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In this work the problem of rewinding of a tape with constant speed is considered. Considering that drums represent bodies of variable weight, the equations of motion of system are formulated. Taking into account parametrical clearing of system of servo-constraints, the structure of force of reaction of servo-constraints which provides steady realization of servo-constraints (a constancy of linear speed of a tape) is defined. For realization of servo-constraints, it is offered to build digital watching system (DWS) and the full system of equations of DWS is formed. Laws of change of the operating influences, systems providing stability under the relation of the variety defined of servo-constraints are defined.

For the first time the concept idea of servo-constraints has been entered into analytical dynamics by H. Beghin [

Appendices of methods of analytical dynamics to a wide range of specific targets demand the account and other features connected with steady realization of servo-constraints, and that for such systems it is impossible to distract from a way of their realization.

S. S. Nugmanova’s attention for the first time has been paid to this circumstance [

In this article the results of works [

Let’s consider process of rewinding of a tape (

where

As drums 2 and 3 represent bodies of variable weight it is necessary to take the general equation of dynamics in the form showed in Bessonov A. P. (1967) [

where for

Process of rewinding of a tape

If relative speed of joining particles is equal to zero, and there is no relative motion of joining particles concerning a drum 2, according to Bessonov A. P. [

On system, according to assumptions, one ideal geometrical constraint

and one kinematical servo-constraint in Beghin H. (1967) [

is imposed, where

Taking into account (3) kinetic energy (1) systems we will lead to a formula

On possible moving constraint (3) imposes restriction [

From a way of action of servo-constraint (4) follows that, moving, on which servo-constraint works do not make reaction, look like in Beghin H. (1967) [

We will consider the right part from a dashed line. Considering that for the right part,

where

where

Considering that for the left part

where

To the received system of the Equations (9), (11) adding one kinematical equation,

will be received system of three Equations (9), (11), (12) concerning unknown persons

Consider a case, when rewinding is carried out with constant brake moment

As it is known [

where

Having for an object steady realization of servo-constraint (4), the received system of the equations will be added to (9), (11), (12) equation [

and reaction compulsion

where

Thus, the system of the Equations (9), (11), (12), (16) describes dynamics of adjustable process of rewinding of a tape concerning variables

and substituting it in (9), the following equation will be received:

Taking into account (12), Equation (17) can be led to the following:

If a reaction of servo-constraint

where

Taking into account a parity (13), last equation will look like:

Considering

Taking into account (20), Equation (19) will look like

where

Apparently, from (22) the Equation (21) represents the differential equation with variable factors. Stability of its zero decision

where

In expression (24) constant number

had a negative material part. This condition, according to Hurvits’s criterion [

which is reduced to a condition

Lyapunov’s definitely positive function

Its full derivative on time, worked out owing to the equation of the indignant motion (23), will look like:

Let’s calculate:

Then the condition of certain positively of the form—

which it is reduced to a condition

Condition (26) show that, for maintenance of steady realization of servo-constraint (4) moment of the electric machine

We will consider a realization problem of servo-constraint (4) by electromechanical digital watching system (DWS) [

where

The law of formation of the operating influences DWM providing stability of motion of system in relation to servo-constraint (4) is defined by the decision of system of the equations:

rather of

Substituting the law (29) in (27), results the Equation (21). Hence, the law (28) provides asymptotically stability of motion of system under the relation of the variety defined by servo-constraint (4).

Along with the generalized model, we will consider the simplified model of watching system [

from (28) we will receive the law of formation of operating influences DWM for simplified model DWS:

Substituting the law (31) in (27) at the same assumptions (30) we will receive the Equation (21), which stability conditions looks like (26).