Non-linear elastic ring tyre model using the absolute nodal coordinate formulation

With the means of rigid-elastic coupled tire model, the in-plane vibration characteristic of heavy truck tire is investigated and the time-domain vibration response with the proposed tire model is discussed. With the means of finite difference method, rigid–elastic coupled tire model is developed from flexible beam on elastic foundation tire model and the nonlinear radial stiffness of sidewall is derived and related with the inflation pressure and geometrical feature of sidewall curved arc. In-plane vibration characteristic of heavy truck tire, including transfer function and modal parameters is investigated by experimental hammer test. Taking the error between the analytical and experimental modal frequency as the object value, Genetic Algorithm is utilized to identify the un-known structural parameters. In-plane time-domain dynamics response of heavy truck tire utilizing the proposed tire model with linear/nonlinear sidewall radial stiffness is researched and compared with experimental result. Experimental and theoretical results shows that in-plane rigid–elastic coupled tire model with nonlinear sidewall stiffness can be used to better the precision on predicting the vibration feature excited by the random irregular signal. Rigid–elastic coupled tire model with nonlinear sidewall stiffness can be extended to other heavy truck tire with a large section ratio or tires under impulsive loading conditions.

Advanced vehicle dynamics control systems, such as integrated chassis control systems, are generally designed in a two layers structure: a tire force determination layer, and a tire force generation layer. The tire force generation layer works as a servo function that follows the force commands from the force determination layer. Therefore, the tire force generation layer should not require a lot of computing resources. This paper presents an inverse tire model that calculates required slip ratios and slip angles when tire forces are commanded. The inverse tire model is based on a nonlinear tire model that is sufficiently complex and, thus, can capture the nonlinear characteristics of tires. Because mathematical inversion of the nonlinear tire model is challenging, recursive methods can be considered for an inverse model. However, recursive methods, such as Newton Raphson method, do not guarantee the fixed time span required for finishing the computation. This paper presents a method for inverse tire model that does not require recursive computations.

This paper examines how changes in the tyre and suspension impact the dynamic spindle force response of an automotive tyre rolling over an obstacle in the roadway. The investigation is conducted with a simple vehicle dynamics simulation coupled to a comprehensive 2D dynamic nonlinear finite tyre model. The tyre model uses an inextensible, circular membrane to approximate the shape, and an empirical, nonlinear term to account for forces that result from the elastic deformation of the sidewall. Both the tyre structure and contact between the tyre and roadway are modelled with the finite element method. This approach allows the tyre model to reproduce the tyre response accurately during large deformations in a computationally efficient manner. This paper presents a brief overview of the models and their limited validation of the tyre model. Included in this discussion is a comparison between measured and simulated tyre contact forces for the case of a slowly rolling tyre. The effect of vehicle speed, suspension stiffness, suspension damping, tyre mass, tyre damping, and tyre inflation pressure on the spindle forces is also presented.

Investigation of the tire in-plane vibration property using an improved ring model

Theoretical modeling, parameters identification and vibration characteristic of heavy-loaded radial tire is investigated with rigid-elastic coupling model with normal damping. The normal damping, including structural damping of flexible carcass and proportion damping of distributed sidewall element is added to enrich the flexible beam on modified elastic foundation tire model. The rigid-elastic coupling tire model with normal damping is investigated and derived with finite difference method. The mass, stiffness and damping matrixes of the proposed tire model are analytically related with the structural and geometrical parameters of heavy-loaded radial tire. Taking the error between the analytical and experimental transfer function as the object value, Genetic Algorithm (GA) is utilized to identify the damping coefficients of flexible carcass and distributed sidewall element. The influence of modal order and tire damping parameters on the in-plane transfer function is discussed. The theoretical and experimental results show that the rigid-elastic coupling tire model with normal damping can achieve the sectional feature of in-plane transfer function resulting from the coupling characteristic between the flexible carcass and distributed sidewall element within the frequency band of 300 Hz.

