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Snow-covered ground severely affects vehicle mobility in cold regions due to low friction coefficients and snow sinkage. Simulation and evaluation of vehicle mobility in cold regions require real-time friendly tire-snow interaction models that are applicable for quasi-real driving conditions. Recently, we have developed tire-snow dynamics models that are snow depth dependent, sinkage dependent and normal load dependent. The number of model parameters is reduced through theoretical analysis of normal indentation, contact pressure and shear stress within the tire-snow interface. In-plane and out-of-plan motion resistances and traction forces (gross traction and net traction) are analytically calculated for combined slip conditions.

There is a scarcity of comprehensive tire-snow interaction models for combined (longitudinal and lateral) slips. Current tire-snow interaction empirical and finite element models mostly focus on force-slip relationships in the longitudinal direction only, following the approach used for tire-soil interaction models. One of the major differences between tire-snow and tire-soil interactions is that the former is typically depth-dependent, especially for shallow snow. Our approach in the modeling of tire-snow interaction is to rely on the underlying physics of the phenomena, wherever we could, and use test data (or finite element simulation results in the absence of test data) to calibrate the required model parameters. We also make contact with on-road models and extend them for off-road applications.

All the frictional forces developed from tire-snow interfaces are closely associated with snow depth and snow sinkage. One of the important differences between tire-soil interaction and tire-snow interaction is that the latter is explicitly snow depth dependent. Based on our established depth-dependent upper bound indentation model, the effects of snow depth on tire-snow interaction are presented in this paper. Snow is considered as a pressure-sensitive Drucker-Prager material. The required snow material parameters of the model are Drucker-Prager material constants only. Snow sinkages, for longitudinal slip close to zero, under different snow depths are numerically solved through the sinkage solver. The comparison between sinkage obtained analytically and the sinkage computed from finite element simulation is very good.

The STI (System Technology Inc.) tire model is one of the most important semi-empirical (steady-state) tire models currently applied in the vehicle dynamics simulation software package of the National Advanced Driving Simulator (NADS). The STI tire model is presented originally based on tire contact length directly and the contact length is required to provide. Based on the concepts of nominal slip in both longitudinal and lateral directions, the STI tire model is analyzed and rewritten. It shows that the STI tire model does not actually depend on the contact length. Meanwhile, the model parameters are partially assigned new physical definitions, for example, static/dynamic stiffness and shape factors. Some simplified expressions are given based on further assumption conditions. The simplified expressions are also obtained regarding longitudinal slip at arbitrary speeds (including low speed, zero speed and stand still), which is originally presented by Bernard.

Vehicle handling and stability performances are greatly determined by non-steady state (NSS) tire cornering properties. Analytical derivation of NSS tire cornering models are presented in this paper based on Pacejka's string-concept assumption, in which carcass is assumed to be a stretched string with lateral deformation and lateral relaxation. The lateral inputs of the models are either displacement-based (lateral displacement and yaw angle) or slip-based (slip angle and turn slip). The transient deformations in spatial domain in both longitudinal and lateral directions are obtained directly from geometric relationship of contact patch. The additional self-aligning moment due to longitudinal deformation of contact patch after effect of tire width is considered is also achieved according to geometric relationship of contact patch in longitudinal direction and two transient geometric conditions of contact point.

Non-steady state (NSS) tire cornering properties show obvious differences from steady state (SS) tire cornering properties. A two-DOF automobile model with steer angle as an input is established based on the known NSS tire model considering complex carcass deformation. The tire model can certainly be applied to modelling of a multi-DOF automobile system. The frequency responses of lateral acceleration and yaw rate are then derived. An evaluation index, amplitude-frequency characteristic of relative error (AFCRE), is used to analyze the influences of NSS front wheels (FW) and/or rear wheels (RW) on automotive handling. The influences of NSS FW are much greater than those of NSS RW only on automotive handling. The established automobile model can also be applied to other similar studies of vehicle dynamics.

Based on the tire cornering properties in steady state condition, a theoretical model of non-steady state tire cornering properties (NSSTCP) with small lateral inputs is presented. The outputs of the model are lateral force and aligning moment, while the inputs are yaw angle and lateral displacement (or turn slip and slip angle). The deformation characteristics of contact patch are analyzed in non-steady state condition. The flexibility of tread and that of carcass are both taken into account. The deformation of carcass is assumed to compose of translating part, bending part and twisting part. The tests of NSSTCP including pure yaw motion and pure lateral motion are realized with step inputs of yaw angle and slip angle respectively and test data is then transformed into frequency domain. The model is validated through comparing the computational results with test frequency response.

Vehicle dynamic control could improve vehicle performance. Vehicle stability is vital to the determination of vehicle dynamic control strategy. The phase plane method is one of the most common methods to judge vehicle stability. To determine the 4WS (four-wheel steering) vehicle stability status faster and more accurately, a novel method to assess the vehicle stability is based on the vehicle sideslip angle and angular velocity (β-β˙) phase plane. At first, the 2 DOF (degree of freedom) model with a nonlinear tire model is established to acquire β-β˙ phase plane. Then the boundary of the stability region generated by the current method is compared. A crosspoint-ellipse method is provided based on the boundary comparison with the ideal boundary. The boundary function determined by the crosspoint-ellipse method is fitted based on vehicle dynamic theory and the boundary analysis with different steering angles, velocity, and road adhesion coefficient.

A generalized theoretical model of tire cornering properties is presented in steady state condition with lateral deflection of tread and complex deformation of carcass under consideration. The model is suitable for full range of vertical load and slip angle. Six parameters are defined to represent the characteristics of tire stiffness, contact pressure distribution and carcass deformation. The model is validated against test data. Some simplified models, e.g. brush model, HSRI model when longitudinal force is zero, Fiala model etc., can be derived as some specific cases of this model. The analytic model provides a sound foundation for semi-empirical expression and gains insight into study of vehicle system dynamics.