D'Alembert's principle allows us to reduce the process of composing dynamic equations to composing static equations.

This principle, which we will present here for a free material point and for a point moving along a surface or along a curve, is applicable to any problem in dynamics. It will allow us to summarize the entire theory of point motion.

Let's consider a material point M of mass that is under the influence of forces, the resultant of which has projections. The equations of motion of this point can be written as follows:

We will consider, along with vectors representing forces applied to point M, a vector with projections - This vector, numerically equal to the product of mass and acceleration and directed opposite to acceleration, is called the force of inertia, although this will in no way be a force applied to the point. The equations then express that the geometric sum of the vectors and is equal to zero, or that at each moment of time there is an equilibrium between the force of inertia and the forces actually applied to the point.

Derivation of the equations of motion from d'Alembert's principle. Based on what has just been said, to find the equations of motion of a point under any conditions, it is enough to express that there is an equilibrium between all the forces applied to the point and the force of inertia. But this can be done using static methods. You can, for example, apply the work opportunity theorem. To do this, it is necessary to distinguish among the forces applied to a point, the forces specified and the reactions of the connections. Let us denote the projections of given forces.

To write that there is an equilibrium between the forces acting on a point and the force of inertia, it is enough to write that at

all possible movements allowed by the connections existing at the moment the sum of the work of the given forces and the inertia force is equal to zero:

Three cases should be distinguished:

1°. Free point. arbitrary. If, as in paragraph 282, an arbitrary coordinate system is used, then, replacing with variations, we get:

where are arbitrary.

Substituting into equality (2) and equating the result to zero for arbitrary, we obtain the equations of motion in the form indicated in paragraph 282, from which we derived the Lagrange equations for a free point.

2°. A point on the surface. Let

is the equation of a surface, which for generality is assumed to be moving. By giving a variable a specific value, we see that we must satisfy the condition

expressing that possible movement is allowed by the connection existing at the moment. If, as in paragraph 263, we express the coordinates of a surface point in functions of two parameters, then we obtain

and relation (2) must take place, no matter what they are. In this way, the equations of motion will be obtained in the form (4) of paragraph 263. 3°. A point on a curve. Let

d'Alembert's principle

The main work of Zh.L. d'Alembert(1717-1783) - "Treatise on Dynamics" - was published in 1743

The first part of the treatise is devoted to the construction of analytical statics. Here d’Alembert formulates the “fundamental principles of mechanics,” including the “principle of inertia,” the “principle of adding motion,” and the “principle of equilibrium.”

The “principle of inertia” is formulated separately for the case of rest and for the case of uniform rectilinear motion. “The force of inertia,” d’Alembert writes, “I, together with Newton, call the property of a body to preserve the state in which it is.”

The “principle of adding motion” is the law of adding velocities and forces according to the parallelogram rule. Based on this principle, d'Alembert solves statics problems.

The “principle of equilibrium” is formulated in the form of the following theorem: “If two bodies moving at speeds inversely proportional to their masses have opposite directions, so that one body cannot move without shifting the other body from place to place, then these bodies will be in a state of equilibrium ". In the second part of the Treatise, d'Alembert proposed a general method for composing differential equations of motion for any material systems, based on reducing the problem of dynamics to statics. He formulated a rule for any system of material points, later called “D’Alembert’s principle,” according to which the forces applied to the points of the system can be decomposed into “active” ones, that is, those that cause acceleration of the system, and “lost” ones, necessary for the equilibrium of the system. D'Alembert believes that the forces that correspond to the "lost" acceleration form a set that does not in any way affect the actual behavior of the system. In other words, if only the totality of “lost” forces is applied to the system, then the system will remain at rest. The modern formulation of d’Alembert’s principle was given by M. E. Zhukovsky in his “Course of Theoretical Mechanics”: “If at any moment in time you stop a system that is moving, and add to it, in addition to its driving forces, all the forces of inertia corresponding to a given moment in time, then equilibrium will be observed, and all the forces of pressure, tension, etc. developing between parts of the system at such equilibrium will be real forces of pressure, tension, etc. during the movement of the system at the considered moment in time." It should be noted that d'Alembert himself, when presenting his principle, did not resort to either the concept of force (considering that it was not clear enough to be included in the list of basic concepts of mechanics), much less to the concept of inertial force. The presentation of d'Alembert's principle using the term "force" belongs to Lagrange, who in his "Analytical Mechanics" gave its analytical expression in the form of the principle of possible displacements. It was Joseph Louis Lagrange (1736-1813) and especially Leonardo Euler (1707-1783) who played a significant role in the final transformation of mechanics into analytical mechanics.

Analytical mechanics of a material point and Euler rigid body dynamics

Leonardo Euler- one of the outstanding scientists who made a great contribution to the development of physical and mathematical sciences in the 18th century. His work amazes with the insight of his research thought, the versatility of his talent and the enormous amount of scientific heritage he left behind.

Already in the first years of scientific activity in St. Petersburg (Euler arrived in Russia in 1727), he drew up a program for a grandiose and comprehensive cycle of work in the field of mechanics. This application is found in his two-volume work “Mechanics or the Science of Motion, Explained Analytically” (1736). Euler's Mechanics was the first systematic course in Newtonian mechanics. It contained the fundamentals of the dynamics of a point - by mechanics Euler understood the science of motion, in contrast to the science of the balance of forces, or statics. The defining feature of Euler's Mechanics was the widespread use of a new mathematical apparatus - differential integral calculus. Briefly describing the main works on mechanics that appeared at the turn of the 17th-18th centuries, Euler noted the son-thetic-geometric style of their writing, which created a lot of work for readers. It is in this manner that Newton’s “Principia” and the later “Phoronomy” (1716) by J. Herman were written. Euler points out that the works of Hermann and Newton were presented “according to the custom of the ancients with the help of synthetic geometric proofs” without the use of analysis, “only through which can a complete understanding of these things be achieved.”

