Formula of possible displacements. The principle of possible movements

Figure 2.4

Decision

Replace the distributed load with a concentrated force Q \u003d q ∙ DH... This force is applied in the middle of the segment DH - at the point L.

Strength F decompose into components, projecting it on 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 equation of work. In support A the reaction is decomposed into components X A, Y A.

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

δS L \u003d BL ∙ δφ 1;
δS C \u003d BC ∙ δφ 1.

In the same time

δS C \u003d CC V1 ∙ δφ 2

δφ 2 \u003d δφ 1 ∙ BC / CC V1.

It is more convenient to form the equation of work through the work of the moments of the 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 \u003d 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 \u003d 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). We will take for a possible movement of a part of the structure BDC turning around a fixed point B — δφ 3.

Figure 2.6

For point C δS C \u003d BC ∙ δφ 3, the instantaneous center of rotation for a part of the structure AKC there will be a point C V2, and moving point C put it.

Establishing the general condition for the equilibrium of a mechanical system. According to this principle, for 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 nonstationary constraints is considered fixed). The projections of possible displacements on the Cartesian coordinate axes are called variations Cartesian coordinates.

Virtual displacementsare 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:

$f\; \_\; (\backslash \backslash \; alpha)\; (\backslash \backslash \; vec\; r,\; t)\; \backslash u003d\; 0,\; \backslash \backslash \; quad\; \backslash \backslash \; alpha\; \backslash u003d\; \backslash \backslash \; overline\; (1,\; l)$

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

$\backslash \backslash \; sum\_\; (i\; \backslash u003d\; 1)\; ^\; (N)\; \backslash \backslash \; frac\; (\backslash \backslash \; partial\; f\; \_\; (\backslash \backslash \; alpha))\; (\backslash \backslash \; partial\; \backslash \backslash \; vec\; (r))\; \backslash \backslash \; cdot\; \backslash \backslash \; Delta\; \backslash \backslash \; vec\; (r)\; +\; \backslash \backslash \; frac\; (\backslash \backslash \; partial\; f\; \_\; (\backslash \backslash \; alpha\; ))\; (\backslash \backslash \; partial\; t)\; \backslash \backslash \; Delta\; t\; \backslash u003d\; 0,\; \backslash \backslash \; quad\; \backslash \backslash \; alpha\; \backslash u003d\; \backslash \backslash \; overline\; (1,\; l)$

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

$\backslash \backslash \; sum\_\; (i\; \backslash u003d\; 1)\; ^\; (N)\; \backslash \backslash \; frac\; (\backslash \backslash \; partial\; f\; \_\; (\backslash \backslash \; alpha))\; (\backslash \backslash \; partial\; \backslash \backslash \; vec\; (r))\; \backslash \backslash \; delta\; \backslash \backslash \; vec\; (r)\; \backslash u003d\; 0,\; \backslash \backslash \; quad\; \backslash \backslash \; alpha\; \backslash u003d\; \backslash \backslash \; overline\; (1\; ,\; l)$

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. Small displacements are needed 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.

An excerpt characterizing the Principle of Possible Displacement

- 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 for this scum,” the princess almost shouted, completely changed. - And why does she rub in here? But I will 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 of the same people. 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 to 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 stairs, 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 clattering, 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 everything was possible now, “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 room.
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.

Elements of Analytical Mechanics

In its attempts to cognize the world around it, human nature tends to reduce the system of knowledge in this area to the smallest number of starting points. This applies primarily to scientific fields. In mechanics, such a striving led to the creation of fundamental principles from which the basic differential equations of motion for various mechanical systems follow. This section of the tutorial is intended to acquaint the reader with some of these principles.

Let us begin the study of the elements of analytical mechanics by considering the issue of the classification of connections that occur not only in statics, but also in dynamics.

Link classification

Communication – any kind of restrictions imposed on the position and speed of points of a mechanical system.

Links are classified:

· By time change:

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

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

By the type of imposed kinematic restrictions:

- geometric relationships impose restrictions on the positions of the points of the system;

- kinematic, or differential connections impose speed limits on system points... 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 dependence between the velocities can be reduced to the dependence between the coordinates. A cylinder rolling without slipping is an example of an integrable differential relationship: the speed of the cylinder axis is related to its angular velocity according to the known formula, or, and after integration is reduced to a geometric relationship between the axis displacement and the angle of rotation of the cylinder in the form.

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

· If possible, "release" from communication:

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

- unstoppable connections - restrictions can be violated for a certain type of system movement, for example, a pendulum suspended from a crumpled thread.

Let's introduce several definitions.

· Possible (or virtual) moving (indicated) is elementary (infinitesimal) and is such that it does not violate the communications imposed on the system.

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

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

· The number of degrees of freedom of the mechanical system – is the number of independent possible movements.

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

For a mechanical system with geometric relationships, the number of independent coordinates defining 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 (rotations along the Euler angles are added), etc.

· Possible job – this is an elementary work of 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 resultant active forces and reaction forces acting on the point. Then the sum of the work of these forces for any displacement is also zero ... Summing up for all points, we get: ... The second term for ideal bonds is equal to zero, whence it is formulated principle of possible movements :

.

(3.82)

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

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

QUESTIONS FOR SELF-CONTROL

1. What movement of a point is called possible?

2. What is called a possible work of force?

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

D'Alembert principle

Let's rewrite the equation of dynamics to-th point of the mechanical system (3.27), moving the left part to the right. Let us introduce 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 come to the formulation d'Alembert principlenamed after the French mathematician and mechanic Jean Leron D'Alembert (1717-1783), Figure 3.13:

Figure 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 static equations can be applied to it.

In fact, this means that from a dynamic system, by adding inertial forces (d'Alembert forces), they pass to a pseudostatic (almost static) system.

