The LuGre tyre model

The LuGre tyre model is a friction based physical tyre model that can withstand any motion situation. In common the tyre-road friction influences the slip with algebraic fuciton which is denoted as the follows

s=sb=r?v-1 if v>r?, v?0 for brakings?=1-vr? if v<r?, ??0 for drivingWhere r is the tyre radius, ? is the angular velocity of the tyre and v is the ground velocity.

Types of LuGre tyre models:

Lumped LuGre model, Distributed lumped LuGre model, Average Lumped LuGre model. Let us consider the Lumped LuGre model.

Lumped LuGre model

center352425000The LuGre model is consider to be the improvement of Dahl model which accounts for the Stribeck effect, zero slip displacement, stick slip, stiction and hysteresis.

Fig. 1: Bristle formation during Acceleration

The above diagram shows the bristle formation during acceleration, that means when the tyre rotates, the bristle will deflect with respect to time and denoted by z can be written as the follow

dzdt=vr-?0|vr|gzg is friction, ?0 is Longitude lumped stiffness, and vr is the relative velocity of the two surfaces. Friction force on bristle deformation can be denoted as the follow

Fs=?0+?1dzdt+?2vrFnWhere ?1 is Longitudinal lumped damping coefficient, ?2 is Viscous relative damping, and Fn is Normal force.

Strength

Capable to show the effect of tyre vertical force and vehicle speed on friction.

Characterize the both steady-state and dynamic tyre performance

High accuracies level.

Takes elastic, viscous and dapping effect on the friction force in account.

Weakness:

At higher slip the steady state longitudinal force curve doesn’t fit well.

Future improvement needed on this tyre model.

RMOD-K7

The RMOD-K7 tyre models are based on the new theoretical work on dynamics of rolling resistance. The RMOD-K7 was influenced from the RMOD-K model from which the base work was grabbed. The RMOD-K7 model has the capability to withstand up to 100Hz of frequency. The slip rate for RMOD-K7 is expressed as the follow.

sx=sx1+sx0+s0ffFzFz, sy=sy1+sy0+s0ff(FzFz)

Types of RMOD-K7:

Flexible Belt and Rigid belt type. Let us consider the Rigid belt type.

The main principle of the rigid belt tyre model is the tyre belt into the rigid belt body. The RB express the contact between the tyre model and the contact area. The main property of the RB is which can withstand the up to 100Hz and ground distance wavelength of 100mm.

RMOD-K7 comes with the MBS system called RMOD-M which has the capability to calculate the steady state behaviour or the sweep slip response of a certain tyres.

Fig. 2: Belt and Rim bushing marker

The dynamic of contact is expressed as follow.

VT?u?x+?u?t=gut, VT?v?x+?v?t=gvt.

Where u and v are displacement, VT is the transport velocity, gv(x,t) and gu(x,t) are the slip velocity.

Strength:

Wide range of applications.

Different complexity of tyre models is available for different situation.

Can with stand up to 100Hz frequency.

Provides the detailed description of tyre model using FEM

Can be applied form the steady state to the complex 3D high frequency short wavelength.

Weakness:

Not suitable for analyse the change of performance parameters.

High computational load.