motion of mechanical systems arising from them;
Be able to:
- correctly formulate problems in terms of theoretical mechanics;
- develop mechanical and mathematical models that adequately reflect the main properties of the phenomena under consideration;
- apply the acquired knowledge to solve relevant specific problems;
Own:
- skills in solving classical problems of theoretical mechanics and mathematics;
- the skills of studying the problems of mechanics and building mechanical and mathematical models that adequately describe a variety of mechanical phenomena;
- skills in the practical use of methods and principles of theoretical mechanics in solving problems: force calculation, determining 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;
- skills to independently master new information in the process of production and scientific activities, using modern educational and information technologies;

20th ed. - M.: 2010.- 416 p.

The book outlines the fundamentals of the mechanics of a material point, the system of material points and a solid body in a volume corresponding to the programs of technical universities. Many examples and tasks are given, the solutions of which are accompanied by appropriate guidelines. For students of full-time and correspondence technical universities.

Format: pdf

The size: 14 MB

Watch, download: drive.google

TABLE OF CONTENTS
Preface to the thirteenth edition 3
Introduction 5
SECTION ONE STATICS OF A SOLID STATE
Chapter I. Basic concepts initial provisions of articles 9
41. Absolutely rigid body; force. Tasks of statics 9
12. Initial provisions of statics » 11
$ 3. Connections and their reactions 15
Chapter II. Composition of forces. System of converging forces 18
§4. Geometrically! Method of combining forces. Resultant of converging forces, decomposition of forces 18
f 5. Force projections on the axis and on the plane, Analytical method for setting and adding forces 20
16. Equilibrium of the system of converging forces_. . . 23
17. Solving problems of statics. 25
Chapter III. Moment of force about the center. Power couple 31
i 8. Moment of force about the center (or point) 31
| 9. A couple of forces. couple moment 33
f 10*. Equivalence and pair addition theorems 35
Chapter IV. Bringing the system of forces to the center. Equilibrium conditions... 37
f 11. Parallel force transfer theorem 37
112. Bringing the system of forces to a given center - . .38
§ 13. Conditions for the equilibrium of a system of forces. Theorem on the moment of the resultant 40
Chapter V. Flat system of forces 41
§ 14. Algebraic moments of force and couples 41
115. Reduction of a flat system of forces to the simplest form .... 44
§ 16. Equilibrium of a flat system of forces. The case of parallel forces. 46
§ 17. Problem solving 48
118. Balance of systems of bodies 63
§ nineteen*. Statically determined and statically indeterminate systems of bodies (structures) 56"
f 20*. Definition of internal forces. 57
§ 21*. Distributed Forces 58
E22*. Calculation of flat trusses 61
Chapter VI. Friction 64
! 23. Laws of sliding friction 64
: 24. Rough bond reactions. Friction angle 66
: 25. Equilibrium in the presence of friction 66
(26*. Thread friction on a cylindrical surface 69
1 27*. Rolling friction 71
Chapter VII. Spatial system of forces 72
§28. Moment of force about the axis. Principal vector calculation
and the main moment of the system of forces 72
§ 29*. Reduction of the spatial system of forces to the simplest form 77
§thirty. Equilibrium of an arbitrary spatial system of forces. The case of parallel forces
Chapter VIII. Center of gravity 86
§31. Center of Parallel Forces 86
§ 32. Force field. Center of gravity of a rigid body 88
§ 33. Coordinates of the centers of gravity of homogeneous bodies 89
§ 34. Methods for determining the coordinates of the centers of gravity of bodies. 90
§ 35. Centers of gravity of some homogeneous bodies 93
SECTION TWO KINEMATICS OF A POINT AND A RIGID BODY
Chapter IX. Point kinematics 95
§ 36. Introduction to kinematics 95
§ 37. Methods for specifying the movement of a point. . 96
§38. Point velocity vector,. 99
§ 39
§40. Determining the speed and acceleration of a point with the coordinate method of specifying movement 102
§41. Solving problems of point kinematics 103
§ 42. Axes of a natural trihedron. Numerical speed value 107
§ 43. Tangent and normal acceleration of a point 108
§44. Some special cases of motion of a point in software
§45. Graphs of movement, speed and acceleration of point 112
§ 46. Problem solving< 114
§47*. Velocity and acceleration of a point in polar coordinates 116
Chapter X. Translational and rotational motions of a rigid body. . 117
§48. Translational movement 117
§ 49. Rotational motion of a rigid body around an axis. Angular Velocity and Angular Acceleration 119
§fifty. Uniform and uniform rotation 121
§51. Velocities and accelerations of points of a rotating body 122
Chapter XI. Plane-parallel motion of a rigid body 127
§52. Equations of plane-parallel motion (motion of a plane figure). Decomposition of motion into translational and rotational 127
§53*. Determination of trajectories of points of a plane figure 129
§54. Determining the velocities of points on a plane figure 130
§ 55. The theorem on the projections of the velocities of two points of the body 131
§ 56. Determination of the velocities of points of a plane figure using the instantaneous center of velocities. The concept of centroids 132
§57. Problem solving 136
§58*. Determination of accelerations of points of a plane figure 140
§59*. Instant center of acceleration "*"*
Chapter XII*. Motion of a rigid body around a fixed point and motion of a free rigid body 147
§ 60. Motion of a rigid body having one fixed point. 147
§61. Kinematic Euler equations 149
§62. Speeds and accelerations of body points 150
§ 63. General case of motion of a free rigid body 153
Chapter XIII. Complex point movement 155
§ 64. Relative, figurative and absolute motions 155
§ 65, Velocity addition theorem » 156
§66. The theorem on the addition of accelerations (Coriols' theorem) 160
§67. Problem solving 16*
Chapter XIV*. Complex motion of a rigid body 169
§68. The addition of translational movements 169
§69. Addition of rotations about two parallel axes 169
§70. Cylindrical gears 172
§ 71. Addition of rotations around intersecting axes 174
