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. So if the dimension [ q] – meter, then the dimension
[Q]= Nm/m = Newton, if [ q] – radian, then [Q] = Nm; If [ q] = m 2 , then [Q] = H/m, etc.
Example 4. A ring slides along a rod swinging in a vertical plane. M weight R(Fig. 10). We consider the rod weightless. Let us define generalized forces.
Fig.10
Solution. The system has two degrees of freedom. We assign two generalized coordinates s And .
Let us find the generalized force corresponding to the coordinate s. We give an increment to this coordinate, leaving the coordinate unchanged, and calculating the work of the only active force R, we obtain the generalized force
Then we increment the coordinate, assuming s= const. When the rod is rotated through an angle, the point of application of force is R, ring M, will move to . The generalized force will be
Since the system is conservative, generalized forces can also be found using potential energy. We get 
And 
. It turns out much simpler.
Lagrange equilibrium equations
By definition (7) generalized forces 
, k = 1,2,3,…,s, Where s– number of degrees of freedom.
If the system is in equilibrium, then according to the principle of possible displacements (1) 
. Here are the movements allowed by the connections, the possible movements. Therefore, when a material system is in equilibrium, all its generalized forces are equal to zero:
Q k= 0, (k=1,2,3,…, s). (10)
These equations equilibrium equations in generalized coordinates or Lagrange equilibrium equations , allow one more method to solve statics problems.
If the system is conservative, then . This means that it is in a position of equilibrium. That is, in the equilibrium position of such a material system, its potential energy is either maximum or minimum, i.e. the function П(q) has an extremum.
This is obvious from the analysis of the simplest example (Fig. 11). Potential energy of the ball in position M 1 has a minimum, in position M 2 – maximum. It can be noticed that in position M 1 equilibrium will be stable; in position M 2 – unstable.
Fig.11
Equilibrium is considered stable if the body in this position is given a low speed or displaced a small distance and these deviations do not increase in the future.
It can be proven (Lagrange-Dirichlet theorem) that if in the equilibrium position of a conservative system its potential energy has a minimum, then this equilibrium position is stable.
For a conservative system with one degree of freedom, the condition for the minimum potential energy, and therefore the stability of the equilibrium position, is determined by the second derivative, its value in the equilibrium position,
Example 5. Kernel OA weight R can rotate in a vertical plane around an axis ABOUT(Fig. 12). Let us find and study the stability of equilibrium positions.
Fig.12
Solution. The rod has one degree of freedom. Generalized coordinate – angle.
Relative to the lower, zero position, potential energy P = Ph or
In the equilibrium position there should be 
. Hence we have two equilibrium positions corresponding to the angles and (positions OA 1 and OA 2). Let's explore their stability. Finding the second derivative. Of course, with , . The equilibrium position is stable. At , 
. The second equilibrium position is unstable. The results are obvious.
Generalized inertial forces.
Using the same method (8) by which the generalized forces were calculated Q k, corresponding to active, specified, forces, generalized forces are also determined S k, corresponding to the inertia forces of the points of the system:
And, since 
That
A few mathematical transformations.
Obviously,
Since a qk = qk(t), (k = 1,2,3,…, s), then
This means that the partial derivative of speed with respect to
In addition, in the last term (14) you can change the order of differentiation:
Substituting (15) and (16) into (14), and then (14) into (13), we get
Dividing the last sum by two and keeping in mind that the sum of derivatives is equal to the derivative of the sum, we get
where is the kinetic energy of the system, and is the generalized speed.
Lagrange equations.
By definition (7) and (12) generalized forces
But based on general equation dynamics (3), the right side of the equality is equal to zero. And since everything ( k = 1,2,3,…,s) are different from zero, then . Substituting the value of the generalized inertia force (17), we obtain the equation
These equations are called differential equations of motion in generalized coordinates, Lagrange equations of the second kind or just – Lagrange equations.
The number of these equations is equal to the number of degrees of freedom of the material system.
If the system is conservative and moves under the influence of potential field forces, when the generalized forces are , the Lagrange equations can be composed in the form
Where L = T– P is called Lagrange function (it is assumed that the potential energy P does not depend on the generalized velocities).
