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Wind Drum Two Webs

Multi-web wind drum component

	Description
	The Wind Drum Two Webs component models simultaneous winding of two webs with variable roll radius and inertia.

	Usage
	The Wind Drum component must be at the end of two web lines. Using the provided 1D rotational flange ( ${flange}_{c}$ ), the Wind Drum component can be equipped with a motor.

Equations

Consider the drum shown below:

$T_{a1}, T_{a2}$	Upstream tension
$v_{a1}, v_{a2}$	Upstream speed
$τ$	Motor torque (zero if modeling an idler roller)
$ω$	Angular velocity (positive means wind)
$R_{c}$	Core radius
$R_{1} (t), R_{2} (t)$	Instantaneous roll radii for web 1 and 2
$d_{B}$	Bearing damping coefficient
${th}_{1}, {th}_{2}$	Web thicknesses
$W_{1}, W_{2}$	Web widths

$T_{a1} R_{1} (t) + T_{a2} R_{2} (t) = τ_{m} - J (t) d \frac{ω}{dt} - d_{B} ω$

$v_{a1} = R_{1} ω v_{a2} = R_{2} ω$

The roll inertia consists of the core inertia, $J_{c}$ , and the approximate roll variable inertia, $J (t)$ :

$J (t) = J_{c} + \frac{π}{2} \overset{&Hat;}{λ} ({R (t)}^{4} - R_{c}^{4})$

where

$\overset{&Hat;}{λ} = \frac{ρ_{1} W_{1} {th}_{1} + ρ_{2} W_{2} {th}_{2}}{{th}_{1} + {th}_{2}}$

Roll Radius

The time-varying roll radius is given by the following differential equations:

When $Use stepwise = false$ , the effective roll radius is given by

$\frac{dR (t)}{dt} = ω \frac{{th}_{1} + {th}_{2}}{2 π}$

with the following initial value:

$R (0) = \frac{1}{2} (D_{init} + {th}_{1} + {th}_{2})$

The radii of the layers are calculated as:

$R_{1} (t) = R (t) - \frac{{th}_{2}}{2}$

$R_{2} (t) = R (t) + \frac{{th}_{1}}{2}$

When $Use stepwise = true$ and $Use multistep = false$ the effective roll radius is given by:

$\frac{dR (t)}{dt} = {\begin{array}{c} ω \frac{{th}_{1} + {th}_{2}}{δ_{φ}} & 0 < φ < δ_{φ} \\ 0 & otherwise \end{array}$

where $φ$ is the rotation of the drum between $0$ and $2 π$ and the roll radius increases only over a portion of each turn defined by the merge angle $δ_{φ}$ .

We also have the following initial value:

$R (0) = \frac{1}{2} (D_{init} + {th}_{1} + {th}_{2})$

The radii of the layers are now calculated as:

$R_{1} (t) = R (t) - \frac{{th}_{2}}{2}$

$R_{2} (t) = R (t) + \frac{{th}_{1}}{2}$

When $Use stepwise = true$ and $Use multistep = true$ the effective layer radii are given by:

${dR}_{1} \frac{t}{dt} = {\begin{array}{c} \frac{ω {th}_{1}}{δ_{φ}} 1 \begin{array}{c} 0 \end{array} < {\overset{&Hat;}{φ}}_{11} < δ_{φ} 1 \\ 0 & otherwise \end{array} + {\begin{array}{c} \frac{ω {th}_{2}}{δ_{φ}} 2 \begin{array}{c} 0 \end{array} < {\overset{&Hat;}{φ}}_{12} < δ_{φ} 2 \\ 0 & otherwise \end{array}$

${dR}_{2} \frac{t}{dt} = {\begin{array}{c} \frac{ω {th}_{1}}{δ_{φ}} 1 \begin{array}{c} 0 \end{array} < {\overset{&Hat;}{φ}}_{21} < δ_{φ} 1 \\ 0 & otherwise \end{array} + {\begin{array}{c} \frac{ω {th}_{2}}{δ_{φ}} 2 \begin{array}{c} 0 \end{array} < {\overset{&Hat;}{φ}}_{22} < δ_{φ} 2 \\ 0 & otherwise \end{array}$

with the following intitial values:

$R_{1} (0) = R (t) + \frac{{th}_{1}}{2}$

$R_{2} (0) = R (t) + \frac{{th}_{2}}{2}$

In the above, $δ_{φ} 1$ and $δ_{φ} 2$ are the merge angles for layer 1 and 2, respectively.

