The principle of possible movements. General dynamics equation

14.07.2023

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Establishing a 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 velocities).

The number of linearly independent equilibrium equations that can be compiled 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 movements imaginary infinitesimal displacements allowed at a given moment by constraints imposed on the system are called non-free mechanical systems (in this case, the time included explicitly in the equations of non-stationary constraints is considered fixed). Projections of possible displacements onto Cartesian coordinate axes are called variations Cartesian coordinates.

virtual movements are called infinitesimal displacements allowed by the connections, with "frozen time". Those. they differ from possible displacements only when the bonds are rheonomic (explicitly dependent on time).

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

$f_(\alpha)(\vec r, t) = 0, \quad \alpha = \overline(1,l)$

Then the possible movements $\Delta \vec r$ are those that satisfy

$\sum_(i=1)^(N) \frac(\partial f_(\alpha))(\partial \vec(r)) \cdot \Delta \vec(r) + \frac(\partial f_(\alpha ))(\partial t) \Delta t = 0, \quad \alpha = \overline(1,l)$

And virtual $\delta \vec r$ :

$\sum_(i=1)^(N) \frac(\partial f_(\alpha))(\partial \vec(r))\delta \vec(r) = 0, \quad \alpha = \overline(1 ,l)$

Virtual displacements, generally speaking, have nothing to do with the process of the system's motion - they are introduced only in order to reveal the relations of forces existing in the system and obtain equilibrium conditions. The smallness of the displacements is needed in order to be able to consider the reactions of ideal bonds as unchanged.

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Literature

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

An excerpt characterizing the Principle of possible displacements

– Nous y voila, [That's the point.] Why didn't you tell me before?
“In the mosaic briefcase he keeps under his pillow. Now I know,” said the princess, without answering. “Yes, if there is a sin for me, a big sin, then it is hatred for this bastard,” the princess almost shouted, completely changed. “And why is she rubbing herself here?” But I will tell her everything, everything. The time will come!

While such conversations were taking place in the waiting room and in the princess's rooms, the carriage with Pierre (for whom it was sent) and Anna Mikhailovna (who found it necessary to go with him) drove into the courtyard of Count Bezukhoy. When the wheels of the carriage sounded softly on the straw laid under the windows, Anna Mikhailovna, turning to her companion with comforting words, convinced herself that he was sleeping in the corner of the carriage, and woke him up. Waking up, Pierre got out of the carriage after Anna Mikhailovna, and then only thought of that meeting with his dying father that awaited him. He noticed that they did not drive up to the front, but to the back entrance. While he was getting off the footboard, two men in bourgeois clothes hurriedly ran away from the entrance into the shadow of the wall. Pausing, Pierre saw in the shadow of the house on both sides several more of the same people. But neither Anna Mikhailovna, nor the footman, nor the coachman, who could not but see these people, paid no attention to them. Therefore, this is so necessary, Pierre decided with himself, and followed Anna Mikhailovna. Anna Mikhailovna with hasty steps walked up the dimly lit narrow stone stairs, calling Pierre, who was lagging behind her, who, although he did not understand why he had to go to the count at all, and still less why he had to go along the back stairs, but , judging by the confidence and haste of Anna Mikhailovna, he decided to himself that this was necessary. Halfway down the stairs they were almost knocked down by some people with buckets, who, clattering with their boots, ran towards them. These people pressed against the wall to let Pierre and Anna Mikhailovna through, and did not show the slightest surprise at the sight of them.
- Are there half princesses here? Anna Mikhailovna asked one of them...
“Here,” the footman answered in a bold, loud voice, as if everything was already possible now, “the door is on the left, mother.”
“Perhaps the count did not call me,” said Pierre, while he went out onto the platform, “I would have gone to my place.
Anna Mikhailovna stopped to catch up with Pierre.
Ah, mon ami! - she said with the same gesture as in the morning with her son, touching his hand: - croyez, que je souffre autant, que vous, mais soyez homme. [Believe me, I suffer no less than you, but be a man.]
- Right, I'll go? asked Pierre, looking affectionately through his spectacles at Anna Mikhailovna.

Figure 2.4

Solution

Let us replace the distributed load with a concentrated force Q = qDH. This force is applied in the middle of the segment D.H.- at the point L.

Strength F decompose into components, projecting it onto the axis: horizontal F x cosα and vertical F y sinα.

