The principle of possible displacements is theoretical mechanics from scratch. Calculation of the reaction of the support according to the principle of possible displacements
As is known from the course of theoretical mechanics, the equilibrium condition of an object can have a force or energy formulation. The first option is the condition of equality to zero of the main vector and the main moment of all forces and reactions acting on the body. The second approach (variational), called the principle of possible displacements, turned out to be very useful for solving a number of problems in structural mechanics.
For a system of absolutely rigid bodies, the principle of possible displacements is formulated as follows: if a system of absolutely rigid bodies is in equilibrium, then the sum of the work of all external forces on any possible infinitesimal displacement is equal to zero. Possible (or virtual) movement is called, which does not violate the kinematic connections and the continuity of the bodies. For the system in Fig. 3.1, only rotation of the rod relative to the support is possible. When turning through an arbitrary small angle, forces and do work 
According to the principle of possible displacements, if the system is in equilibrium, then there must be 
. Substituting here the geometric relations 
we obtain the equilibrium condition in the force formulation
The principle of possible displacements for elastic bodies is formulated as follows: if a system of elastic bodies is in equilibrium, then the sum of the work of all external and internal forces on any possible infinitesimal displacement is equal to zero. This principle is based on the concept of the total energy of an elastic deformed system P. If the structure is loaded statically, then this energy is equal to the work done by external U and internal W forces when the system is transferred from the deformed state to the initial one:
With this translation, external forces do not change their value and do negative work U= -F . In this case, the internal forces decrease to zero and do positive work, since these are the adhesion forces of the particles of the material and are directed in the direction opposite to the external load:
where 
- specific potential energy of elastic deformation; V is the volume of the body. For a linear system , where . According to the Lagrange-Dirichlet theorem, the state of stable equilibrium corresponds to the minimum of the total potential energy of the elastic system, i.e.
The last equality fully corresponds to the formulation of the principle of possible displacements. The energy increments dU and dW can be calculated on any possible displacements (deviations) of the elastic system from the equilibrium state. To calculate structures that meet the requirements of linearity, the infinitely small possible displacement d can be replaced by a very small final displacement , which can be any deformed state of the structure created by an arbitrarily chosen system of forces. With this in mind, the resulting equilibrium condition should be written as
The work of external forces
Consider the method of calculating the work of external forces on the actual and possible displacement. The rod system is loaded with forces and (Fig. 3.2, a), which act simultaneously, and at any time the ratio remains constant. If we consider the generalized force, then by the value at any time you can calculate all other loads (in this case, ). The dashed line shows the actual elastic displacement arising from these forces. We denote this state by index 1. We denote the displacement of the points of application of forces and in the direction of these forces in state 1 by and .
In the process of loading a linear system with forces and, the forces increase and the displacements and increase proportionally to them (Fig. 3.2, c). The actual work of the forces and on the displacements they create is equal to the sum of the areas of the graphs, i.e. 
. By writing this expression as 
, we obtain the product of the generalized force and the generalized displacement . In this form, you can submit
the work of forces under any loading, if all loads change synchronously, i.e. the ratio of their values remains constant.
Next, consider the work of external forces on a possible displacement. As a possible displacement, we will take, for example, the deformed state of the system resulting from the application of a force at a certain point (Fig. 3.2, b). This state, corresponding to the additional displacement of the points of application of forces and by a distance and , will be denoted by 2. Forces and , without changing their value, perform virtual work on displacements and (Fig. 3.2, c):
As you can see, in the displacement notation, the first index shows the state in which the points and directions of these displacements are specified. The second index shows the state in which the forces that cause this movement are acting.
The work of a unit force F 2 on the actual displacement
If we consider state 1 as a possible displacement for the force F 2, then its virtual work on the displacement
The work of internal forces
Let us find the work of the internal forces of state 1, i.e., from the forces and , on virtual displacements of state 2, i.e., resulting from the application of the load F 2 . To do this, select a rod element of length dx (Fig. 3.2 and 3.3, a). Since the system under consideration is flat, only two forces S and Q z and a bending moment Mu act in the sections of the element. These forces for the cut element are external. Internal forces are cohesive forces that provide strength to the material. They are equal to the external ones in value, but are directed in the direction opposite to the deformation, therefore their work under loading is negative (Fig. 3.3, b-d, shown in gray). Let us sequentially calculate the work done by each force factor.
