The alignment of forces is based on the principle of possible displacements. The principle of possible displacements. General equation of dynamics. Work of internal forces

The principle of possible displacements: for the equilibrium of a mechanical system with ideal constraints, it is necessary and sufficient that the sum of the elementary work of all active forces acting on it for any possible displacement is equal to zero. or in projections:.

The principle of possible displacements gives in general form the equilibrium conditions for any mechanical system, gives a general method for solving statics problems.

If the system has several degrees of freedom, then the equation of the principle of possible displacements is made for each of the independent displacements separately, i.e. there will be as many equations as the system has degrees of freedom.

The principle of possible displacements is convenient because, when considering a system with ideal connections, their reactions are not taken into account and it is necessary to operate only with active forces.

The principle of possible displacements is formulated as follows:

To mater. the system, subject to ideal connections, was at rest, it is necessary and sufficient that the sum of elementary work performed by active forces on possible displacements of points of the system is positive

General equation of dynamics- when a system with ideal connections moves at any given moment of time, the sum of the elementary work of all applied active forces and all inertial forces on any possible displacement of the system will be equal to zero. The equation uses the principle of possible displacements and the d'Alembert principle and allows you to compose differential equations of motion for any mechanical system. Gives a general method for solving problems of dynamics.

Sequence of drawing up:

a) applied forces acting on it are applied to each body, and also conventionally applied forces and moments of pairs of inertial forces;

b) inform the system about possible movements;

c) make up the equations of the principle of possible displacements, considering the system to be in equilibrium.

It should be noted that the general equation of dynamics can also be applied to systems with imperfect constraints, only in this case the reactions of imperfect constraints, such as, for example, the friction force or the rolling friction moment, must be classified as active forces.

Work on a possible displacement of both active and inertial forces is sought in the same way as elementary work on an actual displacement:

Possible force work:.

Possible work of the moment (pair of forces):.

Generalized coordinates of a mechanical system are independent parameters q 1, q 2,…, q S of any dimension that uniquely determine the position of the system at any time.

The number of generalized coordinates is S - the number of degrees of freedom of the mechanical system. The position of each ν-th point of the system, that is, its radius vector in the general case can always be expressed as a function of generalized coordinates:

The general equation of dynamics in generalized coordinates looks like a system of S equations as follows:

……..………. ;

………..……. ;

here is the generalized force corresponding to the generalized coordinate:

a is the generalized inertial force corresponding to the generalized coordinate:

The number of independent possible displacements of a system is called the number of degrees of freedom of this system. For example. a ball on a plane can move in any direction, but any of its possible displacements can be obtained as the geometric sum of two displacements along two mutually perpendicular axes. A free rigid body has 6 degrees of freedom.

Generalized forces. For each generalized coordinate, the corresponding generalized force can be calculated Q k.

The calculation is performed according to this rule.

To determine the generalized force Q k corresponding to the generalized coordinate q k, it is necessary to give this coordinate an increment (increase the coordinate by this value), leaving all other coordinates unchanged, calculate the sum of the work of all forces applied to the system on the corresponding displacements of points and divide it by the increment of the coordinate:

where is the displacement i-th point of the system, obtained by changing k-th generalized coordinate.

The generalized force is determined using elementary work. Therefore, this force can be calculated differently:

And since there is an increment of the radius vector due to the increment of the coordinate with the rest of the coordinates and time unchanged t, the relation can be defined as a partial derivative. Then

where the coordinates of the points are functions of the generalized coordinates (5).

If the system is conservative, that is, the motion occurs under the action of the forces of the potential field, the projections of which, where, and the coordinates of the points are functions of generalized coordinates, then

The generalized force of a conservative system is a partial derivative of the potential energy along the corresponding generalized coordinate with a minus sign.

Of course, when calculating this generalized force, the potential energy should be determined as a function of the generalized coordinates

P = P ( q 1 , q 2 , q 3 ,…,q s).

Remarks.

