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Handling tire models like Pacejka (Tire and Vehicle Dynamics, 3rd edn., Elsevier, Amsterdam, 2012) or TMeasy (Rill in Proc. of the XV Int. Symp. on Dynamic Problems of Mechanics, Buzios, RJ, Brazil, 2013) consider the contact patch as one coherent plane. As a consequence, the irregularities of a rough road profile must be approximated by an appropr...
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... Without a doubt, the most famous enveloping model is certainly the SWIFT V R model, which allows the MAGIC FORMULA (MF) model to be used even on rough road surfaces [5]. Simple and parameterless contact models like that used in TMEASY [11,13] can be applied, but do not always guarantee adequate enveloping properties, which can cause sudden and unrealistic variations of the banking and slope angles in the proximity of sharp cleats or asperities. Another family of more sophisticated enveloping models is based on the physical concept of radial and radialinter-radial spring tire [6][7][8]. ...
... Quasi-static simulations are carried out according to the SAE J2731 standard [27] to assess the enveloping capabilities at low speed of the presented model, while dynamic simulations highlight the outcomes of the tireground enveloping model in a dynamic scenario. In both types of simulations, a comparison with the SWIFTV R model [5] and the TMEASY model [13] is provided. Simulations are performed by modeling the tire as a 205/60R15 (2.2 bar inflation pressure) passenger car tire, which is one of the tires used in Ref. [5] to validate the SWIFTV R model. ...
... To evaluate the behavior of the presented tire-ground enveloping model in a typical use case, it is integrated into a complete real-time vehicle simulation framework, and compared with the SWIFTV R [5] and TMEASY [11,13] enveloping models. The used framework consists of a DIL/ SIL/HIL simulator based on a custom high-fidelity vehicle model. ...
Over the past decades, simulation has become vital in vehicle development and virtual testing, especially for autonomous vehicles. High-performance hard real-time simulators, which are crucial for those undertakings, require efficient algorithms to accurately model vehicle behavior within virtual environments. A prime example is tire-ground contact modeling, which is pivotal if we aim to achieve a high level of realism when simulating wheeled vehicles. Contact modeling focuses on an accurate estimation of the parameters needed to compute the forces and torques generated by vehicle-ground interaction. However, the complexity of this task is compounded by the fact that tire-ground contact is a highly non-linear phenomenon, which is further exacerbated by the need to perform tests to fine-tune state-of-the-art tire-ground contact models. To tackle those challenges, we have developed a novel enveloping model that does not require any fitting of experimental data and is based on the 3D geometry of the intersection between undeformed volumes. In this manuscript, we provide a detailed description of the algorithm's formulation, the current software implementation (available as an open-source library), as well as the achieved scalability and real-time performance.
... Since the contact problem is three-dimensional, the description and solution of the contact problem are not straightforward. Rill [1] provides a detailed description of a contact calculation approach for handling tire models. When dealing with higher-frequency vibrations, generated from obstacles that are shorter than the contact patch, it requires a different approach. ...
... The incorporation of the tireroad contact methodology within the multibody framework is explained in detail. The idea of the simple but sophisticated approach proposed by Rill [1], to handle the orientation of the road plane, is used to smooth the transition between the triangular patches that describe the road surface, in the methodology proposed in this work. Emphasis is put on the methods for solving the contact detection problem and their impact on the simulation efficiency and stability as a result of the smoothness of the geometric contact and the tire models. ...
... To avoid or minimize this effect, a set of auxiliary points is used to regularise the transitions between the triangular patches. This strategy proposed by Rill [1] is implemented in this work to obtain a smooth transition between triangular patches describing the road geometry. ...
