## Winding Machine

### Introduction

Winding machines, also called winders, are used in the pulp and paper industry as well as in the textile, steel, and plastic industries.

An important characteristic of most winders is that the force acting on the winding material must remain constant. This is realized by controlling the winder torque proportionally to the roll variable radius. Note that it is assumed here that the material is fed to the winder at constant speed. The latter implies that the winder angular speed is forced to decrease proportionally to the roll radius. Hence the winding machine is a constant power application, because the product of the winder mechanical torque and its angular speed is constant.

### Description of the Winder

The following graphic shows a physical representation of a winder,
where *W* is the roll width, *r _{1 }* the
core radius,

*r*the roll radius, and

_{2 }*MT*the material thickness.

**Physical Representation of a Winder**

Beside the variables described above, the simulation also requires the following parameters and variables:

| Material mass per unit volume |

| Material length |

| Material mass |

| Material inertia |

| Winder core inertia |

| Winder viscous friction coefficient |

**Diagram of the Complete Winding System**

Diagram of the Complete Winding System shows a Simulink^{®} diagram
of the complete winding system. This system consists of four blocks:
the Winder Control block, the DC Motor Drive block,
the Speed Reducer block, and the Winder
Model block.

### Block Description

#### Winder Model Block

This block computes various winder variables using the following equations.

Surface speed *S*

$$S=\omega \cdot {r}_{2}$$

where *ω* is the winder angular speed.

Material length *L*

$$L={\displaystyle \int S}dt$$

Roll radius *r _{2 }*

$${r}_{2}=\sqrt{\frac{L\cdot MT}{\pi}+{r}_{1}^{2}}$$

Material mass *M*

$$M=MV\cdot \pi \cdot W\cdot \left({r}_{2}^{2}-{r}_{1}^{2}\right)$$

Total winder inertia *J _{t }* and
material inertia

*J*

_{ω }$${J}_{t}={J}_{\omega}+{J}_{c}$$

where

$${J}_{\omega}=\frac{1}{2}\cdot M\cdot \left({r}_{2}^{2}+{r}_{1}^{2}\right)$$

The winder angular speed is calculated using the following differential equation

$${T}_{e}={J}_{t}\frac{d\omega}{dt}+B\cdot \omega +{T}_{l}$$

where *T _{l }* is the winder load torque
and

*T*is the motor drive electric torque. The calculation of the tension or force

_{e }*F*applied on the winding material is based on the same differential equation as above, where the load torque is expressed as

*T*=

_{l}*F*·

*r*. Rearranging the equations in term of

_{2}*F*yields

$$F=\frac{{T}_{e}-\left({J}_{t}\cdot \dot{\omega}\right)-\left(B\cdot \omega \right)}{{r}_{2}}$$

This estimated force is fed back to the Winder Control block in order to be regulated.

Note that in the above two equations, the term $$\omega \cdot {\dot{J}}_{t}$$ is omitted because it has been found to be negligible for the case considered here.

#### Winder Control Block

This block contains a PID controller that regulates the tension applied on the winding material. The output of this force controller is a torque reference set point for the winder motor drive. The Winder Control block shown in Winder Control Block also contains the tension versus speed characteristic of the external process supplying the material to the winder at constant speed. This characteristic consists in a straight line of slope equal to the ratio of the reference material tension on the constant surface speed.

**Winder Control Block**

#### DC Motor Drive Block

This block contains a complete two-quadrant three-phase rectifier DC drive with its three-phase voltage source. The DC drive is rated 5 hp, 220 V, 50 Hz and is torque regulated.

#### Speed Reducer Block

The DC motor is connected to the winder by a Speed Reducer block. The speed reduction ratio is 10, allowing the winder to turn 10 times slower than the motor, while the shaft-transmitted torque is almost 10 times higher on the low-speed side. The torque required by the winder in this case study is approximately 200 N.m.

### Simulation Results

The winding machine simulation model is contained in the file `cs_winder`

.
The simulation parameters are those of a paper winding application
where the roll width is 10 m. Open the file and look at the parameters
in the Simulink masks of the Winder Model block,
the Winder Control block, the DC Motor Drive block,
and the Speed Reducer block. In
the Winder Control block, you will see that the tension
set point is 300 N and the surface speed set point is 5 m/s.

The rate of change of the tension set point is limited internally to 25 N/s so that the tension set point requires 12 s to reach its final value. Note that the simulation time step of the complete model is 1 µs in order to comply with the speed reducer, which is the block that requires the smallest simulation time step.

Start the simulation and observe how well the material tension and the surface speed ramp to their prescribed values in Material Tension and Surface Speed respectively. Winder Angular Speed, Mechanical Torque, and Power shows the winder angular speed, mechanical torque, and power. Note that once the operating point is reached (300 N, 5 m/s), the angular speed decreases and the torque increases, both linearly, so that the power is approximately constant. The reason why the mechanical power is not precisely constant but decreases slightly is that the decreasing speed winder own inertia supplies a small part of the constant power required by the winder.

**Material Tension**

**Surface Speed**

**Winder Angular Speed, Mechanical Torque, and
Power**