Tire Response Research Articles

The effect of tyre and suspension dynamics on wheel spindle forces

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.

On Modeling Contact and Friction Calculation of Tyre Response on Uneven Roads

ABSTRACT In this paper, several alternatives in modeling contact and friction within tyre models are discussed. Focusing on existing models for the calculation of ride comfort, the different approaches concerning contact and friction have been used in serveral combinations. These models do not differ very much in results, but in computational effort, which can be influenced by choice of modeling methods. As an example, a tyre model for computation of ride comfort is introduced, in which the model's dynamic characteristics (modes to be modeled) and geometric characteristics (wavelength of road disturbance) are decoupled.

Computational strategies for tire modeling and analysis

Computational strategies are presented for the modeling and analysis of tires in contact with pavement. A procedure is introduced for the simple and accurate determination of tire cross-sectional geometric characteristics from a digitally scanned image. Three new strategies for reducing the computational effort in the finite element solution of tire-pavement contact are also presented. These strategies take advantage of the observation that footprint loads do not usually stimulate a significant tire response away from the pavement contact region. The finite element strategies differ in their level of approximation and required amount of computer resources. The effectiveness of the strategies is demonstrated by numerical examples of frictionless and frictional contact of the Space Shuttle orbiter nose-gear tire. Both an in-house research code and a commercial finite element code are used in the numerical studies.

On Modeling Contact and Friction Calculation of Tyre Response on Uneven Roads

ABSTRACT In this paper, several alternatives in modeling contact and friction within tyre models are discussed. Focusing on existing models for the calculation of ride comfort, the different approaches concerning contact and friction have been used in serveral combinations. These models do not differ very much in results, but in computational effort, which can be influenced by choice of modeling methods. As an example, a tyre model for computation of ride comfort is introduced, in which the model's dynamic characteristics (modes to be modeled) and geometric characteristics (wavelength of road disturbance) are decoupled.

Lateral tyre dynamic characteristics

The ability to model the ride vibration of unsuspended vehicles is at present seriously limited by lack of knowledge about the lateral and longitudinal dynamic characteristics of rolling tyres and the interaction of these characteristics with the ground surface. In this paper, models for the lateral dynamic behaviour of a rolling tyre are reviewed. The beam model, string model and spring-damper models are described and compared. A new lateral tyre model is introduced and tested against experimental results. This model comprises a parallel spring and hysteretic damping unit, representing the tyre carcass, in series with a viscous damper representing the interaction between the tyre and ground. Model justification was made against the measurements of lateral tyre response. The results showed that the proposed model provided more accurate prediction of lateral dynamic tyre response than the other spring-damper models.

Reduced Basis Technique for Evaluating Sensitivity Derivatives of the Nonlinear Response of the Space Shuttle Orbiter Nose-Gear Tire

Abstract A study is made of the sensitivity of the nonlinear tire response to variations in the design variables. The tire is discretized by using three-field mixed finite element models. An efficient reduced basis technique is used for calculating the nonlinear tire response as well as the first-order and second-order sensitivity coefficients (derivatives with respect to design variables). In this technique the vector of structural response and its first-order and second-order sensitivity coefficients are each expressed as a linear combination of a small number of basis (or global approximation) vectors. The Bubnov-Galerkin technique is then used to approximate each of the finite element equations governing the response and the sensitivity coefficients by a small number of algebraic equations in the amplitudes of the vectors. Extensive numerical results are presented for the sensitivity coefficients of the Space Shuttle orbiter nose-gear tire, when subjected to uniform inflation pressure.

Reduced-basis technique for evaluating the sensitivity coefficients of the nonlinear tire response

An efficient reduced-basis technique is presented for calculating the sensitivity of nonlinear tire response to variations in the desin variables. The tire is discretized by using three-field mixed finite element models. The vector of structural response and its first- and second-order sensitivity coefficients (derivatives with respect to design variables) are each expressed as linear combinations of a small number of basis (or global approximation) vectors. The Bubnov-Galerkin technique is then used to approximate each of the finite element equations governing the response and the sensitivity coefficients, by a small number of algebraic equations in the amplitudes of these vectors

Sensitivity of tire response to variations in material and geometric parameters

A computational procedure is presented for evaluating the analytic sensitivity derivatives of the tire response with respect to material and geometric parameters of the tire. The tire is modeled by using a two-dimensional laminated anisotropic shell theory with the effects of variation in material and geometric parameters included. The computational procedure is applied to the case of uniform inflation pressure on the space shuttle nose-gear tire when subjected to uniform inflation pressure. Numerical results are presented showing the sensitivity of the different response quantities to variations in the material characteristics of both the cord and the rubber.

