d'Alembert's principle for a material point. The form of the equation of motion in accordance with Newton's laws is not the only one. These equations can also be written in other forms. One of these possibilities 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, replacing Newton's second law. We use it as a means of solving problems and derive it from Newton's law.

Consider the motion of a material point relative to an inertial frame of reference. For a free material point

we have: that = = I.

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

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

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

• ? ^ + P "n) = 0. (3.47)

Rice. 3.42.

for material point

Equation (3.47) is the d'Alembert principle for a free material point: if the force of inertia is added to the forces applied to the point, then the point will be in a state of equilibrium.

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

d'Alembert considered non-free movement of a point, without using the principle of release from bonds, without introducing a bond 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.

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

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

given) and the reaction of the CU bond: p(a) + n =

This technique is convenient, because for some types of bonds it is possible to compose an equation of motion in such a way that the reactions of these bonds are not included in it. Thus, the d'Alembert principle for a non-free point can be written as (Fig. 3.43):

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

i.e., if an inertial force is applied to a non-free material point, in addition to active forces and the coupling reaction, 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 forces are called active if they retain their values ​​when all bonds are removed.

When considering the curvilinear motion of a point, it is advisable to represent the force of inertia in the form of two components: Г "‘ n) \u003d -ta n- centrifugal and W, p) \u003d -ta x - tangent (Fig. 3.44).

Rice. 3.44.

movement of a material point

Recall that the expressions for the normal and tangential accelerations have the form: a p -U 2 / p and i t = s1U D/L

Then you can write: P^ t) - -t-p Rp p) - -t-t, or finally: R

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

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

Consider a thread of length /, at the end of which is fixed a point of mass 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.

movement of a non-free material point

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

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

/1°") = -ta = -t-= Oh

and the centrifugal force of inertia is determined by the expression P^ m) \u003d mU 2 /p, where p = /Bta.

The application of the d'Alembert principle to this problem allows us to write the equation of motion of the studied material point in the form of a condition for the equilibrium of converging forces: T? + T + Pp n) = 0.

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

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

+ T sin a =

-mg + T cosa = 0,

where do we find T= /u#/coBa; V= Btal/^/Tcosa.

d'Alembert's principle for a system of material points. Consider the motion of a mechanical system of material points. As with the withdrawal of the OZMS, 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 /-th point, and / G (L - the resultant of internal forces applied to the same point. In accordance with the d'Alembert principle, inertial forces must be applied to each material point of the system: Рр n) = -т,а г

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

1?E) + pY) + p0p)

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

Let us express the kinetostatic equilibrium conditions of the system using the static equivalents of inertial forces and external forces. For this purpose, we 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 R "E> + g a X /*") + g a X P t > =0. і = 1,2,..., ".

Then we sum up, as a result we get

// p p

'(E) і G(1)

1l (?) + L (/) + L (, n) \u003d 0;

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

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

ІЯ (?) + Л (/И) = 0;

M (a E) + M(‘n) = 0.

It can be seen from the system of equations (3.50) that the main vector of inertial 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, a direct definition I (sh) And M ( ”" ] geometric summation can be performed relatively simply only when 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 the inertial forces in some particular cases of motion depending on the kinematic characteristics.

Principal vector and principal moment of forces of inertia of a rigid body in various cases of motion. According to the theorem on the movement of the center of mass t with a c \u003d I (E). According to d'Alembert's principle, we have: I (1P) + I (E) = Oh, where do we find: I "1P) = -t with a with. Thus, with any movement of the body the main vector of inertial 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 inertial forces during the rotational motion of the body around a fixed axis perpendicular to the plane of material symmetry of the body (Fig. 3.48). Forces of inertia applied to / -point: R"! n) = m,x op; 2 and R? P)= /u,ep,.

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

m t =?_ C\u003e P (= ?-sh.d x / R. = = -e? / i. p; = - J z (3.51)

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

Rice. 3.48.