Figure 2.5

To solve the problem using the principle of possible displacements, it is necessary that the structure can move and at the same time that there is one unknown reaction in the work equation. in support A The reaction is broken down into components. X A, Y A.

For determining X A change the design of the support A so that the point A could only move horizontally. We express the displacements of the points of the structure through the possible rotation of the part CDB around the dot B on the corner δφ 1, Part AKC construction in this case rotates around the point C V1- instantaneous center of rotation (Figure 2.5) by an angle δφ 2, and moving points L And C- will

δS L = BL∙δφ 1 ;
δS C = BC∙δφ 1.

In the same time

δS C = CC V1 ∙δφ 2

δφ 2 = δφ 1 ∙BC/CC V1.

It is more convenient to compose the work equation through the work of the moments of given forces, relative to the centers of rotation.

Q∙BL∙δφ 1 + F x ∙BH∙δφ 1 + F y ∙ED∙δφ 1 +
+ M∙δφ 2 — X A ∙AC V1 ∙δφ 2 = 0.

Reaction Y A does not do work. Transforming this expression, we get

Q∙(BH + DH/2)∙δφ 1 + F∙cosα∙BD∙δφ 1 +
+ F∙sinα∙DE∙δφ 1 + M∙δφ 1 ∙BC/CC V1 —
— X A ∙AC V1 ∙δφ 1 ∙BC/CC V1 = 0.

Reducing by δφ 1, we obtain an equation from which it is easy to find X A.

For determining Y A support structure A change so that when moving the point A only force did the work Y A(Figure 2.6). Let us take for the possible displacement of a part of the structure bdc rotation around a fixed point B — δφ 3.

Figure 2.6

For point C δS C = BC∙δφ 3, instantaneous center of rotation for part of the structure AKC there will be a point C V2, and moving the point C expressed.

It is necessary and sufficient that the sum of the work , of all active forces applied to the system on any possible displacement of the system, be equal to zero.

The number of equations that can be compiled for a mechanical system, based on the principle of possible displacements, is equal to the number of degrees of freedom of this very mechanical system.

Literature

Targ S. M. A short course in theoretical mechanics. Proc. for technical colleges. - 10th ed., revised. and additional - M.: Higher. school, 1986.- 416 p., ill.
The main course of theoretical mechanics (part one) N. N. Bukhgolts, publishing house "Nauka", Main editorial board of physical and mathematical literature, Moscow, 1972, 468 pages.

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See what the "Principle of possible movements" is in other dictionaries:

principle of possible movements

One of the variational principles of mechanics, which establishes the general condition for the equilibrium of a mechanical systems. According to V. p. p., for the equilibrium of the mechanical. systems with ideal constraints (see MECHANICAL CONNECTIONS) is necessary and sufficient that the sum of works dAi… … Physical Encyclopedia

Big Encyclopedic Dictionary

POSSIBLE MOVEMENTS PRINCIPLE, for the equilibrium of a mechanical system, it is necessary and sufficient that the sum of the work of all forces acting on the system for any possible displacement of the system be equal to zero. The possible displacement principle applies when… … encyclopedic Dictionary

One of the variational principles of mechanics (See Variational principles of mechanics), which establishes a general condition for the equilibrium of a mechanical system. According to V. p. p., for the equilibrium of a mechanical system with ideal connections (see Connections ... ... Great Soviet Encyclopedia

The principle of virtual speeds, the differential variational principle of classical mechanics, which expresses the most general conditions for the equilibrium of mechanical systems constrained by ideal connections. According to V. p. p. mechan. the system is in equilibrium... Mathematical Encyclopedia

For the equilibrium of a mechanical system, it is necessary and sufficient that the sum of the work of all forces acting on the system for any possible displacement of the system be equal to zero. The principle of possible displacements is applied in the study of equilibrium conditions ... ... encyclopedic Dictionary

For mechanical balance system it is necessary and sufficient that the sum of the work of all forces acting on the system for any possible displacement of the system is equal to zero. V. p. p. is used in the study of equilibrium conditions for complex mechan. systems… … Natural science. encyclopedic Dictionary

principle of virtual displacements- virtualiųjų poslinkių principas statusas T sritis fizika atitikmenys: engl. principle of virtual displacement vok. Prinzip der virtuellen Verschiebungen, n rus. the principle of virtual displacements, m; principle of possible movements, m pranc. principe des … Fizikos terminų žodynas

One of the variational principles of mechanics, according to Roma for a given class of mechanical movements compared with each other. system is valid for which physical. value, called action, has the smallest (more precisely, stationary) ... ... Physical Encyclopedia

Books

Theoretical mechanics. In 4 volumes. Volume 3: Dynamics. Analytical mechanics. Texts of lectures. Vulture of the Ministry of Defense of the Russian Federation, Bogomaz Irina Vladimirovna. The textbook contains two parts of a unified course in theoretical mechanics: dynamics and analytical mechanics. In the first part, the first and second problems of dynamics are considered in detail, also ...