The synthetic-geometric method did not have a generalizing nature, but, as a rule, required individual constructions regarding each problem separately. Euler admits that after studying “Phoronomy” and “Principia”, it seemed to him “he quite clearly understood the solutions to many problems, but problems that to some extent deviated from them, he could no longer solve.” Then he tried to “isolate the analysis of this synthetic method and carry out the same proposals analytically for his own benefit.” Euler notes that thanks to this he understood the essence of the issue much better. He developed fundamentally new methods for studying problems in mechanics, created its mathematical apparatus and brilliantly applied it to many complex problems. Thanks to Euler, differential geometry, differential equations, and calculus of variations became tools of mechanics. Euler's method, later developed by his successors, was unambiguous and adequate to the subject.

Euler's work on rigid body dynamics, The Theory of the Motion of Rigid Bodies, has a large introduction of six sections, which again sets out the dynamics of a point. A number of changes have been made to the introduction: in particular, the equations of motion of a point are written using projection on the axes of fixed rectangular coordinates (and not on the tangent, the main normal and the normal, that is, the axes of a fixed natural trihedron associated with the points of the trajectory, as in “Mechanics”) .

Following the introduction, “Treatise on the Motion of Rigid Bodies” consists of 19 sections. The treatise is based on D’Alembert’s principle. Having briefly discussed the translational motion of a rigid body and introducing the concept of the center of inertia, Euler considers rotations around a fixed axis and around a fixed point. Here are the formulas for projections of instantaneous angular velocity, angular acceleration on the coordinate axis, the so-called Euler angles, etc. Next, the properties of the moment of inertia are outlined, after which Euler proceeds to the dynamics of a rigid body. He derives the differential equations for the rotation of a heavy body around its stationary center of gravity at. in the absence of external forces and solves them for a simple particular case. This is how the well-known and equally important problem in the theory of the gyroscope arose: the rotation of a rigid body around a fixed point. Euler also worked on the theory of shipbuilding, in the eyes of hydro- and aeromechanics, ballistics, and the theory of stability. theory of small vibrations, celestial mechanics, etc.

Eight years after the publication of Mechanics, Euler enriched science with the first precise formulation of the principle of least action. The formulation of the principle of least action, which belonged to Maupertuis, was still very imperfect. The first scientific formulation of the principle belongs to Euler. He formulated his principle as follows: the integral has the least value for the real trajectory if we consider

the last in a group of possible trajectories that have a common initial and final position and are carried out with the same energy value. Euler provides his principle with an exact mathematical expression and a strict justification for one material point, testing the actions of central forces. During 1746-1749 pp. Euler wrote several papers on the equilibrium figures of a flexible thread, where the principle of least action was applied to problems in which elastic forces act.

Thus, by 1744 mechanics was enriched with two important principles: d'Alembert's principle and the Maupertuis-Euler principle of least action. Based on these principles, Lagrange built a system of analytical mechanics.

Inertia forces in the dynamics of a material point and a mechanical system

By the force of inertia of a material point is the product of the point’s mass and its acceleration, taken with a minus sign, i.e. Inertial forces in dynamics are applied in the following cases:

• 1. When studying the movement of a material point in non-inertial(moving) coordinate system, i.e. relative motion. These are the transport and Coriolis inertia forces, which are often called Euler forces.
• 2. When solving dynamics problems using the kinetostatic method. This method is based on the d'Alembert principle, according to which the inertial forces of a material point or system of material points moving with some acceleration in inertial reference system. These inertial forces are called d'Alembert forces.
• 3. D'Alembert's inertial forces are also used when solving problems of dynamics using the Lagrange-D'Alembert principle or the general equation of dynamics.

Expression in projections on Cartesian coordinate axes

Where - modules of projections of the acceleration of a point on the Cartesian coordinate axis.

When a point moves in a curvilinear direction, the inertial force can be decomposed into tangent and normal:; , - module of tangential and normal accelerations; - radius of curvature of the trajectory;

V- point speed.

D'Alembert's principle for a material point

If to non-free If a material point moving under the influence of applied active forces and bond reaction forces is applied, then at any moment of time the resulting system of forces will be balanced, i.e. the geometric sum of these forces will be equal to zero.

mechanical point body material

Where - resultant of active forces applied to a point; - resultant of reactions of bonds imposed on a point; inertia force of a material point. Note: In fact, the inertial force of a material point is applied not to the point itself, but to the body that imparts acceleration to this point.

D'Alembert's principle for a mechanical system

Geometric sum the main vectors of external forces acting on the system and the inertia forces of all points of the system, as well as the geometric sum of the main moments of these forces relative to some center for a non-free mechanical system at any moment of time are equal to zero, i.e.

Principal vector and principal moment of inertia forces of a rigid body

The main vector and the main moment of inertia forces of the points of the system are determined separately for each rigid body included in a given mechanical system. Their definition is based on the Poinsot method, known from statics, of bringing an arbitrary system of forces to a given center.

Based on this method, the inertial forces of all points of the body, in the general case, its movements can be brought to the center of mass and replaced by the main vector * and the main moment relative to the center of mass. They are determined by the formulas i.e. for any in the motion of a rigid body, the main vector of inertial forces is equal, with a minus sign, to the product of the mass of the body and the acceleration of the center of mass of the body; ,Where r kc -- radius vector k-th points drawn from the center of mass. These formulas in special cases of motion of a rigid body have the form:

1. Forward movement.

2. Rotation of a body around an axis passing through the center of mass

3. Plane-parallel motion

Introduction to Analytical Mechanics

Basic concepts of analytical mechanics

Analytical mechanics- an area (section) of mechanics in which the motion or equilibrium of mechanical systems is studied using general, unified analytical methods used for any mechanical systems.

Let us consider the most characteristic concepts of analytical mechanics.

1. Connections and their classification.

Connections-- any restrictions in the form of bodies or any kinematic conditions imposed on the movements of points of a mechanical system. These constraints can be written as equations or inequalities.