First. When calculating the generalized forces, the reactions of ideal constraints are not taken into account.

Second. The dimension of the generalized force depends on the dimension of the generalized coordinate.

Lagrange equations of the second kind are derived from the general equation of dynamics in generalized coordinates. The number of equations corresponds to the number of degrees of freedom:

To compose the Lagrange equation of the second kind, generalized coordinates are selected and generalized velocities are found . Find the kinetic energy of the system, which is a function of the generalized velocities , and, in some cases, generalized coordinates. The operations of kinetic energy differentiation, provided by the left-hand sides of the Lagrange equations, are performed. The obtained expressions are equated to generalized forces, to find which, in addition to formulas (26), the following are often used when solving problems:

In the numerator of the right side of the formula - the sum of the elementary work of all active forces on the possible displacement of the system, corresponding to the variation of the i-th generalized coordinate -. With this possible displacement, all other generalized coordinates do not change. The resulting equations are differential equations of motion of a mechanical system with S degrees of freedom.

CLASSIFICATION OF RELATIONS

The concept of bonds introduced in § 3 does not cover all their types. Since the considered even methods for solving problems of mechanics are generally applicable to systems not with any constraints, we will consider the question of constraints and their classification in somewhat more detail.

Constraints of any kind are called constraints that are imposed on the positions and velocities of points of a mechanical system and are satisfied regardless of which forces are applied to the system. Let's see how these links are classified.

Connections that do not change over time are called stationary, and those that change over time are called non-stationary.

The constraints imposing constraints on the positions (coordinates) of the points of the system are called geometric, and those imposing constraints on the velocity (the first derivatives of the coordinates with respect to time) of the points of the system are called kinematic or differential.

If the differential connection can be represented as geometric, that is, the dependence between the velocities established by this connection can be reduced to the dependence between the coordinates, then such a connection is called integrable, and otherwise - non-integrable.

Geometric and integrable differential constraints are called golsnomnshy constraints, and non-integrable differential constraints are called nonholonomic constraints.

By the type of bonds, mechanical systems are also divided into holonomic (with holonomic bonds) and nonholonomic (containing nonholonomic bonds).

Finally, a distinction is made between restraining connections (the constraints imposed by them persist at any position of the system) and non-restraining ones, which do not possess this property (as they say, the system can "free itself" from such connections). Let's look at some examples.

1. All the constraints considered in § 3 are geometric (holonomic) and, moreover, stationary. Moving LPF, shown in Fig. 271, a, will be for the load lying in it, when the position of the load is considered in relation to the axes Oxy, by a non-stationary geometric connection (the floor of the cabin, which implements the connection, changes its position in space over time).

2 The position of a wheel rolling without slipping (see Fig. 328) is determined by the coordinate of the center C of the wheel and the angle of rotation. When rolling, the condition or

This is a differential relationship, but the resulting equation is integrated and gives, that is, it is reduced to the relationship between the coordinates. Therefore, the imposed constraint is holonomic.

3. In contrast to a wheel for a ball rolling without sliding on a rough plane, the condition that the velocity of a point of the ball tangent to the plane is zero cannot be reduced (when the center of the ball is not moving in a straight line) to any dependences between the coordinates, determining the position of the ball. This is an example of a non-halogen communication. Another example is given by the constraints imposed on controlled motion. For example, if a condition (connection) is imposed on the movement of a point (rocket) that its speed at any moment in time should be directed to another moving point (plane), then this condition cannot be reduced to any dependence between the coordinates, and the connection is nonholonomic ...

4. In § 3 the connections shown in fig. are holding, and in Fig. 8 and 9 - non-holding (in Fig. 8, and the ball can leave the surface, and in Fig. 9 - move towards point A, crushing the thread). Taking into account the peculiarities of unstoppable bonds, we encountered in problems 108, 109 (§ 90) and in problem 146 (§ 125).