The work of longitudinal forces on displacement, which is created by forces S 2 that arose as a result of the application of the load F 2 (Fig. 3.2, b, 3.3, b),
We find the elongation of a rod with a length dx using the well-known formula
where A is the sectional area of the rod. Substituting this expression into the previous formula, we find
Similarly, we define the work that the bending moment does on the angular displacement created by the moment (Fig. 3.3, c):
We find the angle of rotation as
where J is the moment of inertia of the rod section relative to the y axis. After substitution, we get
Let's find the work of the transverse force on displacement (Fig. 3.3, d). Tangential stresses and shifts from the shearing force Q z are not distributed linearly over the bar section (in contrast to normal stresses and elongations in the previous loading cases). Therefore, to determine the shear work, it is necessary to consider the work done by shear stresses in the layers of the rod.
Tangential stresses from the force Q z, which act in a layer lying at a distance z from the neutral axis (Fig. 3.3, e), are calculated by the Zhuravsky formula
where Su is the static moment of the part of the cross-sectional area lying above this layer, taken relative to the y-axis; b is the width of the section at the level of the layer under consideration. These stresses create a shear of the layer by an angle, which, according to Hooke's law, is defined as 
- shear modulus. As a result, the end of the layer is displaced by
The total work of shear stresses of the first state, acting on the end of this layer, on displacements of the second state is calculated by integrating the product over the cross-sectional area
After substituting here the expressions for and we get
We take out from under the integral values that do not depend on z, multiply and divide this expression by A, we get
Here, the dimensionless coefficient is introduced,
depending only on the configuration and the ratio of the dimensions of the sections. For a rectangle \u003d 1.2, for I-beam and box sections (A c - sectional area of \u200b\u200bthe wall or in a box section - two walls).
Since the work of each of the considered loading components (S, Q, M) on displacements caused by other components is equal to zero, then the total work of all internal forces for the considered element of the rod of length dx
| (3.3) |
In the section of an element of a spatial rod system, six internal forces act (S, Q, Q z, M x, Mu, M 2), therefore, for it, the expression for the total work of internal forces will look like,
Here M x - torque in the rod; J T is the moment of inertia of the rod in free torsion (geometric torsional rigidity). In the integrand, the indices "and" are omitted.
In formulas (3.3) and (3.4) S v Q yV Q zl , M x1 , M y1 , M g1 denote the analytical expressions of diagrams of internal forces from the action of forces F (and F (, aS 2 , Q y 2 , Q z 2 , M x2 , M y2 , M r2 - descriptions of diagrams of internal forces from the force F 2 .
Theorems on elastic systems
The structure of formulas (3.3) and (3.4) shows that they are “symmetric” with respect to states 1 and 2, i.e. the work of the internal forces of state 1 on the displacements of state 2 is equal to the work of the internal forces of state 2 on the displacements of state 1 But according to (3.2) 
Therefore, if the work of internal forces is equal, then the work of external forces is equal - This statement is called the reciprocity work theorem (Betty's theorem, 1872).
For a rod system loaded with a force F 1 (Fig. 3.4, a), we take as a possible displacement the deformed state that arose when it was loaded with a force F 2 (Fig. 3.4, b). For this system, according to the Betti theorem 1- If we put , then we get
| (3.5) |
This formula expresses Maxwell's theorem (1864) on the reciprocity of displacements: the displacement of the point of application of the first unit force in its direction, caused by the action of the second unit force, is equal to the displacement of the point of application of the second unit force in its direction, caused by the action of the first unit force. This theorem can also be applied to the system in Fig. 3.2. If we set = 1 N (section 3.1.2), then we obtain the equality of generalized displacements 
.
Consider a statically indeterminate system with supports that can be given the required displacement, taken as possible (Fig. 3.4, c, d). In the first state, we shift the support 1 to and in the second - we set the rotation of the embedment by an angle - In this case, reactions will occur in the first state and , and in the second - i . According to the reciprocity of work theorem, we write If we set 
(here the dimension = m, and the value is dimensionless), then we get
This equality is numerical, since the dimension of the reaction = H, a = N-m. Thus, the reaction R 12 in fixed bond 1, which occurs when bond 2 is moved by one, is numerically equal to the reaction that occurs in bond 2 with a unit displacement of bond 1. This statement is called the reaction reciprocity theorem.
