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Description
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The Linear tire component provides a linear relationship between the slip angle and longitudinal slip of the tire, and the lateral and longitudinal forces developed at the contact patch. The aligning moment, rolling resistance and overturning moment are assumed to be zero.
The tire geometry is assumed to be a thin circular disk, which is common in automotive applications. A single point contact is considered for the tire-ground interaction.
The tire kinematics used in this component are described in detail in Tire Kinematics.
Several options are available for defining the surface on which the tire is operating. These options are explained in Surface.
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Details
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Normal Force
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The normal force exerted by the surface to the tire is calculated using the given compliance parameters and surface geometry.
The tire loaded radius is calculated using the distance of the tire center from the surface, (see Surface), and the inclination angle, (see Tire Kinematics).
Using a linear spring and saturated damping forces based on the tire compliance, the normal force, , is calculated as follows
where is the tire center speed with respect to ISO Z, is tire stiffness, is tire damping, and is tire unloaded radius. The use of the min function is to ensure that is continuous at .
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Slip Calculations
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The following equations for longitudinal slip, , and slip angle, , hold true on a flat surface with no inclination angle:
Above, is the tire effective radius and is considered to be equal to the loaded radius, for the tire component. is the tire speed of revolution, and and are the speeds of the tire center with respect to ISO X and ISO Y axes, respectively. The component code implementation is such that the longitudinal slip and slip angle are continuous and differentiable in the neighborhood of .
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Using Time Lags
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A first-order dynamics to the longitudinal slip and slip angle calculation can be introduced in the Time Lags section of the component properties. When active, the following slip formulation will be used:
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Equations
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The formulation for resultant forces/moments of tire-surface interaction at the tire contact patch are summarized below for the linear tire component.
When is true, the longitudinal and lateral forces are linearly dependent on . When is false, the longitudinal and lateral forces are not dependent on , however, a tanh function is used so that the tire does not generate longitudinal or lateral forces when not in contact with the ground, i.e., .
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Connections
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Name
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Description
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Modelica ID
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Multibody frame for tire center
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frame_a
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Signal output for the normal force
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Fz
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Signal output for tire inclination angle or camber
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IncAng
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Signal output for longitudinal slip
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LongSlip
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Signal output for tire effective radius
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r_eff
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Signal output for slip angle
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SlipAng
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Signal output for tire speed of revolution or spin rate
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SpinRate
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[1] Vector signal input for surface normal vector
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en_in
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[1] Signal input for tire center distance from the surface
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rz_in
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[1] Vector signal output for tire center position w.r.t. the inertial frame
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r_c
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[1] Available if Surface parameters Flat surface is false and Defined externally is true.
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Parameters
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Coefficients
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Name
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Default
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Units
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Description
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Modelica ID
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Use Fz dependency
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false
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True (checked) means that longitudinal/lateral forces will also linearly depend on the normal force
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useFz
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Nominal load
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Fz0
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Longitudinal force coefficient
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Clong
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Lateral force coefficient
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Clat
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Inertia
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Name
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Default
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Units
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Description
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Modelica ID
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Use inertia
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True (checked) means use mass and inertia parameters for tire and enable the following two parameters
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useInertia
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Tire mass
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Mass
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[I]
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[1]
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Rotational inertia, expressed in frame_a (center of tire)
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Inertia
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[1]
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Initial Conditions
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Name
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Default
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Units
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Description
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Modelica ID
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Use Initial Conditions
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True (checked) enables the following parameters
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useICs
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Indicates whether to ignore, try to enforce, or strictly enforce the translational initial conditions
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MechTranTree
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Initial displacement of frame_a (tire center) at the start of the simulation expressed in the inertial frame
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InitPos
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Velocity Frame
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Indicates whether the initial velocity is expressed in frame_a or inertial frame
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VelType
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Initial velocity of frame_a (tire center) at the start of the simulation expressed in the frame selected in Velocity Frame
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InitVel
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Indicates whether to ignore, try to enforce, or strictly enforce the rotational initial conditions
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MechRotTree
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Indicates whether the 3D rotations will be represented as a 4 parameter quaternion or 3 Euler angles. Regardless of setting, the initial orientation is specified with Euler angles.
