The speed of the moving loose band v1 is obtained by multiplying speed of the moving tight band by the value a<1. For the band moving between the shafts it is possible to write the following system of the equations [4]:

Here D01 is the initial diameter of coiling bobbin; b is the thickness of the band; F is the tension force; v1 is the linear speed of the coiling; v2 is the linear speed of unwind unit. The first, the fourth and the sixth equations of the system are nonlinear. Therefore the tight band system is nonlinear also. The dynamic of the friction clutch and unwind roll section can be described by set of differential and algebraic equations (4).

Here i_s is the excitation current of friction clutch, Us is supply voltage of the clutch, L_s is inductance of the excitation winding of the clutch, R_s is the active resistance of the excitation winding of the clutch, d2 is the decrease of diameter of unwind roll, b is the thickness of the band, ω2 angular velocity of unwind roll, v2 is the linear velocity of the unwinding band; Dm2 is maximum diameter of unwind roll, R1 and R2 - corresponding radii of wind and unwind rolls, km is transferring coefficient of the clutch, MF is the torque produced by tension force; Ms is the torque developed by the clutch, Mt estimated friction torque of the unwind unit; k_tr is the friction coefficient; F is the tension force of the band, Ma loading torque of the winder, Js is the inertia of the unwinding roll, i reduction coefficient of the gearing, M_sv is the load torque of the motor.

The first equation of this system in (Eq. 5) gives the dependency of the excitation current is of the friction clutch on the supply voltage Us. The second, third and fourth equations gives us angular velocity of unwind roll ω2 decrease of diameter of unwind roll d2, and radius of unwind roll R2. The fifth equation - dependence of the linear speed v2 of unwinding band on the angular velocity of unwind roll ω2 and radius of unwind roll R2. The sixth equation - dependence of the torque MF on the tension force F of the band and radius of unwind roll R2. The seventh equation dependence of the torque developed by clutch Ms on the excitation current is. The ninth equation dependence of the load torque Ma on the tension force of the band F and diameter of winding roll R1.

The dynamics of the rewind roll section have been composed by using structural schemes of coiling process [3] and mathematical description of dynamics of the tight band [4]. In such a way the block diagram of a whole system (Fig. 4) consist of two parts: winding unit (WU) and friction clutch unit (FCU).

According to the third system of equations 5, 6 and 9 equations are nonlinear. Hence, the system of composition of the tension force of the band is not linear. So, the stabilization, analysis of control and the other investigations of such systems are quite complicated. Therefore, often systems are investigated by using only analytical methods i.e. mathematical modeling. The mathematical model of the lab stand is created according to the block diagram (Fig. 4.) with MATLAB/ SIMULINK package [6]. Tight band coiling consist of electrical drive (frequency converter and induction motor), gearing and winding shaft. To stabilize the linear speed of the winding band feedback loop was used according to linear rolling speed.