In this investigation, a nonlinear finite element method for the large deformation and rotation of cable problems is presented. This method is based on finite element absolute nodal coordinate formulation that guarantees the continuity of all displacement gradients and leads to a constant mass matrix. The classical cable theory is first reviewed and the assumptions used in this linear theory are defined in order to demonstrate the basic differences between the linear theory and the nonlinear finite element formulation proposed in this paper for cable applications. The elastic cable forces in the absolute nodal coordinate formulation are obtained using a general continuum mechanics approach that accounts for the effect of all geometric nonlinearities. It is shown in this investigation that the use of the general continuum mechanics approach leads to a simpler and more efficient formulation as compared to the use of the assumptions of the linear theory that employs a local finite element coordinate system. The results obtained using the absolute nodal coordinate formulation show a good agreement with the results obtained using the classical cable theory when linear cable problems are considered. In particular it is shown that the use of perturbation methods to linearize the finite element equations of motion leads to modal characteristics results that are in a good agreement with the linear theory. The results of this investigation obtained using explicit numerical integration also show the potential of the proposed finite element formulation in the nonlinear analysis of cables that experience large rotations and deformations. The generalization of the procedure presented in this paper to three-dimensional cable problems is demonstrated and the computer implementation in multibody algorithms is discussed.

Recent research suggests the various applications of Model Predictive Control on vehicle systems. In numerous cases, nonlinear tire models such as the Magic Formula, which are highly complex and are more detailed than necessary, are used. This paper presents a nonlinear tire model that excludes the region of negative slope but expresses the nonlinear properties of tire well enough for tracking the lane of a racing course. The proposed inverse tire model can also be used to calculate the slip angle from the tire force. Thus, the model can be utilized to design the Model Predictive Controller.

SUMMARYA four wheel steering (4WS) control strategy is derived and evaluated using a nonlinear vehicle model along with a nonlinear tire model. The vehicle model used is a.three degree-of-freedom model. Yaw rate, lateral velocity and longitudinal velocity are considered with the roll effects included in the calculation of the normal loads for each of the four wheels. These normal forces are used in the tire model to calculate the tire forces and moments for each wheel. The results of simulation indicate the potential benefits gained by using 4WS. These benefits include stability and performance at high speeds and maneuverability at low speeds.

A detailed nonlinear finite element analysis (FEA) model of a radial-ply truck tire, 295/75R22.5, has been developed using explicit FEA simulation software, PAM-SHOCK. For the validation of the model, the tire model predictions of contact patch area, vertical stiffness, and cornering characteristics, such as cornering force and aligning moment versus slip angle, at different vertical loads are in good agreement with available physical measurements. For complete vehicle simulations, a simplified rigid ring tire model is required for efficient analysis throughput. The behavior of such a tire model can be verified and improved by comparing responses with the developed FEA model. Moreover, the in-plane and out-of-plane tire parameters needed for the simplified rigid ring tire model could be virtually determined at various vertical loads by testing the FEA tire model instead of performing expensive tire parameters measurements. The in-plane and out-of-plane tire parameters are implemented into a simplified rigid ring tire model to perform durability tests. The durability tests are conducted to examine dynamic behaviors by using the FEA truck tire and the rigid ring tire models during running on a water drainage ditch at various vertical tire loads. The ditch is 12.0-cm (4.72-in) deep and lies in 45-degree angle against tire traveling direction. The dynamic responses such as vertical displacement, forces, and moments at tire center are predicted using both tire models. The results obtained from both models are in reasonable agreement.

The tire forces of vehicles will fall into the non-linear region under extreme handling conditions, which cause poor path tracking performance. In this paper, a model predictive controller based on a nonlinear tire model is designed. The tire forces are characterized with nonlinear composite functions of the magic formula instead of a simple linear relation model. Taylor expansion is used to linearize the controller, the first-order difference quotient method is used for discretization, and the partial derivative of the composite function is used for matrix transformation. Constant velocity and variable velocity conditions are selected to compare the designed controller with the conventional controller in Carsim/Simulink. The results show that when the tire forces fall in the nonlinear region, two controllers have good stability, and the tracking accuracy of the controller designed in this paper is slightly better. However, after the tire forces become nonlinear, the controller with linear tire force becomes worse, the tracking accuracy is far worse than the controller with the nonlinear tire model, and the vehicle stability is also degraded. In addition, an active steering test platform based on LabVIEW-RT is established, and hardware-in-the-loop tests are carried out. The effectiveness of the designed controller is verified.

