The principle of possible displacements is formulated for solving static problems using dynamic methods.
Definitions
Connections all bodies that limit the movement of the body in question are called.
Ideal are called connections, the work of reactions of which on any possible displacement is equal to zero.
Number of degrees of freedom of a mechanical system is the number of such mutually 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 hinge has one degree of freedom.
IN general case the mechanical system may have infinite number degrees of freedom.
Possible movements we will call such movements that, firstly, are allowed by superimposed connections, and, secondly, are infinitesimal.
The crank-slider mechanism has one degree of freedom. The following parameters can be accepted as possible movements: , x and etc.
For any system, the number of possible movements independent of each other is equal to the number of degrees of freedom.
Let some system be in equilibrium and the connections imposed on this system be 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.
Multiply (102) scalarly by the vector of possible movement of the point 
,
since the connections are ideal, it is always 
, what remains is the sum of the elementary works of the active forces acting on the point
.
(103)
Equation (103) can be written for all material points, summing which we obtain
.
(104)
Equation (104) expresses next principle possible movements.
For the equilibrium of a 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 movement of the system is equal to zero.
The number of equations (104) is equal to the number of degrees of freedom of a 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 using dynamic methods; on the other hand, d'Alembert's principle gives general method solving dynamics problems using static methods. By combining these two principles, we can obtain a general method for solving problems in mechanics, which is called the D'Alembert-Lagrange principle.
.
(105)
When a system moves with ideal connections at each moment of time, the sum of the elementary works of all applied active forces and all inertial forces on any possible movement 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) These are parameters that are independent of each other and 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 having any dimension can be chosen as generalized coordinates.
N 
For example, when studying the motion of a mathematical pendulum, which has 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) – angle of rotation.
As the system moves, its generalized coordinates will continuously change over time
Equations (107) are the equations of motion of the system in generalized coordinates.
The derivatives of generalized coordinates with respect to time are called generalized system speeds
.
(108)
The dimension of the generalized speed 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 force understand a quantity equal to the ratio of the sum of the elementary works of all forces acting on the system at a certain elementary increment of the generalized coordinate to this increment
,
(109)
Where S– generalized coordinate index.
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 connections in generalized coordinates, we use differential equations in Lagrange form of the second kind
.
(110)
B (110) kinetic energy T system is expressed through generalized coordinates q S and generalized speeds 
.
Lagrange's equations provide a unified and fairly simple method for solving problems of dynamics. 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 bonds, these equations make it possible to eliminate all previously unknown bond reactions.
1. Generalized coordinates and number of degrees of freedom.
When a mechanical system moves, all its points cannot move arbitrarily, since they are limited by connections. This means that not all point coordinates are independent. The position of the points is determined by specifying only independent coordinates.
generalized coordinates. For holonomic systems (i.e. those whose connections are expressed by equations that depend only on coordinates), the number of independent generalized coordinates of a mechanical system equal to the number of degrees of freedom this system.
Examples:
The position of all points is uniquely determined by the rotation angle
crank.
One degree of freedom.
2. The position of a free point in space is determined by three coordinates independent of each other. That's why three degrees of freedom.
3. Rigid rotating body, position determined by angle of rotation j . One degree of freedom.
4. A free rigid body whose motion is determined by six equations - six degrees of freedom.
2. Possible movements of the mechanical system.
Ideal connections.
Possible displacements are imaginary infinitesimal movements allowed in this moment connections imposed on the system. Possible movements of points of a mechanical system are considered as quantities of the first order of smallness, therefore, curvilinear movements of points are replaced by straight segments plotted tangentially to the trajectories of movement of points and are designated dS.
dS A = dj . O.A.
All forces acting on a material point are divided into specified and reaction forces.
If the sum of the work done by the reactions of the bonds on any possible displacement of the system is equal to zero, then such bonds are called ideal.
3. 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 movement of the system is equal to zero.
Meaning principle of possible movements:
1. Only active forces are taken into account.
2. Gives in general form the equilibrium condition for any mechanical system, whereas in statics it is necessary to consider the equilibrium of each body of the system separately.
