Dynamics theoretical mechanics theory. Basics of Mechanics for Dummies
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 solid), classical problems of dynamics of mechanical systems and dynamics of a rigid body, elements of celestial mechanics, motion of systems of variable composition, 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 motion solid
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 of 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 material point 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
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 address the appropriate specific tasks;
- 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;
- skills in the practical use of methods and principles of theoretical mechanics in solving problems: force calculation, determination of the kinematic characteristics of bodies at different ways assignments of motion, determination of the law of motion of material bodies and mechanical systems under the influence of forces;
- skills to independently master new information in the process of production and scientific activities using modern educational and information technologies;
General theorems of the dynamics of a system of bodies. Theorems about the motion of the center of mass, about changing the momentum, about changing the main moment of momentum, about changing the kinetic energy. D'Alembert's principles and possible displacements. General equation speakers. Lagrange equations.
ContentThe work that power does, is equal to the scalar product of the force vectors and the infinitesimal displacement of the point of its application:
,
that is, the product of the absolute values of the vectors F and ds by the cosine of the angle between them.
The work that the moment of forces does, is equal to the scalar product of the vectors of the moment and the infinitesimal angle of rotation:
.
D'Alembert principle
The essence of the d'Alembert principle is to reduce the problems of dynamics to problems of statics. For this, it is assumed (or it is known in advance) that the bodies of the system have certain (angular) accelerations. Next, inertial forces and (or) moments of inertia forces are introduced, which are equal in magnitude and opposite in direction to forces and moments of forces, which, according to the laws of mechanics, would create specified accelerations or angular accelerations
Let's look at an example. On the way, the body makes a forward motion and external forces act on it. Further, we assume that these forces create the acceleration of the center of mass of the system. According to the theorem on the motion of the center of mass, the center of mass of a body would have the same acceleration if a force acted on the body. Next, we introduce the force of inertia:
.
After that, the dynamics problem:
.
;
.
For rotary motion, proceed in the same way. Let the body rotate around the z-axis and the external moments of forces M e zk act on it. We assume that these moments create an angular acceleration ε z. Next, we introduce the moment of inertia forces M И = - J z ε z. After that, the dynamics problem:
.
Turns into a statics task:
;
.
The principle of possible displacements
The principle of possible displacements is used to solve static problems. In some problems, it gives a shorter solution than the equation of equilibrium. This is especially true for systems with constraints (for example, systems of bodies connected by threads and blocks), consisting of many bodies
The principle of possible displacements.
For balance mechanical system with perfect connections necessary and sufficient for the amount elementary work of all active forces acting on it for any possible displacement of the system was equal to zero.
Possible movement of the system- this is a small displacement, which does not break the connections imposed on the system.
Perfect connections- these are connections that do not perform work when the system is moved. More precisely, the amount of work performed by the links themselves when the system moves is equal to zero.
General equation of dynamics (d'Alembert - Lagrange principle)
The d'Alembert-Lagrange principle is a combination of the d'Alembert principle with the principle of possible displacements. That is, when solving the problem of dynamics, we introduce the forces of inertia and reduce the problem to the problem of statics, which we solve using the principle of possible displacements.
D'Alembert - Lagrange principle.
When a mechanical system with ideal constraints moves at each moment of time, the sum of the elementary work of all applied active forces and all inertial forces on any possible displacement of the system is equal to zero:
.
This equation is called general equation of dynamics.
Lagrange Equations
Generalized coordinates q 1, q 2, ..., q n is a collection of n values that uniquely determine the position of the system.
The number of generalized coordinates n coincides with the number of degrees of freedom of the system.
Generalized speeds are derivatives of generalized coordinates with respect to time t.
Generalized forces Q 1, Q 2, ..., Q n
.
Consider a possible movement of the system, in which the q k coordinate will receive a movement δq k. The rest of the coordinates remain unchanged. Let δA k be the work performed by external forces during such a displacement. Then
δA k = Q k δq k, or
.
