Linear tire formulation
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.
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, rz (see Surface), and the inclination angle, γ (see Tire Kinematics).
Using a linear spring and saturated damping forces based on the tire compliance, the normal force, Fz, is calculated as follows
where Vz is the tire center speed with respect to ISO Z, C is tire stiffness, K is tire damping, and R0 is tire unloaded radius. The use of the min function is to ensure that Fz is continuous at rL=R0.
The following equations for longitudinal slip, κ, and slip angle, α, hold true on a flat surface with no inclination angle:
Above, re is the tire effective radius and is considered to be equal to the loaded radius, rL for the tire component. Ω is the tire speed of revolution, and Vx and Vy 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 Vx=0.
Using Time Lags
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:
The formulation for resultant forces/moments of tire-surface interaction at the tire contact patch are summarized below for the linear tire component.
When use Fz is true, the longitudinal and lateral forces are linearly dependent on Fz. When use Fz is false, the longitudinal and lateral forces are not dependent on Fz, 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., Fz=0.
Multibody frame for tire center
Signal output for the normal force
Signal output for tire inclination angle or camber
Signal output for longitudinal slip
Signal output for tire effective radius
Signal output for slip angle
Signal output for tire speed of revolution or spin rate
 Vector signal input for surface normal vector
 Signal input for tire center distance from the surface
 Vector signal output for tire center position w.r.t. the inertial frame
 Available if Surface parameters Flat surface is false and Defined externally is true.
Use Fz dependency
True (checked) means that longitudinal/lateral forces will also linearly depend on the normal force
Longitudinal force coefficient
Lateral force coefficient
True (checked) means use mass and inertia parameters for tire and enable the following two parameters
Rotational inertia, expressed in frame_a (center of tire)
Use Initial Conditions
True (checked) enables the following parameters
Indicates whether to ignore, try to enforce, or strictly enforce the translational initial conditions
Initial displacement of frame_a (tire center) at the start of the simulation expressed in the inertial frame
Indicates whether the initial velocity is expressed in frame_a or inertial frame
Initial velocity of frame_a (tire center) at the start of the simulation expressed in the frame selected in Velocity Frame
Indicates whether to ignore, try to enforce, or strictly enforce the rotational initial conditions
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.
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)
Initial rotation of frame_a (center of tire) at the start of the simulation (based on Euler Sequence selection)
Angular Velocity Frame
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.
Initial angular velocity of frame_a (center of tire) at the start of the simulation expressed in the frame selected in Angular Velocity Frame
These parameters define the radial compliance of the tire.
Tire radial stiffness
Tire radial damping
Tire's spin axis (local)
Unloaded tire radius
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 e^g
True (checked) means the road surface is defined external to the tire component. Additional input and output signal ports are activated.
Base distance for local surface patch approximation
Data source for the uneven surface. See following table.
Surface data; matrix or attached data set
table or data
Smoothness of table interpolation
Number of iterations to find the contact point candidate, recommended value between 1 and 5
Content of Data source matrix.
Use time lags
True (checked) means use time lags in slip calculation and enable the following two parameters
Time lag for longitudinal slip
Time lag for slip angle
True (checked) creates a tire visualization and enables following three parameters
Tire width (for visualization)
Tire band color
True (checked) means the tire is transparent
Show force arrow
True (checked) display a force vector and enables the following three parameters
True (checked) means three arrows for force components in ISO axes will be shown instead of a single total force arrow
Force arrow color
Specifies the color of the force arrow
Force arrow transparency
True (checked means the force arrow is transparent
Force arrow scale
Scales the length of the force arrow
Show torque arrow
True (checked) displays a torque vector and enables the following three parameters
True (checked) means three arrows for torque components in ISO axes will be shown instead of a single total torque arrow
Torque arrow color
Specifies the color of the torque arrow
Torque arrow transparency
True (checked) means the torque arrow is transparent
Torque arrow scale
Scales the length of the torque arrow
Show tangent plane
True (checked) displays the tangent plane of the contact patch and enables the following four parameters
Patch visualization thickness
Patch visualization radius
Color of the contact patch
True (checked) means contact patch is transparent
Dependency factor of longitudinal/lateral forces on the normal force
Velocity threshold used for singularity avoidance in the slip calculations
Used to smooth sign⁡x as tanh⁡xεsgn
Used to prevent singularity in vector normalization
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