Let us proceed to consider another principle of mechanics, which establishes a general condition for the equilibrium of a mechanical system. By equilibrium (see § 1) we mean that state of the system in which all its points under the action of the applied forces are at rest with respect to the inertial frame of reference (we consider the so-called "absolute" equilibrium). At the same time, we will consider all communications imposed on the system to be stationary, and we will not specify this in the future every time.

Let us introduce the concept of possible work as an elementary work that a force acting on a material point could perform on a displacement that coincides with the possible displacement of this point. We will denote the possible work of the active force by the symbol, and the possible work of the N bond reaction by the symbol

Let us now give a general definition of the concept of ideal constraints, which we have already used (see § 123): constraints are called ideal for which the sum of the elementary workings of their reactions on any possible displacement of the system is equal to zero, that is,

The condition for the ideality of the constraints given in § 123 and expressed by equality (52), when they are simultaneously stationary, corresponds to definition (98), since for stationary constraints, each real displacement coincides with one of the possible ones. Therefore, examples of ideal connections are all examples given in § 123.

To determine the necessary equilibrium condition, we prove that if a mechanical system with ideal constraints is in equilibrium by the action of the applied forces, then for any possible displacement of the system, the equality

where is the angle between force and possible displacement.

Let us denote the resultant of all (both external and internal) active forces and reactions of connections acting on some point of the system, respectively, through. Then, since each of the points of the system is in equilibrium, and therefore, the sum of the work of these forces for any displacement of the point will also be equal to zero, i.e. Having compiled such equalities for all points of the system and adding them term by term, we obtain

But since the connections are ideal, they represent the possible displacements of the points of the system, the second sum by condition (98) will be equal to zero. Then the first sum is also equal to zero, i.e., equality (99) holds. Thus, it has been proved that equality (99) expresses the necessary condition for the equilibrium of the system.

Let us show that this condition is also sufficient, i.e., that if active forces satisfying equality (99) are applied to the points of a mechanical system at rest, then the system will remain at rest. Let us assume the opposite, that is, that the system will start moving and some of its points will actually move. Then the forces will perform work on these displacements and, according to the theorem on the change in kinetic energy, will be:

where, obviously, since in the beginning the system was at rest; therefore, and. But with stationary connections, the actual displacements coincide with some of the possible displacements, and on these displacements there must also be something that contradicts condition (99). Thus, when the applied forces satisfy condition (99), the system cannot leave the state of rest, and this condition is a sufficient condition for equilibrium.

The following principle of possible displacements follows from what has been proved: for the equilibrium of a mechanical system with ideal constraints, it is necessary and sufficient that the sum of elementary works of all active forces acting on it for any possible displacement of the system is equal to zero. The mathematically formulated equilibrium condition is expressed by equality (99), which is also called the equation of possible jobs. This equality can also be represented in analytical form (see § 87):

The principle of possible displacements establishes a general condition for the equilibrium of a mechanical system, which does not require consideration of the equilibrium of individual parts (bodies) of this system and allows, with ideal constraints, to exclude from consideration all previously unknown constraint reactions.

Establishing the general condition for the equilibrium of a mechanical system. According to this principle, for the equilibrium of a mechanical system with ideal constraints, it is necessary and sufficient that the sum of virtual works $A\_i$ only active forces on any possible displacement of the system was equal to zero (if the system is brought to this position with zero speeds).

The number of linearly independent equilibrium equations that can be drawn up for a mechanical system, based on the principle of possible displacements, is equal to the number of degrees of freedom of this mechanical system.

Possible displacements of a non-free mechanical system are called imaginary infinitesimal displacements allowed at a given moment by the constraints imposed on the system (in this case, the time that is explicitly included in the equations of non-stationary constraints is considered fixed). The projections of possible displacements onto the Cartesian coordinate axes are called variations Cartesian coordinates.

Virtual displacements are called the infinitesimal displacements allowed by the bonds in the "frozen time". Those. they differ from possible displacements only when the connections are rheonomical (clearly time-dependent).

