Net traction ratio prediction for high-lug agricultural tyre

A study was conducted to determine the accuracy of Wismer-Luth and Brixius equations in predicting net traction ratio of a high-lug agricultural tyre. The tyre was tested on a sandy clay loam soil in an indoor University Putra Malaysia (UPM) tyre traction testing facility. The experiment was conducted by running the tyre in driving mode. A total of 126 test runs were conducted in a combination consisting of three selected inflation pressures (i.e., 166, 193 and 221 kPa) and two wheel numerics (i.e., 19 and 29) representing two extreme types of soil strength under different levels of travel reduction ranging between 0% and 40%. Regression analysis was conducted to determine the prediction equation describing the tyre torque ratio. Marqurdt’s method used by Wismer-Luth for predicting non-linear equation was not found suitable in predicting the torque ratio of the test tyre awing its low coefficient of determination and inadequacy. The logarithmic model was found suitable in torque ration prediction. From analysis of covariance (ANCOVA) the mean effect of travel speed, tyre inflation pressure and wheel numeric on tyre net traction ratio were found to be highly significant, while the interaction of inflation pressure and wheel numeric was not significant. The 193 kPa inflation pressure was found the best, among the three inflation pressures used, in getting higher net traction ratio and higher maximum efficiency. Finally, two models were formulated for tyre net traction ratio; one in terms of wheel numeric and travel speed reduction and the other in terms of mobility number and travel reduction, to describe the tested tyre performance at different soil strengths.     

Tyre load capacity and energy loss with respect to varying soil support stiffness

An analytical model is developed to predict the tyre deformation and the resultant energy loss due to tyre flexibility relative to various soil stiffnesses ranging from rigid unyielding support to soft soil. The analysis allows for examination of a pneumatic tyre with respect to the wheel load to be carried, the supporting ground rigidity and the rigidity of the tyre casing. The important tyre characteristics in providing for wheel load capability under motion are demonstrated to be the tyre carcass construction, dimension, inflation pressure and the ability of the tyre casing to recover some of the resultant energy losses incurred by the tyre under motion.     

Hopf bifurcation analysis of railway bogie

The railway bogie, the most important running component, has direct association with the dynamic performance of the whole vehicle system. The bifurcation type of the bogie that is affected by vehicle parameters will decide the behavior of the vehicle hunting stability. This paper mainly analyzes the effect of the yaw damper and wheel tread shape on the stability and bifurcation type of the railway bogie. The center manifold theorem is adopted to reduce the dimension of the bogie dynamical model, and the symbolic expression for determining the bifurcation type at the critical speed is obtained by the method of normal form. As a result, the influence of yaw damper on the bifurcation type of the bogie is given qualitatively in contrast to typical wheel profiles with high and low wheel tread effective conicities. Besides, the discriminant of bifurcation type for the wheel tread parameter variation is given which depicts the variation tendency of dynamics characteristics. Finally, numerical analysis is given to exhibit corresponding bifurcation diagrams.     

Digital image correlation techniques for measuring tyre–road interface parameters: Part 2 – Longitudinal tyre slip ratio measurement

Measurement of tyre longitudinal slip-ratio is often estimated from three independent measurements devices namely wheel rotation speed, vehicle speed and tyre rolling radius. This produces an expensive measurement system to indirectly determine the slip-ratio. This paper presents a method by which the slip-ratio is determined from a video camera using digital image correlation techniques. The camera, mounted in such a way that the contact patch region is captured, enables the system to measure the tyre tread speed and ground speed at the contact patch. The slip-ratio is then determined from these two measurements.     

Modelling power requirement for traction tyres with zero sinkage

A model was developed by dimensional analysis to predict the gross traction at zero net traction for traction tyres (11.2–28, 12.4–28, 13.6–28) on a hard surface. Different parameters that affect the torque requirement, namely tyre size, tyre deflection, axle load, and rolling radius, were considered for the analysis. Experiments were conducted to study the effect of various wheel and system parameters on torque and energy consumed per unit distance travelled. The model developed predicts the torque requirement in an acceptable range and can be used as a reference for further traction studies of these tyres in various soils.     