about the axis of rotation

Next, we express the forces of inertia for a plane-parallel motion of the body. Considering the plane-parallel motion of the body (Fig. 3.49) as the sum of the translational motion together with the center of mass and rotation around axis passing through the center of mass perpendicular to the plane of motion, it can be proved, in the presence of a plane of material symmetry coinciding with the plane of motion of the center of mass, that the forces of inertia in plane-parallel motion are equivalent to the main vector /? (" p) applied to the center of mass is 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 the d’Alembert principle allows just write the equation of motion in the form of an equilibrium equation, then it does not give any integrals of the equation of motion.
• 2. We emphasize that inertia force in d'Alembert's principle is fictitious gray, applied in addition to the acting forces with the sole purpose of obtaining an equilibrium system. However, in nature there are forces that are geometrically equal to the forces of inertia, but these forces are applied to other (accelerating) bodies, in interaction with which an accelerating force arises, applied to the considered moving body. For example, when moving a point fixed on a thread rotating at a constant speed around a circle in a horizontal plane, the tension of the thread is exactly equal to force of inertia, those. the reaction force of a point on a thread, while the point moves under the action of the thread's reaction to it.
• 3. As already shown, the above form of the d'Alembert principle differs from that used by d'Alembert himself. The method of compiling differential equations of motion of the system, used here, was developed and expanded by a number of St. Petersburg scientists and received the name kinetostatic method.

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

? movement of a rail vehicle along a curved track. At present, due to the capabilities of computer technology, the analysis of all mechanical phenomena occurring during the movement of a rail vehicle in a curve is carried out using a rather complex model, which takes into account the entire set of individual bodies of the system and the features of the connections between them. This approach makes it possible to obtain all the necessary kinematic and dynamic characteristics of motion.

However, when analyzing the final results and carrying out preliminary estimates in the technical literature, certain distortions of some concepts of mechanics are quite often encountered. Therefore, it is advisable to talk about the most "original foundations" used in describing the movement of the crew in a curve.

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

For a correct, consistent explanation of the characteristics stationary movement of the crew in a circular curve it is necessary:

• choose the method of mechanics used to describe this movement;
• proceed from a clear, from the point of view of mechanics, 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 implies a change in the direction of speed. The 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 p - V 2/p, where p is the radius of the curve.

During the movement, the vehicle interacts with the rail track, resulting in normal and tangential reactive forces applied to the wheelsets. Naturally, equal and opposite pressure forces are applied to the rails. According to the above 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, the forces must be sought in the places of their contact. In addition, the interaction of the crew and the Earth's gravitational field creates gravity.

The description of the movement of the crew in the curve can be made using general theorems of dynamics, which are consequences of the OZMS, or based on principles of mechanics(for example, the d'Alembert 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. The crew will be considered as a material plane with a mass equal to the mass of this system.

The center of mass lying in this plane performs 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 vehicle with the rail track, we can talk about concentrated forces, which are the resultant of all reactions of the rails on individual wheelsets from each of the rails. Moreover, the nature of the occurrence of reactive forces is insignificant;

? carriage movement along the track without elevation of the outer rail. On fig. 3.50 shows the design scheme of the crew moving along a curved path. The outer and inner rails, in this case, are located on the same level. On fig. 3.50 shows the forces acting on the crew and the reactions of the bonds. We emphasize that there are no there are no real centrifugal forces in this scheme.

Within the framework of Newton's geometric mechanics, the motion of a vehicle in a curve is described by general theorems of the system's dynamics.

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

t c a c - I a), (a)

where R) is the main vector of external forces.

Projecting both parts of the expression (A) on the accompanying natural coordinate axes, the center of which is at the center of mass of the vehicle, with unit vectors m, i, b and believe t s = T.

In the projection onto the main normal, we get that n \u003d F n, or

mV / p \u003d Fn (b)

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

In the expression projection (A) on the binormal we get:

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

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

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

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

Designing an expression d on the t axis, where t = nx b - vector product of unit vectors P And b, considering that KCl\u003d U St with t, U St - the moment of inertia of the crew about the axis tangent to the trajectory of the center of mass, we will have

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

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

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

Rice. 3.50.

Thus, the consistent application of the general theorems of dynamics allows us to establish in the problem under consideration all the phenomena associated with the passage of the crew of a curvilinear section of the track.

In fact, both wheels are subject to forces directed inside the curve. The resultant of these forces creates a moment about the center of mass of the vehicle, which can cause rotation and even tipping outward of the curve if V 2 N/p5" > 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 above statement, one can find only 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, during the movement of the carriage, the normal reactions of the outer and inner rails have different values. The outer rail thread is more loaded.

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

In classical mechanics, this state is called overturning, although there is actually no rollover yet. To find out when the state of actual overturning 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. Such a task is of purely academic interest, since, of course, it is unacceptable to bring a real system to such a state.