Geometric connections-- connections whose equations contain only the coordinates of points, i.e., restrictions are imposed only on the coordinates of points. These are connections in the form of bodies, surfaces, lines, etc.

Differential connections-- connections that impose restrictions not only on the coordinates of points, but also on their speed.

Holonomic connections -- all geometric connections and those differential ones whose equations can be integrated.

Non-holonomic connections-- differential non-integrable connections.

Landline connections -- connections whose equations do not explicitly include time.

Non-stationary communications-- connections that change over time, i.e., the equations of which clearly include time.

Two-way (holding) connections -- connections that limit the movement of a point in two opposite directions. Such connections are described by the equations .

Unilateral(non-restraining) connections - connections that limit movement in only one direction. Such connections are described by the inequalities

2. Possible (virtual) and actual movements.

Possible or virtual displacements of points of a mechanical system are imaginary infinitesimal movements that allow connections imposed on the system.

Possible The movement of a mechanical system is a set of simultaneous possible movements of points of the system that are compatible with the connections. Let the mechanical system be a crank mechanism.

Possible movement of the point A is a movement which, due to its smallness, is considered rectilinear and directed perpendicular to OA.

Possible movement of the point IN(slider) is moving in the guides. Possible movement of the crank OA is the rotation angle, and the connecting rod AB -- at an angle around the MCS (point R).

Valid displacements of system points are also called elementary displacements that allow superimposed connections, but taking into account the initial conditions of motion and the forces acting on the system.

Number of degrees freedom S of a mechanical system is the number of its independent possible movements that can be communicated to the points of the system at a fixed point in time.

The principle of possible movements (Lagrange principle)

The principle of possible displacements or the Lagrange principle expresses the equilibrium condition of a non-free mechanical system under the influence of applied active forces. Statement of the principle.

For balance of a non-free mechanical system with two-way, stationary, holonomic and ideal connections, which is at rest under the action of applied active forces, it is necessary and sufficient that the sum of the elementary works of all active forces is equal to the bullet on any possible displacement of the system from the considered equilibrium position:

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

The general equation of dynamics is applied to the study of the motion of non-free mechanical systems, the bodies or points of which move with certain accelerations.

In accordance with d'Alembert's principle, the totality of active forces applied to a mechanical system, coupling reaction forces and inertia forces at all points of the system forms a balanced system of forces.

If we apply the principle of possible displacements (Lagrange’s principle) to such a system, we obtain the combined Lagrange-D’Alembert principle or general equation of dynamics.Statement of this principle.

When moving unfree of a mechanical system with two-way, ideal, stationary and holonomic connections, the sum of the elementary works of all active forces and inertia forces applied to the points of the system at any possible movement of the system is zero:

Lagrange equations of the second kind

Lagrange equations of the second kind are differential equations of motion of a mechanical system in generalized coordinates.

For a system with S degrees of freedom, these equations have the form

Difference the total derivative with respect to time of the partial derivative of the kinetic energy of the system with respect to the generalized speed and the partial derivative of the kinetic energy with respect to the generalized coordinate is equal to the generalized force.

Lagrange equations for conservative mechanical systems. Cyclic coordinates and integrals

For a conservative system, the generalized forces are determined through the potential energy of the system according to the formula

Then the Lagrange equations will be rewritten in the form

Since the potential energy of the system is a function of only generalized coordinates, i.e., then, taking this into account, we present it in the form where T - P = L -- Lagrange function (kinetic potential). Finally, the Lagrange equations for a conservative system

Stability of the equilibrium position of a mechanical system

The question of the stability of the equilibrium position of mechanical systems is of direct importance in the theory of vibration of systems.

The equilibrium position can be stable, unstable and indifferent.

Sustainable equilibrium position - an equilibrium position in which the points of a mechanical system, removed from this position, subsequently move under the action of forces in the immediate vicinity of their equilibrium position.

This movement will have some degree of repeatability in time, i.e. the system will perform an oscillatory movement.

Unstable equilibrium position - an equilibrium position from which, with an arbitrarily small deviation of the points of the system, further acting forces will move the points even further away from their equilibrium position .

Indifferent equilibrium position - an equilibrium position when, for any small initial deviation of the points of the system from this position, in the new position the system also remains in equilibrium. .

There are various methods to determine the stable equilibrium position of a mechanical system.

Let us consider the definition of a stable equilibrium position based on Lagrange-Dirichlet theorems

If in position equilibrium of a conservative mechanical system with ideal and stationary connections, its potential energy has a minimum, then this equilibrium position is stable.

Impact phenomenon. Impact force and impact impulse

The phenomenon in which, over a negligibly small period of time, the velocities of points on a body change by a finite amount is called blow. This period of time is called impact time. During an impact, an impact force is exerted over an infinitesimal period of time. Impact force called a force whose momentum during the impact is a finite value.

If the force is finite in modulus acts over time, beginning its action at a moment in time , then its impulse has the form

Also, when an impact force acts on a material point, we can say that:

the action of non-instantaneous forces during the impact can be neglected;

the movement of the material point during the impact can be ignored;

the result of the action of an impact force on a material point is expressed in the final change in its velocity vector during the impact.

Theorem on the change in momentum of a mechanical system upon impact

the change in the momentum of the mechanical system during the impact is equal to the geometric sum of all external shock pulses applied to the points of the systems, Where - the amount of motion of the mechanical system at the moment of termination of the impact forces, - the amount of motion of the mechanical system at the moment the impact forces begin to act, - external shock impulse.

The methods for solving mechanical problems that have been considered so far are based on equations that follow either directly from Newton's laws or from general theorems that are consequences of these laws. However, this path is not the only one. It turns out that the equations of motion or the equilibrium conditions of a mechanical system can be obtained by basing it on other general principles, called the principles of mechanics, instead of Newton’s laws. In a number of cases, the application of these principles allows, as we will see, to find more effective methods for solving the corresponding problems. This chapter will examine one of the general principles of mechanics, called d'Alembert's principle.