Effect of inflation pressure on motion resistance ratio of a high-lug agricultural tyre

A single wheel tyre facility at University Putra Malaysia (UPM) was used to check the validity of Wismer–Luth and Brixius equations in predicting the motion resistance ratio of a high-lug agricultural tyre and to investigate the effect of inflation pressure. A Bridgestone 5-12, 4 ply, lug M was tested on sandy-clay-loam soil. The experiments were conducted by running the tyre in towing mode. Three inflation pressures (i.e., 166, 193 and 221 kPa) were investigated and wheel numerics ranging between 0 and 70. The analysis of covariance (ANCOVA) revealed that both inflation pressure and wheel numeric have significant effects on tyre motion resistance ratio. Regression analysis was also conducted to determine the closeness of fit for Wismer–Luth’s and Brixius’ equations in predicting the motion resistance ratio of the tested tyre. Finally, three new logarithmic models for tyre motion resistance were formulated. The advantage of reducing tyre inflation pressure from 221 (nominal pressure) to 193 kPa on the motion resistance ratio of the high-lug agricultural tyre was pronounced. However, the tyre’s motion resistance ratio deteriorated with further reduction in tyre inflation pressure from 221 (nominal pressure) to 166 kPa.     

Towed wheel shimmy suppression through a nonlinear tuned vibration absorber

The implementation of the nonlinear tuned vibration absorber (NLTVA) for the suppression of shimmy vibration in towed wheels is addressed in this study. We adopt a modified straight tangent tyre model of a single-degree-of-freedom towed wheel system with an attached NLTVA. Stability analysis illustrated that the NLTVA can significantly improve the stability of the equilibrium of the wheel. Bifurcation analysis highlighted the existence of large bistable regions, which undermines the system’s safety. However, numerical continuation analysis, coupled with a dynamical integrity investigation, revealed that the addition of an intentional softening nonlinearity in the absorber restoring force characteristic enables the complete suppression of the bistable regions, also reducing the amplitude of shimmy oscillations in the unstable region. Quasiperiodic motions were also identified; however, their practical relevance seems marginal.

     

Estimation of a three-dimensional tyre footprint using dynamic soil–tyre contact pressures

A method for estimating the three-dimensional (3D) footprint of a 16.9R38 pneumatic tyre was developed. The method was based on measured values of contact pressure at the soil–tyre interface and wheel contact length determined from the contact pressures and the depths and widths of ruts formed in the soil. The 3D footprint was investigated in an area of the field where the pressure sensors of the tyre passed in a soft clay soil. The tyre was instrumented with six miniature pressure sensors, three on the lug face and the remaining three on the under-tread region between two lugs. The instrumented tyre was run at a constant forward speed of 0.27 m/s and 23% slip on a soft soil, 0.48 MPa cone index, 25.6% d.b. moisture content for four wheel load and tyre pressure combination treatments. The 3D footprint assessment derived from soil–tyre interface stress used in this research is a unique methodology, which could precisely relate the trend profile of the 3D footprint to the measured rut depth. The tyre–soil interface contact pressure distributions results showed that as inflation pressure increased the soil strength increased significantly near the centre of the tyre as a compaction increase sensed with the cone penetrometer.     

Delay effects in shimmy dynamics of wheels with stretched string-like tyres

The dynamics of wheel shimmy is studied when the self-excited vibrations are related to the elasticity of the tyre. The tyre is described by a classical stretched string model, so the tyre-ground contact patch is approximated by a contact line. The lateral deformation of this line is given via a nonholonomic constraint, namely, the contact points stick to the ground, i.e., they have zero velocities. The mathematical form of this constraint is a partial differential equation (PDE) with boundary conditions provided by the relaxation of deformation outside the contact region. This PDE is coupled to an integro-differential equation (IDE), which governs the lateral motion of the wheel. Although the conventional stationary creep force idea is not used here, the coupled PDE-IDE system can still be handled analytically. It can be rewritten as a delay differential equation (DDE) by assuming travelling wave solutions for the deformation of the contact line. This DDE expresses the intrinsic memory effect of the elastic tyre. The linear stability charts and the corresponding numerical simulations of the nonlinear system reveal periodic and quasi-periodic self-excited oscillations that are also confirmed by simple laboratory experiments. The observed quasi-periodic vibrations cannot be explained in single degree-of-freedom wheel models subject to a creep force.     

