Many winding applications in the paper, wire, and plastic industries require a control system to maintain proper tension. Figure P11.37 shows such a system winding paper onto a roll. The paper tension must be held constant to prevent internal stresses that will damage the paper. The pinch rollers are driven at a speed required to produce a paper speed up at the rollers. The paper speed as it approaches the roll is ur. The paper tension changes as the radius of the roll changes or as the speed of the pinch rollers change. The paper has a elastic constant k so that rate of change of tension is
dT / dt = k (ur - up)
For a paper thickness d, the rate of change of the roll radius is
dR/dt = d/2 W
The inertia of the windup roll is I = ( ( W R4/2, where ( is the paper mass density and W is the width of the roll. So R and I are functions of time.
The viscous damping constant for the roll is c. For the armature-controlled motor driving the windup roll, neglect its viscous damping and armature inertia.
Assuming that the paper thickness is small enough so that ( 0 for a short time, develop a state-variable model with the motor voltage e and the paper speed up as the inputs.
Modify the model developed in part (a) to account for R and I being functions of time.
Figure P11.37

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