Dynamic characteristics of large tyres

To optimise the vibrational behaviour of tractors it is necessary to have detailed information on the dynamic properties of tyres. To be able to make statements about the vibrational response of tyres, the vibrational characteristics of the tyres must be ascertained under field and laboratory conditions. A flat-belt tyre test stand has been developed which makes it possible to simulate the essential operating conditions of tractor tyres for speeds of up to 60 km/h and to investigate spring and damping characteristics as well as other performance characteristics. Experiments have been carried out which show that there is a decrease in the tyre stiffness and especially of the damping values with an increase in speed. Non-uniformities in the tyre, such as non-circularity, can significantly affect the vibration characteristics of large tyres.

Exploiting symmetries in the modeling and analysis of tires

Three basic types of symmetry (and their combinations) exhibited by tire response are identified. A simple and efficient computational strategy is presented for reducing both the size of the model and the cost of the analysis of tires in the presence of symmetry-breaking conditions (e.g., unsymmetry of the tire material, geometry, and/or loading). The strategy is based on approximation of the unsymmetric response of the tire with a linear combination of symmetric and antisymmetric global approximation vectors ( or modes). The three main elements of the computational strategy are as follows: (1) use of three-field mixed finite element models having independent shape functions for stress resultants, strain components, and generalized displacements, with the stress resultants and the strain components allowed to be discontinuous at interelement boundaries; (2) use of operator splitting (additive decomposition of some of the matrices and vectors in the finite element model) to delineate the symmetric and antisymmetric contributions to the response; and (3) successive use of the finite element method and the classic Rayleigh-Ritz technique to substantially reduce the number of degrees of freedom. The finite element method is first used to generate a few global approximation vectors (or modes). Then the amplitudes of these modes are computed with the Rayleigh-Ritz technique. The proposed computational strategy is applied to three quasi-symmetric problems of tires, namely, (1) linear analysis of anisotropic tires through the use of semianalytic finite elements, (2) nonlinear analysis of anisotropic tires through the use of two-dimensional shell finite elements, and (3) nonlinear analysis of orthotropic tires subjected to unsymmetric loading. In the first two applications, the anisotropy (nonorthotropy) of the tire is the source of the symmetry breaking; in the third application, the quasi-symmetry is due to the unsymmetry of the loading. The effectiveness of the proposed computational strategy is also demonstrated with numerical examples, and its potential for handling practical tire problems is outlined.

Transient Properties of Truck Tires on Real Road Surfaces

Abstract The cornering, or lateral force response of heavy-duty truck tires, has been evaluated on real road surfaces at speeds of 10–60 km/h. The special mobile truck tire dynamometer has a two-test-tire carriage mounted just ahead of the rear support tires of an articulated truck (tractor) trailer. Equal slip angles may be applied simultaneously to both test tires. The frequency response was evaluated by typical phase angle methods. The phase angle (lag of lateral force behind instantaneous angle) increased with frequency (time rate of application of angle) and decreased with increasing speed.

A Simple Model for Cornering and Belt-edge Separation in Radial Tires

Abstract A mathematical analysis of radial tire cornering was performed to predict tire deflections and belt-edge separation strains. The model includes the effects of pure bending, transverse shear bending, lateral restraint of the carcass on the belt, and shear displacements between belt and carcass. It also provides a description of the key mechanisms that act during cornering. The inputs include belt and carcass cord properties, cord angle, pressure, rubber properties, and cornering force. Outputs include cornering deflections and interlaminar shear strains. Key relations found between tire parameters and responses were the optimum angle for minimum cornering deflections and its dependence on cord modulus, and the effect of cord angle and modulus on interlaminar shear strains.

RECENT RESEARCH IN LATERAL DYNAMICS OF MOTORCYCLES

SUMMARY The mathematical model which has been developed to describe the dynamics of the motorcycle [1–5] has attained a rather complex configuration. Torsion of front and rear frames and lateral flexibilities are options which may be included. Tyre response to certain road irregularities is measured and mathematical models to describe this particular behaviour are being developed. As to the validation of the mathematical motorcycle model it was argued that the commonly performed road tests suffer from disturbances originating from the human rider which are difficult to measure or to control. For that reason a Rider Robot has been developed which replaces the human rider. Each of these three subjects will be discussed in the following chapters.

Dynamic analysis of a radial tire by finite elements and modal expansion

Doubly curved, axisymmetric shell finite elements and the modal expansion approach are used to investigate the linear dynamic response of an automotive radial tire, with particular focus on the frequency response, the travelling sinusoidal load response, and the response spectrum due to a random pavement. For the travelling sinusoidal load response it is shown that, depending upon the natural frequency spectrum of the tire, the vehicle velocity, and the distance-based frequency of the pavement profile, certain modes dominate the response at specific vehicle velocities. For the random pavement the tire response increases with vehicle velocity.