Multibody simulations play an important role in the study of the dynamic behaviour of road vehicles, being one of the key aspects the modelling of the tire-road interaction. The tire-road contact modelling is required to detect the contact between the tire and road and to obtain the tire contact forces from the tire-road relative motion. The first part of the problem is solved by a contact detection algorithm, whereas the second part is handled by a tire model. In both cases, the trade-off between accuracy and computational cost needs to be addressed in the road dynamics scenario. For the contact detection problem, this work proposes a methodology in which the road is described by a mesh of triangles while the tire is described by a toroid. The triangular mesh road input can be directly extracted from any CAD model. To avoid the geometric discontinuities due to the transition between triangular patches, a set of auxiliary points is used to obtain a continuous variation of the road surface normal, based on the method proposed by Rill (Multibody Syst. Dyn. 45:131–153, 2019. In the process of contact detection, all kinematic quantities required by the tire-road contact model are evaluated. The tire–road forces are evaluated using the enhanced UA tire model, which is a numerically stable and efficient development of the original tire model proposed by Gim and Nikravesh (Int. J. Veh. Des. 12, 1991; Int. J. Veh. Des. 12:19–39, 1991; Int. J. Veh. Des. 12:217–228, 1991). Both the contact and tire models have smooth transitions between their internal transitions, thereby reducing the need for the variable-step ODE integrators to unnecessarily decrease time steps. The advances on contact detection and the tire–road contact models, proposed here, are demonstrated with a set of road vehicle simulation scenarios in which the influence of various modelling parameters on the stability and efficiency of the simulations is addressed. The novelties of the work consist not only in the demonstration of the enhanced UA tire model in practical situations but also in the description of the new tire–road contact methodology that employs a simple description of the road geometry, to achieve smooth tire–road contact forces.
... Guo first proposed the Unitire model in 1986, and gradually formed a complete semi-empirical model based on the theoretical model, which not only has good expressive ability for various working conditions, but also has a concise model with outstanding predictive and extrapolation ability [14]. Based on the above tire model, many scholars at home and abroad have carried out a large amount of extrapolation and optimization research on the model after considering the friction coefficient, tire stiffness, contact dynamics, and other factors [15][16][17][18][19]. ...
Self-supporting run-flat tires (SSRFTs) achieve good zero-pressure driving ability by reinforcing the sidewalls, and the structural shape of sidewall insert rubber (SIR) is critical in influencing the mechanical characteristics of SSRFTs. In this paper, an SSRFT contour model is established by combining the radial tire contour theory and the design elements of SIR. The influence of two design parameters (maximum width L and maximum thickness H) of SIR on the tire stiffness characteristics and the contact characteristics is analyzed in depth, and the accuracy of the model is verified by the tire mechanics bench test. The results show that the radial stiffness of SSRFTs is positively correlated with two design parameters; an increase in L affects the stress concentration at the end of SIR, while a change in H has a more drastic effect on the stress distribution of SIR, leading to a large change in both the location of the deformation of SIR and the maximum equivalent stress; under rated pressure conditions, when L is less than 100 mm, the overlap between SIR and the tread decreases, which in turn makes the contact characteristics of SSRFTs closer to that of a normal tire, and obtains better comfort and abrasion resistance; under zero-pressure conditions, the maximum contact stress of the tread is the smallest when the H is 8 mm, but when H is less than 6 mm, the contact characteristics appear to deteriorate uniformly, and the maximum contact stress continues to rise. The results of the research provide a reference value for the selection of the design parameters for SIR and the optimization of the dynamic performance of SSRFTs.
... In the dedicated literature, there is an abundance of pragmatic models that approximate the transient dynamics of the tyre using a system of ordinary differential equations (ODEs), describing the timeevolution of the forces and moment depending upon the slip and spin inputs. Such representations include primarily the single contact point models [24][25][26][27][28][29][30][31][32], the two-regime formulation [4,33,34], and the lumped approximation of the LuGre-brush models [35][36][37][38][39][40][41]. In this context, the single contact point models constitute a standard approach when it comes to full vehicle dynamics simulations, since they can be easily integrated with Pacejka's Magic Formula (MF) [1,42] or other empirical steady-state tyre formulae. ...
... This is in line with the results previously obtained in Sect. 3.1, and may be explained recalling that, according to Eqs. (30), two different expressions for the bristle deflection are valid in the stationary and transient regions of the contact patch P − and P + , respectively. However, as opposed to the case of rigid tyre carcass, the transient does not extinguish immediately after travelling a distance equal to the contact length. ...
... 3.1, input-to-state stability estimates may be derived directly from the integral solutions in Eqs. (30), and read specifically as in the ...
... The basic assumption in the calculation of contact patch is that the tire-road footprint is more close to rectangular or an elliptic shape [6][7][8][9]. Godbole, et al. [10], based this assumption, suggested models for rectangular and elliptic shape of the contact patch in terms of the tire diameter, the tire section height and tire deflection. However, estimation of the tire deflection in dynamic behavior can be considered difficult. ...