If, with a possible movement of the system, all coordinates change, then the work performed by external forces during such a movement has the form:
δA = Q 1 δq 1 + Q 2 δq 2 + ... + Q n δq n.
Then the generalized forces are partial derivatives of the work on displacements:
.
For potential forces
with potential Π,
.
Lagrange Equations are the equations of motion of a mechanical system in generalized coordinates:
Here T is kinetic energy. It is a function of generalized coordinates, velocities, and possibly time. Therefore, its partial derivative is also a function of generalized coordinates, velocities and time. Further, you need to take into account that coordinates and speeds are functions of time. Therefore, to find the total time derivative, it is necessary to apply the differentiation rule complex function:
.
References:
S. M. Targ, A short course in theoretical mechanics, " graduate School", 2010.
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.
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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. 
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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. 
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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.
The state of equilibrium, in statics, is understood as a state in which all parts of a mechanical system are at rest relative to some inertial coordinate system. One of the basic objects of statics are forces and their points of application.
The force acting on a material point with a radius vector from other points is a measure of the influence of other points on the point under consideration, as a result of which it receives acceleration relative to the inertial frame of reference. The magnitude strength determined by the formula:
,
where m is the mass of a point - a value that depends on the properties of the point itself. This formula is called Newton's second law.
Application of statics in dynamics
An important feature of the equations of motion of an absolutely rigid body is that forces can be transformed into equivalent systems. With such a transformation, the equations of motion retain their form, but the system of forces acting on the body can be transformed into a simpler system. Thus, the point of application of the force can be moved along the line of its action; forces can be laid out according to the parallelogram rule; forces applied at one point can be replaced by their geometric sum.
An example of such transformations is the force of gravity. It acts on all points of a rigid body. But the law of motion of the body will not change if the force of gravity distributed over all points is replaced by one vector applied at the center of mass of the body.
It turns out that if we add to the main system of forces acting on the body an equivalent system in which the directions of the forces are reversed, then the body, under the action of these systems, will be in equilibrium. Thus, the problem of determining the equivalent systems of forces is reduced to the problem of equilibrium, that is, to the problem of statics.
The main task of statics is the establishment of the laws of transformation of a system of forces into equivalent systems. Thus, the methods of statics are used not only in the study of bodies in equilibrium, but also in the dynamics of a rigid body, in the transformation of forces into simpler equivalent systems.
Material point statics
Consider a material point that is in equilibrium. And let n forces act on it, k = 1, 2, ..., n.
If a material point is in equilibrium, then the vector sum of the forces acting on it is equal to zero:
(1)
.
In balance geometric sum forces acting on a point is equal to zero.
Geometric interpretation... If the beginning of the second vector is placed at the end of the first vector, and the beginning of the third is placed at the end of the second vector, and then this process is continued, then the end of the last, n -th vector will be aligned with the beginning of the first vector. That is, we get a closed geometric figure, the lengths of the sides of which are equal to the moduli of the vectors. If all vectors lie in the same plane, then we get a closed polygon.
It is often convenient to choose rectangular coordinate system Oxyz. Then the sums of the projections of all force vectors on the coordinate axis are equal to zero:
If you choose any direction given by some vector, then the sum of the projections of the force vectors on this direction is equal to zero:
.
Let us multiply equation (1) scalarly by a vector:
.
Here is the scalar product of vectors and.
Note that the projection of the vector onto the direction of the vector is determined by the formula:
.
Rigid body statics
Moment of force relative to a point
Determination of the moment of force
A moment of power applied to the body at point A, relative to the fixed center O, is called a vector equal to the vector product of vectors and:(2) .
Geometric interpretation
The moment of force is equal to the product of the force F by the shoulder OH.
Let the vectors and be located in the plane of the drawing. According to the property of the vector product, the vector is perpendicular to the vectors and, that is, perpendicular to the plane of the drawing. Its direction is determined by the right screw rule. In the figure, the moment vector is directed at us. Absolute torque value:
.
Since, then
(3)
.
