Multi-Web Roller Out - MapleSim Help

All Products Maple MapleSim

Home : Support : Online Help : MapleSim Toolboxes : MapleSim Web Handling Library : Laminating : Multi-Web Roller Out

Multi-Web Roller Out

Exit condition for the merged web

	Description
	The Multi-Web Roller Out component captures the merging of multiple webs and the exit configuration for the merged web.

	Usage
	A Multi-Web Roller Out component is only used with 2 or more Multi-Web Roller In components. Example: The above arrangement can be replaced by the following:

Equations

Consider a portion of the web between two boundaries $a$ and $b$ (i.e. a control volume). Assume the web velocity goes through a discontinuous change at the left (entry) boundary which is at the roller location.

$T_{a_{1}} \dots T_{a_{n}}$	Upstream tensions
$T_{b}$	Downstream tension
$v_{a}$	Upstream speed
$v_{b}$	Downstream speed
$τ_{1}, τ_{2}$	Motor torque applied to roller 1 and 2
$R_{1}, R_{2}$	Roller effective radius
$ω_{1}, ω_{2}$	Roller angular velocity
$J_{1}, J_{2}$	Moment of inertia or roller 1 and 2
$W$	Web width
$β$	Wrap angle

The incoming velocities match the roller velocity under the no-slip conditions

$v__a = v__b, v__a = R__1 ω__1, v__b = R__2 ω__2$

(1)

The conservation of mass for the control volume can be expressed as

$\frac{\partial}{\partial t} (\int_{a__a}^{x__b} ρ (x, t) A (x, t) &DifferentialD; x) = v__a (t) (\sum_{i = 1}^{n} ρ__a__i (t) A__a__i (t)) - ρ__b (t) A__b (t) v__b (t)$

(2)

Based on the above assumption we can write

$\frac{\partial}{\partial t} (ρ A L) = v__a (\sum_{i = 1}^{n} ρ__a__i (t) A__a__i (t)) - ρ__b (t) A__b (t) v__b (t)$

(3)

where $L = x_{b} - x_{a}$ and we dropped $t$ to simplify the notation.

The mass of an infinitesimal length of the web is

$dm = ρ__u A__u dx__u$

(4)

where subscript $u$ is for the undeformed state.

The element strain in the Machine Direction (MD) can be written as:

$dx = dx__u (1 + ε)$

(5)

Putting (4) and (5) together, we get:

$ρ A = \frac{ρ__u A__u}{1 + ε}$

(6)

Substituting (6) into (3) and simplifying yields:

$\frac{\partial}{\partial t} (\frac{L}{1 + ϵ__b}) = \frac{v__a (\sum_{i = 1}^{n} \frac{ρ__ai A__ai}{1 + ϵ__ai})}{ρ__b A__b} - \frac{v__b}{1 + ϵ__b}$

(7)

What remains is the material's constitutive equation. Assuming linear visco-elastic material (Kelvin–Voigt model), we have, for $i = 1 .. n$ :

$T__a__i = E__a__i A__a__i (ϵ__a__i + d__r__a__i (\frac{&DifferentialD; ϵ_{a__i}}{&DifferentialD; t}))$

$T__b = E__b A__b (ϵ__b + d__r__b (\frac{&DifferentialD; ϵ_{b}}{&DifferentialD; t}))$

(8)

The control volume is selected such that it covers the entire web span from one roller to the next. Under no-slip conditions, the portions of the web in contact with the rollers are neglected.

Finally the tension/torque relationship can be written as:

$J_{1} (\frac{&DifferentialD;}{&DifferentialD; t} ω_{1}) + J_{2} (\frac{&DifferentialD;}{&DifferentialD; t} ω_{2}) + d (ω_{1} + ω_{2}) = \{\begin{array}{c} τ_{1} - R_{1} ((\sum_{i = 1}^{n} T_{a__i}) - T_{b}) & Side = 1 \\ τ_{2} - R_{2} ((\sum_{i = 1}^{n} T_{a__i}) - T_{b}) & Side = 2 \end{array}$

(9)

Connections

Name	Description	Modelica ID
${flange}_{c1}$	Roller rotation flange (enabled when $Side = 1$	flange_c1
${flange}_{c2}$	Roller rotation flange (enabled when $Side = 2$	flange_c2
${frame}_{b}$	Right web 3-D connection point	frame_b
${frame}_{c}$	Roller holder base frame	frame_c
${web}_{b}$	Right (exit) web transfer information port (green)	web_b
$webs$	Multi-Web transfer information port (gray)	webs
$v$	Real output ; velocity	v
$T$	Real output ; tension in web	T

Parameters

Webs

Name	Default	Description	Modelica ID
$n$	$2$	Number of merging webs	n
Incoming Web Properties		Names of Web Properties records defining the incoming webs	wp_i
Combined Web Properties		Names of Web Properties records defining the outgoing web	wp_o

