Multi-Web Roller Out - MapleSim Help

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.

 ${T}_{{a}_{1}}\cdots {T}_{{a}_{n}}$ Upstream tensions ${T}_{b}$ Downstream tension ${v}_{a}$ Upstream speed ${v}_{b}$ Downstream speed ${\mathrm{\tau }}_{1},{\mathrm{\tau }}_{2}$ Motor torque applied to roller 1 and 2 ${R}_{1},{R}_{2}$ Roller effective radius ${\mathrm{\omega }}_{1},{\mathrm{\omega }}_{2}$ Roller angular velocity ${J}_{1},{J}_{2}$ Moment of inertia or roller 1 and 2 $W$ Web width $\mathrm{\beta }$ Wrap angle

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

 $\mathrm{v__a}{=}\mathrm{v__b}{,}\mathrm{v__a}{=}\mathrm{R__1}{}\mathrm{ω__1}{,}\mathrm{v__b}{=}\mathrm{R__2}{}\mathrm{ω__2}$ (1)

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

 $\frac{{\partial }}{{\partial }{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left({{\int }}_{\mathrm{a__a}}^{\mathrm{x__b}}{\mathrm{\rho }}{}\left({x}{,}{t}\right){}{A}{}\left({x}{,}{t}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right){=}\mathrm{v__a}{}\left({t}\right){}\left({\sum }_{{i}{=}{1}}^{{n}}{}\mathrm{ρ__a__i}{}\left({t}\right){}\mathrm{A__a__i}{}\left({t}\right)\right){-}\mathrm{ρ__b}{}\left({t}\right){}\mathrm{A__b}{}\left({t}\right){}\mathrm{v__b}{}\left({t}\right)$ (2)

Based on the above assumption we can write

 $\frac{{\partial }}{{\partial }{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left({\mathrm{\rho }}{}{A}{}{L}\right){=}\mathrm{v__a}{}\left({\sum }_{{i}{=}{1}}^{{n}}{}\mathrm{ρ__a__i}{}\left({t}\right){}\mathrm{A__a__i}{}\left({t}\right)\right){-}\mathrm{ρ__b}{}\left({t}\right){}\mathrm{A__b}{}\left({t}\right){}\mathrm{v__b}{}\left({t}\right)$ (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

 ${\mathrm{dm}}{=}\mathrm{ρ__u}{}\mathrm{A__u}{}\mathrm{dx__u}$ (4)

where subscript $u$ is for the undeformed state.

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

 ${\mathrm{dx}}{=}\mathrm{dx__u}{}\left({1}{+}{\mathrm{\epsilon }}\right)$ (5)

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

 ${\mathrm{\rho }}{}{A}{=}\frac{\mathrm{ρ__u}{}\mathrm{A__u}}{{1}{+}{\mathrm{\epsilon }}}$ (6)

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

 $\frac{{\partial }}{{\partial }{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\left(\frac{{L}}{{1}{+}\mathrm{ϵ__b}}\right){=}\frac{\mathrm{v__a}{}\left({\sum }_{{i}{=}{1}}^{{n}}{}\frac{\mathrm{ρ__ai}{}\mathrm{A__ai}}{{1}{+}\mathrm{ϵ__ai}}\right)}{\mathrm{ρ__b}{}\mathrm{A__b}}{-}\frac{\mathrm{v__b}}{{1}{+}\mathrm{ϵ__b}}$ (7)

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

 $\mathrm{T__a__i}{=}\mathrm{E__a__i}{}\mathrm{A__a__i}{}\left(\mathrm{ϵ__a__i}{+}\mathrm{d__r__a__i}{}\left(\frac{{ⅆ}{{\mathrm{ϵ}}}_{\mathrm{a__i}}}{{ⅆ}{t}}\right)\right)$
 $\mathrm{T__b}{=}\mathrm{E__b}{}\mathrm{A__b}{}\left(\mathrm{ϵ__b}{+}\mathrm{d__r__b}{}\left(\frac{{ⅆ}{{\mathrm{ϵ}}}_{{b}}}{{ⅆ}{t}}\right)\right)$ (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}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\omega }}}_{{1}}\right){+}{{J}}_{{2}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{\mathrm{\omega }}}_{{2}}\right){+}{d}{}\left({{\mathrm{\omega }}}_{{1}}{+}{{\mathrm{\omega }}}_{{2}}\right){=}\left\{\begin{array}{cc}{{\mathrm{\tau }}}_{{1}}{-}{{R}}_{{1}}{}\left(\left({\sum }_{{i}{=}{1}}^{{n}}{}{{T}}_{\mathrm{a__i}}\right){-}{{T}}_{{b}}\right)& {\mathrm{Side}}{=}{1}\\ {{\mathrm{\tau }}}_{{2}}{-}{{R}}_{{2}}{}\left(\left({\sum }_{{i}{=}{1}}^{{n}}{}{{T}}_{\mathrm{a__i}}\right){-}{{T}}_{{b}}\right)& {\mathrm{Side}}{=}{2}\end{array}\right\$ (9)

