where = aileron deflection b = aileron effectiveness

Chapter , Problem 11.37

(choose chapter or problem)

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 \(v_p\) at the rollers. The paper speed as it approaches the roll is \(v_r\). The paper tension changes as the radius of the roll changes or as the speed of the pinch rollers change. The paper has an elastic constant k so that the rate of change of tension is

                                                   \(\frac{d T}{d t}=k\left(v_r-v_p\right)\)

For a paper thickness d, the rate of change of the roll radius is

                                                    \(\frac{d R}{d t}=\frac{d}{2} W\)

The inertia of the windup roll is \(I=\rho \pi W R^4 / 2\), where \(\rho\) 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.

a. Assuming that the paper thickness is small enough so that \(\dot{R} \approx 0\) for a short time, develop a state-variable model with the motor voltage e and the paper speed \(v_p\) as the inputs.

b. Modify the model developed in part (a) to account for R and I being functions of time.