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useQuats
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Euler Sequence
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Indicates the sequence of body-fixed rotations used to describe the initial orientation of frame_a (center of mass). For example, [1, 2, 3] refers to sequential rotations about the x, then y, then z axis (123 - Euler angles)
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RotType
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Initial rotation of frame_a (center of tire) at the start of the simulation (based on Euler Sequence selection)
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InitAng
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Angular Velocity Frame
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Indicates whether the initial angular velocity is expressed in frame_a (body) or the inertial frame. If Euler is chosen, the initial angular velocities are assumed to be the direct derivatives of the Euler angles.
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AngVelType
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Initial angular velocity of frame_a (center of tire) at the start of the simulation expressed in the frame selected in Angular Velocity Frame
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InitAngVel
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Radial Compliance
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These parameters define the radial compliance of the tire.
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Name
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Default
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Units
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Description
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Modelica ID
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Stiffness
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Tire radial stiffness
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C
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Damping
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Tire radial damping
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K
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Settings
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Name
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Default
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Units
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Description
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Modelica ID
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[0,1,0]
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Tire's spin axis (local)
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SymAxis
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Size
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Name
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Default
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Units
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Description
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Modelica ID
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Unloaded tire radius
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R_0
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Surface
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Name
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Default
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Units
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Description
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Modelica ID
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Flat surface
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True (checked) means the road surface is assumed flat. It is defined by a plane passing through (0,0,0) and the normal vector given by
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flatSurface
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Defined externally
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True (checked) means the road surface is defined external to the tire component. Additional input and output signal ports are activated.
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externallyDefined
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Base distance for local surface patch approximation
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deltaL
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Data source
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Data source for the uneven surface. See following table.
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datasourcemode
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Surface data
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Surface data; matrix or attached data set
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table or data
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Smoothness
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Smoothness of table interpolation
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smoothness
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Number of iterations to find the contact point candidate, recommended value between 1 and 5
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nIter
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Content of Data source matrix.
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Surface normal
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First Column
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First Row
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Global Z
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x values
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y values
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Global Y
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z values
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x values
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Global X
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y values
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z values
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Time Lags
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Name
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Default
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Units
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Description
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Modelica ID
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Use time lags
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True (checked) means use time lags in slip calculation and enable the following two parameters
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useTimeLag
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Time lag for longitudinal slip
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Tlong
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Time lag for slip angle
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Tlat
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Visualization
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Name
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Default
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Units
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Description
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Modelica ID
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Show tire
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True (checked) creates a tire visualization and enables following three parameters
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showTire
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Tire width (for visualization)
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D_w
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Tire color
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Tire color
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color00
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Band color
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Tire band color
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color01
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Tire transparency
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True (checked) means the tire is transparent
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transparent0
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Show force arrow
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True (checked) display a force vector and enables the following three parameters
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showForceArrow
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Show components
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True (checked) means three arrows for force components in ISO axes will be shown instead of a single total force arrow
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showForceComponents
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Force arrow color
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Specifies the color of the force arrow
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color1
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Force arrow transparency
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True (checked means the force arrow is transparent
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transparent1
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Force arrow scale
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Scales the length of the force arrow
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scale1
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Show torque arrow
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True (checked) displays a torque vector and enables the following three parameters
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showMomentArrow
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Show components
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True (checked) means three arrows for torque components in ISO axes will be shown instead of a single total torque arrow
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showMomentComponents
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Torque arrow color
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Specifies the color of the torque arrow
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color2
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Torque arrow transparency
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True (checked) means the torque arrow is transparent
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transparent2
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Torque arrow scale
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Scales the length of the torque arrow
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scale2
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Show tangent plane
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True (checked) displays the tangent plane of the contact patch and enables the following four parameters
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ShowTanSurface
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Patch visualization thickness
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th0
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Patch visualization radius
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r_patch
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Patch color
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Green
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Color of the contact patch
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color3
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Patch transparency
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True (checked) means contact patch is transparent
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transparent3
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Advanced Parameters
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Name
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Default
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Units
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Description
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Modelica ID
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Dependency factor of longitudinal/lateral forces on the normal force
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kFz
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Velocity threshold used for singularity avoidance in the slip calculations
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V_x_min
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Used to smooth as
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epsilon_sign
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Used to prevent singularity in vector normalization
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epsilon_norm
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