Model tire dynamics and motion at end of driveline axis
Vehicle Components
The Tire block models a vehicle tire in contact with the road. The driveline port transfers the torque from the wheel axis to the tire. You must specify the vertical load F_{z} and vehicle longitudinal velocity V_{x} as Simulink^{®} input signals. The model provides the tire angular velocity Ω and the longitudinal force F_{x} on the vehicle as Simulink output signals. All signals have MKS units.
The convention for the F_{z} signal is positive downward. If the vertical load F_{z} is zero or negative, the horizontal tire force F_{x} vanishes. In that case, the tire is just touching or has left the ground.
The longitudinal direction lies along the forward-backward axis of the vehicle. See Tire Model following for model details.
Use the blocks of the Vehicle Components library as a starting point for vehicle modeling. To see how a Vehicle Component block models a driveline component, look under the block mask. The blocks of this library serve as suggestions for developing variant or entirely new models to simulate the same components. Break the block's library link before modifying it and creating your own version.
Effective radius r_{e},
in meters (m), at which the longitudinal force is transferred to the
wheel as a torque. The default is 0.3
.
Rated vertical load force F_{z},
in newtons (N). The default is 3000
.
Maximum longitudinal force F_{x},
in newtons (N), the tire exerts on the wheel when the vertical load
equals its rated value. The default is 3500
.
The value of the contact slip, in percent (%), when F_{x} equals
its maximum value and F_{z} equals
its rated value. The default is 10
.
Tire relaxation length σ_{κ},
in meters (m), that determines the transient response of the tire
force F_{x} at the tire's rated
load. The default is 0.2
.
Note: For more about the tire parameters, see Relationship to Block Parameters following. |
The tire is a flexible body in contact with the road surface and subject to slip. When a torque is applied to the wheel axle, the tire deforms, pushes on the ground (while subject to contact friction), and transfers the resulting reaction (including rolling resistance) as a force back on the wheel, pushing the wheel forward or backward.
The Tire block models the tire as a rigid-wheel, flexible-body combination in contact with the road. The model includes only longitudinal motion and no camber, turning, or lateral motion.
At full speed, the tire acts like a damper, and the longitudinal force F_{x} is determined mainly by the slip.
At low speeds, when the tire is starting up from or slowing down to a stop, the tire behaves more like a deformable, circular spring.
The effective rolling radius r_{e} is normally slightly less than the nominal tire radius because the tire deforms under its vertical load.
The tire relaxation length σ_{κ} is the ratio of the slip stiffness to longitudinal force stiffness. It determines the transient response of F_{x} to slip.
This figure and table define the tire model variables. The figure displays the forces from the ground on the tire. The definition here differs from the normal convention in which F_{z} is positive downward, the force from the tire on the ground.
Tire Dynamics and Motion
Tire Model Variables and Constants
Symbol | Meaning and Unit |
---|---|
r_{e} | Effective rolling radius (m) |
I_{w} | Wheel-tire assembly inertia (kg·m^{2}) |
τ_{drive} | Torque applied by the axle to the wheel (N·m) |
V_{x} | Wheel center longitudinal velocity (m/s) |
Ω | Wheel angular velocity (rad/s) |
Ω′ | Contact point angular velocity (rad/s) |
V_{sx} = V_{x} – r_{e}Ω | Wheel slip velocity (m/s) |
V′_{sx} = V_{x} – r_{e}Ω′ | Contact point slip velocity (m/s) |
κ = –V_{sx}/|V_{x}| | Wheel slip |
κ′ = –V′_{sx}/|V_{x}| | Contact patch slip |
u | Tire longitudinal compliance or deformation (m) |
F_{z} | Vertical load on tire (N) |
F_{x} = f(κ′, F_{z}) | Longitudinal force exerted by the tire on the wheel at the contact point (N). Also a characteristic function f of the tire. |
C_{Fx} = (∂F_{x}/∂u)_{0} | Tire longitudinal stiffness (N/m) |
σ_{κ} = (∂F_{x}/∂κ′ )_{0}/(C_{Fx}) | Tire relaxation length (m) |
If the tire were rigid and did not slip, it would roll and translate as V_{x} = r_{e}Ω. In reality, even a rigid tire slips, and a tire develops a longitudinal force F_{x} only in response to slip. The wheel slip velocity V_{sx} = V_{x} – r_{e}Ω ≠ 0. The wheel slip κ = –V_{sx}/|V_{x}| is more convenient. For a locked, sliding tire, κ = –1. For perfect rolling, κ = 0.
The tire is also flexible. Because it deforms, the contact point turns at a slightly different angular velocity Ω′ from the wheel. The contact point slip κ′ = –V′_{sx}/|V_{x}|, where V′_{sx} = V_{x} – r_{e}Ω′.
The tire deformation u directly measures the difference of wheel and contact point slip and satisfies
$$\frac{du}{dt}={{V}^{\prime}}_{sx}-{V}_{sx}$$
A tire model provides a steady-state tire characteristic function F_{x} = f(κ′, F_{z}), the longitudinal force F_{x} the tire exerts on the wheel given
Vertical load F_{z}
Contact slip κ′
The contact slip κ′ in turn depends on the deformation u. The longitudinal force F_{x} is approximately proportional to the vertical load because F_{x} is generated by contact friction and the normal force F_{z}. (The relationship is somewhat nonlinear because of tire deformation and slip.) The dependence of F_{x} on κ′ is more complex.