Often, when studying the motion of material systems, it turns out that some generalized coordinates q j are not included explicitly in the Lagrange function (or in T and P). Such coordinates are called cyclical. The Lagrange equations corresponding to these coordinates are obtained more simply.
The first integral of such equations can be found immediately. It's called a cyclic integral:
Further research and transformations of Lagrange equations are the subject of a special section theoretical mechanics– “Analytical mechanics”.
Lagrange's equations have a number of advantages in comparison with other methods of studying the motion of systems. Main advantages: the method of composing equations is the same in all problems, the reactions of ideal connections are not taken into account when solving problems.
And one more thing - these equations can be used to study not only mechanical, but also other physical systems (electrical, electromagnetic, optical, etc.).
Example 6. Let's continue our study of the movement of the ring M on a swinging rod (example 4).
Generalized coordinates are assigned – and s (Fig. 13). Generalized forces are defined: and 
.
Fig.13
Solution. Kinetic energy of the ring Where a and .
We compose two Lagrange equations
then the equations look like this:
We have obtained two nonlinear second-order differential equations, the solution of which requires special methods.
Example 7. Let's create a differential equation of motion of the beam AB, which rolls without sliding along a cylindrical surface (Fig. 14). Beam length AB = l, weight - R.
In the equilibrium position, the beam was located horizontally and the center of gravity WITH it was located at the top point of the cylinder. The beam has one degree of freedom. Its position is determined by a generalized coordinate - an angle (Fig. 76).
Fig.14
Solution. The system is conservative. Therefore, we will compose the Lagrange equation using the potential energy P=mgh, calculated relative to the horizontal position. At the point of contact there is an instantaneous center of velocities and (equal to the length of the circular arc with angle).
Therefore (see Fig. 76) and .
Kinetic energy (the beam undergoes plane-parallel motion)
We find the necessary derivatives for the equation and 
Let's make an equation
or, finally,
Self-test questions
What is the possible movement of a constrained mechanical system?
How are possible and actual movements of the system related?
What connections are called: a) stationary; b) ideal?
Formulate the principle of possible movements. Write down its formulaic expression.
Is it possible to apply the principle of virtual movements to systems with non-ideal connections?
What are the generalized coordinates of a mechanical system?
What is the number of degrees of freedom of a mechanical system?
In what case Cartesian coordinates points of the system depend not only on generalized coordinates, but also on time?
What are the possible movements of a mechanical system called?
Do possible movements depend on the forces acting on the system?
What connections of a mechanical system are called ideal?
Why is a bond made with friction not an ideal bond?
How is the principle of possible movements formulated?
What types can the work equation have?
Why does the principle of possible displacements simplify the derivation of equilibrium conditions for forces applied to constrained systems consisting of large number tel?
How are work equations constructed for forces acting on a mechanical system with several degrees of freedom?
What is the relationship between driving force and the resistance force in the simplest machines?
How is it formulated? golden rule mechanics?
How are the reactions of connections determined using the principle of possible movements?
What connections are called holonomic?
What is the number of degrees of freedom of a mechanical system?
What are the generalized coordinates of the system?
How many generalized coordinates does a non-free mechanical system have?
How many degrees of freedom does a car's steering wheel have?
What is generalized force?
Write down a formula expressing the total elementary work of all forces applied to the system in generalized coordinates.
How is the dimension of the generalized force determined?
How are generalized forces calculated in conservative systems?
Write down one of the formulas expressing the general equation of the dynamics of a system with ideal connections. What physical meaning this equation?
What is the generalized force of active forces applied to a system?
What is the generalized inertial force?
Formulate d'Alembert's principle in generalized forces.
What is the general equation of dynamics?
What is called the generalized force corresponding to some generalized coordinate of the system, and what dimension does it have?
What are the generalized reactions of ideal bonds?
Derive the general equation of dynamics in generalized forces.
What form are the equilibrium conditions for forces applied to a mechanical system obtained from the general equation of dynamics in generalized forces?
What formulas express generalized forces through projections of forces onto fixed axes Cartesian coordinates?
