Winders and web tension: constant power

The paper and film industries depend on precise web-tension control. As material is unwound, processed and rewound, the tension in the moving web must stay constant even though the roll diameters change all the time. This lesson is course section 6.1.4.

The winder law

A constant surface speed at a changing radius means a changing angular speed; a constant tension at a changing radius means a changing torque. The winder is therefore the archetypal constant-power load of Chapter 1: torque times speed is held constant, with torque commanded in proportion to the instantaneous roll radius. Requiring a constant web tension $F$ at a constant surface speed $v$ gives the winder law:

T_\text{wind} = F\,r(t), \qquad \omega(t) = \frac{v}{r(t)}, \qquad T_\text{wind}\,\omega = F v = \text{constant} \tag{6.2}

Worked example: torque and speed range of a winder (course §6.1.4)

A rewind roll grows from a 0.15 m core to 0.75 m full diameter while holding $F = 600$ N of web tension at $v = 8$ m/s. At the two extremes:

$T_\text{core} = 600 \times 0.075 = 45$ N·m and $\omega_\text{core} = 8 / 0.075 \approx 107$ rad/s;
$T_\text{full} = 600 \times 0.375 = 225$ N·m and $\omega_\text{full} = 8 / 0.375 \approx 21$ rad/s.

Torque rises five-fold while speed falls five-fold; the product, and so the power $F v = 4.8$ kW, is constant throughout. This is Chapter 1's constant-power load in its purest industrial form, and it demands a drive with a 5:1 constant-power (field-weakening) range.

Try it: a rewind roll that grows

Roll diameter now0.150 m

Web tension F600 N

Surface speed v8.0 m/s

Core diameter0.15 m

Full-roll diameter0.75 m

Torque
Speed

Torque F r: 45.0 N·m
Speed v / r: 106.7 rad/s · 1,019 rpm
Power F v: 4.80 kW
Constant-power range: 5.0 : 1

T = F r ; ω = v / r ; T ω = F v = const

Predict first

The same winder is re-specified for a 1.5 m full roll on the same 0.15 m core. What constant-power range must the drive now cover?

Still 5:1: the tension and speed have not changed10:125:1

Closing the loop: dancer or load cell

Commanding $T = F r$ open-loop needs to know the radius, which the drive estimates from the ratio of line speed to motor speed. Errors in that estimate, in friction or in the roll's inertia during speed changes all show up as tension errors. So a dancer roll (a weighted or pneumatic roll whose position measures the web's slack) or a load cell closes a tension loop around the coordinated unwind and rewind drives. The unwind stand works in the mirror image: the web pulls the roll, so the unwind drive brakes, generating in quadrant II, and on a common DC bus its energy feeds the rewind.

The glass and textile industries pose similar coordinated-drive and tension problems: a small number of control principles, correctly combined, spans the whole of manufacturing.

FoundationStart here if this is new to you

Wind a garden hose onto a reel at a steady walking pace. At first the reel spins fast and is easy to turn; as the coil grows, it turns more slowly and needs more effort to keep the hose taut. Your pace and the pull on the hose never changed, so neither did your effort multiplied by the reel's speed. A winder does the same with paper at 8 m/s.

ExplorerGo deeper: derivations and open questions

Acceleration. During a line speed-up, the rewind must also accelerate the roll's own inertia, which grows as $r^4$ . Why does this make the tension loop's job hardest near the full roll?

Unwind. Mirror the worked example for an unwind stand going from 0.75 m to 0.15 m. In which quadrant is its drive, and where can its power go?