Span Merge - MapleSim Help

All Products Maple MapleSim

Home : Support : Online Help : MapleSim Toolboxes : MapleSim Web Handling Library : Laminating : Basic : Span Merge

Span Merge

Exit condition for the merged web

	Description
	The Span Merge component captures the merging of multiple webs from a Node Webs Merge component and the exit configuration for the merged web. A Span component must be connected to rollers, drums, sources, or sinks from the Web Handling Library.

	Usage
	A Span Merge components is only used with one of the Merge Node components (2, 3 or 4 Nodes). The component requires the definition of the number of the ingoing webs and the reference for the web properties. Example:

Equations

The equation base is comparable with a MultiWebRollerOut component without considering the nip rollers. 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 node location.

$T_{a_{1}} \dots T_{a_{n}}$	Upstream tensions
$T_{b}$	Downstream tension
$v_{a}$	Upstream speed
$v_{b}$	Downstream speed
$W$	Web width
$β$	Wrap angle

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

$\frac{\partial}{\partial t} (\int_{x_{a}}^{x_{b}} ρ (x, t) A (x, t) &DifferentialD; x) = v_{a} (\sum_{i = 1}^{n} ρ_{a_{i}} (t) A_{a_{i}} (t)) - ρ_{b} (t) A_{b} (t) v_{b} (t)$

(1)

Based on the above assumption we can write

$\frac{\partial}{\partial t} (ρ A L) = v_{a} (\sum_{i = 1}^{n} ρ_{a_{i}} (t) A_{a_{i}} (t)) - ρ_{b} (t) a_{b} (t) v_{b} (t)$

(2)

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

The mass of an infinitesimal length of the web is

$dm = ρ_{u} A_{u} {dx}_{u}$

(3)

where subscript $u$ is for the undeformed state.

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

$dx = {dx}_{u} (1 + ε)$

(4)

Putting (3) and (4) together, we get:

$ρ A = \frac{ρ_{u} A_{u}}{1 + ε}$

(5)

Substituting (5) into (2) and simplifying yields:

$\frac{\partial}{\partial t} (\frac{L}{1 + ε_{u}}) = \frac{v_{a} (\sum_{i = 1}^{n} \frac{ρ_{ai} A_{ai}}{1 + ε_{ai}})}{ρ_{b} A_{b}} - \frac{v_{b}}{1 + ε_{b}}$

(6)

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

$T_{a_{i}} = E_{a_{i}} A_{a_{i}} (ε_{a_{i}} + d_{r_{a_{i}}} (\frac{&DifferentialD;}{&DifferentialD; t} ε_{a_{i}}))$

$T_{b} = E_{b} A_{b} (ε_{b} + d_{r_{b}} (\frac{&DifferentialD;}{&DifferentialD; t} ε_{b}))$

(7)

Connections

Name	Description	Modelica ID
${frame}_{a}$	Left web 3-D connection point	frame_a
${frame}_{b}$	Right web 3-D connection point	frame_b
${web}_{a}$	Entrance web transfer information port (left, green)	web_a
${web}_{b}$	Exit web transfer information port (right, green)	web_b
$v$	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

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

Summary Variables

Summary variables provide easy access to quantities that maybe important in simulations to measure performance and other key metrics.

Name	Units	Description	Modelica ID
summary_Width	[1]	Web width considering the Poisson's ratio	summary_Width
summary_DeltaWidth	[1]	Changed web width regarding the Poisson's ratio	summary_DeltaWidth

[1] Units are set in the corresponding Web Properties block (under the Summary variables section).

Maple

Add-ons Maple

Math Success Platform

Améliorer les taux de retention

Maple Flow

MapleSim

Services-conseils

Produits d’enseignement en ligne

Enseignement

Industrie

Automobile et aéronautique

Robotique

Conception de machine & automatismes industriels

Autres

Domaines d’application

Prix et d'achats

Achats

Licences institutionnelles Etudiants

Vue d’ensemble du service EMP

Support

Formation produit

Aide en ligne Produit

Webinaires et évènements

Publications

Hubs de contenu

Exemples et applications

Communauté

A propos de Maplesoft

Ressources presse

Communauté

Contacts

Online Help

All Products Maple MapleSim

Maple

Logiciel de mathématiques puissant facile à utiliser

Add-ons Maple

Math Success Platform

Améliorer les taux de retention

Maple Flow

Engineering calculations & documentation

MapleSim

Modélisation avancée au niveau système

Services-conseils

Produits d’enseignement en ligne

Enseignement

Industrie

Automobile et aéronautique

Robotique

Conception de machine & automatismes industriels

Autres

Domaines d’application

Prix et d'achats

Achats

Licences institutionnelles Etudiants

Vue d’ensemble du service EMP

Support

Formation produit

Aide en ligne Produit

Webinaires et évènements

Publications

Hubs de contenu

Exemples et applications

Communauté

A propos de Maplesoft

Ressources presse

Communauté

Contacts

Online Help

All Products Maple MapleSim