How are generalized forces determined in the case of conservative and in the case of non-conservative forces?
What connections are called geometric?
Give a vector representation of the principle of possible displacements.
Name what you need and sufficient condition equilibrium of a mechanical system with ideal stationary geometric connections.
What property does the force function of a conservative system have in a state of equilibrium?
Record the system differential equations Lagrange of the second kind.
How many Lagrange equations of the second kind can be constructed for a constrained mechanical system?
Does the number of Lagrange equations of a mechanical system depend on the number of bodies included in the system?
What is the kinetic potential of a system?
For which mechanical systems is there a Lagrange function?
What arguments are the function of the velocity vector of a point belonging to a mechanical system with s degrees of freedom?
What is the partial derivative of the velocity vector of a point in the system with respect to some generalized velocity?
The function of which arguments is the kinetic energy of a system subject to holonomic non-stationary constraints?
What form do Lagrange equations of the second kind have? What is the number of these equations for each mechanical system?
What form do Lagrange equations of the second kind take in the case when the system is simultaneously acted upon by conservative and non-conservative forces?
What is the Lagrange function, or kinetic potential?
What form do the Lagrange equations of the second kind have for a conservative system?
Depending on what variables should the kinetic energy of a mechanical system be expressed when composing the Lagrange equations?
How is the potential energy of a mechanical system under the influence of elastic forces determined?
Tasks for independent decision
Task 1. Using the principle of possible displacements, determine the reactions of connections of composite structures. Structural diagrams are shown in Fig. 15, and the data necessary for the solution are given in table. 1. In the pictures, all dimensions are in meters.
Table 1
Option 1 Option 2
Option 3 Option 4
Option 5 Option 6
Option 7 Option 8
Fig.16 Fig.17
Solution. It is easy to verify that in this problem all the conditions for applying the Lagrange principle are met (the system is in equilibrium, the connections are stationary, holonomic, confining and ideal).
Let's free ourselves from the connection corresponding to the reaction X A (Fig. 17). To do this, at point A, the fixed hinge should be replaced, for example, with a rod support, in which case the system receives one degree of freedom. As already noted, possible relocation system is determined by the constraints imposed on it and does not depend on the applied forces. Therefore, determining possible displacements is a kinematic problem. Since in this example the frame can only move in the plane of the picture, its possible movements are also planar. In plane motion, the movement of the body can be considered as a rotation around the instantaneous center of velocities. If the instantaneous center of velocities lies at infinity, then this corresponds to the case of instantaneous translational motion, when the displacements of all points of the body are the same.
To find the instantaneous center of velocities, it is necessary to know the directions of velocities of any two points of the body. Therefore, determining the possible displacements of a composite structure should begin with finding the possible displacements of the element for which such velocities are known. In this case, you should start with the frame CDB, since its point IN is motionless and, therefore, the possible movement of this frame is its rotation through an angle around an axis passing through hinge B. Now, knowing the possible movement of the point WITH(it simultaneously belongs to both frames of the system) and possible movement of the point A(a possible movement of point A is its movement along the axis X), find the instantaneous velocity center C 1 of the frame AES. Thus, possible movement of the frame AES is its rotation around point C 1 by an angle . The connection between the angles and is determined through the movement of point C (see Fig. 17)
From the similarity of triangles EC 1 C and BCD we have
As a result, we get the dependencies:
According to the principle of possible movements
Let us sequentially calculate the possible jobs included here:
Q=2q – resultant of the distributed load, the application point of which is shown in Fig. 79; the possible work done by it is equal.
Fig.71
Fig.70
Fig.69
The position of the points of the crank mechanism (Fig. 70) can be determined by setting the angle of rotation of the crank or the distance s, which determines the position of the slider IN(at ).
The position of the spherical pendulum (Fig. 71) is determined by specifying two parameters, angles and .
Minimum quantity generalized coordinates independent from each other, which are sufficient to completely and unambiguously determine the position of all points of the system, are called number of degrees of freedom this system.