We emphasize once again that in explaining all the phenomena, we proceeded from the fact the movement of the car under the action 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. Choosing this axis at a different point leads to a change in the form of the left side of the equation of the moment theorem. 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 normal reactions in this case. However, this approach will lead to the wrong result: I osh \u003d M 1Sh1 \u003d 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 motion g x x ta s: J Cl? t+ T(g ks xx d)=^ M Kh.

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

(8 )

/ St? t+ t[g ks X a c) t = -teB + N ipp 25,

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

Let us show that the successive implementation of the necessary procedures allows us to find s w from the resulting equation). From fig. 3.50 shows that

g ks - bp + Hb 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 the expression (/).

In conclusion of the presentation of the problem, we point out that the consideration of the car in movement using Newton's methods of geometrical mechanics allows solving the problem without the introduction of fictitious and this inertia. It is only necessary to correctly use all the provisions of the mechanics. However, it should be noted that the use of this method may be associated with 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 the d'Alembert principle in the generally accepted form of the kinetostatics method. In this case, it is necessary to apply an additional

threading fictitious force of inertia: G* = -ta sp = -T-P. And eki-

page stops, i.e. now the acceleration of its center of mass a c= 0. In fig. 3.51 shows such resting system. All forces applied to it, including the force of inertia, must satisfy the kinetostatic equations balance, not movement, as in the previous case.

This circumstance allows us to find all unknown quantities from balance equation. In this case, the choice of the form of the equilibrium equations and the points with respect 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 = about.

1 at MP

Rice. 3.51. The design scheme of the forces acting on the crew under the same conditions as in Fig. 3.50 when using the d'Alembert principle

It is easy to see that the solutions of this system of equations coincide with the corresponding formulas obtained using the theory of dynamics. Thus, in the example under consideration, the application of the d'Alembert 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. no opposition.

In general, when solving many problems of mechanics, including the problem of crew movement in a curve, it is convenient to apply the d'Alembert principle. However, one should not associate any phenomena with action this force of inertia. For example, to 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 vehicle to overturn. This is not only illiterate, but also meaningless.

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

? movement of the carriage along a curve with an elevation of the outer rail. As it was shown, the processes that occur when the vehicle passes through curves without elevation of the outer rail are associated with undesirable consequences - uneven vertical loading of the rails, a significant normal horizontal response of the rail to the wheel, accompanied by increased wear of both wheels and rails, the possibility of overturning 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 by raising the outer rail 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 \u003d arcsin (L / 25), or at small angles

F A * L/2 S.

Rice. 3.52.

with an elevation of the 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 own plane in the same way as when fitting into a curve without raising the outer rail.

Consider a technique for solving the problem using general theorems of dynamics. We will assume that the center of mass of the vehicle moves along a circular curve of 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:

H sin cf L ~ H f A "r.

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

Projections of acceleration on the axis Su, and Cz^ are equal respectively:

a ux = a p sovf,; I. \u003d a „smy h.

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

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

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

-sof A = mgs ipf A + F ;

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

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 axis Cy, and Cr, 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 projections of accelerations 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 ω = ω (+ 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 cp L was not envisaged. However, in a practical design

wtf A ~ /g/25.

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

= -g^+ LG, „ + M gsh,;

T- = /yy#--1- r, ;

O \u003d + L / -5 - / U 0I / 5 + R p N.

Solving these equations, we get:

N...... =

mg + TU/G

Fri/77 K AND /77 „

• - +--+-n
• 2r 25 25

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

Let us now turn to the analysis of the results of solving the problem for 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 axis Su, not only the force //, but also the component of gravity takes part. Moreover, for a certain value AND\u003d 25K 2 / p? force R becomes zero:

Bearing in mind that

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

• (3.42)

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

When discussing the problem under consideration, sometimes there is a sophistical reasoning that the acceleration a p is directed horizontally, and gravity is vertical (see Fig. 3.52), and therefore it cannot form the considered acceleration a p at R= 0. This reasoning contains an error, since in the formation of horizontal acceleration, in addition to the force R, the normal reactions D r w u and / V o r also take part. The sum of these two reactions at small f A is equal to 1H tp + 1U oig \u003d mg. Therefore, gravity still participates in the formation of horizontal acceleration a p, but through the action of reactions N m And S oiG

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

Note that, in contrast to 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 rigorous reasoning, omit this term for expressions and N w do not do it.

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

the reaction of the inner rail is less than 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 IV oSH = mg|2(for small AND), those. the elevation of the outer rail allows not only to obtain RU= 0, but also equalize the pressure on the outer and outer rails. These circumstances make it possible to achieve more uniform wear values ​​for both rails.