Using geometry, you can give a different interpretation of the moment of force. To do this, draw a straight line AH through the force vector. From the center O we drop the perpendicular OH to this line. The length of this perpendicular is called shoulder of strength... Then
(4)
.
Since, then formulas (3) and (4) are equivalent.
Thus, absolute value of the moment of force with respect to the center O equals force per shoulder this force relative to the selected center O.
When calculating the moment, it is often convenient to decompose the force into two components:
,
where . The force passes through point O. Therefore, its moment is zero. Then
.
Absolute torque value:
.
Moment components in a rectangular coordinate system
If we choose a rectangular coordinate system Oxyz centered at point O, then the moment of force will have the following components:
(5.1)
;
(5.2)
;
(5.3)
.
Here are the coordinates of point A in the selected coordinate system:
.
The components represent the values of the moment of force about the axes, respectively.
Properties of the moment of force relative to the center
The moment about the center O, from the force passing through this center, is equal to zero.
If the point of application of the force is moved along a line passing through the force vector, then the moment will not change with this movement.
The moment from the vector sum of the forces applied to one point of the body is equal to the vector sum of the moments from each of the forces applied to the same point:
.
The same applies to forces whose continuation lines intersect at one point.
If the vector sum of the forces is zero:
,
then the sum of the moments of these forces does not depend on the position of the center, relative to which the moments are calculated:
.
A couple of forces
A couple of forces- these are two forces, equal in absolute value and having opposite directions, applied to different points of the body.
A pair of forces is characterized by the moment they create. Since the vector sum of the forces included in the pair is equal to zero, the moment created by the pair does not depend on the point relative to which the moment is calculated. From the point of view of static balance, the nature of the forces included in the pair is irrelevant. A pair of forces is used to indicate that a moment of forces is acting on the body, which has a certain value.
Moment of force about a given axis
There are often cases when we do not need to know all the components of the moment of force relative to a selected point, but only need to know the moment of force relative to the selected axis.
The moment of force about the axis passing through the point O is the projection of the vector of the moment of force, relative to the point O, onto the direction of the axis.
The properties of the moment of force about the axis
The moment about the axis from the force passing through this axis is equal to zero.
The moment about an axis from a force parallel to this axis is zero.
Calculation of the moment of force about the axis
Let a force act on the body at point A. Let's find the moment of this force about the O'O '' axis.
Let's build a rectangular coordinate system. Let the Oz axis coincide with O′O ′ ′. From point A we drop the perpendicular OH to O′O ′ ′. Draw the Ox axis through points O and A. Draw the Oy axis perpendicular to Ox and Oz. Let us decompose the force into components along the axes of the coordinate system:
.
The force crosses the O′O ′ ′ axis. Therefore, its moment is zero. The force is parallel to the O'O '' axis. Therefore, its moment is also zero. By formula (5.3) we find:
.
Note that the component is directed tangentially to the circle whose center is point O. The direction of the vector is determined by the right screw rule.
Equilibrium conditions for a rigid body
In equilibrium, the vector sum of all forces acting on the body is zero and the vector sum of the moments of these forces relative to an arbitrary stationary center is zero:
(6.1)
;
(6.2)
.
We emphasize that the center O, relative to which the moments of forces are calculated, can be chosen arbitrarily. Point O can either belong to the body or be outside it. Usually the center O is chosen to make the calculations simpler.
Equilibrium conditions can be formulated in another way.
In equilibrium, the sum of the projections of forces on any direction given by an arbitrary vector is equal to zero:
.
The sum of the moments of forces about an arbitrary axis O′O ′ ′ is also equal to zero:
.
Sometimes these conditions are more convenient. There are times when, by choosing the axes, you can make the calculations simpler.
Body center of gravity
Let's consider one of the most important forces - the force of gravity. Here forces are not applied at certain points of the body, but are continuously distributed over its volume. For each part of the body with an infinitely small volume Δ V, the force of gravity acts. Here ρ is the density of the substance of the body, is the acceleration of gravity.
Let be the mass of an infinitesimal part of the body. And let the point A k determine the position of this section. Let us find the quantities related to the force of gravity, which are included in the equilibrium equations (6).