The maximum acceleration that can be achieved by the car is limited by two main factors of maximum torque in the drive wheels and the maximum traction force in the contact patch. The first factor depends on the performance of the engine and the transmission system in the drivetrain dynamics, and the second factor depends on the friction between the tire and the contact patch. The importance of investigating the contact patch is valuable from perspective of increasing traction efficiency to reducing fuel consumption. In this study, a novel method based on image processing techniques was utilized to dynamically measure contact patch. Then, contact patch was modeled by the Bat algorithm considering the load on the tire and inflation pressure. Finally, the model was evaluated visually and statistically. Coefficient of determination (R ² ) for this model was 0.95.
... There are several models of tyreroad contacts that are available for researchers to study, as well as available in the used Simpack software. In our case, a tyreroad model called the Pajecka contact model has been used [37,38]. The parameters are defined through a text file, which has been used in conjunction with a setting win-dow applicable to this modelling element ...
... While the tyre model of the trailer wheels has been very different in comparison with the tyre models of the vehicle, the tyre models of the front and rear wheels have differed only in having stiffness values and damping coefficients that are slightly different from each other, and this minute difference was caused by variations in the tyres' air pressure values. The friction coefficient has been set to the value of 0.75 for all tyre-road contacts [37][38][39][40]. ...
Passenger cars are a means of transportation used widely for various purposes. The category that a vehicle belongs to is largely responsible for determining its size and storage capacity. There are situations when the capacity of a passenger vehicle is not sufficient. On the one hand, this insufficient capacity is related to a paucity in the space needed for stowing luggage. It is possible to mount a rooftop cargo carrier or a roof basket on the roof of a vehicle. If a vehicle is equipped with a towbar, a towbar cargo carrier can be used for improving its space capacity. These accessories, however, offer limited additional space, and the maximal load is determined by the maximal payload of the concerned vehicle. If, on the other hand, there is a requirement for transporting a load with a mass or dimensions that are greater than what could be supported using these accessories, then, provided the vehicle is equipped with a towbar, a trailer represents an elegant solution for such demanding requirements. A standard flat trailer allows the transportation of goods of various characters, such as goods on pallets, bulk material, etc. However, the towing of a trailer changes the distribution of the loads, together with changes of loads of individual axes of the vehicle–trailer axles. The distribution of the loads is one of the key factors affecting the driving properties of a vehicle–trailer combination in terms of driving stability, which is mainly a function of the distribution of the load on the trailer. This research introduces a study into how the distribution of the load on a trailer influences the driving stability of a vehicle–trailer combination. The research activities are based on simulation computations performed in a commercial multibody software. While the results presented in the article are reached for a particular vehicle–trailer combination as well as for a particular set of driving conditions, the applicability of the findings can also be extended more generally to the impact that the load distributions corresponding to various vehicle–trailer combinations have on the related parameters and other driving properties.
... More complex but still real-time capable vehicle models including the drive train, the steering system, and dynamic force elements are described in [11] and [14]. Appropriate road models provide the road height z and the friction coefficient µ as a function of the contact point coordinates x and y [15]. ...
... The time histories of the steer inputs deliver also their time derivativesρ 1 toρ 4 . The steer motions of the knuckles s 1 to s 4 and their derivativesṡ 1 toṡ 4 are part of the vectors of generalized coordinates y and generalized velocities z as defined in (15). Table 2 approximates the step by a continuous ramp, where the rack is moved in the time interval 0.5 ≤ τ S ≤ 0.6 s from its center position s 1 = s 2 = 0 to the right s 1 = s 2 = −5 mm and then (τ S > 0.6 s) kept constant. ...
Virtual testing procedures have become a standard in vehicle dynamics. The increasing complexity of driver assistance systems demand for more and more virtual tests, which are supposed to produce reliable results even in the limit range. As a consequence, simplified vehicle models, like the classical bicycle model or 4-wheel vehicle models, have to be replaced by a fully three-dimensional and nonlinear vehicle model, which also encompasses the details of the suspension systems. This paper presents a passenger car model, where the chassis, the four knuckles, and the four wheels are described by rigid bodies, the suspension system is modeled by the generic design kinematics, and the TMeasy tire model provides the tire forces and torques in all driving situations.