The theorems presented in this section are used for the analytical calculation of statically indeterminate systems.
Definition of displacements
General displacement formula
To calculate the displacements that occur in the rod system under the action of a given load (state 1), it is necessary to form an auxiliary state of the system in which one unit force acts, doing work on the desired displacement (state 2). This means that when determining the linear displacement, it is necessary to specify a unit force F 2 = 1 N applied at the same point and in the same direction in which the displacement is to be determined. If it is required to determine the angle of rotation of any section, then a single moment F 2 = 1 N m is applied in this section. After that, the energy equation (3.2) is compiled, in which state 2 is taken as the main one, and the deformed
 |
state 1 is treated as a virtual move. From this equation, the desired displacement is calculated.
Let us find the horizontal displacement of point B for the system in fig. 3.5, a. In order for the desired displacement D 21 to fall into the equation of works (3.2), we take as the main state the displacement of the system under the action of a unit force F 2 - 1 N (state 2, Fig. 3.5, b). We will consider the actual deformed state of the structure as a possible displacement (Fig. 3.5, a).
The work of external forces of state 2 on the displacements of state 1 is found as According to (3.2),
therefore, the desired displacement
Since (section 3.1.4), the work of the internal forces of state 2 on the displacements of state 1 is calculated by formula (3.3) or (3.4). Substituting into (3.7) expression (3.3) for the work of the internal forces of a flat rod system, we find
For further use of this expression, it is advisable to introduce the concept of single diagrams of internal force factors, i.e. of which the first two are dimensionless, and the dimension . The result will be
These integrals should be substituted with expressions for the distribution diagrams of the corresponding internal forces from the acting load and and from forces F 2 = 1. The resulting expression is called Mohr's formula (1881).
When calculating spatial bar systems, formula (3.4) should be used to calculate the total work of internal forces, then it will turn out
It is quite obvious that expressions for diagrams of internal forces S, Q y , Q z , M x, M y, M g and the values of the geometric characteristics of the sections A, J t, Jy, J, for the corresponding n-th section are substituted into the integrals. To shorten the notation in the notation of these quantities, the index "i" is omitted.
3.2.2. Particular cases of determining displacements
Formula (3.8) is used in the general case of a planar rod system, but in some cases it can be significantly simplified. Consider special cases of its implementation.
1. If deformations from longitudinal forces can be neglected, which is typical for beam systems, then formula (3.8) will be written as
2. If a flat system consists only of bent thin-walled beams with a ratio l / h> 5 for consoles or l / h> 10 for spans (I and h are the beam length and section height), then, as a rule, the bending strain energy significantly exceeds deformation energy from longitudinal and transverse forces, so they can be ignored in the calculation of displacements. Then formula (3.8) takes the form
3. For trusses, the rods of which, under nodal loading, experience mainly longitudinal forces, we can assume M = 0 and Q = 0. Then the displacement of the node is calculated by the formula
Integration is performed over the length of each rod, and summation is performed over all rods. Keeping in mind that the force S u in the i-th rod and the cross-sectional area do not change along its length, we can simplify this expression:
For all the apparent simplicity of this formula, the analytical calculation of displacements in trusses is very laborious, since it requires determining the forces in all truss rods from the acting load () and from a unit force () applied at the point whose displacement needs to be found.
3.2.3. Methodology and examples for determining displacements
Consider the calculation of the Mohr integral by the method of A. N. Vereshchagin (1925). Mohr's integral has the form (3.8), where as D 1 , D 2 may appear diagrams of bending moments, longitudinal or transverse forces. At least one of the diagrams () in the integrand is linear or piecewise linear, since it is built from a single load. Therefore, for
 |
The first of the integrals is numerically equal to the area of the subgraph (shaded in Fig. 3.6), and the second is the static moment of this area relative to the axis. The static moment can be written as , where is the coordinate of the position of the center of gravity of the area (point A). In view of what has been said, we get
(3.13)
Vereshchagin's rule is formulated as follows: if at least one of the diagrams is linear on the plot, then the Mohr integral is calculated as the product of the area of an arbitrarily
When calculating structures in the Mathcad environment, there is no need to use the Vereshchagin rule, since you can calculate the integral by numerical integration.
