Fiala Tire

Fiala tire formulation

 Description The Fiala Tire component models a tire based on the Fiala model. 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.

Details

 Normal Force 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, $\mathrm{rz}$ (see Surface), and the inclination angle, $\mathrm{\gamma }$ (see Tire Kinematics). ${r}_{L}=\frac{\mathrm{rz}}{\mathrm{cos}\left(\mathrm{\gamma }\right)}$ Using a linear spring and saturated damping forces based on the tire compliance, the normal force, ${F}_{z}$, is calculated as follows ${F}_{z}^{C}=\left\{\begin{array}{cc}C\left({R}_{0}-{r}_{L}\right)& {r}_{L}<{R}_{0}\\ 0& \mathrm{otherwise}\end{array}$ ${F}_{z}^{K}=\left\{\begin{array}{cc}K{V}_{z}& {r}_{L}<{R}_{0}\\ 0& \mathrm{otherwise}\end{array}$ ${F}_{z}\phantom{\rule[-0.0ex]{1.0ex}{0.0ex}}=\left\{\begin{array}{cc}{F}_{z}^{C}+\mathrm{min}\left({F}_{z}^{C},{F}_{z}^{K}\right)& 0<{F}_{z}^{C}+{F}_{z}^{K}\\ 0& \mathrm{otherwise}\end{array}$ where ${V}_{z}$  is the tire center speed with respect to ISO Z, $C$ is tire stiffness, $K$ is tire damping, and ${R}_{0}$  is tire unloaded radius. The use of the min function is to ensure that ${F}_{z}$ is continuous at ${r}_{L}={R}_{0}$.

Slip Calculations

The following equations for longitudinal slip, $\mathrm{\kappa }$, and slip angle, $\mathrm{\alpha }$, hold true on a flat surface with no inclination angle:

$\mathrm{\kappa }=\frac{\mathrm{\Omega }{r}_{e}-{V}_{x}}{\left|{V}_{x}\right|}$

$\mathrm{tan}\left(\mathrm{\alpha }\right)=\frac{{V}_{y}}{\left|{V}_{x}\right|}$

Above, ${r}_{e}$ is the tire effective radius and is considered to be equal to the loaded radius, ${r}_{L}$ for the tire component. $\mathrm{\Omega }$  is the tire speed of revolution, and ${V}_{x}$ and ${V}_{y}$ 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 ${V}_{x}=0$.

 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: ${T}_{\mathrm{long}}\frac{\mathrm{d}\mathrm{\kappa }}{\mathrm{d}t}={r}_{e}\mathrm{\Omega }-{V}_{x}-\mathrm{\kappa }\left|{V}_{x}\right|$ ${T}_{\mathrm{lat}}\frac{\mathrm{d}}{\mathrm{d}t}\left(\mathrm{tan}\left(\mathrm{\alpha }\right)\right)={V}_{y}-\mathrm{tan}\left(\mathrm{\alpha }\right)\left|{V}_{x}\right|$
 Equations The formulation for resultant forces/moments of tire-surface interaction at the tire contact patch are summarized below for the Fiala tire component. The longitudinal force is ${F}_{x}=\left\{\begin{array}{cc}{C}_{\mathrm{long}}\mathrm{\kappa }& |\mathrm{\kappa }|<{\mathrm{\kappa }}_{c}\\ \frac{\mathrm{\mu }{F}_{z}-{\left(\mathrm{\mu }\frac{{F}_{z}}{2}\right)}^{2}}{{C}_{\mathrm{long}}{\mathrm{\kappa }}^{2}}& \mathrm{otherwise}\end{array}$ where ${\mathrm{\kappa }}_{c}$ is the critical longitudinal slip, given by ${\mathrm{\kappa }}_{c}=\frac{1}{2}|\frac{\mathrm{\mu }{F}_{z}}{{C}_{\mathrm{long}}}|$ where $\mathrm{\mu }$ and $\mathrm{\beta }$ are given by $\mathrm{\mu }=\mathrm{max}\left({\mathrm{\mu }}_{2}-\mathrm{\beta }\left({\mathrm{\mu }}_{2}-{\mathrm{\mu }}_{1}\right),0\right)$ $\mathrm{\beta }=\sqrt{{\mathrm{\kappa }}^{2}+{\mathrm{tan}\left(\mathrm{\alpha }\right)}^{2}}$ The lateral force is ${F}_{y}=-\mathrm{\mu }{F}_{z}\mathrm{signum}\left(\mathrm{\alpha }\right)\left\{\begin{array}{cc}1-{H}^{3}& |\mathrm{\alpha }|<{\mathrm{\alpha }}_{c}\\ 1& \mathrm{otherwise}\end{array}$ where $H=1-\frac{1}{3}{C}_{\mathrm{lat}}\frac{\left|\mathrm{tan}\left(\mathrm{\alpha }\right)\right|}{\mathrm{\mu }{F}_{z}}$ and the critical slip angle, ${\mathrm{\alpha }}_{c}$, is ${\mathrm{\alpha }}_{c}=\mathrm{arctan}\left(3\mathrm{\mu }\frac{{F}_{z}}{{C}_{\mathrm{lat}}}\right)$ The Fiala formulation does not consider the overturning couple, thus ${M}_{x}=0$ The equation for rolling resistance moment is ${M}_{y}=-\mathrm{tanh}\left({\mathrm{kC}}_{\mathrm{rr}}\mathrm{\Omega }\right){C}_{\mathrm{rr}}{F}_{z}$ The equation for the self-aligning torque is ${M}_{z}=\left\{\begin{array}{cc}2\mathrm{\mu }{F}_{z}\mathrm{r2}\left(1-H\right){H}^{3}\mathrm{signum}\left(\mathrm{\alpha }\right)& |\mathrm{\alpha }|<{\mathrm{\alpha }}_{c}\\ 0& \mathrm{otherwise}\end{array}$