However, due to the elevation of the outer rail, there is a possibility of a negative value R", which in a real system with non-retaining constraints corresponds to the process of sliding the vehicle along the axis y g those. inside the curve. Due to the same slope of the path, a redistribution of reactions can occur N w And N oh! dominant M sh.

Thus, studies of the motion of a vehicle in a curve along a path with an elevation of the outer rail, carried out using Newton's methods of geometrical mechanics, make it possible to analyze the state of the system without additional terminological hypotheses. There are no forces of inertia in the reasoning.

Let us now consider how the motion of the carriage in the same curve is described using the d'Alembert 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 the normal (centrifugal) force of inertia to the center of mass Р„ n), directed in the direction opposite to normal acceleration (Fig. 3.53):

Wherein system again stops, i.e. the crew is not moving along the track. Therefore, all equations of kineto-static equilibrium are valid:

I To= °-X r* = O.

/L^ypf, - G‘ p sovf* + G U[ = 0;

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

Substituting the value here, we obtain 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 the d'Alembert principle are identical.

Note, however, that in the final results do not include any inertial forces. This is understandable, since the d'Alembert principle, which underlies the method of kinetostatics, 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 the d'Alembert principle made it possible to simplify the calculations and can be recommended for practical calculations.

However, we emphasize once again that in reality there is no power 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).

In conclusion, we point out that the "Newton method" and "D'Alembert's method" in the problem under consideration were used only for the purpose of compiling differential equations of motion. At the same time, at the first stage, we do not receive any information, except for the differential equations themselves. The subsequent solution of the obtained equations and the analysis carried out 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.

d'Alembert 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 analytic statics. Here d'Alembert formulates the "basic principles of mechanics", among which are the "principle of inertia", the "principle of adding motions" 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, - writes d'Alembert, I, together with Newton, call the property of the body to maintain the state in which it is."

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

The "principle of equilibrium" is formulated as 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 from place to another body, then these bodies will be in equilibrium ". In the second part of the Treatise, d'Alembert proposed a general method for compiling 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 the "d'Alembert principle", according to which the forces applied to the points of the system can be decomposed into "acting", that is, those that cause the acceleration of the system, and "lost", necessary for the equilibrium of the system. d'Alembert believes that the forces that correspond to the "lost" acceleration form such a combination that does not affect the actual behavior of the system. In other words, if only a set of "lost" forces is applied to the system, then the system will remain at rest. The modern formulation of the d'Alembert principle was given by M. E. Zhukovsky in his "Course of Theoretical Mechanics": "If at any point in time the system is stopped, it is moving, and we add to it, in addition to its driving forces, all the forces of inertia corresponding to a given point in time, then an equilibrium will be observed, while all the forces of pressure, tension, etc. developing between the parts of the system at such an equilibrium, will be real forces of pressure, tension, etc. when the system moves at the considered moment of time ". It should be noted that d'Alembert himself, when presenting his principle, did not resort either to the concept of force (considering that it is 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 the d'Alembert 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 an essential role in the final transformation of mechanics into analytical mechanics.

Analytical mechanics of a material point and Euler's 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 XVIII century. His work is striking in the insight of research thought, the universality of talent and the huge amount of scientific heritage left behind.

Already in the first years of his scientific activity in St. Petersburg (Euler arrived in Russia in 1727), he drew up a program of a grandiose and comprehensive cycle of work in the field of mechanics. This appendix is ​​found in his two-volume work "Mechanics or the science of motion, stated analytically" (1736). Euler's Mechanics was the first systematic course in Newtonian mechanics. It contained the basics of the dynamics of a point - by mechanics, Euler understood the science of movement, in contrast to the science of the balance of forces, or statics. The defining feature of Euler's "Mechanics" was the wide use of a new mathematical apparatus - differential and integral calculus. Briefly characterizing the main works on mechanics that appeared at the turn of the 17th-18th centuries, Euler noted the son-tethiko-geometric style of their work, which created a lot of work for readers. It is in this manner that Newton's Elements and the later Foronomia (1716) by J. Herman were written. Euler points out that the works of Hermann and Newton are stated "according to the custom of the ancients with the help of synthetic geometric proofs" without the use of analysis, "only through which can one achieve a complete understanding of these things."

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

Euler's work on the dynamics of a rigid body "Theory of motion of rigid bodies" has a large introduction of six sections, where the dynamics of a point is again outlined. A number of changes have been made to the introduction: in particular, the equations of motion of a point are written using the projection on the axis of fixed rectangular coordinates (and not on the tangent, main normal and normal, that is, the axis of an immovable natural trihedron associated with trajectory points, as in "Mechanics") .