Also, ${\overset{&Hat;}{φ}}_{ij}$ are angles dependent on the initial web starting point angle parameters ( $φ_{2}$ and $φ_{2 \to 1}$ ) and the variable angle between the tangent point of web 1 and web 2, $β_{12}$ :

In this case, we also define $R (t) = \frac{1}{2} (R_{1} (t) + R_{2} (t))$

	Example: Using the multistep option
	Consider this simple model which only covers the winding of two webs: Relevant model parameters and layout is shown below: The multistep winding results in the following sequence of changes to the radii:

Variables

Name	Value	Units	Description	Modelica ID
$FF1$				FF1
$RBF00$				RBF00
$RBF01$				RBF01
$RBF03$				RBF03
$ARc$				ARc
$ATc$				ATc
${PR}_{a}$				PR_a
${RBF}_{a}$				RBF_a
$RB1$				RB1
$RB2$				RB2
$RB3$				RB3
${PR}_{v}$				PR_v
$Φ$				Phi
${RBF}_{ecc} [θ0]$				RBF_ecc_theta0
${RBF}_{ecc}$				RBF_ecc
${PT}_{ecc}$				PT_ecc
$m_{web}$				m_web
${tanPoint}_{a}$				tanPoint_a
$ATa$				ATa
$AWM1$				AWM1
$frameVis$				frameVis
$rollVis$				rollVis
$rollVisColor$				rollVisColor
$drumVis$				drumVis
$RBFF1$				RBFF1
$RBFF2$				RBFF2
$R1$				R1
$FAM2$				FAM2
$TT1$				TT1
${pt}_{R}$				pt_R
${pt}_{Rmax}$				pt_Rmax
${pt}_{D}$				pt_D
${pt}_{T}$				pt_T
${pt}_{RotMatc}$				pt_RotMatc
${pt}_{ey}$				pt_ey
${pt}_{ez}$				pt_ez
${reaction}_{correction}$				reaction_correction
${CAD}_{torque}$				CAD_torque
$BPT$				BPT
$w$		$\frac{rad}{s}$		w
$FAM1$				FAM1
$FA1$				FA1
$TA1$				TA1

Connections

Name	Description	Modelica ID
${frame}_{a1}$	Left web 3D connection point	frame_a1
${frame}_{a2}$	Left web 3D connection point	frame_a2
${frame}_{c}$	Roller holder base frame	frame_c
${frame}_{d}$	External mass/inertia	frame_d
${web}_{a1}$	Web #1 entrance transfer information port	web_a1
${web}_{a2}$	Web #2 entrance transfer information port	web_a2
${flange}_{c}$	Drum rotation flange	flange_c
$Rmax$	Real output; drum and web radius $[m]$ [1]	Rmax
${ecc}_{in}$	Real input; eccentricity signal $[m]$	ecc_in
$reset$	Boolean input to trigger roll radius reset	reset

[1] Available if $Enable radius sensor = true$ .

Parameters

Web Properties

Name	Default	Units	Description	Modelica ID
Web Properties for layer 1	$WP1$		Name of the Web Properties record for layer 1 (buttom)	wp1
Web Properties for layer 2	$WP2$		Name of the Web Properties record for layer 2 (top)	wp2
Comp. Factor	$1$		Ratio of the thickness of the mixed web to the sum of thicknesses of component webs	CompressionFactor

Settings

Name	Default	Units	Description	Modelica ID
Use alternative configuration	$false$		Choose between two possible solutions for the web/roll configuration	useAltConfig
Flip direction	$false$		Use this option when building web lines that go from right to left. See documentation for more information	flipDirection
Use fixed base	$false$		When checked (true), frame is fixed in the inertial coordinate frame	useFixedBase
$r$	$[0, 0]$	$m$	Location of frame_b with respect to ground [1]	InitPos
$θ$	$0$	$\deg$	Rotation about z axis [1]	InitAng
Enable radius sensor	$false$		When checked (true), drum outer radius is provided via an output signal	useSensor

[1] Available when $Use fixed base = true$ .

Drum

Name	Default	Units	Description	Modelica ID
$D_{init}$	$0.1$	$m$	Initial outer roll diameter	D_init
$D$	$0.10$	$m$	Core diameter	D
$D_{i}$	$0$	$m$	Core inner diameter	Di
$L$	$1.2$	$m$	Drum length	L
Use cylindrical geometry	$true$		When checked (true), drum core inertia is calculated assuming a uniform cylindrical geometry	useCylindricalGeometry
$ρ$	$2.7 \cdot 10^{3}$	$\frac{kg}{m^{3}}$	Drum core density [1]	rho_roll
$m$	$1$	$kg$	Drum core mass (excluding the holder frame) [2]	m
$J$	$0.004$	$kg m^{2}$	Drum core rotational inertia [2]	J
$d$	$0$	$N m \frac{s}{rad}$	Bearing viscous damping constant	d
Use CAD	$false$		When checked (true), mass and inertia are defined from an externally attached CAD component or a rigid body component	useCAD
Use reset	$false$		The roller diameter is reset to D_init with an external signal	useReset