If, for example, the system is imposed $l$ holonomic rheonomic bonds:

$f\; \_\; (\backslash \; alpha)\; (\backslash \; vec\; r,\; t)\; =\; 0,\; \backslash \; quad\; \backslash \; alpha\; =\; \backslash \; overline\; (1,\; l)$

That is the possible movement $\backslash \; Delta\; \backslash \; vec\; r$ are those that satisfy

$\backslash \; sum\_\; (i\; =\; 1)\; ^\; (N)\; \backslash \; frac\; (\backslash \; partial\; f\; \_\; (\backslash \; alpha))\; (\backslash \; partial\; \backslash \; vec\; (r))\; \backslash \; cdot\; \backslash \; Delta\; \backslash \; vec\; (r)\; +\; \backslash \; frac\; (\backslash \; partial\; f\; \_\; (\backslash \; alpha\; ))\; (\backslash \; partial\; t)\; \backslash \; Delta\; t\; =\; 0,\; \backslash \; quad\; \backslash \; alpha\; =\; \backslash \; overline\; (1,\; l)$

And virtual $\backslash \; delta\; \backslash \; vec\; r$:

$\backslash \; sum\_\; (i\; =\; 1)\; ^\; (N)\; \backslash \; frac\; (\backslash \; partial\; f\; \_\; (\backslash \; alpha))\; (\backslash \; partial\; \backslash \; vec\; (r))\; \backslash \; delta\; \backslash \; vec\; (r)\; =\; 0,\; \backslash \; quad\; \backslash \; alpha\; =\; \backslash \; overline\; (1\; ,\; l)$

Generally speaking, virtual displacements have nothing to do with the process of motion of the system - they are introduced only in order to reveal the relations of forces existing in the system and to obtain equilibrium conditions. The smallness of the displacements is necessary so that the reactions of ideal connections can be considered unchanged.

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Literature

• Bukhgolts N.N. Basic course of theoretical mechanics. Part 1. 10th ed. - SPb .: Lan, 2009 .-- 480 p. - ISBN 978-5-8114-0926-6.
• Targ S.M. A short course in theoretical mechanics: Textbook for universities. 18th ed. - M .: Higher school, 2010 .-- 416 p. - ISBN 978-5-06-006193-2.
• A.P. Markeev Theoretical Mechanics: A Textbook for Universities. - Izhevsk: Research Center "Regular and Chaotic Dynamics", 2001. - 592 p. - ISBN 5-93972-088-9.

Excerpt describing the Principle of Possible Movements

- Nous u voila, [This is the point.] Why didn't you tell me anything before?
“In the mosaic briefcase he keeps under his pillow. Now I know, - said the princess without answering. “Yes, if there is a sin behind me, a great sin, then it’s hatred of this scum,” the princess almost shouted, completely changed. - And why is she rubbing herself in here? But I'll tell her everything, everything. The time will come!

While such conversations were taking place in the reception room and in the princess's rooms, the carriage with Pierre (for whom it was sent) and with Anna Mikhailovna (who found it necessary to go with him) drove into the courtyard of Count Bezukhoi. When the wheels of the carriage softly sounded on the straw laid under the windows, Anna Mikhailovna, turning to her companion with comforting words, made sure that he was sleeping in the corner of the carriage, and woke him up. Waking up, Pierre followed Anna Mikhailovna out of the carriage and then only thought of the meeting with his dying father that awaited him. He noticed that they had arrived not at the front entrance, but at the back entrance. While he was stepping off the step, two people in bourgeois clothes hurriedly ran away from the entrance to the shadow of the wall. Pausing, Pierre saw in the shadow of the house on both sides several more people of the same kind. But neither Anna Mikhailovna, nor the footman, nor the coachman, who could not help seeing these people, paid attention to them. Therefore, this is so necessary, Pierre decided with himself, and followed Anna Mikhailovna. Anna Mikhailovna hurried up the dimly lit narrow stone staircase, beckoning Pierre who was behind her, who, although he did not understand why he had to go to the count at all, and even less why he had to go up the back staircase, but judging by Anna Mikhailovna's confidence and haste, he decided to himself that it was necessary. Halfway down the stairs, they were nearly knocked off their feet by some people with buckets, who, with their boots knocking, ran to meet them. These people pressed against the wall to let Pierre and Anna Mikhailovna pass, and did not show the slightest surprise at the sight of them.
- Are half princesses here? - Anna Mikhailovna asked one of them ...
“Here,” the footman answered in a bold, loud voice, as if now everything was possible, “the door is to the left, mother.
“Maybe the count didn’t call me,” said Pierre as he walked out onto the platform, “I would go to my place.
Anna Mikhailovna stopped to catch up with Pierre.
- Ah, mon ami! - she said with the same gesture as with her son in the morning, touching his hand: - croyez, que je souffre autant, que vous, mais soyez homme. [Believe me, I suffer as much as you do, but be a man.]
- Right, I'll go? - asked Pierre, affectionately looking through his glasses at Anna Mikhailovna.