Let's find the sum of the forces of gravity formed by all parts of the body:
,
where is the body weight. Thus, the sum of the gravity forces of individual infinitesimal parts of the body can be replaced by one vector of the gravity of the whole body:
.
Let us find the sum of the moments of gravity, relative to the chosen center O in an arbitrary way:
.
Here we have introduced point C which is called center of gravity body. The position of the center of gravity, in a coordinate system centered at point O, is determined by the formula:
(7)
.
So, when determining static balance, the sum of the forces of gravity of individual parts of the body can be replaced by the resultant
,
applied to the center of mass of the body C, the position of which is determined by formula (7).
Center of gravity position for different geometric shapes can be found in the respective reference books. If the body has an axis or plane of symmetry, then the center of gravity is located on this axis or plane. So, the centers of gravity of a sphere, circle or circle are in the centers of the circles of these figures. Centers of gravity rectangular parallelepiped, rectangle or square are also located in their centers - at the intersection points of the diagonals.
Uniformly (A) and linearly (B) distributed load.
There are also cases similar to gravity when forces are not applied at certain points of the body, but are continuously distributed over its surface or volume. Such forces are called distributed forces or .
(Figure A). Also, as in the case of gravity, it can be replaced by the resultant force of the quantity applied at the center of gravity of the plot. Since the diagram in Figure A is a rectangle, the center of gravity of the diagram is at its center - point C: | AC | = | CB |. (Figure B). It can also be replaced with a resultant. The value of the resultant is equal to the area of the diagram:.
The application point is at the center of gravity of the plot. The center of gravity of a triangle with height h is at a distance from the base. That's why .
Friction forces
Sliding friction... Let the body be on a flat surface. And let be the force perpendicular to the surface from which the surface acts on the body (pressure force). Then the sliding friction force is parallel to the surface and directed to the side, preventing the movement of the body. Its largest value is equal to:
,
where f is the coefficient of friction. The friction coefficient is dimensionless.
Rolling friction... Let the rounded body roll or can roll on the surface. And let be the pressure force perpendicular to the surface from which the surface acts on the body. Then a moment of friction forces acts on the body, at the point of contact with the surface, which prevents the body from moving. Greatest value the moment of friction is equal to:
,
where δ is the rolling friction coefficient. It has the dimension of length.
References:
S. M. Targ, A short course in theoretical mechanics, "High School", 2010.
20th ed. - M .: 2010.- 416 p.
The book describes the basics of the mechanics of a material point, a system of material points and a rigid body in the volume corresponding to the programs of technical universities. Many examples and problems are given, the solutions of which are accompanied by appropriate guidelines... For students of full-time and part-time technical universities.
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TABLE OF CONTENTS
Preface to Thirteenth Edition 3
Introduction 5
SECTION ONE SOLID STATIC
Chapter I. Basic concepts background provisions of Articles 9
41. Absolutely solid; force. Statics problems 9
12. Initial positions of statics "11
$ 3. Bonds and their reactions 15
Chapter II. The addition of forces. System of converging forces 18
§4. Geometrically! The way of adding forces. Resultant of converging forces, decomposition of forces 18
f 5. Projections of the force on the axis and on the plane, Analytical method of 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 relative to the center. A pair of forces 31
i 8. Moment of force relative to 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. Theorem on parallel force transfer 37
112. Bringing the system of forces to this center -. , 38
§ 13. Conditions of equilibrium of the system of forces. The resultant moment theorem 40
Chapter V. Flat system of forces 41
§ 14. Algebraic moments of force and pairs 41
115. Bringing a flat system of forces to the simplest form .... 44