Let us first find the expression of the principle for one material point. Let a material point with mass be acted upon by a system of active forces, the resultant of which will be denoted by the coupling reaction N (if the point is not free). Under the influence of all these forces, the point will move relative to the inertial frame of reference with some acceleration a.

Let us introduce into consideration the quantity

having the dimension of force. A vector quantity equal in magnitude to the product of the mass of a point and its acceleration and directed opposite to this acceleration is called the inertial force of the point.

Then it turns out that the motion of a point has the following property: if at any moment of time the force of inertia is added to the active forces acting on the point and the coupling reaction, then the resulting system of forces will be balanced, i.e.

This position expresses d'Alembert's principle for a material point. It is easy to see that it is equivalent to Newton's second law and vice versa. In fact, Newton's second law for the point under consideration gives. Transferring here the value m to the right side of the equality and taking into account notation (84), we arrive at relation (85). On the contrary, transferring the quantity in equation (85) to the other part of the equality and taking into account notation (84), we obtain the expression for Newton’s second law.

Let us now consider a mechanical system consisting of material points. Let us select one of the points of the system with mass . Under the influence of external and internal forces applied to it (which include both active forces and coupling reactions), the point will move relative to the inertial frame of reference with some acceleration. By introducing the inertial force for this point, we obtain according to equality (85) that

i.e., that they form a balanced system of forces. Repeating such reasoning for each of the points of the system, we arrive at the following result, expressing D’Alembert’s principle for the system: if at any moment of time the corresponding inertial forces are added to each of the points of the system, in addition to the external and internal forces acting on it, then the resulting system of forces will be balanced and all equations of statics can be applied to it.

Mathematically, D’Alembert’s principle for a system is expressed by vector equalities of the form (85), which are obviously equivalent to the differential equations of motion of the system (13), obtained in § 106. Consequently, from D’Alembert’s principle, as well as from equations (13), one can obtain all general theorems speakers.

The significance of d'Alembert's principle lies in the fact that when directly applied to problems of dynamics, the equations of motion of the system are compiled in the form of well-known equilibrium equations; this makes the approach to solving problems uniform and often simplifies the corresponding calculations. In addition, in combination with the principle of possible displacements, which will be discussed in the next chapter, d'Alembert's principle allows us to obtain a new general method for solving problems of dynamics (see § 141).

It is known from statics that the geometric sum of forces in equilibrium and the sum of their moments relative to any center O are equal to zero, and, as shown in § 120, this is true for forces acting not only on a rigid body but also on any variable mechanical system .

Then, based on D'Alembert's principle, it should be:

Let us introduce the following notation:

The quantities represent the main vector and the main moment relative to the center O of the system of inertial forces. As a result, taking into account that the geometric sum of internal forces and the sum of their moments are equal to zero, we obtain from equalities (86):

The use of equations (88), resulting from d'Alembert's principle, simplifies the process of solving problems, since these equations do not contain internal forces. Essentially, equations (88) are equivalent to equations expressing theorems on changes in momentum and the main angular momentum of the system, and differ from them only in form.

Equations (88) are especially convenient to use when studying the motion of a rigid body or a system of rigid bodies. For a complete study of the motion of any variable system, these equations will not be enough, just as the equations of statics are not enough to study the equilibrium of any mechanical system (see § 120).

In projections onto the coordinate axes, equalities (88) give equations similar to the corresponding static equations (see § 16, 30). To use these equations when solving problems, you need to know the expressions for the main vector and the main moment of inertia forces.

In conclusion, it should be emphasized that when studying motion with respect to an inertial reference frame, which is considered here, inertial forces are introduced only when d'Alembert's principle is applied to solve problems

D'Alembert's principle for a material point. The form of writing the equation of motion in accordance with Newton's laws is not the only one. These equations can be written in other forms. One such possibility is d'Alembert's principle, which formally allows the differential equations of motion to take the form of equilibrium equations.

This principle can be considered as an independent axiom that replaces Newton's second law. Let's use it as a means of solving problems and derive it from Newton's law.

Let us consider the motion of a material point relative to an inertial reference frame. For a free material point

we have: that = = I.

Transferring vector that to the right side of the equality, this relationship can be represented as an equilibrium equation: I am the one 0.

Let's introduce the concept inertia forces. Let's call a vector directed opposite to the acceleration and equal to the product of the mass of a point and its acceleration inertial force of a material point: = -ta.

Using this concept, we can write (Fig. 3.42):

• ? ^ + P"n) = 0. (3.47)

Rice. 3.42.

for a material point

Equation (3.47) is D’Alembert’s principle for a free material point: If we add the force of inertia to the forces applied to the point, then the point will be in a state of equilibrium.

Strictly speaking, the stated position is not d'Alembert's principle in the form in which it was formulated by the author.

d'Alembert considered non-free movement of the point, without using the principle of being released from connections, without introducing a connection reaction. Noting that in the presence of a connection, the acceleration of a point does not coincide in direction with the force and ta F R, he introduced the concept lost power P - that and stated that the application of a lost force to a point does not disturb its state of equilibrium, since the lost force is balanced by the reaction of the connection.

Relationship (3.47) is basic equation of kinetostatics, or equation of Hermann's Petersburg principle-Euler. The kinetostatics method can be considered as a modification of the notation of d'Alembert's principle, including for a free material point, more convenient for practical use. Therefore, in most literature sources, equation (3.47) is called d'Alembert's principle.

If the point is not free, i.e. a connection is imposed on it, it is convenient to divide the forces that act on the point into active ones 1, R°(ask-

e) and the reaction of the connection of the control unit: p(a) + N =

This technique is convenient because with some types of connections it is possible to create an equation of motion so that the reactions of these connections are not included in it. Thus, d’Alembert’s principle for a non-free point can be written as (Fig. 3.43):

R(a)+ /V + R Sh) = 0, (3.48)

i.e., if, in addition to active forces and coupling reactions, a force of inertia is applied to a non-free material point, then the resulting system of forces will be in equilibrium at any time.

