To Problem of the Rewinding of the Tape with Automatically Adjustable Influences

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.

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 [11]. Following the theory of parametrical clearing [12], and the theory of the compelled motions [13] constructed the theory, allowing to de-velop the area of practical applicability of methods of analytical mechanics of systems with servo-constraints, including questions of their steady realization [14] [15]. In works [16] [17] the equations of motion of the systems, interfered by constraints of the first and second sort are deduced, and also the obvious kind of forces of reactions of servo-constraints is defined.
In this article the results of works [16] [17] are illustrated to the problem of rewinding of a tape.

Forming the Equation of Motion and Refining the Servo-Constraints' Forces of Reaction
Let's consider process of rewinding of a tape (Figure 1). Rewinding of a tape from the drum 3 on a drum 2 is carried out by means of electric machine (EM) of a direct current of the independent excitation operating through a reducer with transfer number 1 i on a drum 2. On where 1 2 3 , , J J J -the moments of inertia of a reducer both corresponding drums; 2 3 , ϕ ϕ -angles of rotation of drums 2 and 3.
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) [19]: where for i q co-ordinates 1 2 3 , , ϕ ϕ ϕ are designated; i Q -the generalized force appropriated to co-ordinate i q ; i R -the generalized reactive force, which looks like -relative speed of joining particles; r v a -particle acceleration in motion concerning the system of co-ordinates connected by a link, and the symbol "*" above specifies that differentiation is made according to a hardening principle.  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. [16] (1967) we conclude that: On system, according to assumptions, one ideal geometrical constraint and one kinematical servo-constraint in Beghin H. (1967) [1] is imposed, where 20 ρ -radius of an empty drum 2; h -a thickness of a tape.
Taking into account (3) kinetic energy (1) systems we will lead to a formula 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) [1]: We will consider the right part from a dashed line. Considering that for the right part, 3 3 3 where эл M -the constant moment put EM under the influence of direct current; F -force of a tension of a tape, from (2), (5)-(8), by method (A)-moving [6] [7], we will receive: where λ -reaction of servo-constraint (4).
Considering that for the left part where 3 ρ -radius of no to wind off drum 3; тр M -the brake moment of a drum 3, from (5), (10) we will receive 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 2 3 , , , , тр F M ϕ ϕ λ . Consider a case, when rewinding is carried out with constant brake moment сonst тр M = , and speed of a tape υ is regulated only.
As it is known [9] [10], servo-constraints are carried out not precisely and therefore, along with (4) the have occurrence parity where ξ -the parameter, characterizing clearing of system from servo-constraint (4).

Stability
If a reaction of servo-constraint λ to form under the law, showed in [17] [18]: In expression (24) constant number 1 е we will choose such, that a root λ of the characteristic Equation

The Realization of Servo-Constraints
We will consider a realization problem of servo-constraint (4) by electromechanical digital watching system (DWS) [20] for which executive element we accept the engine of a direct current of independent excitation. Its full system of the equations will look like [17]:  DWM  DWM  RC  CCP  2  2  1 , , , , ,