In general, for any material system you can assign several generalized coordinates. For example, the crank mechanism (Fig. 70) has two generalized coordinates and . But this does not mean that the mechanism has two degrees of freedom, since one coordinate can be determined through the other:
But the pendulum (Fig. 71) has two degrees of freedom, because its position is determined by two independent generalized coordinates. By the way, if the length of the pendulum changes, then to determine the position of the point M one more parameter is required - a generalized coordinate l, thread length. And the pendulum will have three degrees of freedom.
In the general case, we will denote generalized coordinates by the letter q.
Let material system has s degrees of freedom. Its position is determined by generalized coordinates: q 1 , q 2 , q 3 ,…, q k,…, qs. .
It is easy to verify that the Cartesian coordinates n points of the system can be defined as functions of generalized coordinates and time:
So the pendulum (Fig. 71) has the coordinates of the point M
there are coordinate functions l, and , and time t, If l = l(t).
Accordingly, the radius vector of system points can be defined as a function of generalized coordinates and time:
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 the 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 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. So if the dimension [ q] – meter, then the dimension
Nm/m = Newton, if [ q] – radian, then = Nm; If [ q] = m 2 , then, etc.
Example 23. A ring slides along a rod swinging in a vertical plane. M weight R(Fig. 72). We consider the rod weightless. Let us define generalized forces.
Let us consider a mechanical system with ideal connections. Let be the active forces of the system. Let's give the mechanical system a virtual displacement and calculate the elementary work of the system forces on this displacement:
.
Using equality (17.2) we express the variation 
radius vector 
points M k through variations 
generalized coordinates:
hence,
. (17.6)
Let us change the order of summation in equality (17.6):
. (17.7)
Let us denote in expression (17.7)
. (17.8)
.
Generalized forces Q j name the coefficients for variations of generalized coordinates in the expression of the elementary work of system forces.
Depending on the dimension of variations of generalized coordinates 
generalized forces Q j may have dimensions of force, moment, etc.
Methods for calculating generalized forces
Let's consider three ways to calculate generalized forces.
1. Determination of generalized forces using the basic formula(17.8)
. (17.9)
Formula (17.9) is rarely used in practice. When solving problems, the second method is most often used.
2. A method of “freezing” generalized coordinates.
Let us give the mechanical system a virtual displacement such that all variations of generalized coordinates except 
are equal to zero:
Let's calculate the work for this movement 
all active forces applied to the system
.
By definition, the multiplier for variation 
equal to the first generalized force Q 1 .
and define the second generalized force Q 2, having calculated the virtual work of all forces of the system
.
Let us similarly calculate all other generalized forces of the system.
3. The case of a potential force field.
Suppose the potential energy of a mechanical system is known
Then 
and according to formula (32.8)
The principle of virtual movements of statics in generalized coordinates
According to the principle of virtual displacements of statics, for the equilibrium of a system with ideal restraining holonomic, stationary connections, the condition is necessary and sufficient:
at zero initial speeds.
Passing to generalized coordinates, we get
. (17.11)
Since the variations of generalized coordinates are independent, the equality to zero of expression (17.11) is possible only in the case when all coefficients for variations of generalized coordinates are equal to zero:
Thus, In order for a mechanical system with ideal, holonomic, stationary and restraining connections to be in equilibrium, it is necessary and sufficient that all generalized forces of the system are equal to zero (at zero initial velocities of the system).
Lagrange equations in generalized coordinates (Lagrange equations of the second kind)
Lagrange's equations are derived from the general equation of dynamics by replacing virtual displacements with their expressions through variations of generalized coordinates. They represent a system of differential equations of motion of a mechanical system in generalized coordinates:
. (17.13)
Where 
- generalized speeds,
T kinetic energy of the system, presented as a function of generalized coordinates and generalized velocities
Q j- generalized forces.
The number of equations of the system (17.13) is determined by the number of degrees of freedom and does not depend on the number of bodies included in the system. With ideal connections, only active forces will enter into the right-hand sides of the equations. If the connections are not ideal, then their reactions should be classified as active forces.
In the case of potential forces acting on the mechanical system, equations (17.13) take the form
.