Rice. 3.43.

material point

A- from English, active- active. Recall that active forces are those that retain their values ​​when all connections are removed.

When considering the curvilinear movement of a point, it is advisable to represent the inertial force in the form of two components: G "‘ p) = -ta p- centrifugal and Shch,n) = -ta x - tangent (Fig. 3.44).

Rice. 3.44.

motion of a material point

Let us recall that the expressions for the values ​​of normal and tangential accelerations have the form: a p -U 2 / p and i t = s1U D/L

Then we can write: Р^t) - -t-p Рр p) - -t-t, or finally: R

rt + p(t) + p(a) + yy = o (3.49)

Equality (3.49) expresses d'Alembert's principle for the curvilinear motion of a non-free point.

Consider a thread of length /, at the end of which a point of mass is attached T. The thread rotates around a vertical axis, describing a conical surface with a constant angle of inclination of the generatrix A. Determine the corresponding constant speed of the point and the tension of the thread T(Fig. 3.45).

Rice. 3.45.

motion of a non-free material point

Yes, but: /and,/, a = const. Find: T, V.

Let us apply inertia forces to the point, directed oppositely to the corresponding acceleration components. Note that the tangential force of inertia is zero, since according to the condition the speed is constant:

/1°") = -ta = -t-= Oh,

and the centrifugal force of inertia is determined by the expression Р^т) = ТУ 2 /р, where p = /Bta.

Application of d'Alembert's principle to this problem allows us to write down the equation of motion of the material point under study in the form of an equilibrium condition for converging forces: T? + T + PP p) = 0.

In this case, all equilibrium equations are valid in projection onto the natural coordinate axes:

X^„=0, - FJ" 1+ Tsina = 0; ^ F h = 0, - mg + T cosa = 0,

+ T sin a =

-mg + T cosa = 0,

where do we find it from? T= /u#/soBa; V= Btal/^/Tsosa.

D'Alembert's principle for a system of material points. Let us consider the movement of a mechanical system of material points. As with the derivation of OMS, we divide the forces applied to each point into external and internal (Fig. 3.46).

Rice. 3.46.

Let ’ be the resultant of external forces applied to the i-th point, and /Γ(L) be the resultant of internal forces applied to the same point. In accordance with d’Alembert’s principle, inertial forces must be applied to each material point of the system: Рр p) = -t,a g

Then the forces applied to each point of the system satisfy the relation:

1?E) + рУ) + р0п)

those. the system of material points will be in equilibrium if additional inertial forces are applied to each of its points. Thus, with the help of d'Alembert's principle, it is possible to give the equations of motion of a system the form of equilibrium equations.

Let us express the kinetostatic conditions of equilibrium of the system using static equivalents of inertial forces and external forces. For this purpose, let’s sum over all P equations (A), describing the forces applied to individual points of the system. Then we calculate the moments of all external and internal forces and inertia forces applied to individual points, relative to an arbitrary point ABOUT:

g a X P" E> +g a X /*") +g a X R t > =0. і = 1,2,...,«.

Then we carry out the summation, as a result we get

// p p

'(E) і G(1)

1l (?) +L (/) +L (,p) = 0;

[M (0 E) + M (0 p + M% a) = 0.

Because the K i)= 0 and M 1 0 p = 0, then we finally have:

II (?) + L (/I) =0;

M (a E) + M(‘n) = 0.

From the system of equations (3.50) it is clear that the main vector of inertia forces is balanced by the main vector of external forces, and the main moment of inertia forces relative to an arbitrary point is balanced by the main moment of external forces relative to the same point.

When solving problems, it is necessary to have expressions for the main vector and the main moment of inertia forces. The magnitudes and directions of these vectors depend on the distribution of accelerations of individual points and their masses. As a rule, direct determination I (sh) And M(""] geometric summation can be performed relatively simply only with P - 2 or P= 3. At the same time, in the problem of the motion of a rigid body, it is possible to express the static equivalents of inertia forces in some special cases of motion depending on the kinematic characteristics.

The main vector and the main moment of inertia forces of a rigid body in various cases of motion. According to the theorem on the motion of the center of mass t c a c = I (E). According to d'Alembert's principle we have: I (1P) + I (E) = Oh, where do we find: I" 1P) = -t s a s. Thus, with any movement of the body the main vector of inertia forces is equal to the product of the body mass and the acceleration of the center of mass and is directed opposite to the acceleration of the center of mass(Fig. 3.47).

Rice. 3.47.

Let us express the main moment of inertia forces during the rotational motion of a body around a fixed axis perpendicular to the plane of material symmetry of the body (Fig. 3.48). Inertia forces applied to the point: R"! n) = m, x op; 2 and R? P)= /u,er,.

Since all centrifugal inertial forces intersect the axis of rotation, the main moment of these inertial forces is equal to zero, and the main moment of the tangential inertial forces is equal to:

m t =?_ C > P(= ?-sh.d x/P. = = -e?/i.p; = - J z g. (3.51)

Thus, the main moment of tangential forces of inertia relative to the axis of rotation is equal to the product of the moment of inertia relative to this axis and angular acceleration, and the direction of the main moment of tangential forces of inertia is opposite to the direction of angular acceleration.

Rice. 3.48.

relative to the axis of rotation

Next, we express the inertial forces during plane-parallel motion of the body. Considering the plane-parallel motion of a body (Fig. 3.49) as the sum of translational motion along with the center of mass and rotational movement around axis passing through the center of mass perpendicular to the plane of motion, it can be proven in the presence of a plane of material symmetry coinciding with the plane of motion of the center of mass that the inertial forces during plane-parallel motion are equivalent to the main vector /? ("n) applied to the center of mass opposite to the acceleration of the center of mass, and the main moment of inertia forces M^n) relative to the central axis, perpendicular to the plane of motion, directed in the direction opposite to the angular acceleration:

Rice. 3.49.