Connections

 Name Description Modelica ID ${\mathrm{frame}}_{a}$ Multibody frame for tire center frame_a $\mathrm{Fz}$ Signal output for the normal force Fz $\mathrm{IncAng}$ Signal output for tire inclination angle or camber IncAng $\mathrm{LongSlip}$ Signal output for longitudinal slip LongSlip ${r}_{\mathrm{eff}}$ Signal output for tire effective radius r_eff $\mathrm{SlipAng}$ Signal output for slip angle SlipAng $\mathrm{SpinRate}$ Signal output for tire speed of revolution or spin rate SpinRate ${\mathrm{en}}_{\mathrm{in}}$ [1] Vector signal input for surface normal vector en_in ${\mathrm{rz}}_{\mathrm{in}}$ [1] Signal input for tire center distance from the surface rz_in ${r}_{c}$ [1] Vector signal output for tire center position w.r.t. the inertial frame r_c

[1] Available if Surface parameters Flat surface is false and Defined externally is true.

Parameters

Coefficients

 Name Default Units Description Modelica ID ${C}_{\mathrm{long}}$ $1.15·{10}^{5}$ $N$ Longitudinal force coefficient Clong ${C}_{\mathrm{lat}}$ $1.17·{10}^{5}$ $N$ Lateral force coefficient Clat ${\mathrm{\mu }}_{1}$ $0.2$ Dynamic coefficient of friction mu1 ${\mathrm{\mu }}_{2}$ $0.75$ Static coefficient of friction mu2 ${C}_{\mathrm{rr}}$ $0.01$ Rolling resistance moment coefficient Crr ${\mathrm{kC}}_{\mathrm{rr}}$ $10$ Smoothing factor for rolling resistance moment zero-crossing kCrr

Inertia

 Name Default Units Description Modelica ID Use inertia $\mathrm{false}$ True (checked) means use mass and inertia parameters for tire and enable the following two parameters useInertia $m$ $28$ $\mathrm{kg}$ Tire mass Mass [I] $\mathrm{kg}{m}^{2}$ Rotational inertia, expressed in frame_a (center of tire) Inertia