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

According to V. p. p. mechan. the system is in equilibrium in a certain position if and only if the sum of the elementary works of the 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 point in time.

Possible (virtual) movements of the system are called. elementary (infinitely small) displacements of the points of the system, allowed at a given moment of time by the constraints imposed on the system. If the bonds are holding (two-way), then the possible displacements are reversible, and in the condition (*) one should take the equal sign; if the bonds are non-retaining (one-sided), then among the possible displacements there are irreversible ones. When the system moves under the action of active forces, the bonds act on the points of the system with certain reaction forces (passive forces), in the definition of which it is assumed that mechanical forces are fully taken into account. the action of bonds on the system (in the sense that bonds can be replaced by reactions caused by them) (liberability axiom). Connections called ideal, if the sum of the elementary works of their reactions, and the equal sign takes place for reversible possible displacements, and the equal signs or greater than zero - for irreversible displacements. The 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, it is assumed that the constraint equations are satisfied for any t values Active forces are generally assumed to be given functions and in condition (*) should be considered

The condition (*) contains all the equations and laws of equilibrium for systems with ideal connections, thanks to which we can say that all statics is reduced to one general formula (*).

The law of equilibrium, expressed by V. p. p., was first established by Guido Ubaldi on a lever and on moving blocks or chain hoists. G. Galilei established it for inclined planes and considered this law as a general property of the equilibrium of simple machines. J. Wallis put it at the foundation of statics and derived the theory of machine equilibrium from it. R. Descartes (R. Descartes) reduced all statics to a single principle, which, in essence, coincides with the principle of Galileo. J. Bernoulli was the first to understand the great generality of V.p.p. and its usefulness in solving problems of statics. J. Lagrange expressed V. p. p. in a general form and thereby reduced all statics to a single general formula; he gave a proof (not completely rigorous) of the VP for systems constrained by two-way (retaining) constraints. The general formula of statics 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 properties of the equilibrium of a system of bodies and solve various problems of statics, including problems of equilibrium of incompressible, as well as compressible and elastic fluids. J. Lagrange considered V. p. p. the basic principle for all mechanics. A rigorous proof of V. p. p., as well as its extension to one-sided (non-retaining) connections, was given by J. Fourier, M. V. Ostrogradsky.

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

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Elements of analytical mechanics

In its attempts to cognize the surrounding world, human nature tends to strive to reduce the system of knowledge in a given area to the smallest number of initial positions. This primarily applies to scientific fields. In mechanics, this desire has led to the creation of fundamental principles from which follow the basic differential equations of motion for various mechanical systems. This section of the tutorial is intended to introduce the reader to some of these principles.

Let us begin the study of the elements of analytical mechanics by examining the question of the classification of relations that occur not only in statics, but also in dynamics.

Relationship classification

Connection – any kind of restrictions imposed on the positions and speeds of the points of a mechanical system.

Relationships are classified:

By change over time:

- non-stationary communications, those. changing over time. A support moving in space is an example of a non-stationary connection.

- fixed communications, those. not changing over time. Stationary links include all links discussed in the "Statics" section.

By the type of imposed kinematic restrictions:

- geometric connections impose restrictions on the positions of points in the system;

- kinematic, or differential connections impose restrictions on the speed of points in the system. If possible, reduce one type of relationship to another:

- integrable, or holonomic(simple) connection, if the kinematic (differential) connection can be represented as a geometric. In such connections, the dependences between the velocities can be reduced to the dependence between the coordinates. A cylinder rolling without slipping is an example of an integrable differential constraint: the speed of the cylinder axis is related to its angular velocity by the well-known formula , or , and after integration it is reduced to a geometric relationship between the axis displacement and the cylinder rotation angle in the form

- non-integrable, or nonholonomic connection– if the kinematic (differential) connection cannot be represented as a geometric. An example is the rolling of a ball without slipping during its non-rectilinear motion.