Connections

 Name Description Modelica ID ${\mathrm{flange}}_{\mathrm{c1}}$ Roller rotation flange (enabled when $\mathrm{Side}=1$ flange_c1 ${\mathrm{flange}}_{\mathrm{c2}}$ Roller rotation flange (enabled when $\mathrm{Side}=2$ flange_c2 ${\mathrm{frame}}_{b}$ Right web 3-D connection point frame_b ${\mathrm{frame}}_{c}$ Roller holder base frame frame_c ${\mathrm{web}}_{b}$ Right (exit) web transfer information port (green) web_b $\mathrm{webs}$ 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 ${\mathrm{D}}_{1}$ $0.2$ $m$ Diameter of roller 1 D1 ${\mathrm{D}}_{\mathrm{i1}}$ $0$ $m$ Inner diameter of roller 1 Di1 ${\mathrm{D}}_{2}$ ${\mathrm{D}}_{1}$ $m$ Diameter of roller 2 D2 ${\mathrm{D}}_{\mathrm{i2}}$ $0$ $m$ Inner diameter of roller 2 Di2 $L$ $1.2$ $m$ Roller length L Use cylindrical geometry $\mathrm{true}$ When checked (true), roller inertia is calculated assuming a uniform cylindrical geometry useCylindricalGeometry $\mathrm{\rho }$ $2.7·{10}^{3}$ $\frac{\mathrm{kg}}{{m}^{3}}$ Roller density; enabled when using cylindrical geometry rho_roll ${m}_{1}$ $1$ $\mathrm{kg}$ Roller 1 mass; enabled when not using cylindrical geometry m1 ${m}_{2}$ $1$ $\mathrm{kg}$ Roller 2 mass; enabled when not using cylindrical geometry m2 ${J}_{1}$ $0.005$ $\mathrm{kg}{m}^{2}$ Roller 1 rotational inertia; enabled when not using cylindrical geometry J1 ${J}_{2}$ $0.005$ $\mathrm{kg}{m}^{2}$ Roller 2 rotational inertia; enabled when not using cylindrical geometry J2 $d$ $0$ $\frac{Nms}{\mathrm{rad}}$ 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 ${\mathrm{frame}}_{c}$ (holder). There are three options available: Middle of the gap, Center of roller 1, and Center of roller 2 Ref_pos ${m}_{f}$ $1$ $\mathrm{kg}$ Frame mass m_frame ${J}_{f}$ $0.01$ $\mathrm{kg}{m}^{2}$ 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 $\mathrm{true}$ When checked (true), a visualization of roller 1 is created showVisualization_roller Show roller 2 $\mathrm{true}$ When checked (true), a visualization of roller 2 is created showVisualization_second Transparent roller $\mathrm{false}$ 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 $20$ $\mathrm{deg}$ Band angle band_angle Show frame $\mathrm{false}$ When checked (true), a visualization of the frame is created showVisualization_frame Transparent frame $\mathrm{false}$ 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 $\mathrm{false}$ 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 $\mathrm{Use Sensor}=\mathrm{true}$ toUnitV Force output unit $N$ Selects units of the force output; enabled when $\mathrm{Use Sensor}=\mathrm{true}$ toUnitF

 Name Default Description Modelica ID Use effective radius $\mathrm{false}$ When checked (true), the roller radius is padded with half the web thickness useEffectiveRadius
 Name Units Description Modelica ID summary_WrapAngle $\mathrm{rad}$ Wrap angle summary_WrapAngle summary_Length $m$ Length of the web in contact with the roller summary_Length