Initial Conditions

 Name Default Units Description Modelica ID Use Initial Conditions $\mathrm{false}$ True (checked) enables the following parameters useICs ${\mathrm{IC}}_{r,v}$ $\mathrm{Ignore}$ Indicates whether MapleSim will ignore, try to enforce, or strictly enforce the translational initial conditions MechTranTree ${\stackrel{&conjugate0;}{r}}_{0}$ $\left[0,0,0\right]$ $m$ Initial displacement of frame_a (tire center) at the start of the simulation expressed in the inertial frame InitPos Velocity Frame $\mathrm{Inertial}$ Indicates whether the initial velocity is expressed in frame_a or inertial frame VelType ${\stackrel{&conjugate0;}{v}}_{0}$ $\left[0,0,0\right]$ $\frac{m}{s}$ Initial velocity of frame_a (tire center) at the start of the simulation expressed in the frame selected in Velocity Frame InitVel ${\mathrm{IC}}_{\mathrm{\theta },\mathrm{\omega }}$ $\mathrm{Ignore}$ Indicates whether MapleSim will ignore, try to enforce, or strictly enforce the rotational initial conditions MechRotTree $\mathrm{Quaternions}$ $\mathrm{false}$ 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. useQuats Euler Sequence $\left[1,2,3\right]$ 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) RotType ${\stackrel{&conjugate0;}{\mathrm{\theta }}}_{0}$ $\left[0,0,0\right]$ $\mathrm{rad}$ Initial rotation of frame_a (center of tire) at the start of the simulation (based on Euler Sequence selection) InitAng Angular Velocity Frame $\mathrm{Euler}$ 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. AngVelType ${\stackrel{&conjugate0;}{\mathrm{\omega }}}_{0}$ $\left[0,0,0\right]$ $\frac{\mathrm{rad}}{s}$ Initial angular velocity of frame_a (center of tire) at the start of the simulation expressed in the frame selected in Angular Velocity Frame InitAngVel

These parameters, which define the radial compliance of the tire, are enabled if the Settings parameter Calculate Fz internally is true.

 Name Default Units Description Modelica ID $C$ $3.04·{10}^{5}$ $\frac{N}{m}$ Stiffness C $K$ $500$ $N\frac{s}{m}$ Damping K

Settings

 Name Default Units Description Modelica ID ${\stackrel{^}{e}}_{\mathrm{spin}}$ [0,1,0] Tire's spin axis (local) SymAxis

Size

 Name Default Units Description Modelica ID ${R}_{0}$ $0.355$ $m$ Unloaded tire radius R_0 $\mathrm{r2}$ $0.16$ $m$ Half of tire width r2

Surface

 Name Default Units Description Modelica ID Flat surface $\mathrm{true}$ 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 ${\stackrel{^}{e}}_{g}$ flatSurface Defined externally $\mathrm{false}$ True (checked) means the road surface is defined external to the tire component. Additional input and output signal ports are activated. externallyDefined ${\mathrm{\delta }}_{L}$ $0.01$ $m$ Base distance for local surface patch approximation deltaL Data source $\mathrm{inline}$ Data source for the uneven surface.  See following table. datasourcemode Surface data Surface data; matrix or attached data set table or data Smoothness $\mathrm{linear}$ Smoothness of table interpolation smoothness ${n}_{\mathrm{Iter}}$ $2$ Number of iterations to find the contact point candidate, recommended value between 1 and 5 nIter

Content of Data source matrix.

 Surface normal First Column First Row Global Z x values y values Global Y z values x values Global X y values z values

Time Lags

 Name Default Units Description Modelica ID Use time lags $\mathrm{false}$ True (checked) means use time lags in slip calculation and  enable the following two  parameters useTimeLag ${T}_{\mathrm{long}}$ $0.3$ $s$ Time lag for longitudinal slip Tlong ${T}_{\mathrm{lat}}$ $0.3$ $s$ Time lag for slip angle Tlat

Visualization

 Name Default Units Description Modelica ID Show tire $\mathrm{false}$ True (checked) creates a tire visualization and enables following three parameters showTire ${\mathrm{D}}_{w}$ $0.1$ $m$ Tire width (for visualization) D_w Tire color $\mathrm{black}$ Tire color color00 Band color $\mathrm{yellow}$ Tire band color color01 Tire transparency $\mathrm{false}$ True (checked) means the tire is transparent transparent0 Show force arrow $\mathrm{false}$ True (checked) display a force vector and enables the following three parameters showForceArrow Show components $\mathrm{false}$ True (checked) means three arrows for force components in ISO axes will be shown instead of a single total force arrow showForceComponents Force arrow color $\mathrm{red}$ Specifies the color of the force arrow color1 Force arrow transparency $\mathrm{false}$ True (checked means the force arrow is transparent transparent1 Force arrow scale $1$ $\frac{N}{m}$ Scales the length of the force arrow scale1 Show torque arrow $\mathrm{false}$ True (checked) displays a torque vector and enables the following three parameters showMomentArrow Show components $\mathrm{false}$ True (checked) means three  arrows for torque components in ISO axes will be shown instead of a single total torque arrow showMomentComponents Torque arrow color $\mathrm{blue}$ Specifies the color of the torque arrow color2 Torque arrow transparency $\mathrm{false}$ True (checked) means the torque arrow is transparent transparent2 Torque arrow scale $1$ $\frac{Nm}{m}$ Scales the length of the torque arrow scale2