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Roller Detailed

Roller with slippage

	Description
	The Roller Detailed component models a pull roller with slippage.

	Usage
	The Roller component must be between two Span components. Using the provided 1D rotational flange (flange_c), the Roller component models a pull roller. Without any applied torque or setpoint velocity, the Roller models an idler roller.

Equations

The Roller Detailed component uses a series of spans internally to capture the changing nature of the web/roller interaction over the contact arc. Each span is subjected to friction on its nodes which are calculated from the span tension resultant $(f_{n})$ and the coefficient of friction.

Consider one of the discretizing spans that cover $\frac{1}{n}$ of the contact arc (where $n$ is a component parameter).

	Tension
	$t_{b} - t_{a} - μ f_{n} = 0$

Torque Transfer

$τ = R \cdot (t_{b} - t_{a})$

where

$t_{a}$	Upstream tension
$t_{b}$	Downstream tension
$v_{a}$	Upstream speed
$v_{b}$	Downstream speed
$τ$	Friction torque applied to the roller by the span
$R$	Roller effective radius
$ω_{r}$	Roller angular velocity
$v_{0}$	Smoothing velocity value

	Coefficient of Friction
	The signed coefficient of friction is defined as $μ = μ_{d} v_{rel} + μ_{c} \tanh (\frac{v_{rel}}{v_{0}})$ where $v_{rel} = R ω_{r} - v_{a}$ See Span and Roller for further details.

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
${frame}_{c}$	Roller holder base frame; enabled when Use fixed base = false	frame_c
${web}_{a}$	Left (entry) web transfer information port (blue)	web_a
${web}_{b}$	Right (exit) web transfer information port (green)	web_b
${flange}_{c}$	Roller rotation flange	flange_c1
$v_{out}$	Sensor output [1]	v_out

[1] Enabled when Use sensor is true.

Parameters

Web Properties

Name	Default	Description	Modelica ID
Use Default Properties (WP1)	true	When checked (true), web properties are defined by the closest Web Properties block found	useDefault
Web Properties	WP1	Name of the record containing the web properties (enabled when Use default properties = false)	wp

Settings Parameters

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	flipDirection
Use fixed base	$false$		When checked (true), frame_c is disabled and the roller frame is fixed in the inertial coordinate frame (addition position and orientation parameters are enabled instead)	useFixedBase
Use sensor	$false$		When checked (true), a signal output is enabled for the web speed	useSensor
Output type	$1$		Select measured quantity. 1 means Web speed; 2 means Roller surface speed	sensorType
Velocity output unit	[1]			toUnitV
$r$	$[0, 0]$	$m$	Location of frame_b with respect to ground. Available when Use fixed base = true	InitPos
$θ$	$0$	$\deg$	Rotation about z-axis. Available when Use fixed base = true	InitAng

Roller

Name	Default	Units	Description	Modelica ID
$D$	$0.2$	$m$	Diameter of roller	D
Use CAD	$false$		When checked (true), mass and inertia are defined from an externally attached CAD component or a rigid body component	useCAD
$D_{i}$	$0$	$m$	Inner diameter of roller	Di
$L$	$1.2$	$m$	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
$m$	$1$	$kg$	Roller mass; enabled when not using cylindrical geometry	m
$J$	$0.005$	$kg m^{2}$	Roller rotational inertia; enabled when not using cylindrical geometry	J
$d$	$0$	$\frac{N m s}{rad}$	Bearing viscous damping constant	d

Friction Parameters

Name	Default	Units	Description	Modelica ID
$μ_{c}$	$0.2$		Coulomb coefficient of friction between web and roller	mu_c
$μ_{d}$	$0$		Viscous coefficient of friction between web and roller [1]	mu_d
$v_{0}$	$1 \cdot 10^{−4}$	$\frac{m}{s}$	Smoothing function scale, i.e. tanh(v_rel/v0)	v0
$n_{div}$	$5$		Number of segments	nDiv
$peak$	$1$		Maximum friction force at zero sliding velocity is given by Peak.u_c [1]	peak
$v_{s}$	$0.01$	$\frac{m}{s}$	Stribeck sliding friction [1]	vs
$n_{Decay}$	$1$		Decay exponent [1]	nDecay
$P$	$0$	$Pa$	Surface pressure (positive is suction)	P
Use constant CoF	$true$		When unchecked (false), a velocity-dependent formulation is used for the coefficient of friction.	useConstantCoF

[1] Enabled when Use constant CoF is false.

Frame

Name	Default	Units	Description	Modelica ID
$m_{f}$	$0$	$kg$	Frame mass	m_frame
$J_{f}$	$0$	$kg m^{2}$	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
$r_{f}$	$[0, 0]$	$m$	Offset in the XY plane; enabled when Add frame offset is true	rf

Visualization Parameters

Name	Default	Units	Description	Modelica ID
Show roller	$true$		When checked (true), a visualization of the roller is created	showVisualization_roller
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 contact forces	$false$		When checked (true), arrows are added to visualize discretized web/roller contact forces (normal + friction)	showContactForce
Scale	$100$	$\frac{m}{N}$	Force scale	forceScale
Color	Red		Force arrow color	forceColor
Transparency	$false$		Force arrow transparency	forceTransparent
Show sliding velocity arrows	$false$		When checked (true), an arrow is added to visualize local web/roller sliding velocity	showSlidingVelocity
Scale	$0.01$	$s$	Speed scale	speedScale
Color	Green		Speed arrow color	speedColor
Transparency	$false$		Speed arrow transparency	speedTransparent

Advanced Settings

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

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_WrapAngle	[1]	Wrap angle	summary_WrapAngle
summary_Length	[1]	Length of the web in contact with the roller	summary_Length
summary_Limit		0 to 1; 0 means tension limit is less than capstan limit or include slippage is true; 1 means tension limit exceeds capstan limit (web will slip)	summary_Limit
summary_SlipVelocity	[1]	Slip velocity: web speed - roller surface speed	summary_SlipVelocity
summary_SlipRatio		(roller surface speed - web speed)/(roller surface speed)	summary_SlipRatio
summary_RollerAngularVelocity	[1]	Roller angular velocity	summary_RollerAngularVelocity
summary_WebSpeed	[1]	Web speed	summary_WebSpeed
summary_RollerSurfaceSpeed	[1]	Roller surface speed	summary_RollerSurfaceSpeed
summary_CoF		The traction/friction coefficient.	summary_CoF

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

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