The "Treatise on the Motion of Rigid Bodies" following the introduction consists of 19 sections. The treatise is based on the d'Alembert principle. Briefly dwelling on 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 the instantaneous angular velocity, angular acceleration on the coordinate axes, the so-called Euler angles are used, etc. Next, the properties of the moment of inertia are described, after which Euler proceeds to the dynamics of a rigid body proper.He derives differential equations for the rotation of a heavy body around its immovable 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 about 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, the theory of stability and theory of small vibrations, celestial mechanic and 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 smallest value for a real trajectory, if we consider

the last in the 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 rigorous justification for one material point, tests the actions of central forces. During 1746-1749 pp. Euler wrote several papers on the figures of equilibrium 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: the d'Alembert principle and the Maupertuis-Euler principle of least action. Based on these principles, Lagrange built a system of analytical mechanics.

When a material point moves, its acceleration at each moment of time is such that the given (active) forces applied to the point, the reactions of the bonds and the fictitious d'Alembert force Ф = - that form a balanced system of forces.

Proof. Consider the motion of a non-free material point with a mass T in an inertial frame of reference. According to the basic law of dynamics and the principle of release from bonds, we have:

where F is the resultant of the given (active) forces; N is the resultant of the reactions of all bonds imposed on the point.

It is easy to transform (13.1) to the form:

Vector Ф = - that called the d'Alembert force of inertia, the force of inertia, or simply d'Alembert's power. In what follows, we will use only the last term.

Equation (13.3), expressing the d'Alembert principle in symbolic form, is called kinetostatics equation material point.

It is easy to obtain a generalization of the d'Alembert principle for a mechanical system (system P material points).

For any To th point of the mechanical system, equality (13.3) is satisfied:

Where ? To - resultant of given (active) forces acting on To-th point; N To - resultant of the reactions of the bonds superimposed on k-th point; F k \u003d - that k- d'Alembert force To-th point.

Obviously, if the equilibrium conditions (13.4) are met for each triple of forces F*, N* : , Ф* (To = 1,. .., P), then the whole system 3 P forces

is balanced.

Consequently, during the movement of a mechanical system at each moment of time, the active forces applied to it, the reactions of the bonds and the d'Alembert forces of the points of the system form a balanced system of forces.

The forces of the system (13.5) are no longer convergent, therefore, as is known from statics (section 3.4), the necessary and sufficient conditions for its equilibrium have the following form:

Equations (13.6) are called the equations of the kinetostatics of a mechanical system. For calculations, the projections of these vector equations on the axes passing through the moment point are used ABOUT.

Remark 1. Since the sum of all internal forces of the system, as well as the sum of their moments with respect to any point, are equal to zero, then in equations (13.6) it is sufficient to take into account only the reactions external connections.

The equations of kinetostatics (13.6) are usually used to determine the reactions of the constraints of a mechanical system when the motion of the system is given, and therefore the accelerations of the points of the system and the d'Alembert forces that depend on them are known.

Example 1 Find support reactions A And IN shaft with its uniform rotation at a frequency of 5000 rpm.

Point masses are rigidly connected to the shaft gp= 0.1 kg, t 2 = 0.2 kg. Sizes known AC - CD - DB = 0.4 m h= 0.01 m. Consider the mass of the shaft to be negligible.

Solution. To use the d'Alembert principle for a mechanical system consisting of two point masses, we indicate in the diagram (Fig. 13.2) the given forces (gravity) Gi, G 2, the reaction of the bonds N4, N # and the d'Alembert forces Ф|, Ф 2.

The directions of the Dalambres forces are opposite to the accelerations of point masses T b t 2y which uniformly describe circles of radius h around the axis AB shaft.

We find the magnitudes of the forces of gravity and Dalambres forces:

Here the angular velocity of the shaft co- 5000* l/30 = 523.6 s Ah, ah, Az, we obtain the equilibrium conditions for a flat system of parallel forces Gi, G 2 , 1Chd, N tf , Ф ь Ф 2:

From the equation of moments we find N in = - + - 1 - - - 2 --- =

(0.98 + 274) 0.4 - (548 -1.96) 0.8 w "

272 N, and from the projection equation on

axis Ay:Na \u003d -N B + G, + G 2 + F, -F 2 \u003d 272 + 0.98 + 1.96 + 274-548 \u003d 0.06 N.

