Generalized coordinates and generalized forces. Generalized coordinates and generalized forces What does the work of forces look like in generalized coordinates

Analytical mechanics. Classification of connections. Examples. Possible movements.

Connection– this is the relationship between the coordinates and velocities of the points of the system, presented in the form of equalities or inequalities.

Classification:

Geometric– imposes restrictions only on the coordinates of system points (velocities are not included)

Kinematic– velocities enter into the equations. If you can get rid of speeds, then the connection is integrated.

Holonomic connections– geometric and integrable differential connections.

The connection is called holding(imposed or restrictions remain in any position of the system) and unrestrained, which do not possess this property (from such connections, as they say, the system can be “freed”

Possible relocation

Any mental

Infinitesimal

Moving system points allowed

At this moment in time

Connections imposed on the system.

Actual movement– depends on forces, time, connections, initial conditions.

Possible movement depends only on connections.

For stationary connections, actual movement is one of the possible ones.

Ideal connections. The principle of possible movements.

Ideal are called connections for which the sum of the elementary works of all their reactions on any possible displacement is equal to 0.

The principle of possible movements.

For the equilibrium of a mechanical system with ideal stationary connections, it is necessary and sufficient that the sum of the elementary work of all active forces on any possible displacement is equal to 0. In this case, for sufficiency, the initial velocity must be equal to zero. Necessary balance => Sufficient => balance.

Generalized coordinates. The number of degrees of freedom of the system. Generalized forces, methods for calculating them. Equilibrium conditions for a system with holonomic constraints, expressed in terms of generalized forces.

Generalized coordinates– an independent parameter that completely determines the position of the system and through which all Cartesian coordinates of points in the system can be expressed.

The number of degrees of freedom is determined by the number of generalized coordinates

The number of mutually independent scalar quantities that uniquely determine the position of a mechanical system in space is called the number of degrees of freedom.

Generalized coordinates of a mechanical system are any geometric quantities independent of each other that uniquely determine the position of the system in space.

Q i = δA j /δq j or δA j = Q i ⋅ δq j .

Generalized force- this is a force that does the same work on a possible displacement along its generalized coordinate as all the forces applied to the system on the corresponding displacement of the points of their application.

To find the generalized force, we give the possible displacement along its generalized coordinate, leaving the other coordinates unchanged. Then we find the work done by all forces applied to the system and divide by the possible displacement.

The principle of possible displacements in terms of generalized forces.

Since in equilibrium the sum of elementary work on any possible displacement ( bA=bq j , which do not depend on each other, then for this the following must be true: Q 1 =0; Q 2 =0; Q K =0

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 a force on a possible displacement is equal to the 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 reaction forces of connections, then the resulting set of forces will be balanced and the dynamic 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 force of inertia, 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

we 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 Qj 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 xA(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 at 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 xA, 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 the 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 xA,

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 of the ball, its module is determined by the formula, 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 generalized forces Q 1 , Q 2 , … , Q n.

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.

In analytical mechanics, along with the concept of force as a vector quantity characterizing the impact on a given body from other material bodies, they use the concept of generalized force. For determining generalized power Let's consider the virtual work of forces applied to points of the system.

If a mechanical system with holonomic restraining forces imposed on it h has connections s =3n-h degrees of freedom , then the position of this system is determined ( i = s)

generalized coordinates and (2.11) : According to (2.13), (2.14) virtual displacement k – th points

(2.13)

(2.14)

Substituting (2.14): into the formula for the virtual work of forces

(2.24), we get

Scalar quantity = (2.26)

called generalized force, corresponding i th generalized coordinate.

Generalized forcecorresponding to i-th generalized coordinate is a quantity equal to the multiplier for the variation of a given generalized coordinate in the expression of the virtual work of forces acting on a mechanical system.

Virtual work determined from

¾ specified active forces independent of restrictions and

¾ coupling reactions (if the couplings are not ideal, then to solve the problem it is necessary to additionally set the physical dependence T j from N j , ( T j ¾ these are, as a rule, friction forces or moments of resistance to rolling friction, which we can determine).

In general generalized force is a function of generalized coordinates, velocities of system points and time. From the definition it follows that generalized force¾ is a scalar quantity that depends on the generalized coordinates chosen for a given mechanical system. This means that when the set of generalized coordinates that determine the position of a given system changes, the generalized forces.

Example 2.10. For a disk with radius r and mass m, which rolls without sliding on an inclined plane (Fig. 2.9), can be taken as a generalized coordinate:

¾ or q = s¾ movement of the center of mass of the disk,

¾either q= j ¾ angle of rotation of the disk. If we neglect rolling resistance, then:

¾ in the first case generalized force will

Rice. 2.9 Q s = mg sina, a

¾ in the second case ¾ Q j = mg r cosa.

The generalized coordinate also determines the unit of measurement of the corresponding generalized power. From expression (2.25)

(2.27)

it follows that the unit of measurement generalized power equal to the unit of work divided by the unit of generalized coordinate.

If, as a generalized coordinate q accept q = s¾ movement of any point, then the unit of measurement generalized power Q s ¾ will be [newton] ,

If, as a q= j ¾ the angle of rotation (in radians) of the body will be taken, then the unit of measurement generalized power Q j 2 will be [ newton ´ meter].

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 elementary work (226"), i.e.

3. The most appropriate method for calculating generalized forces, which is obtained from (226 ""), is if the system is given such a possible movement that 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 valid: 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; – strength of reaction 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. To do this, apply the corresponding formulas for elementary work obtained for ordinary forces. Let us consider their application to the inertial forces of a rigid body in particular cases of its motion.

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 inertia 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 .

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 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; pregnant 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 the general dynamics equation (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 simply – 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 studies and transformations of Lagrange’s equations form the subject of a special section of 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 horizontal 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 called?

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 do the Cartesian coordinates of points in 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 a large number of bodies?

How are work equations constructed for forces acting on a mechanical system with several degrees of freedom?

What is the relationship between the driving force and the resisting force in simple machines?

How is the golden rule of mechanics formulated?

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 is the physical meaning of 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 the fixed axes of 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 the necessary and sufficient condition for the 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?

Write down a system of Lagrange differential equations 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 does the Lagrange function exist?

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?

Problems to solve independently

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, the possible movement of the 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 point of application of which is shown in Fig. 79; the possible work done by it is equal.