Notes

• 1. Note that, since d'Alembert's principle allows just write down the equation of motion in the form of an equilibrium equation, then it does not give any integrals of the equation of motion.
• 2. Let us emphasize that inertial force d'Alembert's principle is fictitious gray applied in addition to the acting forces for the sole purpose of obtaining an equilibrium system. However, in nature there are forces geometrically equal to the forces of inertia, but these forces are applied to other (accelerating) bodies, in interaction with which an accelerating force appears, applied to the moving body in question. For example, when moving a point attached to a thread rotating at a constant speed in a circle in a horizontal plane, the tension in the thread is exactly equal to force of inertia, those. the force of reaction of the point on the thread, while the point moves under the influence of the thread's reaction to it.
• 3. As has already been shown, the given form of D’Alembert’s principle differs from the one that D’Alembert himself used. The method of composing differential equations of motion of a system, used here, was developed and expanded by a number of St. Petersburg scientists and was called kinetostatic method.

Application of mechanics methods to some problems of the dynamics of rail vehicles:

? movement of a rail vehicle along a curved track. Currently, due to the capabilities of computer technology, the analysis of all mechanical phenomena occurring when a rail vehicle moves in a curve is carried out using a fairly complex model, which takes into account the entire set of individual bodies of the system and the characteristics of the connections between them. This approach allows us to obtain all the necessary kinematic and dynamic characteristics of movement.

However, when analyzing the final results and carrying out preliminary rough calculations in the technical literature, certain distortions of some concepts of mechanics are quite often encountered. Therefore, it is advisable to talk about the very “primordial fundamentals” used in describing the movement of the crew in a curve.

Let us present some mathematical descriptions of the processes under consideration in an elementary formulation.

For a correct, consistent explanation of characteristics stationary crew movement in a circular curve you need:

• select the mechanics method used to describe this movement;
• proceed from a clear, from a mechanical point of view, concept of “force”;
• do not forget the law of equality of action and reaction.

The process of movement of the crew in a curve inevitably involves a change in the direction of speed. A characteristic of the speed of this change is the normal acceleration directed to the center of curvature of the curvilinear trajectory of the center of mass: a n - V 2/р, where р is the radius of the curve.

During the movement, the crew interacts with the rail track, resulting in normal and tangential reactive forces applied to the wheel pairs. Naturally, equal and opposite pressure forces are applied to the rails. According to the presented mechanical concepts, force is understood as the result of the interaction of bodies, or a body and a field. In the problem under consideration, there are two physical systems: a carriage with wheelsets and a rail track, therefore, forces must be sought at the points of their contact. In addition, the interaction of the crew and the Earth's gravitational field creates gravity.

A description of the crew's movement in a curve can be made using general theorems of dynamics, which are consequences of health insurance, or based on principles of mechanics(for example, d'Alembert's principle), which is the basis kinetostatic method.

Wanting to explain equal features methods for taking into account the curvature of the track axis on the characteristics of the movement of the crew, we first use the simplest idealized model. We will consider the crew as a material plane with a mass equal to the mass of this system.

The center of mass lying in this plane makes a given movement along a trajectory congruent to the axis of the path, with a speed V. Contact with the rail track is carried out at two points of intersection of the moving plane with the rail threads. Therefore, speaking about the interaction of the crew with the rail track, we can talk about concentrated forces, which represent the resultant of all reactions of the rails on individual wheel pairs from each of the rails. Moreover, the nature of the occurrence of reactive forces is unimportant;

? movement of the crew along the track without raising the outer rail. In Fig. Figure 3.50 shows a design diagram of a crew moving along a curved path. The outer and inner rails, in this case, are located on the same level. In Fig. 3.50 indicates the forces and reactions of the connections acting on the crew. Let us emphasize that there are no There are no real centrifugal forces in this scheme.

Within the framework of Newtonian geometric mechanics, the movement of the vehicle in a curve is described by general theorems of system dynamics.

In this case, according to the theorem on the motion of the center of mass,

t s a s - I a) , (a)

where R) is the main vector of external forces.

Designing both sides of the expression (A) to accompanying natural coordinate axes, the center of which is located at the center of mass of the crew, with unit vectors m, i, b and count t s = T.

In the projection onto the main normal we get mаn = Fn, or

mV /p = F„ (b)

Where Fn - real force reactions of the rail to the wheelsets, which is the sum of the projections of the reactions of the rails to the normal to the trajectory. These can be directing pressure forces of the rails on the flanges of the wheels. There are no other external forces in this direction.

In the projection of the expression (A) for the binormal we get:

O = -mg + N out + N inn. (With)

Here are the indices out 1 correspond to the outer one, a inn - the inner rail of the curve. The left side in expression (c) is equal to zero, since the projection of acceleration onto the binormal is equal to zero.

We obtain the third equation using the theorem on the change in angular momentum relative to the center of mass:

dK c /dt = ^M c . (d)

Designing Expression d on the t axis, where t = nx b - vector product of unit vectors P And b, taking into account the fact that K Cl=U St so t, U St is the moment of inertia of the crew relative to the tangent axis to the trajectory of the center of mass, we will have

J a *i=NJS-N m S + F K H = 0, (f)

since the angular acceleration relative to the m axis in steady motion along a circular curve is zero.

Expressions ( b), (c) and (f) represent a system of linear algebraic equations for three unknown quantities M-tp> solving which, we get:

Rice. 3.50.

Thus, the consistent application of general theorems of dynamics makes it possible to establish in the problem under consideration all the phenomena associated with the passage of a curved section of the path by the crew.

In fact, both wheels are subject to forces directed towards the inside of the curve. The resultant of these forces creates a moment about the center of mass of the crew, which can cause rotation and even tipping outward of the curve if V 2 N/р5" > g. The action of this force leads to wear of the wheels. Naturally, the oppositely directed force acting on the rail -R p causes rail wear.

