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 is equal to zero. or in projections: .
The principle of possible displacements provides in general form the equilibrium conditions for any mechanical system and provides a general method for solving statics problems.
If the system has several degrees of freedom, then the equation of the principle of possible movements is compiled for each of the independent movements separately, i.e. there will be as many equations as the system has degrees of freedom.
The principle of possible displacements is convenient in that 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:
In order to mater. a system subject to ideal connections is in a state of rest; it is necessary and sufficient that the sum of elementary work performed by active forces on possible displacements of points in the system is positive
General equation of dynamics- when a system moves with ideal connections at any given moment in 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. The equation uses the principle of possible displacements and D'Alembert's principle and allows you to compose differential equations of motion of any mechanical system. Gives a general method for solving dynamics problems.
Compilation sequence:
a) the specified forces acting on it are applied to each body, and forces and moments of inertial force pairs are also conditionally applied;
b) inform the system of possible movements;
c) draw up equations for the principle of possible movements, 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 connections, only in this case the reactions of non-ideal connections, such as the friction force or rolling friction moment, must be classified as active forces.
Work on possible displacement of both active and inertial forces is sought in the same way as elementary work on actual displacement:
Possible work of force: .
Possible work of the moment (force pair): .
Generalized coordinates of a mechanical system are parameters q 1 , q 2 , ..., q S, independent of each other, of any dimension, which uniquely determine the position of the system at any time.
The number of generalized coordinates is equal to 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 inertial force corresponding to the generalized coordinate:
The number of mutually independent possible movements of a system is called the number of degrees of freedom of this system. For example. a ball on a plane can move in any direction, but any possible movement of it 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 points and divide it by the increment of the coordinate:
where is displacement i-that point of the system, obtained by changing k-that generalized coordinate.
The generalized force is determined using elementary work. Therefore, this force can be calculated differently:
And since there is an increment of the radius vector due to the increment of the coordinate with other constant coordinates and time t, the relation can be defined as a partial derivative. Then
where the coordinates of the points are functions of generalized coordinates (5).
If the system is conservative, that is, the movement occurs under the influence of potential field forces, the projections of which are , where , and the coordinates of points are functions of generalized coordinates, then
The generalized force of a conservative system is the partial derivative of the potential energy along the corresponding generalized coordinate with a minus sign.
Of course, when calculating this generalized force, the potential energy should be determined as a function of the generalized coordinates
P = P( q 1 , q 2 , q 3 ,…,qs).
Notes.
First. When calculating the generalized reaction forces, ideal connections 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 compile the Lagrange equation of the 2nd kind, generalized coordinates are selected and generalized velocities are found . The kinetic energy of the system is found, which is a function of generalized velocities , and, in some cases, generalized coordinates. The operations of differentiation of kinetic energy provided by the left sides of the Lagrange equations are performed. The resulting expressions are equated to generalized forces, for finding which, in addition to formulas (26), the following are often used when solving problems:
In the numerator on the right side of the formula is the sum of the elementary works of all active forces on the possible displacement of the system corresponding to the variation of the i-th generalized coordinate - . With this possible movement, all other generalized coordinates do not change. The resulting equations are differential equations of motion of a mechanical system with S degrees of freedom.
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, a mechanical system can have an infinite number of 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 the following principle of 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 makes it possible to solve static problems using dynamic methods; on the other hand, d'Alembert's principle provides a general method for solving dynamic 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, differential equations in the Lagrange form of the second kind are used
.
(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.
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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See what the “Principle of Possible Displacements” is in other dictionaries:
principle of possible movements
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Big Encyclopedic Dictionary
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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. The textbook contains 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...
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.
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 at a given moment by the 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 a 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 ; Mp
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;