Full-field strain measurements of the inside of an agricultural tyre using digital image correlation

This paper investigates strain measured on the inside of an agricultural tyre with large tread-blocks during a series of static tests using a novel measuring system. The full field strain measurements may be used in the development of a tyre which is capable of estimating tyre forces from strain measurements. The strain measurement system makes use of a calibrated stereo camera system on a mechanical stabilizing system that keeps the cameras pointed at the inside surface of the tyre in contact with the road while the wheel rotates. A static tyre test rig is used to displace the road surface relative to the tyre in the vertical and longitudinal direction. The large tread-blocks caused strain concentrations on the inner surface as the tyre deforms to comply with the road surface. Vertical and longitudinal tests each produce unique strain patterns in the contact patch region. Relationships between the applied forces and strain measurements were developed and showed that these relationships are near linear with R² values above 0.97. The strain measurements also show that the location where strain gauges, for single point strain measurements, are placed inside the tyre is very important on large lugged tyres.     

Parameterisation,validation and implementation of an all-terrain SUV FTire tyre model

This paper focuses on the parameterisation, validation and implementation of an FTire model of a Michelin LTX A/T² 235/85R16 tyre. This tyre is designed for both on- and off-road use and is commonly used on all wheel drive SUVs. Quasi-static laboratory and dynamic field tests were conducted to acquire parameterisation and validation test data for the FTire model. Quasi-static parameterisation tests include acquiring vertical tyre stiffness over a flat plate and cleats, tyre footprint sizes and shapes, longitudinal, lateral and torsional tyre stiffness for various tyre normal loads, as well as vibrational tyre responses. Dynamic parameterisation tests include dynamic cleat test data. An Adams model of the tyre testing equipment is implemented to simulate the FTire model and validate it against dynamic validation test results. Finally, the model is implemented on a fully nonlinear multi-body dynamics model of a Land Rover Defender. It is found that the FTire model is able to predict the lateral tyre behaviour well on a smooth road surface. The vertical and longitudinal tyre behaviour on a smooth road surface and on a rough surface are predicted accurately.     

3D contact patch measurement inside rolling tyres

This paper presents a novel method for capturing the 3D profile of the inside of a rolling off-road vehicle tyre at the tyre-road contact region. This method captures the contact region at all times as the vehicle negotiates obstacles. The system uses a pair of inexpensive digital cameras (capable of capturing up to 300 frames per second) and features a purely mechanical stabilisation system to ensure that the cameras capture the contact region at any wheel speed or vehicle acceleration.The captured images are processed using 3D computer vision techniques using an open source computer vision library called OpenCV. Stereo image pairs are used to create clouds of 3D points showing the profile of the inside surface with good accuracy. Various obstacles were traversed with the deformed tyre profile being compared to the undeformed profile. The system improves on current measurement techniques used to measure the contact patch by capturing a large region of the contact patch, providing full 3D surface geometry, as well as remaining centred on the contact patch irrespective of wheel rotation. The system also enables other imaging techniques to be used such as digital image correlation to determine velocity profiles as well as strain measurements.     