§ 16. Equilibrium of a plane system of forces. The case of parallel forces. 46
§ 17. Solving problems 48
118. Equilibrium of systems of bodies 63
§ 19*. Statically definable and statically indeterminate systems of bodies (structures) 56 "
f 20 *. Definition of internal efforts. 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 *. Friction of a thread on a cylindrical surface 69
1 27 *. Rolling friction 71
Chapter VII. Spatial Force System 72
§28. The moment of force about the axis. Calculation of the principal vector
and the main moment of the system of forces 72
§ 29 *. Reducing the spatial system of forces to the simplest form 77
§thirty. Equilibrium of an arbitrary spatial system of forces. Parallel Forces Case
Chapter VIII. Center of gravity 86
§31. Center of Parallel Forces 86
§ 32. Force field. Center of gravity of 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 THE POINT AND SOLID BODY
Chapter IX. Point kinematics 95
§ 36. Introduction to kinematics 95
§ 37. Methods of specifying the movement of a point. ... 96
§38. Point velocity vector ,. 99
§ 39. The vector of "cutting point 100
§40. Determination of the speed and acceleration of a point in the coordinate method of specifying movement 102
§41. Solving the problems of kinematics point 103
§ 42. Axes of the natural trihedron. Numerical value of speed 107
§ 43. Tangent and normal acceleration of point 108
§44. Some special cases of movement of the PO point
§45. Graphs of movement, speed and acceleration of point 112
§ 46. Problem solving< 114
§47 *. Point velocity and acceleration in polar coordinates 116
Chapter X. Translational and rotational motion of a rigid body. ... 117
§48. Translational motion 117
Section 49. Rotational motion a rigid body around an axis. Angular velocity and angular acceleration 119
§50. Uniform and equal rotation 121
§51. Velocities and accelerations of points of a rotating body 122
Chapter XI. Plane-parallel movement 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 *. Defining the trajectories of points of a flat figure 129
§54. Determination of the velocities of points of a flat figure 130
§ 55. A theorem on the projections of the velocities of two points of a body 131
§ 56. Determination of the velocities of the points of a flat figure using the instantaneous center of velocities. Understanding centroids 132
§57. Problem Solving 136
§58 *. Determining the acceleration of points of a flat figure 140
§59 *. Instant acceleration center "*" *
Chapter XII *. The movement of a rigid body around a fixed point and the movement of a free rigid body 147
§ 60. Motion of a rigid body having one fixed point. 147
§61. Euler's Kinematic Equations 149
§62. Velocities and accelerations of body points 150
§ 63. The general case of motion of a free rigid body 153
Chapter XIII. Difficult Point Movement 155
§ 64. Relative, figurative and absolute motion 155
§ 65, The theorem on the addition of velocities "156
§66. The theorem on the addition of accelerations (Coriolns' theorem) 160
§67. Problem solving 16 *
Chapter XIV *. Complex motion of a rigid body 169
§68. Addition of translational movements 169
§69. Adding Rotations Around Two Parallel Axes 169
§70. Spur gears 172
§ 71. Addition of rotations around intersecting axes 174
§72. Addition of translational and rotational movements. Screw movement 176
SECTION THREE POINT DYNAMICS
Chapter XV: Introduction to Dynamics. The laws of dynamics 180
§ 73. Basic concepts and definitions 180
§ 74. The laws of dynamics. Problems of the dynamics of a material point 181
Section 75. Systems of units 183
§76. Basic forces 184
Chapter XVI. Differential Equations point movement. Solving the problems of the dynamics of a point 186
§ 77. Differential equations, motion of a material point No. 6
§ 78. Solution of the first problem of dynamics (determination of forces for a given motion) 187
§ 79. Solution of the main problem of dynamics for straight motion points 189
§ 80. Examples of solving problems 191
§81 *. The fall of the body in a resisting environment (in the 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 point movement. Force Impulse 201
§ S4. The theorem on the change in the momentum of a point 202
§ 85. A theorem on the change in the angular momentum of a point (the theorem of moments) "204
§86 *. Movement under the influence of a central force. The Law of Areas .. 266
§ 8-7. Work of strength. Power 208
§88. Examples of calculating work 210
§89. Theorem on the change in the kinetic energy of a point. "... 213J