If we introduce the Lagrange function L = T P, then taking into account that the potential energy does not depend on the generalized velocities, we obtain the Lagrange equations of the second kind for the case of potential forces in the following form
.
When composing Lagrange equations of the second kind, you need to perform the following steps:
Set the number of degrees of freedom of the mechanical system and select its generalized coordinates.
Compose an expression kinetic energy system and represent it as a function of generalized coordinates and generalized velocities.
Using the methods outlined above, find the generalized active forces of the system.
Perform all differentiation operations necessary in the Lagrange equations.
Example.
Where J z moment of inertia of the body relative to the axis of rotation z,
- angular velocity of the body.
3. Let's define the generalized force. Let's give the body a virtual displacement and calculate the virtual work of all active forces of the system:
Hence, Q = M z the main moment of the active forces of the system relative to the axis of rotation of the body.
4. Let us perform differentiation operations in the Lagrange equation
:
(17.14)
.
(17.15)
Substituting equalities (17.15) into equation (173
14) we obtain the differential equation of the rotational motion of the body
.
Let us write down the sum of the elementary works of forces acting on points of the system on the possible displacement of the system:
Let the holonomic system have 
degrees of freedom and, therefore, its position in space is determined 
generalized coordinates 
.
Substituting (225) into (226) and changing the order of summation by indices 
And 
, we get
. (226")
where is the scalar quantity
called generalized force related to the generalized coordinate
. Using the well-known expression for the scalar product of two vectors, the imparted force can also be represented as
– projections of force on the coordinate axes; 
– coordinates of the force application point.
The dimension of the generalized force in accordance with (226") depends on the dimension as follows 
, coinciding with the dimension 
:
, (228)
that is, the dimension of the generalized force is equal to the dimension of the work of the force (energy) or the moment of the force, divided by the dimension of the generalized coordinate to which the generalized force is assigned. It follows from this that a generalized force can have the dimension of force or moment of force.
Calculation of generalized force
1. The generalized force can be calculated using formula (227), which defines it, i.e.
2. Generalized forces can be calculated as coefficients for the corresponding variations of generalized coordinates in the expression for basic work(226"), i.e.
3. The most appropriate method for calculating the generalized forces, which is obtained from (226 ""), is if the system is given such a possible movement in which only one generalized coordinate changes, while the others do not change. So, if 
, and the rest 
, then from (179") we have
.
Index 
indicates that the sum of elementary works is calculated on a possible displacement, during which only the coordinate changes (varies) 
. If the variable coordinate is 
, That
. (227")
Equilibrium conditions for a system of forces in terms of generalized forces
System equilibrium conditions are derived from the principle of possible movements. They apply to systems for which this principle is true: for the equilibrium of a mechanical system subject to holonomic, stationary, ideal and non-releasing constraints, at the moment when the velocities of all points of the system are equal to zero, it is necessary and sufficient that all generalized forces be equal to zero
. (228")
3.6.7. General equation of dynamics
General equation of dynamics for a system with any connections (combined d'Alembert-Lagrange principle or general equation of mechanics):
, (229)
Where 
– active force applied to 
-th point of the system; 
– reaction strength of bonds; 
– point inertia force; 
– possible movement.
In the case of equilibrium of the system, when all inertial forces of the points of the system vanish, it turns into the principle of possible displacements. It is usually used for systems with ideal connections, for which the condition is satisfied
In this case (229) takes one of the forms:
,
,
. (230)
Thus, according to the general equation of dynamics, at any moment of motion of a system with ideal connections, the sum of the elementary works of all active forces and inertia forces of points of the system is equal to zero at any possible movement of the system allowed by the connections.
The general equation of dynamics can be given other, equivalent forms. Expanding the scalar product of vectors, it can be expressed as
Where 
– coordinates 
-th point of the system. Considering that the projections of inertia forces on the coordinate axes through the projections of accelerations on these axes are expressed by the relations
,
the general equation of dynamics can be given the form
In this form it is called general equation of dynamics in analytical form.