Task.
For a given position of the crank-slider mechanism in equilibrium, find the relationship between moment and force if OA = ℓ.
General equation of dynamics.
The principle of possible displacements provides a general method for solving statics problems. On the other hand, d'Alembert's principle allows the use of statics methods to solve dynamic problems. Therefore, by applying these two principles simultaneously, a general method for solving dynamics problems can be obtained.
Let us consider a mechanical system on which ideal constraints are imposed. If the corresponding forces of inertia are added to all points of the system, except for the active forces and coupling reactions acting on them, then according to d'Alembert's principle, the resulting system of forces will be in equilibrium. Applying the principle of possible movements, we obtain:
Since the connections are ideal, then:
This equality represents general equation of dynamics.
It follows from it d'Alembert-Lagrange principle– when a system moves with ideal connections at each moment of time, the sum of the elementary works of all applied active forces and all inertial forces at any possible movement of the system will be equal to zero.
Task.
In the lift to the gear 2 weight 2G with radius R 2 =R torque applied M=4GR.
Determine the acceleration of the lifted load A weight G, neglecting the weight of the rope and friction in the axles. A drum on which the rope is wound, and a gear rigidly attached to it 1 , have total weight 4G and radius of gyration r = R. Drum radius R A = R and gears 1
R 1 =0.5R.
Let us depict all the acting forces, the direction of accelerations and possible displacements.
________________
Let us substitute into the general equation of dynamics
Let's express the displacement in terms of the rotation angle δφ 1
Let's substitute the values
δφ 1 ≠0
Let us express all accelerations through the required a A and equate the expression in brackets to zero
Let's substitute the values
The principle of possible movements.
a = 0.15 m
b = 2a = 0.3 m
m = 1.2 Nm _________________
x B; at B; N A ; M p
Solution: Let's find the reaction of the movable support A why let’s mentally discard this connection, replacing its action with a reaction N A
Possible movement of the rod AC is its rotation around the hinge WITH at an angle dj. Kernel Sun remains motionless.
Let's create an equation of work, taking into account that the work of forces when turning a body is equal to the product of the moment of force relative to the center of rotation and the angle of rotation of the body.
To determine reactions of rigid fastening in a support IN first find the moment of reaction M r. To do this, let’s discard the connection that prevents the rotation of the rod Sun, replacing the rigid fastening with a hinged-fixed support and applying a moment M r .
Let's tell the rod a possible rotation by an angle DJ 1.
Let's create an equation of work for the rod Sun:
Let's define the displacements:
To determine the vertical component of the reaction of rigid fastening, we discard the connection that prevents the vertical movement of the point IN, replacing the rigid fastening with a sliding one (rotation is impossible) and applying the reaction:
Let's tell the left side (rod) Sun with slider IN) possible speed V B forward movement down. Kernel AC will rotate around a point A .
Let's create a work equation:
To determine the horizontal component of the reaction of rigid fastening, we discard the connection that prevents the horizontal movement of the point IN replacing the rigid seal with a sliding one and applying the reaction:
Let's tell the left side (slider) IN together with the rod Sun) possible speed V B forward movement to the left. Since the support A on rollers, then the right side will move forward at the same speed. Hence 
.
Let's create a work equation for the entire structure.
To check the correctness of the solution, we draw up the equilibrium equations for the entire system:
The condition is met.
Answer: y B = -14.2 H; X B = -28.4 H; N A = 14.2 H; V P =3.33 Nm.
Generalized speeds. Generalized forces.
Independent quantities that uniquely determine the position of all points of a mechanical system are called generalized coordinates. – q
If the system has S degrees of freedom, then its position will be determined S generalized coordinates:
q 1 ; q 2 ; ...; qs.
Since the generalized coordinates are independent of each other, the elementary increments of these coordinates will also be independent:
dq 1 ; dq 2 ; ...; dq S .
Moreover, each of the quantities dq 1 ; dq 2 ; ...; dq S determines the corresponding possible movement of the system, independent of others.
As the system moves, its generalized coordinates will continuously change over time; the law of this motion is determined by the equations:
, …. ,
These are the equations of motion of the system in generalized coordinates.