Settings

Name	Default	Description	Modelica ID
Side	1	Select actuated roller: 1 or 2	Side
Use alternative configuration	false	Choose between two possible solutions for the web/roll configuration for the merged web	useAltConfig
Flip direction	false	Use this option when building web lines that go from right to left	flipDirection

Roller

Name	Default	Units	Description	Modelica ID
$D_{1}$	$0.2$	$m$	Diameter of roller 1	D1
$D_{i1}$	$0$	$m$	Inner diameter of roller 1	Di1
$D_{2}$	$D_{1}$	$m$	Diameter of roller 2	D2
$D_{i2}$	$0$	$m$	Inner diameter of roller 2	Di2
$L$	$1.2$	$m$	Roller length	L
Use cylindrical geometry	$true$		When checked (true), roller inertia is calculated assuming a uniform cylindrical geometry	useCylindricalGeometry
$ρ$	$2.7 \cdot 10^{3}$	$\frac{kg}{m^{3}}$	Roller density; enabled when using cylindrical geometry	rho_roll
$m_{1}$	$1$	$kg$	Roller 1 mass; enabled when not using cylindrical geometry	m1
$m_{2}$	$1$	$kg$	Roller 2 mass; enabled when not using cylindrical geometry	m2
$J_{1}$	$0.005$	$kg m^{2}$	Roller 1 rotational inertia; enabled when not using cylindrical geometry	J1
$J_{2}$	$0.005$	$kg m^{2}$	Roller 2 rotational inertia; enabled when not using cylindrical geometry	J2
$d$	$0$	$\frac{N m s}{rad}$	Bearing viscous damping constant (both rollers)	d

Frame

Name	Default	Units	Description	Modelica ID
Reference frame	Middle of the gap		Select option for the location of the ${frame}_{c}$ (holder). There are three options available: Middle of the gap, Center of roller 1, and Center of roller 2	Ref_pos
$m_{f}$	$1$	$kg$	Frame mass	m_frame
$J_{f}$	$0.01$	$kg m^{2}$	Frame inertia	J_frame
$Y$	0	$m$	Offset from the reference point in the Y direction	Y

Visualization

Name	Default	Units	Description	Modelica ID
Show roller 1	$true$		When checked (true), a visualization of roller 1 is created	showVisualization_roller
Show roller 2	$true$		When checked (true), a visualization of roller 2 is created	showVisualization_second
Transparent roller	$false$		When checked (true) the roller visualization is transparent	transparent_roller
Roller color	Blue		Roller color	color_roller
Band color	Yellow		Band color	color_band
Band angle	$20$	$\deg$	Band angle	band_angle
Show frame	$false$		When checked (true), a visualization of the frame is created	showVisualization_frame
Transparent frame	$false$		When checked (true), the roller visualization is transparent	transparent_frame
Frame color	Orange		Frame color	color_frame

Web Sensor

Name	Default	Description	Modelica ID
Use sensor	$false$	When checked (true), two signal outputs are enabled for the tension in the span and the material transfer speed	useSensor
Velocity output unit	$\frac{m}{s}$	Selects units of the velocity output; enabled when $Use Sensor = true$	toUnitV
Force output unit	$N$	Selects units of the force output; enabled when $Use Sensor = true$	toUnitF

Advanced Settings

Name	Default	Description	Modelica ID
Use effective radius	$false$	When checked (true), the roller radius is padded with half the web thickness	useEffectiveRadius

Summary Variables

Name	Units	Description	Modelica ID
summary_WrapAngle	$rad$	Wrap angle	summary_WrapAngle
summary_Length	$m$	Length of the web in contact with the roller	summary_Length

Maple

Maple Add-Ons

MapleSim

MapleSim Add-Ons

Systems Engineering

Consulting Services

Maple T.A. and Möbius

Education

Industries

Automotive and Aerospace

Robotics

Machine Design & Industrial Automation

Other

Application Areas

Product Pricing

Purchasing

Institutional Student Licensing

Maplesoft Elite Maintenance (EMP)

Support

Product Training

Online Product Help

Webinars & Events

Publications

Content Hubs

Examples & Applications

Community

About Maplesoft

Media Center

User Community

Contact

Online Help

All Products Maple MapleSim

Maple

Powerful math software that is easy to use

Maple Add-Ons

MapleSim

Advanced System Level Modeling

MapleSim Add-Ons

Systems Engineering

Consulting Services

Maple T.A. and Möbius

Education

Industries

Automotive and Aerospace

Robotics

Machine Design & Industrial Automation

Other

Application Areas

Product Pricing

Purchasing

Institutional Student Licensing

Maplesoft Elite Maintenance (EMP)

Support

Product Training

Online Product Help

Webinars & Events

Publications

Content Hubs

Examples & Applications

Community

About Maplesoft

Media Center

User Community

Contact

Online Help

All Products Maple MapleSim