[1] Enabled when $Use cylindrical geometry = true$

[2] Enabled when $Use cylindrical geometry = false$

Winding

Name	Default	Units	Description	Modelica ID
Use stepwise	$false$		When checked (true), the roll diameter is changed over a portion of each rotation (instead of changing continuously)	useStepwise
Use multi-step	$false$		When checked (true), the roll diameter is changed over multiple steps	useMultistep
$φ_{0}$	$0$	$\deg$	Angle offset of start of the combined web w.r.t local x-axis; $φ_{0} < 360$ [1]	phi0
$δ φ$	$0$	$\deg$	Angle offset of start of web 2 (top layer) w.r.t initial tangent point; $δ φ < 360$ [1]	delta_phi
$φ_{2}$	$0$	$\deg$	Angle offset of start of web 2 (top layer) w.r.t. initial tangent point; $φ_{2} + φ_{2 \to 1} < 360 - β_{12}$ [2]	phi12
$φ_{2 \to 1}$	$0$	$\deg$	Angle offset of start of web 1 (bottom layer) w.r.t. web 2 (top layer); $φ_{2} + φ_{2 \to 1} < 360 - β_{12}$ [2]	phi12
$δ φ_{1}$	$0$	$\deg$	Merge angle web 1 (bottom layer); $δ φ_{1} < 360$ [2]	delta_phi1
$δ φ_{2}$	$0$	$\deg$	Merge angle web 2 (top layer); $δ φ_{2} < 360$ [2]	delta_phi2

[1] Available when $Use stepwise = true$ and $Use multistep = false$ .

[2] Available when $Use stepwise = true$ and $Use multistep = true$ .

Eccentricity

Name	Default	Units	Description	Modelica ID
Add eccentricity	$false$		When checked (true), a parameter is enabled for the roller eccentricity	addEccentricity
Use input	$false$		When checked (true), eccentricity is given by an input signal in meters	useInputEccentricity
$e$	$0$	$m$	Roller eccentricity	ecc
$θ_{0}$	$0$	$rad$	Initial eccentricity angle	theta_ecc0

Frame

Name	Default	Units	Description	Modelica ID
$m_{f}$	$1$	$kg$	Frame mass	m_frame
$J_{f}$	$\cdot 1 e - 1$	$kg m^{2}$	Frame inertia	J_frame
Add frame offset	$false$		When checked (true), frame connection point is placed at an offset w.r.t. the roll center	addFrameOffset
$r_{f}$	$[0, \cdot 5 D]$	$m$	Offset in the XY plane [1]	rf [1] Enabled if Add frame offset is true.

Visualization

Name	Default	Units	Description	Modelica ID
Show drum visualization	$true$		When checked (true), a visualization is produced for the drum	showVisualization_drum
Transparent drum	$false$		When checked (true), the drum visualization is transparent [1]	transparent_drum
Drum color	Blue		Color of the drum [1]	color_drum
Band color	Yellow		Color of a radial band on the drum [1]	color_band
Band angle	$20$	$\deg$	Band angle [1]	band_angle
Show frame	$true$		When checked (true), a visualization of the frame is created [4]	showVisualization_frame
Transparent frame	$false$		When checked (true) the frame visualization is transparent [4] and [5]	transparent_frame
Frame color	Orange		Color of the frame; [4] and [5]	color_frame
Show reaction force arrow	$false$		When checked (true), ground reaction force is visualized with an arrow	showReactionForce
Force arrow scale	$1000$	$1$	Force scale (N to m) [2]	forceScale
Force arrow color	Cyan		Force arrow color [2]	forceColor
Force arrow transparency	$false$		Force arrow transparency [2]	forceTransparent
Show reaction torque arrow	$false$		When checked (true), ground reaction force is visualized with an arrow	showReactionTorque
Torque arrow scale	$100$	$1$	Torque scale (N to m) [3]	torqueScale
Torque arrow color	Magenta		Torque arrow color [3]	torqueColor
Torque arrow transparency	$false$		Torque arrow transparency [3]	torqueTransparent

[1] Enabled if Show drum visualization is true.

[2] Enabled if Show reaction force arrow is true.

[3] Enabled if Show reaction torque arrow is true.

[4] Enabled if Add frame offset is true.

[5] Enabled if Show frame is true

Summary Variables

Name	Units	Description	Modelica ID
summary_Radius	$m$	Roll radius	summary_Radius
summary_Layers		Number of layers radius	summary_Layers

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