Example 3.1(Fig. 3.7, a). The beam is loaded with two symmetrically located forces. Find the displacements of the points of application of forces.
1. Let's build a diagram of bending moments M 1 from forces F 1 . Support reactions 
Maximum bending moment under force
2. Since the system is symmetrical, the deflections under the forces will be the same. As an auxiliary state, we take the loading of the beam by two unit forces F 2 = 1 N, applied at the same points as the forces F 1
(Fig. 3.7, b). The diagram of bending moments for this loading is similar to the previous one, and the maximum bending moment M 2max = 0.5 (L-b).
3. The loading of the system by two forces of the second state is characterized by the generalized force F 2 and the generalized displacement , which create the work of external forces on the displacement of state 1, equal to 
. Let us calculate the displacement using the formula (3.11). Multiplying the diagrams by sections according to the Vereshchagin rule, we find
After substituting the values 
we get
Example 3.2. Find the horizontal displacement of the movable support of the U-shaped frame loaded with the force F x (Fig. 3.8, a).
1. Let's build a diagram of bending moments from the force F 1 Support reactions 
. Maximum bending moment under force F 1
2. As an auxiliary state, we take the loading of the beam with a unit horizontal force F 2 applied at point B (Fig. 3.8, b). We build a diagram of bending moments for this loading case. Support reactions A 2y \u003d B 2y \u003d 0, A 2x \u003d 1. Maximum bending moment.
3. We calculate the displacement according to the formula (3.11). On vertical sections, the product is zero. On a horizontal section, the plot M 1 is not linear, but the plot is linear. Multiplying the diagrams by the Vereshchagin method, we get
The product is negative, since the diagrams lie on opposite sides. The obtained negative displacement value indicates that its actual direction is opposite to the direction of the unit force.
Example 3.3(Fig. 3.9). Find the angle of rotation of the section of the two-support beam under the force and find the position of the force at which this angle will be maximum.
1. Let's build a diagram of the bending moments M 1 from the force F 1. To do this, we will find the support reaction A 1. From the equilibrium equation for the system as a whole 
find. The maximum bending moment under the force Fj
2. As an auxiliary state, we take the loading of the beam with a single moment F 2 \u003d 1 Nm in the section whose rotation must be determined (Fig. 3.9, b). We build a diagram of bending moments for this loading case. Support reactions A 2 \u003d -B 2 \u003d 1 / L, bending moments
Both moments are negative, since they are directed clockwise. Diagrams are built on a stretched fiber.
3. We calculate the angle of rotation according to the formula (3.11), performing the multiplication over two sections,
Denoting , you can get this expression in a more convenient form:
The graph of the dependence of the angle of rotation on the position of the force F 1 is shown in fig. 3.9, c. Differentiating this expression, from the condition we find the position of the force at which the angle of inclination of the beam under it will be the largest in absolute value. This will happen at values equal to 0.21 and 0.79.
Let's move on to the consideration of another principle of mechanics, which establishes a general condition for the equilibrium of a mechanical system. By equilibrium (see § 1) we mean the state of the system in which all its points under the action of 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 superimposed on the system to be stationary, and we will not specifically stipulate this every time in the future.
Let us introduce the concept of possible work as elementary work that the force acting on a material point could do at 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 bonds, which we have already used (see § 123): bonds are called ideal if the sum of the elementary works of their reactions on any possible displacement of the system is equal to zero, i.e.
Given in § 123 and expressed by equality (52), the condition of the ideality of the bonds, when they are simultaneously stationary, corresponds to the definition (98), since with stationary bonds, each real displacement coincides with one of the possible ones. Therefore, examples of ideal connections will be all the 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 the force and the possible displacement.