§72. Addition of translational and rotational movements. Screw movement 176
SECTION THREE DYNAMICS OF A POINT
Chapter XV: Introduction to dynamics. Laws of dynamics 180
§ 73. Basic concepts and definitions 180
§ 74. Laws of dynamics. Problems of the dynamics of a material point 181
§ 75. Systems of units 183
§76. Basic types of forces 184
Chapter XVI. Differential equations of motion of a point. Solving problems of point dynamics 186
§ 77. Differential equations, motions of a material point No. 6
§ 78. Solution of the first problem of dynamics (determination of forces from a given motion) 187
§ 79. Solution of the main problem of dynamics in the rectilinear motion of a point 189
§ 80. Examples of problem solving 191
§81*. Fall of a body in a resisting medium (in air) 196
§82. Solution of the main problem of dynamics, with curvilinear motion of a point 197
Chapter XVII. General theorems of point dynamics 201
§83. The amount of movement of the point. Force Impulse 201
§ S4. Theorem on the change in the momentum of a point 202
§ 85. The theorem on the change in the angular momentum of a point (theorem of moments) "204
§86*. Movement under the action of a central force. Law of areas.. 266
§ 8-7. Force work. Power 208
§88. Work Calculation Examples 210
§89. Theorem on the change in the kinetic energy of a point. ". . . 213J
Chapter XVIII. Non-free and relative motion of a point 219
§90. Non-free movement of a point. 219
§91. Relative movement of a point 223
§ 92. Influence of the Earth's rotation on the balance and motion of bodies... 227
Section 93*. Deviation of the incident point from the vertical due to the rotation of the Earth "230
Chapter XIX. Rectilinear fluctuations of a point. . . 232
§ 94. Free vibrations without taking into account the forces of resistance 232
§ 95. Free oscillations with viscous resistance (damped oscillations) 238
§96. Forced vibrations. Resonance 241
Chapter XX*. Motion of a body in the field of gravity 250
§ 97. Movement of a thrown body in the Earth's gravitational field "250
§98. Artificial satellites of the Earth. Elliptical trajectories. 254
§ 99. The concept of weightlessness. "Local reference systems 257
SECTION FOUR DYNAMICS OF A SYSTEM AND A RIGID BODY
G i a v a XXI. Introduction to system dynamics. moments of inertia. 263
§ 100. Mechanical system. Forces external and internal 263
§ 101. Mass of the system. Center of gravity 264
§ 102. Moment of inertia of a body about an axis. Radius of inertia. . 265
$ 103. Moments of inertia of a body about parallel axes. Huygens' theorem 268
§ 104*. centrifugal moments of inertia. Concepts about the main axes of inertia of the body 269
$105*. Moment of inertia of a body about an arbitrary axis. 271
Chapter XXII. The theorem on the motion of the center of mass of the system 273
$ 106. Differential equations of system motion 273
§ 107. The theorem on the motion of the center of mass 274
$ 108. Law of conservation of motion of the center of mass 276
§ 109. Problem solving 277
Chapter XXIII. Theorem on the change in the quantity of a movable system. . 280
$ BUT. Number of movement system 280
§111. Theorem on change of momentum 281
§ 112. Law of conservation of momentum 282
$113*. Application of the theorem to the motion of a liquid (gas) 284
§ 114*. Body of variable mass. Rocket movement 287
Gdawa XXIV. The theorem on the change in the moment of momentum of the system 290
§ 115. The main moment of the quantities of motion of the system 290
$ 116. Theorem on the change of the main moment of the momentum of the system (theorem of moments) 292
$117. The law of conservation of the main moment of momentum. . 294
$ 118. Problem solving 295
$119*. Application of the moment theorem to the motion of a liquid (gas) 298
§ 120. Equilibrium conditions for a mechanical system 300
Chapter XXV. Theorem on the change in the kinetic energy of the system. . 301.
§ 121. Kinetic energy of the system 301
$122. Some cases of calculating work 305
$ 123. Theorem on the change in the kinetic energy of the system 307
$ 124. Problem solving 310
$125*. Mixed tasks "314
$ 126. Potential force field and force function 317
$127, Potential Energy. Law of conservation of mechanical energy 320
Chapter XXVI. "Application of General Theorems to the Dynamics of a Rigid Body 323
$12&. Rotational motion of a rigid body around a fixed axis ". 323"
$ 129. Physical pendulum. Experimental determination of moments of inertia. 326
$130. Plane-parallel motion of a rigid body 328
$131*. Elementary theory of the gyroscope 334
$132*. Motion of a rigid body around a fixed point and motion of a free rigid body 340
Chapter XXVII. d'Alembert principle 344
$ 133. d'Alembert's principle for a point and a mechanical system. . 344
$ 134. Principal vector and principal moment of inertia forces 346
$ 135. Problem solving 348
$136*, Didemic reactions acting on the axis of a rotating body. Balancing of rotating bodies 352
Chapter XXVIII. The principle of possible displacements and the general equation of dynamics 357
§ 137. Classification of connections 357
§ 138. Possible displacements of the system. Number of degrees of freedom. . 358
§ 139. The principle of possible movements 360
§ 140. Solving problems 362
§ 141. General equation of dynamics 367
Chapter XXIX. Equilibrium conditions and equations of motion of the system in generalized coordinates 369
§ 142. Generalized coordinates and generalized velocities. . . 369
§ 143. Generalized forces 371
§ 144. Equilibrium conditions for a system in generalized coordinates 375
§ 145. Lagrange's equations 376
§ 146. Solving problems 379
Chapter XXX*. Small oscillations of the system around the position of stable equilibrium 387
§ 147. The concept of equilibrium stability 387
§ 148. Small free vibrations of a system with one degree of freedom 389
§ 149. Small damped and forced oscillations of a system with one degree of freedom 392
§ 150. Small summary oscillations of a system with two degrees of freedom 394
Chapter XXXI. Elementary Impact Theory 396
§ 151. Basic equation of the theory of impact 396
§ 152. General theorems of the theory of impact 397
§ 153. Impact recovery factor 399
§ 154. Impact of the body on a fixed barrier 400
§ 155. Direct central impact of two bodies (impact of balls) 401
§ 156. Loss of kinetic energy during an inelastic impact of two bodies. Carnot's theorem 403
§ 157*. A blow to a rotating body. Impact Center 405
Index 409