When using the general equation of dynamics, it is necessary to be able to calculate the elementary work of the inertial forces of the system on possible displacements. For this purpose, the corresponding formulas for elementary work obtained for ordinary forces are used. Let's consider their application for inertial forces solid in special cases of its movement.
During forward motion. In this case, the body has three degrees of freedom and, due to the imposed constraints, can only perform translational motion. Possible movements of the body that allow connections are also translational.
Inertial forces during translational motion are reduced to the resultant 
. For the sum of elementary works of inertial forces on the possible translational movement of a body, we obtain
Where 
– possible movement of the center of mass and any point of the body, since the translational possible movement of all points of the body is the same: the accelerations are also the same, i.e. 
.
When a rigid body rotates around a fixed axis.
The body in this case has one degree of freedom. It can rotate around a fixed axis 
. Possible movement, which is allowed by superimposed connections, is also a rotation of the body by an elementary angle 
around a fixed axis.
Inertia forces reduced to a point 
on the axis of rotation, are reduced to the main vector 
and the main point 
. The main vector of inertial forces is applied to a fixed point, and its elementary work on possible displacement is zero. For the main moment of inertial forces, non-zero elementary work will be performed only by its projection onto the axis of rotation 
. Thus, for the sum of the work of inertia forces on the possible displacement under consideration we have
,
if the angle 
report in the direction of the arc arrow of angular acceleration 
.
In flat motion.
In this case, the constraints imposed on the rigid body allow only possible planar movement. In the general case, it consists of a translational possible movement together with the pole, for which we choose the center of mass, and a rotation through an elementary angle 
around the axis 
, passing through the center of mass and perpendicular to the plane parallel to which the body can perform plane motion.
Since the inertial forces in the plane motion of a rigid body can be reduced to the main vector 
and the main point 
(if we choose the center of mass as the center of reduction), then the sum of the elementary work of inertia forces on a plane possible displacement will be reduced to the elementary work of the inertia force vector 
on the possible movement of the center of mass and the elementary work of the main moment of inertia forces on an elementary rotational movement around an axis 
, passing through the center of mass. In this case, non-zero elementary work can only be performed by the projection of the main moment of inertia forces onto the axis 
, i.e. 
. Thus, in the case under consideration we have
if the rotation is by an elementary angle 
direct in an arcing arrow to 
.
Definition of generalized forces
For a system with one degree of freedom, a generalized force corresponding to the generalized coordinate q, is called the quantity determined by the formula
where d q– small increment of the generalized coordinate; – the sum of the elementary works of the forces of the system on its possible movement.
Let us recall that the possible movement of the system is defined as the movement of the system to an infinitely close position allowed by the connections at a given moment in time (for more details, see Appendix 1).
It is known that the sum of the work done by the reaction forces of ideal bonds on any possible displacement of the system is equal to zero. Therefore, for a system with ideal connections, only the work of the active forces of the system should be taken into account in the expression. If the connections are not ideal, then their reaction forces, for example, friction forces, are conventionally considered active forces (see below for instructions on the diagram in Fig. 1.5). This includes the elementary work of active forces and the elementary work of moments of active pairs of forces. Let's write down formulas to determine these works. Let's say the force ( F kx ,F ky ,F kz) applied at the point TO, whose radius vector is ( x k ,y k ,z k), and possible displacement – (d xk, d y k , d z k). The elementary work of force on possible displacement is equal to scalar product, which in analytical form corresponds to the expression
d A( ) = F to d r to cos(), (1.3a)
and in coordinate form – the expression
d A( ) = F kx d x k + F ky d y k + F kz d z k. (1.3b)
If a couple of forces with a moment M applied to a rotating body, the angular coordinate of which is j, and the possible displacement is dj, then the elementary work of the moment M on the possible displacement dj is determined by the formula
d A(M) = ± M d j. (1.3v)
Here the sign (+) corresponds to the case when the moment M and possible movement dj coincide in direction; sign (–) when they are opposite in direction.
In order to be able to determine the generalized force using formula (1.3), it is necessary to express the possible movements of bodies and points in through a small increment of the generalized coordinate d q, using dependencies (1)…(7) adj. 1.