If possible, "release" from communication:

- holding ties, under which the restrictions imposed by them are always preserved, for example, a pendulum suspended from a rigid rod;

- non-retaining ties - restrictions can be violated for a certain type of system motion, for example, a pendulum suspended on a crumpled thread.

Let us introduce several definitions.

· Possible(or virtual) moving(denoted) is elementary (infinitely small) and is such that it does not violate the constraints imposed on the system.

Example: a point, being on the surface, as possible has a set of elementary displacements in any direction along the reference surface, without breaking away from it. The movement of a point, leading to its detachment from the surface, breaks the connection and, in accordance with the definition, is not a possible movement.

For stationary systems, the usual real (real) elementary displacement is included in the set of possible displacements.

· Number of degrees of freedom of a mechanical system – is the number of its independent possible displacements.

So, when a point moves on a plane, any possible movement of it is expressed in terms of its two orthogonal (and hence independent) components.

For a mechanical system with geometric constraints, the number of independent coordinates that determine the position of the system coincides with the number of its degrees of freedom.

Thus, a point on a plane has two degrees of freedom. Free material point - three degrees of freedom. A free body has six (turns at Euler angles are added), etc.

· Possible work – is the elementary work of a force on a possible displacement.

The principle of possible movements

If the system is in equilibrium, then for any of its points the equality holds, where are the resultants of the active forces and reaction forces acting on the point. Then the sum of the work of these forces for any displacement is also equal to zero . Summing up for all points, we get: . The second term for ideal bonds is equal to zero, whence we formulate principle of possible movements :

(3.82)

Under conditions of equilibrium of a mechanical system with ideal connections, the sum of the elementary works of all active forces acting on it for any possible displacement of the system is equal to zero.

The value of the principle of possible displacements lies in the formulation of equilibrium conditions for a mechanical system (3.81), in which unknown reactions of constraints do not appear.

QUESTIONS FOR SELF-CHECKING

1. What movement of a point is called possible?

2. What is called the possible work of the force?

3. Formulate and write down the principle of possible movements.

d'Alembert principle

Let's rewrite the equation of dynamics To th point of the mechanical system (3.27), transferring the left side to the right. Let us introduce into consideration the quantity

The forces in equation (3.83) form a balanced system of forces.

Extending this conclusion to all points of the mechanical system, we arrive at the formulation d'Alembert principle, named after the French mathematician and mechanic Jean Leron D'Alembert (1717–1783), Fig. 3.13:

Fig.3.13

If all the forces of inertia are added to all the forces acting in a given mechanical system, the resulting system of forces will be balanced and all the equations of statics can be applied to it.

In fact, this means that from a dynamic system, by adding inertia forces (D'Alembert forces), one passes to a pseudostatic (almost static) system.

Using the d'Alembert principle, one can obtain the estimate principal vector of inertial forces And main moment of inertia about the center as:

Dynamic reactions acting on the axis of a rotating body

Consider a rigid body rotating uniformly with an angular velocity ω around the axis fixed in bearings A and B (Fig. 3.14). Let us connect with the body the axes Axyz rotating with it; the advantage of such axes is that with respect to them the coordinates of the center of mass and the moments of inertia of the body will be constant values. Let the given forces act on the body. Let us denote the projections of the main vector of all these forces on the axis Axyz through ( etc.), and their main moments about the same axes - through ( etc.); meanwhile, because ω = const, then = 0.

Fig.3.14

To determine dynamic responses X A, Y A, Z A, X B , Y B bearings, i.e. reactions that occur during the rotation of the body, we add to all the given forces acting on the body and the reactions of the bonds of the inertial force of all particles of the body, bringing them to the center A. Then the forces of inertia will be represented by one force equal to and applied at point A , and a pair of forces with a moment equal to . Projections of this moment on the axis To And at will be: , ; here again , because ω = const.

Now, composing equations (3.86) in accordance with the d’Alembert principle in projections on the Axyz axis and setting AB =b, we get

(3.87)

Last Equation is satisfied identically, since .

The main vector of inertial forces , Where T - body weight (3.85). At ω =const center of mass C has only normal acceleration , where is the distance of point C from the axis of rotation. Therefore, the direction of the vector coincide with the direction of the OS . Computing projections on the coordinate axes and taking into account that , where - coordinates of the center of mass, we find:

To determine and , consider some particle of the body with mass m k , spaced from the axis at a distance h k . For her at ω =const the force of inertia also has only a centrifugal component , projections of which, as well as vectors R", are equal.