The vehicle sideslip angle is one of the most important state variables for vehicle motion control. Its accurate estimation is necessary for many vehicle control systems. In this paper, a novel switched Takagi–Sugeno (T-S) fuzzy observer is proposed to estimates the vehicle sideslip angle. Firstly, to approximate the nonlinear tire model well, a new piecewise affine (PWA) tire model is established. Different from the classic two-segment PWA tire model, the lateral tire force is divided into three regions, i.e. linear region, nonlinear region and saturated region in the proposed model. Substituting it into the bicycle model, the switched lateral dynamic model is obtained whose switching signal is the tire slip angle. Then, the T-S fuzzy modelling technique is applied to represent the vehicle lateral dynamics with a varying speed. Combined with the aforementioned model, a switched T-S fuzzy model is obtained to describe the lateral dynamics. Based on the measured yaw rate, a switched T-S observer is designed. Simulation results of different maneuvers based on the high- fidelity simulation software veDYNA show that the designed observer provides a fast response and has a good estimation result.

This paper focuses on the tire force characteristics related to the vehicle speed. We constructed a mechanical tire model using a Voigt-Type viscoelastic spring system for the tread rubber. And, we analyze the force distribution along the contact length, the dependence of vehicle speed to the tire forces and the moments to a change in sideslip angle. Finally, it is shown that the tire model has good agreement with actual measurements results qualitatively, and also it is found that the characteristics of lateral force have independence to the vehicle speed.

ABSTRACTIn recent years, the ever-increasing interest in intelligent tyre technology has led to the formulation of different empirical models correlating deformation measurements provided by the sensors with tyre dynamics. In this paper, a real-time physical model, suitable for describing the dynamics of intelligent tyres based on measurements of strains and/or displacements of the tyre carcass, is presented. The proposed flexible ring model can reproduce the tyre dynamics for both concentrated and distributed forces by introducing a discrete approach that also allows to analyse the longitudinal dynamics of the tyre in real-time. The analytical description of the problem allows to obtain solutions in closed form and to quickly identify model parameters from experimental data. The comparison between the simulated results and the ones provided by indoor tests of two intelligent tyre concepts highlighted that the proposed tyre model can estimate with an acceptable precision both the carcass deformations and the forces acting on the tyre.

Based on the multi-spoke, single-plane, steady-state tyre model, a transient multi-spoke, three-plane tyre model is presented. This model involves updating the states of all the spokes under consideration to t+dt from the corresponding states at t. Also, a spoke base, which has lateral and longitudinal flexibilities relative to wheel hub, is included. By adding two extra planes of spokes to the original one, the effect of tyre width is built into the model. The three planes are equally spaced across the width of the tyre. The interaction with the ground of the spokes on these three planes is used to represent that of tyre elements at different locations across the width. Analytical results show good qualitative agreement with published experimental data. This model can be used to illustrate transient tyre behaviour and in simulations in which a generic tyre will suffice.

This study outlines an approach for speeding up the simulation of the dynamic response of vehicle models that include hysteretic nonlinear tire components. The method proposed replaces the hysteretic nonlinear tire model with a surrogate model that emulates the dynamic response of the actual tire. The approach is demonstrated via a dynamic simulation of a quarter vehicle model. In the proposed methodology, training information generated with a reduced number of harmonic excitations is used to construct the tire hysteretic force emulator using a Neural Network (NN) element. The proposed approach has two stages: a learning stage, followed by an embedding of the learned model into the quarter car model. The learning related main challenge stems from the attempt to capture with the NN element the behavior of a hysteretic element whose response depends on its loading history. The methodology is demonstrated in conjunction with a simple nonlinear quarter vehicle system as well as an ADAMS based model that uses a complex tire element. The results obtained with the surrogate model prove to be accurate and are obtained at a fraction of the CPU time required to handle the original models. The approach proposed is anticipated to be useful for reducing the duration of vehicle simulations, or when a tire model is not available but experimental data can be used to generate a surrogate model.