Definition of generalized force Q, corresponding to the selected generalized coordinate q, it is recommended to do it in the following order.
· Draw on the design diagram all the active forces of the system.
· Give a small increment to the generalized coordinate d q> 0; show on the calculation diagram the corresponding possible displacements of all points at which forces are applied, and the possible angular displacements of all bodies to which the moments of pairs of forces are applied.
· Compose an expression for the elementary work of all active forces of the system on these movements, express possible movements in through d q.
· Determine the generalized force using formula (1.3).
Example 1.4 (see condition to Fig. 1.1).
Let us define the generalized force corresponding to the generalized coordinate s(Fig. 1.4).
Active forces act on the system: P– cargo weight; G– drum weight and torque M.
The rough inclined plane is for the load A imperfect connection. Sliding friction force F tr, acting on the load A from this connection, is equal to F tr = f N.
To determine the strength N normal pressure of a load on a plane during movement, we use d’Alembert’s principle: if a conditional inertial force is applied to each point of the system, in addition to the active active forces and coupling reaction forces, then the resulting set of forces will be balanced and the dynamics equations can be given the form of static equilibrium equations. Following the well-known method of applying this principle, we will depict all the forces acting on the load A(Fig. 1.5), – and , where is the tension force of the cable.
Rice. 1.4 Fig. 1.5
Let's add the inertia force, where is the acceleration of the load. Equation of d'Alembert's principle in projection onto the axis y looks like N–Pcos a = 0.
From here N = Pcos a. The sliding friction force can now be determined by the formula F tr = f P cos a.
Let's give the generalized coordinate s small increment d s> 0. In this case, the load (Fig. 1.4) will move up the inclined plane to a distance d s, and the drum will turn counterclockwise by the angle dj.
Using formulas like (1.3a) and (1.3c), let us compose an expression for the sum of elementary torque works M, strength P And F tr:
Let's express dj in this equation through d s: , Then
let's define the generalized force using formula (1.3)
Let's take into account the previously written formula for F tr and we will finally get
If in the same example we take the angle j as the generalized coordinate, then the generalized force Q j expressed by the formula
1.4.2. Determination of generalized system forces
with two degrees of freedom
If the system has n degrees of freedom, its position is determined n generalized coordinates. Each coordinate qi(i = 1,2,…,n) corresponds to its generalized force Qi, which is determined by the formula
where is the sum of elementary works of active forces on i-th possible movement of the system when d q i > 0, and the remaining generalized coordinates are unchanged.
When determining, it is necessary to take into account the instructions for determining generalized forces according to formula (1.3).
It is recommended to determine the generalized forces of a system with two degrees of freedom in the following order.
· Show on the design diagram all the active forces of the system.
· Determine the first generalized force Q 1. To do this, give the system the first possible movement when d q 1 > 0, and d q 2 =q 1 possible movements of all bodies and points of the system; compose - an expression of the elementary work of the forces of the system on the first possible displacement; possible movements in expressed through d q 1; find Q 1 according to formula (1.4), taking i = 1.
· Determine the second generalized force Q 2. To do this, give the system a second possible movement when d q 2 > 0, and d q 1 = 0; show the corresponding d on the design diagram q 2 possible movements of all bodies and points of the system; compose - an expression of the elementary work of the system forces on the second possible displacement; possible movements in expressed through d q 2; find Q 2 according to formula (1.4), taking i = 2.
Example 1.5 (see condition to Fig. 1.2)
Let's define Q 1 And Q 2, corresponding to generalized coordinates xD And x A(Fig. 1.6, A).
There are three active forces acting on the system: P A = 2P, P B = P D = P.
Definition Q 1. Let's give the system the first possible movement when d xD> 0, d x A = 0 (Fig. 1.6, A). At the same time, the load D xD, block B will rotate counterclockwise by angle dj B, cylinder axis A will remain motionless, cylinder A will rotate around an axis A to the angle dj A clockwise. Let's compile the sum of work on the indicated movements:
let's define
Let's define Q 2. Let's give the system a second possible movement when d x D = 0, d xA> 0 (Fig. 1.6, b). In this case, the cylinder axis A will move vertically down a distance d x A, cylinder A will rotate around an axis A clockwise to angle dj A, block B and cargo D will remain motionless. Let's compile the sum of work on the indicated movements:
let's define
Example 1.6 (see condition to Fig. 1.3)
Let's define Q 1 And Q 2, corresponding to the generalized coordinates j, s(Fig. 1.7, A). There are four active forces acting on the system: the weight of the rod P, ball weight, spring elastic force and .