Experimental observations of tyre deformation characteristics on heavy mining vehicles under static and quasi-static loading

Due to large sidewall and bead thicknesses, multi-piece rims are necessary for use with large off-the-road (OTR) tyres. This paper presents the testing protocol and observed load/deflection and vertical/sidewall deflection characteristics of three Goodyear OTR tyre assemblies, namely, (1) a radial 29.5R29 (2) a bias-ply 29.5-29, and (3) a bias-ply 26.5-26. Localized tyre deformations and rim displacements were measured using optical displacement transducers and post-processing high-speed camera images using digital image analysis software. A validation analysis illustrated a maximum difference of 4.05% of vertical wheel displacements between the aforementioned methods. Quasi-static tests show the maximum values of vertical rim displacement and lateral tyre deflection are in the range of 72.2–78.9 mm and 23.3–27.1 mm, respectively, for a severe excitation condition. Differences ranging from 0.2% to 21.5% for maximum vertical and lateral tyre deflections were found between static load tests and engineering data provided by the tyre manufacturer. Linear relationships were observed for both vertical wheel displacement and lateral tyre deflection versus load for all tests. This study demonstrates a thorough methodology to study deflection characteristics of heavy duty OTR tyres and the collected data could be very useful in the development of numerical models of wheel and tyre assemblies for mining vehicles.     

The performance of free rolling rigid and flexible wheels on sand

Experimental results are presented for a towed 6–16 smooth tyre and the same size rigid steel wheel in three types of sands covering a wide range of particle size distribution, two dry and one submerged sands. Their performance was compared at high and low tyre inflation pressures, two vertical loads and a wide range of soil compaction for each sand. The sand performance prediction number, Ns, proposed by the U.S. Army Engineer Waterways Experiment Station (W.E.S.) was then applied to compare with the measured results for the tyre. It was found that in all the three sands the coefficient of rolling resistance was substantially underestimated by the W.E.S. method. However Ns = 10–20 was found to be very important overall criterion for towed tyres on sand. The correlation between the skid and the fractional sinkage of the rigid wheel and the tyre was also examined.     

The slip and drift model of a wheel with tyre compared to some other attempts in this field

The Slip and Drift Model (SDM) of a wheel with tyre, introduced in 1969, has brought the formally correct fundamental solution to the two-dimensional problem of a wheel motion: longitudinal force-lateral force-slip-drift (slip angle) relationship. It has taken into account the tyre elasticity and the specifics of soil behaviour. In some recent attempts to compile a modified model, the SDM was either ignored or presented inadequately. The analysis of two such attempts from 1981 and 1987 has revealed that their kinematic formulae reproduce the basic formulae of the SDM and that their static equilibrium formulae, as far as they are based on the resolution of the soil shear stress into x, y components, are not satisfactory as a matter of principle.     

Tyre flexibility and mobility on soft soils

Five model tyres were tested in the soil bin to investigate the effects of wheel flexibility on the tyre-soil performances. Two different soil types were used together with various inflation pressures which governed the tyre flexibility. The results confirm that tyre flexibility contributes significantly to the development of all the energy components [equation (1)] in the tyre-soil system. As can be seen from the contrasting performances shown, increasing the inflation pressure may allow for a favourable increase in the drawbar pull in one soil (frictional soil) so long as the input energy available can be increased, whilst the reverse may be true in the case of the other (clay) soil. The finite element model used satisfactorily confirms the measered values obtained and is seen to be able to account for tyre flexibility as shown in Figs. 11–14.     

Hopf Bifurcation in Symmetric Networks of Coupled Oscillators with Hysteresis

The standard approach to study symmetric Hopf bifurcation phenomenon is based on the usage of the equivariant singularity theory developed by M. Golubitsky et?al. In this paper, we present the equivariant degree theory based method which is complementary to the equivariant singularity approach. Our method allows systematic study of symmetric Hopf bifurcation problems in non-smooth/non-generic equivariant settings. The exposition is focused on a network of eight identical van der Pol oscillators with hysteresis memory, which are coupled in a cube-like configuration leading to S ₄-equivariance. The hysteresis memory is the source of non-smoothness and of the presence of an infinite dimensional phase space without local linear structure. Symmetric properties and multiplicity of bifurcating branches of periodic solutions are discussed in the context showing a direct link between the physical properties and the equivariant topology underlying this problem.     

