Pacejka 2012 Tire
Tire component with Pacejka 2012 formulation and visualization
The Pacejka 2012 Tire component employs the 2012 formulation of the Pacejka tire model, presented in .
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.
Tire Parameters Block
The Pacejka 2012 tire model has about 180 parameters. Unlike the Linear and the Fiala tire components, where the required parameters are defined in the MapleSim GUI, to facilitate parameter handling process the Pacejka Parameters App should be used to generate a parameter block which contains the necessary tire parameters. To open this app, browse to Add Apps or Templates > Pacejka Parameters in MapleSim. The generated parameter block will be located in the Local Components panel on the left side of the MapleSim GUI.
The user should place the generated parameter block into the MapleSim workspace at the same or higher level as the Pacejka tire components that it defines.
There is an Override checkbox in the Inertia, Radial Compliance, and Scaling Factors sections of the Pacejka tire component properties.
Enabling one of these checkboxes allows the user to override the associated parameters otherwise defined in the tire parameters block. For example, the user can override the inertia properties as shown below.
Checking an Override checkbox also exposes the associated parameters to MapleSim apps such as the Parameter Sweep app and the FMU Generation app.
Unlike the Linear and Fiala tire models, the Pacejka tire model is typically asymmetric, that is Fx⁡−κ≠−Fx⁡κ or Fy⁡−α≠−Fy⁡α. To ensure the correct formulation, the ISO X axis of a tire should point towards the heading of the vehicle. The Show ISO axis option in the visualization section of the tire parameters can be helpful to visually confirm that the ISO axes have been assigned correctly.
If not assigned correctly, the user can change the integer parameter of ISO from 0 to 1 to rotate the ISO axis 180 degrees around ISO Z.
The Pacejka tire parameters apply to a specific tire side. This denotes the side of the vehicle where the tire should be mounted. The Side parameter in the properties can be used to mirror the tire. For example, if the parameters of the generated parameter block are for a right side tire, then the tire components mounted on the right side of the vehicle model in MapleSim should be used with Side=0, and those on the left side should have Side=1.
Normal Force and Effective Radius
The normal force exerted by the surface to the tire is calculated using the given compliance parameters and surface geometry. There are two implemented formulation in the Pacejka tire component for calculating the normal force: Pacejka formulation and Linear spring-damper.
The Pacejka formulation option uses the following equation for the normal force 
Note that with this option selected for the normal force, the Pacejka effective radius formulation will also be used internally. This formulation is as follows 
where the nominal load, Fz0 and the rest of the parameters used in these equations are defined in the tire parameters block.
The Linear spring-damper option is the same formulation used for normal force calculation in the Linear and the Fiala tire components as explained below.
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.
With this option selected for Fz, the user can choose between the tire unloaded radius, R0, and the loaded radius, rL, to assign to the effective radius, reff.
Three options are available for tire slip calculation, Quasi-static, Constant time lags, and Stretched String.
With the choice of Quasi-static, the following equations for longitudinal slip, κ, and slip angle, α, hold true on a flat surface with no inclination angle
where re is the tire effective radius, 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.
Constant time lags
A first-order dynamics to the longitudinal slip and slip angle calculation can be introduced using the Constant time lags option. When active, the following slip formulation is used:
With this option active, the relaxation lengths will be used in slip calculation as follows
Parameters in the above equations should be inserted using the MapleSim GUI.
The load ratio, dfz, is defined as
The formulation for resultant forces/moments of tire-surface interaction at the tire contact patch are summarized below for the Pacejka 2012 tire component.
The longitudinal force is
The lateral force is
The normal force, Fz, has been discussed in the Normal Force and Effective Radius section.
The overturning couple is
The rolling resistance moment is
The self-aligning torque is
where M′z is the torque due to pneumatic trail, t, Mzr is the residual torque, and s⁢Fx is the longitudinal force contribution to the self-aligning torque. Each of these terms has a specific expression, discussed in  in more detail.
All the employed parameters in the equations above need to be defined in the tire parameters block and be accessible to the Pacejka tire components.
Multibody frame for tire center
Signal output for the normal force
Signal output for longitudinal slip
Signal output for slip angle
Signal output for tire speed of revolution or spin rate
Signal output for tire effective radius
Signal output for tire inclination angle or camber
 Vector input for surface normal vector
 Vector output for tire center position w.r.t. the inertial frame
 Signal input for tire center distance from the surface
 Signal input for tire inflations pressure
 Available if Surface parameters Flat surface is false and Defined externally is true.
 Available if Pressure Override parameter is true and Constant Pressure is false.
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 MapleSim will 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 MapleSim will 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
Tire's spin axis (local)
0: default, 1: mirrored
0: Keep ISO, 1: Rotate ISO pi radians around Z axis
Normal force equation
Choose type of slip calculation (Quasi-static, Constant time lags, or Stretched-string)
Time lag for longitudinal slip
Time lag for slip angle
Parameters for stretched-string formulation: [LSkappa, LSalpha, p_Tx1, p_Tx2, p_Tx3, p_Ty1, p_Ty2, p_Ty3]
Minimum longitudinal relaxation length
Minimum lateral relaxation length
True (checked) overrides override the pressure parameters and enables the following parameters
Nominal tire pressure
True (checked) uses constant pressure; false provides an input port for the tire pressure
True (checked) override the scaling factors and enables the following parameter
Nominal load scaling factor
Peak friction coefficient (x) scaling factor
Peak friction coefficient (y) scaling factor
Slip speed decaying friction scaling factor
Brake slip stiffness scaling factor
Cornering stiffness scaling factor
Shape factor (x) scaling factor
Shape factor (y) scaling factor
Curvature factor (x) scaling factor
Curvature factor (y) scaling factor
Horizontal shift (x) scaling factor
Horizontal shift (y) scaling factor
Vertical shift (x) scaling factor
Vertical shift (y) scaling factor
Camber force stiffness scaling factor
Camber torque stiffness scaling factor
Pneumatic trail scaling factor
Residual torque scaling factor
Alpha influence on Fx (kappa) scaling factor
Kappa influence on Fy (alpha) scaling factor
Kappa induces ply-steer Fy scaling factor
Mz moment arm of Fx scaling factor
Radial tire stiffness scaling factor
Overturning couple stiffness scaling factor
Rolling resistance moment scaling factor
Overturning couple vertical shift scaling factor
True (checked) means theroad 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.