In-Plane and Out-of-Plane Dynamics of Pneumatic Tyres

SUMMARY During the last decade research has been conducted on the dvnamics of the tvre considered as a vehiclecomponent. Asurvey is given of these developments where tyre mass plays an important role. The underlying theoretical considerations concerning the massless tyre have been discussed as well. Two groups of tyre response have been distinguished: the lateral (out-of-plane) response and the vertical/longitudinal (in-plane) response to motions of the wheel. In both categories tyre compliance, slip and inertia have influence. The dynamic properties of the rolling tyre have been presented in the form of transfer functions and/or differential equations. The frequency of the imposed oscillations is assumed to be below ca. 40 Hz. Non-linear effects and modelling have been briefly touched on.

Dynamic Response of Tires

Abstract Tire tests were performed under time-varying inputs on the Calspan Tire Research Facility with a G78-14 belted tire at low speeds and path frequencies up to 6 rad/ft (20 rad/m). Both slip angle and vertical load were varied, either separately and periodically or in combination. The combination simulated actual time histories of slip angle and load measured in road accident avoidance tests. For path frequencies up to 0.2 rad/ft (0.7 rad/m), attenuation of lateral force and aligning torque amplitudes was negligibly small. However, both lagged static values substantially, leading to dynamic offsets in these quantities. The offsets appeared to be the most significant factors for tires. Their effect on the dynamic response of a vehicle remains to be investigated.

Tire Cornering Properties

Abstract This paper provides a critical review of tire test techniques, test equipment, data processing methods, and mathematical models for obtaining data and describing tire cornering properties. It examines applicability of this data for evaluation of vehicle directional response properties and shows how test and data processing methods influence tire parameters used in simulations with models. Mathematical relationships between tire parameters and vehicle directional response parameters such as steering sensitivity are shown. Fundamental differences between conditions prevailing in testing, hypothetical assumptions used in mathematical models, and actual tire operating conditions on a vehicle are discussed. The need for transient tire measurements and modeling for transient conditions is also shown.

On the dynamic response of rolling tires according to thin shell approximations

The response of the rolling tire is formulated in terms of the natural modes and frequencies of the non-contacting tire for contact loading due to rolling, braking, accelerating or cornering. The tire is viewed as an equivalent thin shell. Formulation is by way of the three-dimensional dynamic Green function of this shell. As the application example, the response of the tire during pure rolling is evaluated. The critical rolling speed at which the tire develops its first large amplitude standing waves is calculated in a novel fashion, showing the relationship between the standing wave phenomenon and natural tire modes and frequencies. Findings underline the need for measuring or calculating higher order tire modes than those considered in what seems to be the present standard practice.

Frequency Response of Tires Using the Point Contact Theory

The theoretical frequency response of a rolling tire is obtained here for angular oscillations about a vertical axis through the axle in the wheel plane. The point contact theory, also known as the DeCarbon or Moreland theory, is used to describe the side (cornering) force and the torsional (self-aligning) moment as a function of the oscillating frequency. These results are compared to previously published experimental results and found to show reasonable correlation especially when compared to theoretical results obtained using the von Schlippe or stretched string theory. It is noted, both from experimental and theoretical results, that the frequency response is considerably influenced by rolling speed which is contrary to implications given in previous papers. It is also pointed out that an upper limit of the forcing frequency exists above which the empirical data is not likely to represent natural shimmy oscillations. The results indicate that the point contact theory is adequate for use in shimmy analysis over both the tire yaw and structural-torsion shimmy modes of vibration if appropriate tire parameters are used. Nomenclature b = half width of tire footprint or contact area c = tire yaw coefficient CL = tire lateral damping coefficient ct = tire time constant D = tire diameter Airum = drum diameter d = effective tire width F = magnitude of dimensionless, complex lateral force ampli_ tude

Force and Moment Response of Pneumatic Tires to Lateral Motion Inputs

Equations of motion applicable to a “running-band” model of the pneumatic tire, as developed by von Schlippe, are transformed to the frequency domain, and side-force and aligning-moment responses are determined for a tire undergoing sinusoidal (a) lateral oscillations of the wheel center plane; (b) steering oscillations with no lateral motion of the wheel center plane; and (c) lateral and yawing motions phased so that the wheel center plane is always aligned with the direction of motion, i.e., the slip angle is zero. The analytical results obtained for the postulated model show that the side force and aligning moment caused by a steering oscillation (namely, the motion described by (b) above) is equal to that produced by pure side-slipping (i.e., (a) above) plus that caused by pure yawing (i.e., (c) above). Numerical evaluation of the derived frequency-response expressions yields an aligning-moment response to steering judged to be extremely significant in view of the unstable behavior often exhibited by wheels with a steering degree of freedom. Comparison of theoretical predictions with experimental data obtained recently by Saito indicates that the “running-band” model is a good first-order approximation of the lateral nonstationary characteristics of pneumatic tires.