Note that in the stated formulation we can only find the resultant of the horizontal reactions of two rails R. To determine the distribution of this force between the inner and outer rails, it is necessary to solve a statically indeterminate problem using additional conditions. In addition, when the carriage moves, the normal reactions of the outer and inner rails have different values. The outer rail thread is more loaded.

The reaction of the internal thread to the carriage is less and at a certain speed value can even be equal to zero.

In classical mechanics this state is called capsizing, although there is no actual capsizing yet. To find out when a state of actual capsizing occurs, one should consider the rotation of the car around an axis parallel to m and passing through the point of contact of the wheel with the outer rail at? T F 0. This problem is of purely academic interest, since, of course, bringing a real system to such a state is unacceptable.

Let us emphasize once again that when explaining all phenomena we proceeded from the fact movement of the car under the influence of only real forces.

Note that the differential equation of rotation around the m axis, even at = 0, is written with respect to the central axis m. Selecting this axis at another point leads to a change in the form of the left side of the equation of the theorem of moments. Therefore, it is impossible, for example, to write this equation in the same form relative to the axis passing through the point of contact of the wheel with the rail, although it would seem that it would be easier to find the value of the normal reactions. However, this approach will lead to the wrong result: And osh = M 1Sh1 = mg| 2.

It can be shown that the point is that the equation of rotation about an axis passing, for example, through a point TO, must be written taking into account the moment of momentum of the body from the translational part of the movement g x x t s: J Cl? t + T(g ks xy g)=^ M Kh.

Therefore, instead of equation (c) in projection onto the St axis, we obtain the expression

(8 )

/ St? t + t[g ks X and with) t = -tёB + N іпп 25,

where in parentheses the value of the projection onto the C axis of the vector product is written ? ks ha s.

Let us show that sequential implementation of the necessary procedures allows us to find Y sp from the resulting equation). From Fig. 3.50 it is clear that

g ks - Bp + NH And a c =

Let's calculate the vector product:

It is taken into account here that php = 0 And bxn = - t. Therefore,

TNU 2

2L g/lp 5’,

where we find the reaction of the inner rail:

which is the same as the result obtained in expression (/).

To conclude the presentation of the problem, we point out that considering a car in movement using Newtonian geometric mechanics methods allows us to solve the problem without introducing fictitious inertia. You just need to use all the mechanics correctly. It should, however, be noted that the use of this method may involve a larger amount of calculations than, for example, when using the d'Alembert principle.

Let us now show how the same problem is solved based on the use of d'Alembert's principle in the generally accepted form of the kinetostatic method. In this case, it is necessary to apply additional

personal fictitious inertia force: G* = -ta Sp = -T-P. And equi-

page stops, i.e. now the acceleration of its center of mass and with= 0. In Fig. 3.51 shows this resting system. All forces applied to it, including the force of inertia, must satisfy the kinetic-static equations balance, not movement, as in the previous case.

This circumstance allows us to find all unknown quantities from equilibrium equation. In this case, the choice of the form of the equilibrium equations and the points relative to which the moments are calculated becomes arbitrary. The latter circumstance allows us to find all unknowns independently of each other:

I M. = oh, I m,_= oh

-n = o.

1 at MP

Rice. 3.51. Calculation diagram of the forces acting on the crew under the same conditions as in Fig. 3.50 using d'Alembert's principle

It is easy to see that the solutions to this system of equations coincide with the corresponding formulas obtained using the theory of dynamics. Thus, in the example under consideration, the application of d'Alembert's principle made it possible to somewhat simplify the solution of the problem.

However, when interpreting the results, it should be borne in mind that the additionally applied inertial force is fictitious in the sense that in reality there is no such force acting on the crew. In addition, this force does not satisfy Newton’s third law - there is no “second end” of this force, i.e. there is no opposition.

In general, when solving many problems of mechanics, including the problem of vehicle movement in a curve, it is convenient to apply the d'Alembert principle. However, one should not associate any phenomena with action this inertial force. For example, we can say that this centrifugal force of inertia additionally loads the outer rail and unloads the inner one, and moreover, that this force can cause the carriage to capsize. This is not only ignorant, but also pointless.

Let us recall once again that the external applied forces acting on the carriage in a curve and changing the state of its motion are gravity, vertical and horizontal reactions of the rails;

? movement of the carriage along a curve with an elevation of the outer rail. As has been shown, the processes that arise when a vehicle passes in curves without raising the outer rail are associated with undesirable consequences - uneven vertical load of the rails, significant normal horizontal reaction of the rail to the wheel, accompanied by increased wear of both wheels and rails, the possibility of capsizing when the speed is exceeded movement of a certain limit, etc.

To a large extent, the unpleasant phenomena that accompany the passage of curves can be avoided if the outer rail is raised above the inner one. In this case, the carriage will roll along the surface of the cone with the angle of inclination of the generatrix to the horizontal axis (Fig. 3.52): f L = arcsin (L/25), or at small angles

F A * L/2 S.

Rice. 3.52.

with raised outer rail

In the stationary case, when V- const and φ A = const, we can consider the movement of a flat section of the carriage in its plane in the same way as when it fits into a curve without raising the outer rail.

Let's consider a technique for solving the problem using general theorems of dynamics. We will assume that the center of mass of the crew moves along a circular curve with radius p, although in the case under consideration, strictly speaking, the radius of curvature of the track axis differs from the radius of curvature of the trajectory of the center of mass by a small amount:

N sin avg L ~ N f A « r.

Therefore, compared to p, the latter value can be neglected. The movement of the “flat section” of the vehicle will be attributed to the accompanying axes SuSi x(see Fig. 3.52), where the axis Su ] parallel to the plane of the path. At a constant speed of movement, the projection of the acceleration of the center of mass onto the main normal of the trajectory of its movement can be written in the same way as when moving in a curve without an elevation, i.e. a p = V i/R.