Derivatives of generalized coordinates with respect to time are called generalized velocities of the system:
Size depends on size q.
Consider a mechanical system consisting of n material points on which forces act F 1 , F 2 , F n. Let the system have S degrees of freedom and its position is determined by generalized coordinates q 1 ; q 2 ; q 3. Let us inform the system of a possible movement at which the coordinate q 1 gets increment dq 1, and the remaining coordinates do not change. Then the radius vector of the point receives an elementary increment (dr k) 1. This is the increment that the radius vector receives when only the coordinate changes q 1 by the amount dq 1. The remaining coordinates remain unchanged. That's why (dr k) 1 calculated as a partial differential:
Let us calculate the elementary work of all applied forces:
Let's put it out of brackets dq 1, we get:
Where 
- generalized power.
So, generalized force – this is the coefficient for increments of the generalized coordinate.
The calculation of generalized forces comes down to the calculation of possible elementary work.
If everyone changes q, That:
According to the principle of possible displacements, for the system to be in equilibrium it is necessary and sufficient that SdА а к = 0. In generalized coordinates Q 1. dq 1 + Q 2 . dq 2 + … + Q s . dq s = 0 hence, For system equilibrium it is necessary and sufficient that the generalized forces corresponding to the possible displacements selected for the system, and therefore the generalized coordinates, were equal to zero.
Q 1 = 0; Q2 = 0; … Q s = 0.
Lagrange equations.
Using the general dynamic equation for a mechanical system, the equations of motion of the mechanical system can be found.
4) determine the kinetic energy of the system, express this energy through generalized velocities and generalized coordinates;
5) find the corresponding partial derivatives of T by and and substitute all values into the equation.
Impact theory.
The movement of a body under the action of ordinary forces is characterized by a continuous change in the modules and directions of the velocities of this body. However, there are cases when the velocities of points of the body, and therefore the momentum of the rigid body, undergo finite changes in a very short period of time.
Phenomenon, in which, in a negligibly small period of time, the velocities of points on the body change by a finite amount is called blow.
Strength, under the action of which an impact occurs, are called drums.
Short period of time t, during which the impact occurs is called impact time.
Since the impact forces are very large and change within significant limits during the impact, in the theory of impact, not the impact forces themselves, but their impulses are considered as a measure of the interaction of bodies.
Impulses of non-impact forces over time t will be very small values and can be neglected.
Theorem about the change in the momentum of a point upon impact:
Where v– speed of the point at the beginning of the impact,
u– speed of the point at the end of the impact.
Basic equation of impact theory.
The displacement of points in a very short period of time, that is, during the impact, will also be small, and therefore, we will consider the body motionless.
So, we can draw the following conclusions about the action of shock forces:
1) the action of non-impact forces during the impact can be neglected;
2) the displacements of points of the body during the impact can be neglected and the body can be considered motionless during the impact;
Figure 2.4
Solution
Let's replace the distributed load with a concentrated force Q = q∙DH. This force is applied in the middle of the segment D.H.- at the point L.
Strength F Let's decompose it into components, projecting it onto the axis: horizontal Fxcosα and vertical F y sinα.
Figure 2.5
To solve a 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. Let us express the displacement of the points of the structure through a possible rotation of the part CDB around the point B at an angle δφ 1, Part A.K.C. the 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 = 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 construct the work equation through the work of 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 doesn't do the 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.
Reduced by δφ 1, we obtain an equation from which we can easily find X A.
For determining Y A support structure A Let's change it so that when moving the point A only force did the work Y A(Figure 2.6). Let's take the possible movement of part of the structure as BDC rotation around a fixed point B — δφ 3.
Figure 2.6
For a point C δS C = BC∙δφ 3, the instantaneous center of rotation for a part of the structure A.K.C. there will be a point C V2, and moving the point C will express itself.
The principle of possible displacements makes it possible to solve a wide variety of problems on the equilibrium of mechanical systems - to find unknown active forces, to determine the reactions of connections, to find the equilibrium positions of a mechanical system under the influence of an applied system of forces. Let us illustrate this with specific examples.