Chapter XVIII. Not free and relative to the movement of a point 219
§90. Non-free movement of a point. 219
§91. Relative point movement 223
§ 92. Influence of the Earth's rotation on the balance and motion of bodies ... 227
§ 93 *. Deviation of the falling point from the vertical due to the rotation of the Earth "230
Chapter XIX. Rectilinear point vibrations. ... ... 232
§ 94. Free vibrations without taking into account the forces of resistance 232
§ 95. Free vibrations with viscous resistance (damped vibrations) 238
§96. Forced vibrations. Rezonayas 241
Chapter XX *. Body movement in a gravitational field 250
§ 97. The motion of a thrown body in the gravitational field of the Earth "250
§98. Artificial satellites Earth. Elliptical trajectories. 254
§ 99. The concept of weightlessness. "Local frames of reference 257
SECTION FOUR SYSTEM AND SOLID BODY DYNAMICS
Chapter XXI. Introduction to system dynamics. Moments of inertia. 263
§ 100. Mechanical system. External forces and internal forces 263
§ 101. The mass of the system. Center of gravity 264
§ 102. Moment of inertia of a body about an axis. Radius of gyration. ... 265
$ 103. Moments of inertia of a body relative to parallel axes. Huygens' theorem 268
§ 104 *. Centrifugal moments of inertia. Concepts about the main axes of inertia of a body 269
$ 105 *. The moment of inertia of a body about an arbitrary axis. 271
Chapter XXII. The theorem on the motion of the center of mass of a system 273
$ 106. Differential equations of motion of the system 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. The theorem on the change in the number of movable systems. ... 280
$ BUT. System movement amount 280
§111. Momentum Change Theorem 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
Gdava XXIV. Theorem on the change in the moment of quantities of motion of the system 290
§ 115. The main moment of the quantities of motion of the system 290
$ 116. The theorem on the change of the main moment of the quantities of motion of the system (the theorem of moments) 292
$ 117. The law of conservation of the main moment of the quantities of motion. ... 294
$ 118. Problem solving 295
$ 119 *. Application of the theorem of moments to the motion of a liquid (gas) 298
§ 120. Conditions of equilibrium of 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. The theorem on the change in the kinetic energy of the system 307
$ 124. Problem solving 310
$ 125 *. Mixed problems "314
$ 126. Potential Force Field and Force Function 317
$ 127, Potential Energy. Mechanical energy conservation law 320
Chapter XXVI. "Application of general theorems to rigid body dynamics 323
$ 12 &. Rotational motion of a rigid body about a fixed axis ". 323"
$ 129. Physical pendulum. Experimental determination of the moments of inertia. 326
$ 130. Plane-parallel motion of a rigid body 328
$ 131*. Elementary theory gyro 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 principle for a point and a mechanical system. ... 344
$ 134. The main vector and main point forces of inertia 346
$ 135. Problem solving 348
$ 136 *, Didemic reactions acting on the axis of a rotating body. Balancing non-rotating bodies 352
Chapter XXVIII. The principle of possible displacements and the general equation of dynamics 357
§ 137. Classification of ties 357
§ 138. Possible movements of the system. The number of degrees of freedom. ... 358
Section 139. The principle of possible movements 360
§ 140. Problem solving 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
Section 143. Generalized forces 371
§ 144. Equilibrium conditions of the system in generalized coordinates 375
§ 145. Lagrange's equations 376
§ 146. Problem solving 379
Chapter XXX *. Small oscillations of the system about a stable equilibrium position 387
§ 147. The concept of stability of equilibrium 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 combined oscillations of a system with two degrees of freedom 394
Chapter XXXI. Elementary Impact Theory 396
§ 151. The basic equation of the theory of impact 396
§ 152. General theorems of the theory of impact 397
§ 153. Coefficient of recovery on impact 399
§ 154. Impact of the body against a fixed obstacle 400
§ 155. Direct central blow of two bodies (blow of balls) 401
§ 156. Loss of kinetic energy during inelastic collision of two bodies. Carnot's theorem 403
§ 157 *. A blow to a rotating body. Impact center 405
Index 409
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