Let's take into account that. The modulus of elastic forces is determined by formula (a).
Note that the point of application of force F 2 is motionless, therefore the work of this force on any possible displacement of the system is zero, in the expression of generalized forces the force F 2 won't go in.
Definition Q 1. Let's give the system the first possible movement when dj > 0, d s = 0 (Fig. 1.7, A). In this case, the rod AB will rotate around an axis z counterclockwise by angle dj, possible movements of the ball D and center E the rods are directed perpendicular to the segment AD, the length of the spring will not change. Let's put it in coordinate form [see. formula (1.3b)]:
(Please note that , therefore, the work done by this force on the first possible displacement is zero).
Let us express the displacements d x E and d xD via dj. To do this, we first write
Then, in accordance with formula (7) adj. 1 we will find
Substituting the found values into , we get
Using formula (1.4), taking into account that , we determine
Definition Q 2. Let's give the system a second possible movement when dj = 0, d s> 0 (Fig. 1.7, b). In this case, the rod AB will remain motionless, and the ball M will move along the rod by a distance d s. Let's compile the sum of work on the indicated movements:
let's define
substituting the force value F 1 from formula (a), we get
1.5. Expressing the kinetic energy of a system
in generalized coordinates
The kinetic energy of a system is equal to the sum of the kinetic energies of its bodies and points (Appendix 2). To get for T Expression (1.2) should express the velocities of all bodies and points of the system through generalized velocities using kinematics methods. In this case, the system is considered to be in an arbitrary position, all its generalized velocities are considered positive, i.e., directed towards increasing generalized coordinates.
Example 1. 7 (see condition to Fig. 1.1)
Let us determine the kinetic energy of the system (Fig. 1.8), taking the distance as a generalized coordinate s,
T = T A + T B.
According to formulas (2) and (3) adj. 2 we have: .
Substituting this data into T and taking into account that , we get
Example 1.8(see condition to Fig. 1.2)
Let us determine the kinetic energy of the system in Fig. 1.9, taking as generalized coordinates the quantities xD And x A,
T = T A + T B + T D.
According to formulas (2), (3), (4) adj. 2 we'll write down
Let's express V A , V D , w B and w A through :
When determining w A it is taken into account that the point O(Fig. 1.9) – instantaneous center of cylinder speeds A And V k = V D(see the corresponding explanations for example 2 appendix 2).
Substituting the results obtained into T and given that
let's define
Example 1.9(see condition to Fig. 1.3)
Let us determine the kinetic energy of the system in Fig. 1.10, taking j and as generalized coordinates s,
T = T AB + T D.
According to formulas (1) and (3) adj. 2 we have
Let us express w AB And V D via and :
where is the transfer speed of the ball D, its modulus is determined by the formula
Directed perpendicular to the segment AD in the direction of increasing angle j; – relative speed ball, its module is determined by the formula , is directed towards increasing coordinates s. Note that is perpendicular, therefore
Substituting these results into T and given that
1.6. Drawing up differential equations
movement of mechanical systems
To obtain the required equations, it is necessary to substitute into the Lagrange equations (1.1) the previously found expression for the kinetic energy of the system in generalized coordinates and the generalized forces Q 1 , Q 2 , … , Qn.
When finding partial derivatives T using generalized coordinates and generalized velocities, it should be taken into account that the variables q 1 , q 2 , … , q n; are considered independent of each other. This means that when defining the partial derivative T for one of these variables, all other variables in the expression for T should be considered as constants.
When performing an operation, all variables included in the variable must be differentiated in time.
We emphasize that the Lagrange equations are written for each generalized coordinate qi (i = 1, 2,…n) systems.