Content

Kinematics

Kinematics of a material point

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 velocity, full, 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 \u003d 12 cm, R 3 \u003d 16 cm; s 5 \u003d t 3 - 6t (cm).

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

Kinematic analysis of a flat mechanism

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

The flat mechanism consists of rods 1, 2, 3, 4 and slider 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 of link AB; positions of instantaneous centers of speeds P 2 and P 3 of links 2 and 3 of the mechanism.

Determining the absolute speed and absolute acceleration of a point

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

The point M moves along the straight line BD along 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 the position where 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, the load is affected by a constant force T (its direction is shown in the figure) and the force R of the resistance of the medium (the module of this force is R = μV 2, the vector R is directed opposite to the velocity V of the load).

The load, having completed its movement in section AB, at point B of the pipe, without changing the value of its velocity modulus, passes to section BC. On the 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 motion on the section BC, i.e. x = f(t), where x = BD. Ignore the friction of the load on the pipe.

Download solution

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 pulleys; sections of threads are parallel to the corresponding planes. The roller (solid homogeneous cylinder) rolls along the reference plane without slipping. The radii of the steps of 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 supporting planes of weights 1 and 2 are rough, the coefficient of sliding friction for each weight is f = 0.1.

Under the action of 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, 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 velocity of pulley 4 at the moment when the displacement s of the point of application of force F becomes equal to s 1 = 1.2 m.

Download solution

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

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

Download solution

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

The vertical shaft AK, rotating uniformly with an angular velocity ω = 10 s -1 , is fixed with 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 of mass 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.