Acceleration projections on the Su axis, and Cz^ are equal respectively:

a uh = a p sovf; I. =a„smy h .

The equations of motion of a flat section based on the theorem on the movement of the center of mass and the theorem on the change in angular momentum relative to the Cx axis are as follows:

Taking into account the fact that = 0, after substitution we obtain a system of three linear algebraic equations for three unknowns F Vi, N iiw, N(nil:

/i-si Pf l = -mg cos V/ , + N mn + N out; P

-soєf A = mgs ipf A + F ;

0 = +N ilw S-N oul S + F y H.

Let us note that the inclination of the plane of the track axis due to the elevation of the outer rail leads to a change in the projection of the acceleration of the center of mass on the Cy and Cr axis, which is associated with a change in the reactions of the rails compared to those in the absence of elevation, when A. - 0, a l These changes in the acceleration projections can be explained if we consider the rotation of the vehicle around the binormal passing through the center of curvature of the curve as the geometric sum of two rotations with A = co (+ b) around the axes ?, y, passing through the same center of the curve.

When compiling a system of equations (To) the smallness of the angle sr L was not provided for. However, in a practically feasible design

wtf A ~ /g/25.

Thus, in the case of small f L, the system of equations for determining the reactions of the track on the crew has the following form:

= -g^+ LG,„ + M gsh,;

T- = /gg#--1- g, ;

O = + L/-5 - /U 0I/ 5 + R p N.

Solving these equations, we get:

N...... =

mg + tU/G

Fri/77 K AND /77 „

• - +--+-n
• 2р 25 25

In the special case when there is no elevation (AND= 0), these expressions coincide with those obtained earlier (/).

Now let's move on to analyzing the results of solving the problem with I F 0.

It should be noted that in this case the transverse reaction of the rail, directed in the plane of the track, decreases. This is explained by the fact that in the formation of the acceleration of the center of mass in the direction of the Cy axis, not only the force // takes part, but also the component of gravity. Moreover, at a certain value AND= 25K 2 /r? force R becomes equal to zero:

Bearing in mind that

t g - T,= X A,%>+X A[

• (3.42)

The value in brackets is called outstanding acceleration. State when P = 0, corresponds to the case in which the normal acceleration A is formed only by the projection onto the d> axis of the gravity force of the crew.

When discussing the problem under consideration, sometimes a sophistic argument arises that acceleration a p is directed horizontally, and the force of gravity is vertical (see Fig. 3.52), and therefore it cannot form the acceleration in question a p at R= 0. This reasoning contains an error, since in the formation of horizontal acceleration, in addition to force R, the normal reactions D g shya and /U o uG also take part. The sum of these two reactions for small f L is equal 1H tp + 1U oig = mg. Consequently, gravity still participates in the formation of horizontal acceleration a p, but through the action of reactions N tp And S oiG

Let us now discuss how the normal reactions of rails perpendicular to the track surface change.

Note that, unlike the case /7 = 0, the reactions increase by the same value TU 2 I/2r28, which is neglected because ///25 - the value is small. However, in strict reasoning, omit this term for expressions and N w do not do it.

When - > -2-, i.e. with positive undamped acceleration, p 25

the reaction of the inner rail is less than that of the outer one, however, the difference between them is not as significant as with AND = 0.

If the outstanding acceleration is equal to zero, the reaction values ​​become equal to /U /yal = IV osh = mg|2(at small AND), those. raising the outer rail allows not only to obtain RU= 0, but also to equalize the pressure on the outer and outer rails. These circumstances make it possible to achieve more uniform wear values ​​for both rails.

At the same time, due to the elevation of the outer rail, the possibility of a negative value arises R", which in a real system with non-restraining connections corresponds to the process of the vehicle sliding along the axis y g those. inside the curved path. Due to the same slope of the path, a redistribution of reactions may occur N w And N oi! with predominant meaning M sh.

Thus, studies of the movement of the vehicle in a curve along a path with an elevation of the outer rail, carried out using the methods of Newtonian geometric mechanics, make it possible to analyze the state of the system without additional terminological hypotheses. There are no inertial forces present in the reasoning.

Let us now consider how the movement of the crew in the same curve is described using D'Alembert's principle.

Applying this principle in the formulation of the kinetostatics method in the same way as in the previous case, it is necessary to apply a normal (centrifugal) inertial force to the center of mass Р„п) , directed in the direction opposite to normal acceleration (Fig. 3.53):

Wherein system again stops, i.e. the crew does not move along the track. Therefore, all kinetic-static equilibrium equations are valid:

I To= °-X g* = O.

/L^ypf, - G' p sovf* + G U[ = 0;

- /L?S08f/; - BIPf, + + N^1

Substituting the value here we get the same system of equations as the system (/) for any f /(or (To) at small AND.

Thus, the use of both methods leads to exactly the same results. System of equations ( To) and the system obtained on the basis of d'Alembert's principle are identical.

Let us note at the same time that in The final results do not include any inertial forces. This is understandable, since the d'Alembert principle, which underlies the kinetostatics method, is only a means of compiling differential equations of motion of the system. At the same time, we see that in the problem under consideration, the application of d'Alembert's principle made it possible to simplify the calculations and can be recommended when carrying out practical calculations.

However, let us emphasize once again that in reality there is no force TU 2/p applied to the center of mass of the moving vehicle. Therefore, all phenomena associated with movement in a curve should be explained as it was done on the basis of an analysis of the results of solving the system (/), or (To).

Let us point out in conclusion that the “Newton method” and the “D’Alembert method” in the problem under consideration were used only for the purpose of composing differential equations of motion. In this case, at the first stage we do not receive any information other than the differential equations themselves. The subsequent solution of the resulting equations and the analysis performed are not related to the method of obtaining the equations themselves.

Rice. 3.53.

• out - from English, outer - external.
• inn - from English, inner - interior.
• inn - from English, inner - interior.