virtual speeds principle, - differential the variational principle of classical mechanics, expressing the most general conditions for the equilibrium of mechanical systems constrained by ideal connections.

According to V. p. P. Mechanic. the system is in equilibrium in a certain position if and only if the sum of elementary work of given active forces on any possible displacement that takes the system out of the considered position is equal to zero or less than zero:

at any given time.

Possible (virtual) movements of the system are called. elementary (infinitesimal) displacements of points of the system, allowed at a given time by the constraints imposed on the system. If the ties are holding (two-way), then the possible displacements are reversible, and in the condition (*) one should take the equal sign; if the connections are non-holding (one-way), then among the possible movements there are irreversible ones. When the system moves under the action of active forces of connection, they act on points of the system with certain reaction forces (passive forces), in the definition of which it is assumed that mechanic is fully taken into account. the effect of links on the system (in the sense that links can be replaced by the reactions caused by them) (the axiom of release). Connections called. ideal if the sum of the elementary works of their reactions, with the equal sign taking place for reversible possible displacements, and equal signs or greater than zero for irreversible displacements. Equilibrium positions of the system are such positions in which the system will remain all the time if it is placed in these positions with zero initial velocities; in this case, it is assumed that the constraint equations are satisfied for any t values.The active forces in the general case are assumed to be given functions and in condition (*) it should be considered

Condition (*) contains all the equations and laws of equilibrium of systems with ideal constraints, due to which we can say that all statics are reduced to one general formula (*).

The law of equilibrium, expressed by the VP, was first established by Guido Ubaldi on a lever and on moving blocks or pulleys. G. Galilei established it for inclined planes and considered this law as a general property of equilibrium of simple machines. J. Wallis put it in the basis of statics and from it he deduced the theory of equilibrium of machines. R. Descartes reduced all statics to a single principle, which essentially coincides with Galileo's principle. J. Bernoulli was the first to understand the great generality of the VP and its usefulness in solving problems of statics. J. Lagrange expressed the general form of paradigm, and thus reduced all statics to a hell of a general formula; he gave a (not completely rigorous) proof of the generalized semiconductor system for systems constrained by two-sided (retaining) constraints. The general statics formula for the equilibrium of any system of forces and the method of applying this formula developed by J. Lagrange were systematically used by him to derive the general equilibrium properties of a system of bodies and to solve various problems of statics, including the equilibrium problems of incompressible, as well as compressible and elastic fluids. J. Lagrange considered the principle of the general principle for all mechanics. J. Fourier and M.V. Ostrogradskii gave a rigorous proof of the invoice, as well as its extension to one-way (non-holding) connections.

Lit.: Lagrange J., Mecanique analytiquc, P., 1788 (Russian translation: Lagrange J., Analytical mechanic, M.-L., 1950); Fourier J., "J. de 1" Ecole Polytechnique ", 1798, t. II, p. 20; Ostrogradskiy MV, Lectures on analytical mechanics, Collected works, vol. 1 , Part 2, M.-L., 1946.