Using the d'Alembert principle, one can obtain the estimate the main vector of inertia forces and the 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 АиВ (Fig. 3.14). Let's connect the axes Axyz rotating together with the body; the advantage of such axes is that in relation 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. We denote the projections of the main vector of all these forces on the Axyz axis through ( etc.), and their main points relative to the same axes - through ( etc.); moreover, since ω \u003d const, then = 0.

Figure 3.14

To determine dynamic reactions X A, Y A, Z A, X B, Y Bbearings, i.e. reactions arising 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 forces of all particles of the body, bringing them to the center A. Then the inertial forces will be represented by one force equal to and applied at point A , and a pair of forces with a moment equal to . The projections of this moment on the axis toand atwill be: , ; here again , because ω \u003d const.

Now, composing, according to the d'Alembert principle, equations (3.86) in projections on the Axyz axis and setting AB \u003d b,get

.

(3.87)

The last equation is satisfied identically, since .

The main vector of inertia forces , where t -body weight (3.85). When ω \u003d const center of mass С has only normal acceleration , where is the distance of point C from the axis of rotation. Therefore, the direction of the vector match the direction of the OS . Calculating projections on the coordinate axes and taking into account that, where - coordinates of the center of mass, we find:

To define and, consider some particle of a body with mass m k spaced from the axis at a distance h k.For her with ω \u003d const the inertial force also has only a centrifugal component , whose projection, like the vector R ",are equal.

The principle of possible displacements makes it possible to solve a variety of problems on the equilibrium of mechanical systems - to find unknown active forces, to determine the reactions of bonds, to find the equilibrium positions of a mechanical system under the action of an applied system of forces. Let us illustrate this with specific examples.

Example 1. Find the magnitude of the force P, which holds heavy smooth prisms with masses in equilibrium. The bevel angle of the prisms is equal (Fig. 73).

Decision. Let's use the principle of possible displacements. Let us inform the system about the possible displacement and calculate the possible work of active forces:

The possible work of the force of gravity is zero, since the force is perpendicular to the vector of elementary displacement of the point of application of the force. Substituting the value here and equating the expression to zero, we get:

Since, then the expression in brackets is equal to zero:

From here we find

Example 2. A homogeneous beam AB of length and weight P, loaded by a pair of forces with a given moment M, is fixed as shown in Fig. 74 and is at rest. Determine the reaction of the bar BD if it makes an angle a with the horizon.

Decision. The task differs from the previous one in that here it is required to find the reaction of the ideal connection. But in the equation of works expressing the principle of possible displacements, the reactions of ideal connections are not included. In such cases, the principle of possible displacement should be applied in conjunction with the principle of release from ties.

We mentally discard the rod BD, and consider its reaction S an active force of unknown magnitude. After that, we will inform the system of a possible movement (provided that this connection is completely absent). This will be an elementary rotation of the beam AB by an angle around the axis of the hinge A in one direction or the other (counterclockwise in Fig. 74). The elementary displacements of the points of application of active forces and the reaction S referred to them are equal:

We make the equation of work

Equating to zero the expression in parentheses, from here we find

Example 3. A homogeneous rod OA with a weight is fixed by means of a cylindrical hinge O and a spring AB (Fig. 75). Determine the positions in which the rod can be in equilibrium, if the stiffness of the spring is equal to k, the natural length of the spring - and point B is on the same vertical line with point O.

Decision. Two active forces are applied to the rod ОА - its own weight and the elastic force of the spring where is the angle formed by the rod with the vertical line ОВ. The superimposed connections are ideal (in this case there is only one connection - the O hinge).

Let us tell the system a possible movement - an elementary rotation of the rod around the axis of the hinge O by an angle, calculate the possible work of active forces and equate it to zero:

Substituting here the expression for the force F and the value

after simple transformations, we obtain the following trigonometric equation for determining the angle (p at equilibrium of the rod:

The equation defines three values \u200b\u200bfor the angle:

Consequently, the rod has three equilibrium positions. Since the first two equilibrium positions exist if the condition is met. Equilibrium at always exists.

In conclusion, we note that the principle of possible displacements can be applied to systems with imperfect connections. The emphasis on the ideality of the connections is made in the formulation of the principle with one single purpose - to show that the equilibrium equations of mechanical systems can be composed without including the reactions of ideal connections in them, thereby simplifying the calculations.

For systems with imperfect connections, the principle of possible displacements should be reformulated as follows: for equilibrium of a mechanical system with restraining connections, among which there are imperfect connections, it is necessary and sufficient that the possible work of active forces and reactions of imperfect connections be equal to zero. It is possible, however, to dispense with the reformulation of the principle, conditionally referring the reactions of imperfect connections to the number of active forces.

Self-test questions

1. What is the main feature of a non-free mechanical system in comparison with a free one?

2. What is called a possible movement? Give examples.

3. How are the variations in the coordinates of points of the system determined during its possible movement (indicate three ways)?

4. How are links classified according to the form of their equations? Give examples of restraining and non-restraining connections, stationary and non-stationary.

5. When is the connection called ideal? Imperfect?

6. Give the verbal formulation and mathematical record of the principle of possible displacements.

7. How is the principle of possible displacements formulated for systems containing imperfect connections?

8. List the main types of problems that can be solved using the principle of possible displacements.

Exercises

Using the principle of possible displacements, solve the following problems from the collection of I.V. Meshchersky 1981 edition: 46.1; 46.8; 46.17; 2.49; 4.53.

Let us proceed to consider another principle of mechanics, which establishes the 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 each 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 movement that coincides with the possible movement 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, i.e.

The condition given in § 123 and expressed by equality (52) for the ideality of the constraints, 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 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 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; hence, 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 work 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.