The equations of kinetostatics (13.6) can also be used to obtain differential equations of motion of the system, if they are composed in such a way that the reactions of the bonds are excluded and, as a result, it becomes possible to obtain the dependences of the accelerations on the given forces.

Forces of inertia 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 mass of a point and its acceleration, taken with a minus sign, i.e. Inertial forces in dynamics are used in the following cases:

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

Expression in projections on the axes of Cartesian coordinates

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

With a curvilinear motion of a point, the force of inertia can be decomposed into tangential and normal:; , - modulus of tangential and normal accelerations; - radius of curvature of the trajectory;

V- point speed.

d'Alembert's principle for a material point

If to not free to a material point moving under the action of applied active forces and reaction forces of bonds, apply its inertia force, then at any 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 - the resultant of the active forces applied to the point; - the resultant of the reactions of the bonds imposed on the point; force of inertia of a material point. Note: In fact, the force of inertia of a material point is not applied 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 inertial forces of all points of the system, as well as the geometric sum of the main moments of these forces relative to a certain center for a non-free mechanical system at any time are equal to zero, i.e.

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

The main vector and the main moment of the forces of inertia of the points of the system are determined separately for each rigid body included in this mechanical system. Their definition is based on the Poinsot method known from statics about 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 of its motion can be brought to the center of mass and replaced by the main vector * and the main moment about the center of mass. They are determined by the formulas i.e. for any motion of a rigid body, the main vector of inertial forces is equal with a minus sign to the product of the body mass and the acceleration of the center of mass of the body; ,Where r kc -- radius vector k-th point drawn from the center of mass. These formulas in particular cases of motion of a rigid body have the form:

1. Progressive movement.

2. Rotation of a body about 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 movement or balance 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 movement of points of a mechanical system. These constraints can be written as equations or inequalities.

Geometric links-- connections, the equations of which 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.

Nonholonomic constraints-- differential non-integrable connections.

Stationary communications -- connections, the equations of which do not explicitly include time.

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

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

Unilateral(non-retaining) links - links that restrict 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 displacements that are allowed by constraints imposed on the system.

Possible The displacement of a mechanical system is a set of simultaneous possible displacements of the points of the system that are compatible with constraints. Let the mechanical system be a crank mechanism.

Possible moving point A is a displacement which, due to its smallness, is considered rectilinear and directed perpendicular to OA.

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

Valid The displacements of the points of the system are also called elementary displacements, which 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 displacements that can be communicated to the points of the system at a fixed point in time.

Principle of possible displacements (Lagrange principle)

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

For balance For a non-free mechanical system with bilateral, stationary, holonomic and ideal constraints, 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 equals a 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 the d'Alembert principle, the totality of the active forces applied to the mechanical system, the reaction forces of the bonds and the forces of inertia of all points of the system forms a balanced system of forces.

If the principle of possible displacements (the Lagrange principle) is applied to such a system, then we obtain the combined Lagrange-D'Alembert principle or general equation of dynamics.formulation of this principle.

When moving not free of a mechanical system with two-way, ideal, stationary and holonomic constraints, the sum of elementary works of all active forces and inertia forces applied to the points of the system on any possible displacement of the system is equal to 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 time derivative of the partial derivative of the kinetic energy of the system with respect to the generalized velocity 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 in terms of the potential energy of the system by the formula

Then the Lagrange equations are 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 represent it in the form where T - P \u003d 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 oscillations of systems.

The equilibrium position can be stable, unstable and indifferent.

sustainable equilibrium position - a position of equilibrium at which the points of a mechanical system, derived from this position, subsequently move under the action of forces in the immediate vicinity near their equilibrium position.

This movement will have a varying degree of repetition in time, i.e., the system will perform an oscillatory movement.

unstable equilibrium position - a position of equilibrium from which, with an arbitrarily small deviation of the points of the system, in the future, the acting forces will further remove the points from their equilibrium position .

indifferent equilibrium position - the 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 for determining the stable equilibrium position of a mechanical system.

Consider the definition of a stable equilibrium based on Lagrange-Dirichlet theorems

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

Impact phenomenon. Impact force and impact impulse

The phenomenon in which the velocities of the points of the body change by a finite amount in a negligibly small period of time is called blow. This period of time is called impact time. During an impact, an impact force acts for an infinitely small period of time. strike force is called a force whose momentum during the impact is a finite value.