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The principle of possible displacements is formulated for solving statics problems using dynamic methods.

Definitions

Links all bodies that limit the movement of the body in question are called.

Ideal the connections are called, the work of reactions of which on any possible displacement is equal to zero.

By the number of degrees of freedom a mechanical system is the number of such independent parameters, with the help of which the position of the system is uniquely determined.

For example, a ball located on a plane has five degrees of freedom, and a cylindrical joint has one degree of freedom.

In the general case, a mechanical system can have an infinite number of degrees of freedom.

Possible movements we will call such displacements, which, firstly, are allowed by imposed constraints, and, secondly, are infinitesimal.

The crank-slider mechanism has one degree of freedom. The parameters can be taken as possible displacements -  ,  x and etc.

For any system, the number of independent possible displacements is equal to the number of degrees of freedom.

Let some system be in equilibrium and the connections imposed on this system are ideal. Then, for each point of the system, we can write the equation

, (102)

where
- resultant of active forces applied to a material point;

- resultant of bond reactions.

We multiply (102) scalar by the vector of the possible displacement of the point

,

since the connections are perfect, then always
, the sum of elementary works of active forces acting on the point will remain

. (103)

Equation (103) can be written for all material points, summing up which we obtain

. (104)

Equation (104) expresses the following principle of possible displacements.

For the equilibrium of a system with ideal constraints, it is necessary and sufficient that the sum of the elementary work of all active forces acting on it for any possible displacement of the system is equal to zero.

The number of equations (104) is equal to the number of degrees of freedom of the given system, which is an advantage of this method.

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

The principle of possible displacements allows solving problems of statics by methods of dynamics, on the other hand, the principle of d'Alembert provides a general method for solving problems of dynamics by methods of statics. Combining these two principles, you can get a general method for solving problems in mechanics, which is called the d'Alembert-Lagrange principle.

. (105)

When a system with ideal constraints moves at each moment of time, the sum of the elementary work of all applied active forces and all inertial forces on any possible displacement of the system will be equal to zero.

In analytical form, equation (105) has the form

Lagrange equations of the second kind

Generalized coordinates (q) are called such parameters independent of each other that uniquely determine the behavior of a mechanical system.

The number of generalized coordinates is always equal to the number of degrees of freedom of the mechanical system.

Any parameters of any dimension can be selected as generalized coordinates.

H
For example, when studying the motion of a mathematical pendulum with one degree of freedom, as a generalized coordinate q parameters can be accepted:

x(m), y(m) - point coordinates;

s(m) - arc length;

 (m 2) - sector area;

 (rad) - rotation angle.

When the system moves, its generalized coordinates will continuously change over time

Equations (107) are the equations of motion of the system in generalized coordinates.

Derivatives of generalized coordinates with respect to time are called generalized system speeds

. (108)

The dimension of the generalized velocity depends on the dimension of the generalized coordinate.

Any other coordinates (Cartesian, polar, etc.) can be expressed through generalized coordinates.

Along with the concept of a generalized coordinate, the concept of a generalized force is introduced.

Under generalized power understand the value equal to the ratio of the sum of elementary work of all forces acting on the system at some elementary increment of the generalized coordinate to this increment

, (109)

where S Is the index of the generalized coordinate.

The dimension of the generalized force depends on the dimension of the generalized coordinate.

To find the equations of motion (107) of a mechanical system with geometric constraints in generalized coordinates, differential equations in the Lagrange form of the second kind are used

. (110)

In (110) kinetic energy T system expressed in terms of generalized coordinates q S and generalized speeds .

The Lagrange equations provide a unified and fairly simple method for solving dynamic problems. The type and number of equations does not depend on the number of bodies (points) included in the system, but only on the number of degrees of freedom. With ideal constraints, these equations make it possible to exclude all previously unknown constraint reactions.