Example 1. Find the magnitude of the force P holding heavy smooth prisms with masses in a state of equilibrium. The bevel angle of the prisms is equal (Fig. 73).
Solution. Let's use the principle of possible movements. Let us inform the system of possible displacement and calculate the possible work of active forces:
The possible work done by 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 rod BD if it makes an angle a with the horizontal.
Solution. The task differs from the previous one in that here it is required to find the reaction of an ideal connection. But the reaction of ideal connections is not included in the equation of work expressing the principle of possible movements. In such cases, the principle of possible movements should be applied in conjunction with the principle of release from ties.
Let us mentally discard the rod BD, and consider its reaction S as an active force of unknown magnitude. After this, we will inform the system of the possible movement (provided that this connection is completely absent). This will be an elementary rotation of the beam AB at an angle around the hinge axis A in one direction or another (in Fig. 74 - counterclockwise). The elementary displacements of the points of application of active forces and the reaction S attributed to them are equal to:
We create an equation of work
Equating the expression in parentheses to zero, we find
Example 3. A homogeneous rod OA is fixed by weight using a cylindrical hinge O and a spring AB (Fig. 75). Determine the positions in which the rod can be in equilibrium if the spring stiffness is equal to k, the natural length of the spring - and point B is on the same vertical as point O.
Solution. Two active forces are applied to the rod OA - its own weight and the elastic force of the spring where is the angle formed by the rod with the vertical OB. The superimposed connections are ideal (in this case there is only one connection - hinge O).
Let's inform the system of possible movement - an elementary rotation of the rod around the hinge axis O by an angle , calculate the possible work of the active forces and equate it to zero:
Substituting here the expression for the force F and the value
after simple transformations we get the following trigonometric equation to determine the angle (p when the rod is in equilibrium:
The equation defines three values for the angle:
Consequently, the rod has three equilibrium positions. Since the first two equilibrium positions exist if the condition is satisfied. Equilibrium at always exists.
In conclusion, we note that the principle of possible movements can also be applied to systems with non-ideal connections. The emphasis on the ideality of connections is made in the formulation of the principle with one sole purpose - to show that the equilibrium equations of mechanical systems can be compiled without including the reactions of ideal connections, thereby simplifying the calculations.
For systems with non-ideal connections, the principle of possible displacements should be reformulated as follows: for the equilibrium of a mechanical system with restraining connections, among which there are non-ideal connections, it is necessary and sufficient that possible work active forces and reactions of non-ideal connections was equal to zero. It is possible, however, to do without reformulating the principle, conditionally classifying the reactions of non-ideal connections among the active forces.
Self-test questions
1. What is the main feature of a non-free mechanical system compared to a free one?
2. What is possible movement? Give examples.
3. How are variations in the coordinates of points in the system determined during its possible movement (indicate three methods)?
4. How are connections classified according to the type of their equations? Give examples of confining and non-containing connections, stationary and non-stationary.
5. In what case is the connection called ideal? Imperfect?
6. Give a verbal formulation and mathematical notation of the principle of possible movements.
7. How is the principle of possible displacements formulated for systems containing non-ideal connections?
8. List the main types of problems solved using the principle of possible movements.
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.
It is necessary and sufficient that the sum of work , all active forces applied to the system for any possible movement of the system, is 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. Short course in theoretical mechanics. Textbook for colleges - 10th ed., revised. and additional - M.: Higher. school, 1986.- 416 p., ill.
- Basic course in theoretical mechanics (part one) N. N. Buchgolts, Nauka Publishing House, Main Editorial Office of Physics and Mathematics Literature, Moscow, 1972, 468 pp.
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principle of possible movements
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Books
- Theoretical mechanics. In 4 volumes. Volume 3: Dynamics. Analytical mechanics. Lecture texts. Vulture of the Ministry of Defense of the Russian Federation, Bogomaz Irina Vladimirovna. IN textbook two parts of a single course on theoretical mechanics: dynamics and analytical mechanics. The first part discusses in detail the first and second problems of dynamics, also...