Converting Span
Web span component to convert one web to another
Description
Usage
Equations
Connections
Parameters
Summary Variables
The Converting Span component models a span where the web properties change from one set to another.
A Converting Span component must be connected to rollers, drums, sources, or sinks from the Web Handling Library.
The Converting Span component may only be used as a connector between rollers, drums, a source, or a sink.
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.
Ta
Upstream tension
Tb
Downstream tension
va
Upstream speed
vb
Downstream speed
τm
Motor torque (= zero if modeling an idler roller)
R
Roller effective radius
ωr
Roller angular velocity
b
Bearing damping coefficient
W
Web width
β
Wrap angle
The conservation of mass for the control volume can be expressed as:
∂∂t∫x__ax__bρx,tAx,tⅆx=ρ__atA__atv__at−ρ__btA__btv__bt
Based on the above assumption we can write
∂∂tρAL=A__aρ__av__a−A__bρ__bv__b
where L=xb−xa and the explicit dependence on t was dropped to simplify the notation.
The mass of an infinitesimal length of the web is
dm=ρ__uA__udx__u
where the subscript u indicates the undeformed state.
The element strain in the Machine Direction (MD) can be written as:
dx=dx__u1+ε
Putting (3) and (4) together, we get:
ρ__xA__x=ρ__uxA__ux1+ε,x∈a,b
Substituting (5) into (2) and simplifying yields:
∂∂tL1+ϵ__b=v__a1+ϵ__a−v__b1+ϵ__b
What remains is the material's constitutive equation. Assuming linear visco-elastic material (Kelvin–Voigt model), we have:
T__i=EAϵ__i+d__rⅆϵiⅆt,i=a,b
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.
When the option Include mass (MD) is used (see Web Properties), the conservation of momentum is added to the span equations. Redefining Tb:
T__b−T__a=∂∂t∫x__ax__bρx,tAx,tvx,tⅆx−ρ__atA__atv__at2+ρ__btA__btv__bt2
where,
T__a=EAϵ__b+d__rⅆⅆtεb
As above, we can simplify to:
T__b−T__a=∂∂tρAvL−ρ__aA__av__a2+ρ__bA__bv__b2
and using the strains and the undeformed properties:
T__b−T__a=ρ__ubA__ub∂∂tLv__b1+ϵ__b−ρ__auV__uav__a2ρ__ubV__ub1+ϵ__a+v__b21+ϵ__b
Name
Modelica ID
framea
Left web 3-D connection point
frame_a
frameb
Right web 3-D connection point
frame_b
weba
Entrance web transfer information port
web_a
webb
Exit web transfer information port
web_b
v
Real output; velocity
T
Real output; tension in web
Web Properties
Default
Upstream Web Properties
Name of the record containing the upstream web properties
wp1
Downstream Web Properties
Name of the record containing the downstream web properties
wp2
Web Sensor
Units
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
ms
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_Tension
Uniform tension in the web span
summary_Length
m
Length of the web span
See Also
Web Handling Library
Webs
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