The tire model incorporates transient as well as steady-state behavior and is thus appropriate for starting from, and coming to, a stop.
Because a rolling, stressed tire is not in a steady state, the contact slip κ′ and deformation u are not constant. Before they can be used in the characteristic function, their time evolution must be accounted for. In this model, u and κ′ are moderate to small. The relationships of F_{x} to u and u to κ′ are then linear:
$$\begin{array}{l}{F}_{x}={C}_{Fx}\cdot u={C}_{F\kappa}\cdot {\kappa}^{\prime},{C}_{Fx}={\left(\frac{\partial {F}_{x}}{\partial u}\right)}_{u=0},{C}_{F\kappa}={\left(\frac{\partial {F}_{x}}{\partial {\kappa}^{\prime}}\right)}_{{\kappa}^{\prime}=0}\\ u={\sigma}_{\kappa}\cdot {\kappa}^{\prime},{\sigma}_{\kappa}={\left(\frac{\partial {F}_{x}}{\partial {\kappa}^{\prime}}\right)}_{{\kappa}^{\prime}=0}/{\left(\frac{\partial {F}_{x}}{\partial u}\right)}_{u=0}={C}_{F\kappa}/{C}_{Fx}\end{array}$$
These properties are taken from empirical tire data.
The deformation u evolves according to
$$\frac{du}{dt}+\left(\frac{1}{\sigma \kappa}\right)\cdot \text{}\left|{V}_{x}\right|u=-{V}_{sx}$$
The slip κ′ follows from σ_{κ} and u. The tire behaves like a driven damper of damping rate |V_{x}|/σ_{κ}.
At low speeds, the slip remains finite, and the tire behaves more like a circular spring of stiffness C_{Fx}. In this limit, the linear approximation relating contact slip κ′ and deformation u becomes singular if damping is not explicitly included. This relationship can be modified:
$${\kappa}^{\prime}=\frac{u}{\sigma \kappa}\to {\kappa}^{\prime}=\left(\frac{u}{\sigma \kappa}-\frac{{k}_{V,\text{low}}}{{C}_{F\kappa}}{V}_{sx}\right)$$
where smooth transition from zero speed is provided by
$${k}_{V,\text{low}}=\{\begin{array}{cc}{\scriptscriptstyle \frac{1}{2}}{k}_{V,\text{low}}(0)\left\{1+\mathrm{cos}\left(\pi \frac{\left|{V}_{x}\right|}{{V}_{\text{low}}}\right)\right\}& ,\left|{V}_{x}\right|\le {V}_{\text{low}}\\ 0& ,\left|{V}_{x}\right|>{V}_{\text{low}}\end{array}$$
A nonsingular tire evolution valid at vanishing speeds is
$$\begin{array}{l}\left(\frac{1}{{C}_{Fx}}\right)\cdot \frac{d{F}_{x}}{dt}+\text{}\left|{V}_{x}\right|{\kappa}^{\prime}=-{V}_{sx},\\ \left(\frac{1}{{C}_{Fx}}\right)\cdot \frac{\partial {F}_{x}}{\partial {\kappa}^{\prime}}\frac{d{\kappa}^{\prime}}{dt}+\text{}\left|{V}_{x}\right|{\kappa}^{\prime}=-{V}_{sx}-\left(\frac{1}{{C}_{Fx}}\right)\cdot \frac{\partial {F}_{x}}{\partial {F}_{z}}\frac{d{F}_{z}}{dt}\end{array}$$
The second form explicitly shows the dependence on a varying vertical load F_{z}.
With the tire characteristic function f(κ′, F_{z}), the vertical load F_{z}, and the evolved u and κ′, you can find the longitudinal force F_{x} and wheel velocity Ω. From these, the equations of motion determine the wheel angular motion (the angular velocity Ω) and longitudinal motion (the wheel center velocity V_{x}):
$$\begin{array}{l}{I}_{w}\frac{d\Omega}{dt}={\tau}_{\text{drive}}-{r}_{e}{F}_{x}\\ m\frac{d{V}_{x}}{dt}={F}_{x}-\text{}mg\cdot \mathrm{sin}\beta \end{array}$$
where β is the slope of the incline upon which the vehicle is traveling (positive for uphill), and m and g are the wheel load mass and the gravitational acceleration, respectively. τ_{drive} is the driveshaft torque applied to the wheel axis.
The effective rolling radius is r_{e}. The rated load normalizes the tire characteristic function f(κ′, F_{z}), and the peak force, slip at peak force, and relaxation length fields determine the peak and slope of f(κ′, F_{z}) and thus C_{Fx} and σ_{κ}.
The example model drive_4wd_dynamics combines two differentials with four tire-wheel assemblies to model the contact of tires with the road and the longitudinal vehicle motion.
The example model drive_vehicle models an entire one-wheel vehicle, including Tire and Longitudinal Vehicle Dynamics blocks.
Centa, G., Motor Vehicle Dynamics: Modeling and Simulation, Singapore, World Scientific, 1997.
Pacejka, H. B., Tire and Vehicle Dynamics, Society of Automotive Engineers and Butterworth-Heinemann, Oxford, 2002.