If the modulo finite force acts over time, starting its action at a point in time , then its momentum has the form

Also, when the 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 a material point during the impact can be ignored;

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

Theorem on the change in the 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 impulses applied to the points of the systems, Where - the amount of motion of the mechanical system at the moment of termination of the action of impact forces, - the amount of motion of the mechanical system at the moment the impact forces begin to act, - external shock impulse.

The d'Alembert principle makes it possible to formulate the problems of the dynamics of mechanical systems as problems of statics. In this case, the dynamic differential equations of motion are given the form of equilibrium equations. Such a method is called kinetostatic method .

d'Alembert's principle for a material point: « At each moment in time of the movement of a material point, the active forces actually acting on it, the reactions of the bonds and the force of inertia conditionally applied to the point form a balanced system of forces»

point inertia force called a vector quantity that has the dimension of a force equal in absolute value to the product of the mass of a point and its acceleration and directed opposite to the acceleration vector

. (3.38)

Considering a mechanical system as a set of material points, each of which is affected, according to the d'Alembert principle, by balanced systems of forces, we have consequences from this principle in relation to the system. The main vector and the main moment relative to any center of external forces applied to the system and the forces of inertia of all its points are equal to zero:

(3.39)

Here external forces are active forces and reactions of bonds.

The main vector of inertial forces of a mechanical system is equal to the product of the mass of the system and the acceleration of its center of mass and is directed in the direction opposite to this acceleration

. (3.40)

The main moment of inertia forces system relative to an arbitrary center ABOUT equal to the time derivative of its angular momentum with respect to the same center

. (3.41)

For a rigid body rotating about a fixed axis Oz, we find the main moment of the forces of inertia about this axis

. (3.42)

3.8. Elements of analytical mechanics

The section "Analytical Mechanics" considers the general principles and analytical methods for solving problems in the mechanics of material systems.

3.8.1. Possible movements of the system. Classification

some connections

Possible point movements
any imaginary, infinitely small displacements of them, allowed by the constraints imposed on the system, at a fixed point in time, are called mechanical systems. A-priory, number of degrees of freedom of a mechanical system is the number of its independent possible displacements.

The connections imposed on the system are called ideal , if the sum of the elementary works of their reactions to any of the possible displacements of the points of the system is equal to zero

. (3. 43)

Connections for which the restrictions imposed by them are preserved at any position of the system are called holding back . Relations that do not change in time, the equations of which explicitly do not include time, are called stationary . The connections that limit only the displacements of the points of the system are called geometric , and the limiting speeds are kinematic . In the future, we will consider only geometric relationships and those kinematic ones that can be reduced to geometric ones by integration.

3.8.2. The principle of possible movements

For the equilibrium of a mechanical system with confining ideal and stationary constraints, it is necessary and sufficient that

the sum of the elementary works of all active forces acting on it, on any possible displacements of the system, was equal to zero

. (3.44)

In projections on the coordinate axes:

. (3.45)

The principle of possible displacements allows us to establish in a general form the conditions for the equilibrium of any mechanical system, without considering the equilibrium of its individual parts. In this case, only the active forces acting on the system are taken into account. Unknown reactions of ideal bonds are not included in these conditions. At the same time, this principle makes it possible to determine unknown reactions of ideal bonds by discarding these bonds and introducing their reactions into the number of active forces. When the bonds whose reactions must be determined are discarded, the system additionally acquires the corresponding number of degrees of freedom.

Example 1 . Find the relationship between forces And jack, if it is known that with each turn of the handle AB = l, screw WITH extends to the extent h(Fig. 3.3).

Solution

The possible movements of the mechanism are the rotation of the handle  and the movement of the load  h. The condition of equality to zero of elementary work of forces:

pl– Qh = 0;

Then
. Since h 0, then

3.8.3. General variational equation of dynamics

Consider the motion of a system consisting of n points. Active forces act on it and bond reactions .(k = 1,…,n) If we add to the acting forces the inertia forces of the points
, then, according to the d'Alembert principle, the resulting system of forces will be in equilibrium and, therefore, the expression written on the basis of the principle of possible displacements (3.44) is valid:

. (3.46)

If all connections are ideal, then the 2nd sum is equal to zero and in projections on the coordinate axes, equality (3.46) will look like this:

The last equality is a general variational equation of dynamics in projections on the coordinate axes, which allows one to compose differential equations of motion of a mechanical system.

The general variational equation of dynamics is a mathematical expression d'Alembert-Lagrange principle: « When a system is in motion, subject to stationary, ideal, restraining constraints, at any given moment of time, the sum of the elementary works of all active forces applied to the system and the forces of inertia on any possible displacement of the system is equal to zero».