A feedback linearization design for the control of vehicle’s lateral dynamics

In this paper, the feedback linearization scheme is applied to the control of vehicle’s lateral dynamics. Based on the assumption of constant driving speed, a second-order nonlinear lateral dynamical model is adopted for controller design. It was observed in (Liaw, D.C., Chung, W.-C. in 2006 IEEE International Conference on Systems, Man, and Cybernetics, 2006) that the saddle-node bifurcation would appear in vehicle dynamics with respect to the variation of the front wheel steering angle, which might result in spin and/or system instability. The vehicle dynamics at the saddle node bifurcation point is derived and then decomposed as an affine nominal model plus the remaining term of the overall system dynamics. Feedback linearization scheme is employed to construct the stabilizing control laws for the nominal model. The stability of the overall vehicle dynamics at the saddle-node bifurcation is then guaranteed by applying Lyapunov stability criteria. Since the remaining term of the vehicle dynamics contains the steering control input, which might change system equilibrium except the designed one. Parametric analysis of system equilibrium for an example vehicle model is also obtained to classify the regime of control gains for potential behavior of vehicle’s dynamical behavior.     

Chaos in a non-autonomous nonlinear system describing asymmetric water wheels

We derive a water wheel model from first principles under the assumption of an asymmetric water wheel for which the water inflow rate is in general unsteady (modeled by an arbitrary function of time). Our model allows one to recover the asymmetric water wheel with steady flow rate, as well as the symmetric water wheel, as special cases. Under physically reasonable assumptions, we then reduce the underlying model into a non-autonomous nonlinear system. In order to determine parameter regimes giving chaotic dynamics in this non-autonomous nonlinear system, we consider an application of competitive modes analysis. In order to apply this method to a non-autonomous system, we are required to generalize the competitive modes analysis so that it is applicable to non-autonomous systems. The non-autonomous nonlinear water wheel model is shown to satisfy competitive modes conditions for chaos in certain parameter regimes, and we employ the obtained parameter regimes to construct the chaotic attractors. As anticipated, the asymmetric unsteady water wheel exhibits more disorder than does the asymmetric steady water wheel, which in turn is less regular than the symmetric steady state water wheel. Our results suggest that chaos should be fairly ubiquitous in the asymmetric water wheel model with unsteady inflow of water.     

Bifurcation of a modified railway wheelset model with nonlinear equivalent conicity and wheel–rail force

This paper presents the bifurcation behaviors of a modified railway wheelset model to explore its instability mechanisms of hunting motion. Equivalent conicity data measured from China high-speed railway vehicle are used to modify the wheelset model. Firstly, the relationships between longitudinal stiffness, lateral stiffness, equivalent conicity and critical speed are taken into account by calculating the real parts of the eigenvalues of the Jacobian matrix and Hurwitz criterion for the corresponding linear model. Secondly, measured equivalent conicity data are fitted by a nonlinear function of the lateral displacement rather than are considered as a constant as usual. Nonlinear wheel–rail force function is used to describe the wheel–rail contact force. Based on these modifications, a modified railway wheelset model with nonlinear equivalent conicity and wheel–rail force is set up, and then, some instability mechanisms of China high-speed train vehicle are investigated based on Hopf bifurcation, fold (limit point) bifurcation of cycles, cusp bifurcation of cycles, Neimark–Sacker bifurcation of cycles and 1:1 resonance. In particular, fold bifurcation of cycles can produce a vast effect on the hunting motion of the modified wheelset model. One of the main reasons leading to hunting motion is due to the fold bifurcation structure of cycles, in which stable limit cycles and unstable limit cycles may coincide, and multiple nested limit cycles appear on a side of fold bifurcation curve of cycles. Unstable hunting motion mainly depends on the coexistence of equilibria and limit cycles and their positions; if the most outward limit cycle is stable, then the motion of high-speed vehicle should be safe in a reasonable range. Otherwise, if the initial values are chosen near the most outward unstable limit cycle or the system is perturbed by noises, the high-speed vehicle will take place unstable hunting motion and even lead to serious train derailment events. Therefore, in order to control hunting motions, it may be the easiest way in theory to guarantee the coexistence of the inner stable equilibrium and the most outward stable limit cycle in a wheelset system.