Let us designate the resultants of all (both external and internal) active forces and reactions of the bonds acting on some point of the system, respectively, through . Then, since each of the points of the system is in equilibrium, and, consequently, the sum of the work of these forces for any movement of the point will also be equal to zero, i.e. . Compiling 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, then the second sum according to condition (98) will be equal to zero. Then the first sum is also equal to zero, i.e., equality (99) holds. Thus, we have 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 equation (99) are applied to the points of a mechanical system at rest, then the system will remain at rest. Let us assume the opposite, i.e., that the system will begin to move and some of its points will make real displacements. Then the forces will do work on these displacements and, according to the theorem on the change in kinetic energy, it will be:
where, obviously, since the system was initially at rest; hence, and . But with stationary connections, the actual displacements coincide with some of the possible displacements, and these displacements must also have 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 the proved: for the equilibrium of a mechanical system with ideal constraints, it is necessary and sufficient that the sum of the elementary works of all active forces acting on it for any possible displacement of the system be 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 bonds, to exclude from consideration all previously unknown reactions of bonds.
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 single course in theoretical mechanics: dynamics and analytical mechanics. In the first part, the first and second problems of dynamics are considered in detail, also ...
The principle of possible movements: for the equilibrium of a mechanical system with ideal connections, it is necessary and sufficient that the sum of the elementary works of all active forces acting on it for any possible displacement be equal to zero. or in projections: .
The principle of possible displacements gives in a general form the equilibrium conditions for any mechanical system, gives a general method for solving problems of statics.
If the system has several degrees of freedom, then the equation of the principle of possible displacements is made up 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 movements is formulated as follows:
To the mother. the system, subject to ideal constraints, was at rest, it is necessary and sufficient that the sum of elementary works performed by active forces on possible displacements of the points of the system be positive
General dynamics equation- when a system moves with ideal connections at any given moment of time, the sum of elementary works of all applied active forces and all inertia forces on any possible movement 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 of any mechanical system. Gives a general method for solving problems of dynamics.
Compilation sequence:
a) the specified forces acting on it are applied to each body, and also the forces and moments of pairs of inertia forces are conditionally applied;
b) inform the system of possible movements;
c) compose 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 non-ideal bonds, only in this case the reactions of non-ideal bonds, such as, for example, the friction force or the rolling friction moment, must be classified as active forces.
The work on the possible displacement of both active and inertia forces is sought in the same way as the elementary work on the actual displacement:
Possible work of force: .
Possible work of the moment (pair of forces): .
Generalized coordinates of a mechanical system are mutually independent parameters q 1 , q 2 , …, q S of any dimension, which 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 νth 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 inertia force corresponding to the generalized coordinate :
The number of independent possible displacements of the system is called the number of degrees of freedom of this system. For example. the ball on the plane can move in any direction, but any possible movement can be obtained as the geometric sum of two movements along two mutually perpendicular axes. A free rigid body has 6 degrees of freedom.
Generalized forces. For each generalized coordinate, one can calculate the corresponding generalized force Q k.
The calculation is made according to this rule.
To determine the generalized force Q k corresponding to the generalized coordinate q k, you need to give this coordinate an increment (increase the coordinate by this amount), leaving all other coordinates unchanged, calculate the sum of the work of all forces applied to the system on the corresponding displacements of the points and divide it by the increment of the coordinate:
where is displacement i-that 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 coordinates with the remaining coordinates and time unchanged t, the ratio can be defined as a partial derivative of . Then
where the coordinates of the points are functions of the generalized coordinates (5).
If the system is conservative, that is, the movement occurs under the action of potential field forces, the projections of which are , 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 with respect to the corresponding generalized coordinate with a minus sign.
Of course, when calculating this generalized force, the potential energy should be defined as a function of the generalized coordinates
P = P( q 1 , q 2 , q 3 ,…,qs).
Remarks.
First. When calculating the generalized reaction forces, ideal bonds are not taken into account.
Second. The dimension of the generalized force depends on the dimension of the generalized coordinate.
Lagrange equations of the 2nd 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 2nd kind, generalized coordinates are chosen and generalized velocities are found . The kinetic energy of the system is found, which is a function of the generalized velocities , and, in some cases, generalized coordinates. The operations of differentiation of the kinetic energy are performed, provided for by the left-hand sides of the Lagrange equations. The resulting expressions are equated to generalized forces, for 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.
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's start the study of the elements of analytical mechanics by considering the problem of classifying the connections 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 connection: the velocity of the cylinder axis is related to its angular velocity according to 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 movement, 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 Axyz axis 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 inertia 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 ,
as 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.
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