... The inherent complexity of modelling the tyre carcass as a distributed system has legitimated the adoption of approximated formulations derived directly from the stretched string models. Amongst these, the single contact point models [35][36][37][38][39][40][41][42][43] constitute a standard approach when it comes to full vehicle dynamics simulations, since they can be easily integrated with Pacejka's Magic Formula (MF) [1,44] or other empirical steady-state tyre formulae. The basic assumption of the single contact point formulation is that the tyre dynamics may be approximated as the one of a linear system, whose main parameter is the so-called relaxation length. ...
This paper presents a novel tyre model which combines the LuGre formulation with the exact brush theory recently developed by the authors, and which accounts for large camber angles and turning speeds. Closed-form solutions for the frictional state at the tyre-road interface are provided for the case of constant slip inputs, considering rectangular and elliptical contact patches. The steady-state tyre characteristics resulting from the proposed approach are compared to those obtained by employing the standard formulation of the LuGre-brush tyre models and the exact brush theory for large camber angles. Then, to cope with the general situation of time-varying slips and spins, two approximated lumped models are developed that describe the aggregate dynamics of the tyre forces and moment. In particular, it is found that the transient evolution of the tangential forces may be approximated by a system of two coupled ordinary differential equations (ODEs), whilst the dynamics of the self-aligning moment may described by combining two systems of two coupled ODEs. Given its stability properties and ease of implementation, the lumped one may be effectively employed for vehicle state estimation and control purposes.
... However, they led to fundamental results that represent the conceptual basis for simplified models, like the single contact point ones [22][23][24]. In this category also fall some variations that are grounded directly on the simpler brush theory [1,[25][26][27][28][29]. In any case, these models are all based on the assumption that the tyre dynamics may be described by a first-order dynamic model, whose main parameter is the so-called relaxation length, that is the distance that the tyre needs to travel to develop the 63% of the steady-state forces. ...
... From Eq. (25), it may be inferred that the above Assumption 3.1 ensures that S F is also positive definite. ...
This paper refines the two-regime transient theory developed by Romano et al. [Romano L, Bruzelius F, Jacobson B. Unsteady-state brush theory. Vehicle Syst Dyn. 2020;59:11-29..] to include the effect of combined slip. A nonlinear system is derived that describes the non-steady generation of tyre forces and considers the coupling between the longitudinal and lateral characteristics. The proposed formulation accounts for both the carcass and the bristle dynamics, and represents a generalisation of the single contact point models. A formal analysis is conducted to investigate the effect of the tyre carcass anisotropy on the properties of the system. It is concluded that a fundamental role is played by the ratio between the longitudinal and lateral relaxation lengths. In particular, it is demonstrated that the maximum slip that guarantees (partial) adhesion conditions does not coincide with the stationary value and decreases considerably for highly anisotropic tyres. The dissipative nature of the model is also analysed using elementary tools borrowed from the classic theory for nonlinear systems. A comparison is performed against the single contact point models , showing a good agreement especially towards the full-nonlinear one. Furthermore, compared to the single contact point models, the two-regime appears to be able to better replicate the exact dynamics of the tyre forces predicted by the complete brush theory. Finally, the transient model is partially validated against experimental results. ARTICLE HISTORY
... This approach was introduced in [19,20] as an approximation of the stretched-string tyre model and then extended towards more complex applications to include camber-related effects. Analogous pragmatic formulations may be also found in [39][40][41][42]. The effectiveness of this approach has also motivated its integration with the LuGre formulation in [43,44]. ...
... For the problems at hand, well-posed solutions are considered functions at least C 0 (P × R ≥0 ; R 2 ) solving Eqs. (39) weakly in the adhesion zone P (a) , satisfying the BC and IC given respectively by Eqs. (40), (41) and of the form (43) in P (s) . ...
This paper establishes new analytical results in the mathematical theory of brush tyre models. In the first part, the exact problem which considers large camber angles is analysed from the perspective of linear dynamical systems. Under the assumption of vanishing sliding, the most salient properties of the model are discussed with some insights on concepts as existence and uniqueness of the solution. A comparison against the classic steady-state theory suggests that the latter represents a very good approximation even in case of large camber angles. Furthermore, in respect to the classic theory, the more general situation of limited friction is explored. It is demonstrated that, in transient conditions, exact sliding solutions can be determined for all the one-dimensional problems. For the case of pure lateral slip, the investigation is conducted under the assumption of a strictly concave pressure distribution in the rolling direction.