Example 2 . For a mechanical system (Fig. 3.4), consisting of three bodies, determine the acceleration of the load 1 and the tension of the cable 1-2 if: m 1 = 5m; m 2 = 4m; m 3 = 8m; r 2 = 0,5R 2; radius of gyration of block 2 i = 1,5r 2. Roller 3 is a continuous homogeneous disk.

Solution

Let's depict the forces that do elementary work on a possible displacement  s cargo 1:

We write the possible displacements of all bodies through the possible displacement of load 1:

We express the linear and angular accelerations of all bodies in terms of the desired acceleration of load 1 (the ratios are the same as in the case of possible displacements):

.

The general variational equation for this problem has the form:

Substituting the previously obtained expressions for active forces, inertial forces and possible displacements, after simple transformations, we obtain

Since  s 0, therefore, the expression in brackets containing the acceleration is equal to zero A 1 , where a 1 = 5g/8,25 = 0,606g.

To determine the tension of the cable holding the load, we release the load from the cable, replacing its action with the desired reaction . Under the influence of given forces ,and the inertial force applied to the load
he is in balance. Therefore, the d’Alembert principle is applicable to the considered load (point), i.e. we write that
. From here
.

3.8.4. Lagrange equation of the 2nd kind

Generalized coordinates and generalized velocities. Any mutually independent parameters that uniquely determine the position of a mechanical system in space are called generalized coordinates . These coordinates, denoted q 1 ,....q i , can have any dimension. In particular, the generalized coordinates may be displacements or rotation angles.

For the systems under consideration, the number of generalized coordinates is equal to the number of degrees of freedom. The position of each point of the system is a single-valued function of the generalized coordinates

Thus, the motion of the system in generalized coordinates is determined by the following dependencies:

The first derivatives of generalized coordinates are called generalized speeds :
.

Generalized forces. Expression for the elementary work of a force on a possible move
looks like:

.

For the elementary work of the system of forces, we write

Using the obtained dependencies, this expression can be written as:

,

where is the generalized force corresponding to i-th generalized coordinate,

. (3.49)

Thus, generalized force corresponding i-th generalized coordinate, is the coefficient of variation of this coordinate in the expression of the sum of elementary works of active forces on the possible displacement of the system . To calculate the generalized force, it is necessary to inform the system of a possible displacement, in which only the generalized coordinate changes q i. Coefficient at
and will be the desired generalized force.

Equations of system motion in generalized coordinates. Let a mechanical system be given with s degrees of freedom. Knowing the forces acting on it, it is necessary to compose differential equations of motion in generalized coordinates
. We apply the procedure for compiling the differential equations of motion of the system - the Lagrange equations of the 2nd kind - by analogy with the derivation of these equations for a free material point. Based on Newton's 2nd law, we write

We obtain an analogue of these equations, using the notation for the kinetic energy of a material point,

Partial derivative of kinetic energy with respect to the projection of velocity on the axis
is equal to the projection of the amount of motion on this axis, i.e.

To obtain the necessary equations, we calculate the derivatives with respect to time:

The resulting system of equations is the Lagrange equations of the 2nd kind for a material point.

For a mechanical system, we represent the Lagrange equations of the 2nd kind in the form of equations in which instead of projections of active forces P x , P y , P z use generalized forces Q 1 , Q 2 ,...,Q i and take into account in the general case the dependence of the kinetic energy on the generalized coordinates.

The Lagrange equations of the 2nd kind for a mechanical system have the form:

. (3.50)

They can be used to study the motion of any mechanical system with geometric, ideal, and confining constraints.

Example 3 . For the mechanical system (Fig. 3.5), the data for which are given in the previous example, draw up a differential equation of motion using the Lagrange equation of the 2nd kind,

Solution

The mechanical system has one degree of freedom. For the generalized coordinate we take the linear movement of the load q 1 = s; generalized speed - . With this in mind, we write the Lagrange equation of the 2nd kind

.

Let us compose an expression for the kinetic energy of the system

.

We express all angular and linear velocities in terms of the generalized velocity:

Now we get

Let us calculate the generalized force by composing the expression for elementary work on a possible displacement  s all active forces. Without friction forces, work in the system is performed only by the gravity of the load 1
We write the generalized force at  s, as a coefficient in elementary work Q 1 = 5mg. Next we find

Finally, the differential equation of motion of the system will have the form: