Static Turning Analysis of Vehicles Subject to Externally Applied Forces - A Moment Arm Ratio Formulation

SUMMARY

A formulation for representing the static turning response of a two-axle vehicle due to applied external or control forces is expressed in terms of a simple ratio of two distances along the vehicle longitudinal axis. The two distancescoincide with points on the vehicle at which externally applied/ control forces and their reactive inertial forces act with respect to the vehicle neutral steer point. The resulting formulation is equivalent to the rotational equilibrium equation written with respect to the neutral steer point. The method allows a simple “visual analysis” of the steady turning process by showing how key forces and associated moment arms can change with respect to one another due to vehicle modifications or different operatingconditions, thereby affecting the static turning response of the vehicle.     

Force control for path following of a 4WS4WD vehicle by the integration of PSO and SMC

The aim of this paper is to present a novel control method for a four-wheel steer and four-wheel drive (4WS4WD) vehicle. The novelty is in the integration of sliding mode control (SMC) and particle swarm optimization (PSO) that is proposed to solve the control problem caused by the nonlinear, highly coupled and over-actuated characteristics of the four-wheel steer and four-wheel drive (4WS4WD) vehicle. The validity of the control method is evaluated by two criterions, namely path following performance assessed by the vehicle's position errors with respect to the reference path, and motion quality reflected by the smoothness of vehicle's velocities and accelerations. In vehicle modelling, a kinematic model and a dynamic model considering all slip forces are proposed for the controller design. Simulation results are provided to demonstrate the applicability of the proposed methodology and its robustness.     

Vibrations Generated by Rail Vehicles: A Mathematical Model in the Frequency Domain

Movement of railway vehicles generates mechanical vibrations of a wide range of frequency. Depending on track materials, dissipation in form of viscous and hysteretic damping is present, and stiffness depends on strain-rate. In a previous paper (Castellani et al., 1998), a mathematical model to describe track materials has been developed in the frequency domain. The present paper applies this model, and attempts an analytical formulation of vehicle-track and soil interaction in the frequency domain. Rail vibrations during the passage of a vehicle are generated by three families of forces: a) the weight of the moving vehicle, b) the inertial reaction of the vehicle under the effect of corrugations over an undeformable rail, and, c) the vehicle inertial forces due to displacements of the rail. The first two groups of forces do not depend on the rail displacement, and the related mathematical formulation is a simple problem of forces at a mobile point of application. Formulation of the vehicle inertial forces, related to the rail vibration, requires reference to the acceleration of the rail, as seen by an observer in motion with the vehicle itself. Moreover, it is necessary to express the equilibrium equation of two dynamic systems, the vehicle and the track, at a the movable point of contact. There is no straight numerical procedure to solve this equation in the frequency domain. In the paper two theoretical propositions (Fryba, 1988; Grassie et al., 1982) are revisited with reference to the effect of the transit of a single wheel. Fryba infers that, in the absence of corrugations, the forces c) are null. Grassie et al. (1982) present a mathematical formulation of the interaction between wheel and rail, at mobile point of contact. At each position, the interaction force is of impulsive type. They presume that for a corrugation of harmonic type, of wavelength ?, the wheel is subject to a harmonic motion, of the frequency f = V/?, where V is the wheel velocity. All other frequency components, due to the impulse, are disregarded. Both these assumptions are shown to be inconsistent from a theoretical point of view, however they suggest suitable approaches to the solution.     

Vehicle Dynamic Control for In-Wheel Electric Vehicles Via Temperature Consideration of Braking Systems

?Vehicle dynamic control (VDC) systems play an important role with regard to vehicle stability and safety when turning. VDC systems prevent vehicles from spinning or slipping when cornering sharply by controlling vehicle yaw moment, which is generated by braking forces. Thus, it is important to control braking forces depending on the driving conditions of the vehicle. The required yaw moment to stabilize a vehicle is calculated through optimal control and a combination of braking forces used to generate the calculated yaw moment. However, braking forces can change due to frictional coefficients being affected by variations in temperature. This can cause vehicles to experience stability problems due an improper yaw moment being applied to the vehicle. In this paper, a brake temperature estimator based on the finite different method (FDM) was proposed with a friction coefficient estimator in order to solve this problem. The developed braking characteristic estimation model was used to develop a VDC cooperative control algorithm using hydraulic braking and the regenerative braking of an in-wheel motor. Performance simulations of the developed cooperative control algorithm were performed through cosimulation with MATLAB/Simulink and CarSim. From the simulation results, it was verified that vehicle stability was ensured despite any changes in the braking characteristics due to brake temperatures.     

Simplified Analysis of Steady-state Turning Behaviour of Motor Vehicles. Part 1. Handling Diagrams of Simple Systems.

An approximate method is presented which produces a handling diagram useful for the study of steady-state turning behaviour at different values of steer angle, path radius and speed In three successive parts the steady state response of simple and more elaborate vehicle models and the stability of the resulting motion are discussed.     

Direct yaw moment control and power consumption of in-wheel motor vehicle in steady-state turning

Driving force distribution control is one of the characteristic performance aspects of in-wheel motor vehicles and various methods have been developed to control direct yaw moment while turning. However, while these controls significantly enhance vehicle dynamic performance, the additional power required to control vehicle motion still remains to be clarified. This paper constructed new formulae of the mechanism by which direct yaw moment alters the cornering resistance and mechanical power of all wheels based on a simple bicycle model, including the electric loss of the motors and the inverters. These formulation results were validated by an actual test vehicle equipped with in-wheel motors in steady-state turning. The validated theory was also applied to a comparison of several different driving force distribution mechanisms from the standpoint of innate mechanical power.     

On the objective assessment and quantification of the transient-handling response of a vehicle

This article suggests a new methodology for the objective assessment and quantification of the response of a vehicle subjected to transient-handling manoeuvres. For this purpose, a non-dimensional measure is defined, namely the normalized yaw impulse. This measure appears in two variations. In its general or dynamic form, it represents the difference between the yaw moment due to the front-tyre forces and the yaw moment due to the rear-tyre forces, divided by the sum of the aforementioned yaw moments. By employing a linear, two-degree-of-freedom bicycle model, it is shown that the general form of the normalized yaw impulse can be written as a function of the steer angle and the forward, lateral and yaw velocities of the vehicle. This form is referred to as the kinematic yaw impulse. It is demonstrated that the combined application of the dynamic and kinematic expressions of the yaw impulse not only facilitates the explicit assessment and quantification of the transient behaviour of a vehicle, but also reveals the influence of parameters such as the yaw moment of inertia, which traditionally leave the steady-state behaviour unaffected.     

On the objective assessment and quantification of the transient-handling response of a vehicle

This article suggests a new methodology for the objective assessment and quantification of the response of a vehicle subjected to transient-handling manoeuvres. For this purpose, a non-dimensional measure is defined, namely the normalized yaw impulse. This measure appears in two variations. In its general or dynamic form, it represents the difference between the yaw moment due to the front-tyre forces and the yaw moment due to the rear-tyre forces, divided by the sum of the aforementioned yaw moments. By employing a linear, two-degree-of-freedom bicycle model, it is shown that the general form of the normalized yaw impulse can be written as a function of the steer angle and the forward, lateral and yaw velocities of the vehicle. This form is referred to as the kinematic yaw impulse. It is demonstrated that the combined application of the dynamic and kinematic expressions of the yaw impulse not only facilitates the explicit assessment and quantification of the transient behaviour of a vehicle, but also reveals the influence of parameters such as the yaw moment of inertia, which traditionally leave the steady-state behaviour unaffected.     

Lane-keeping benefits of practical rear axle steer

The classic two-degree-of-freedom yaw-plane or ‘bicycle’ vehicle model is augmented with two additional states to describe lane-keeping behaviour and further augmented with an additional control input to steer the rear axle. A simple driver model is hypothesised where the driver closes a loop on a projected lateral lane position. The driver can select the preview distance to compensate driver/vehicle dynamics, consistent with the ‘cross-over’ model found in the literature. A rear axle steer control law is found to be a function of the front axle steering input and vehicle speed that exhibits stability similar to a positive-real system, while at the same time improving the ability of the driver/vehicle system to track a complex curved lane and improving steady-state manoeuvrability. The theoretically derived control law bears similarity to practical embodiments allowing a deeper understanding of the functional value of steering a rear axle.     

Stationar- und Ubergangsverhalten von Sattel- und Lasttugen bei der Kreisfahrt: Lineare Berechnungen

ABSTRACT

Steady and Transient Turning of Tractor-Semitrailer and Truck-Trailer Combinations: A Linear Analysis

A simplified analysis is made of the yaw stability and control of the two types of the commercial vehicle combinations (tractor-semitrailer, truck-trailer) at a constant forward velocity during steady and transient turning. The combined vehicle is treated as a linear dynamic system (Fig. 2). The steer angle at the front wheels of the tractor (or truck) and the steady-state responses if the road verhicle train (yaw rate, articulation angles and sideslip angle) are calculated (Equations 18 to 25). Exploratory calculations are performed to determine the influence of the cornering stiffness of the tires for the two types of the vehicle combinations upon the steady-state responses (Figs. 7 to 10). For a linear simplified model of articulated vehicle the steady-state turning behaviour is stable also under conditions of rather high driving speed (70 km/h). A simplified analysis of the transient turning behaviour of the two types of road trains has shown the tractor-semitrailer to preserve stability even under driving speeds exceeding 70 km/h (Fig. 13), whereas the truck-trailer combinations appear to become oscillatory unstable if the driving speed rises above the 60 km/h margin (Fig. 14).     

A Pitch-Plane Model of a Vehicle with Controlled Suspension

A vehicle model incorporating front and rear wheel suspensions and seat suspension is presented. The suspension control includes algorithms to provide both dynamic and steady state (levelling) control. Vehicle response to (a) vertical inputs due to ground disturbances at the wheels and (b) longitudinal inputs due to the inertial forces during braking and accelerating, are investigated. It is shown that the static (self-levelling) control causes a slight deterioration in dynamic performance. The active ride control produces improvements of ride comfort under dynamic conditions compared to an equivalent passively suspended vehicle. In steady state the proposed control eliminates the error heave of the body caused by tilting of the vehicle with active suspension.     

On Inverse Equations for Vehicle Dynamics

Abstract

Instead of writing equations which when solved yield the response of a vehicle to an input such as the front wheel steer angle, one can often invert the equations so that a response quantity is specified as an input and a new set of equations is solved yielding the steer angle required as an output. Using these equations one can discover the input steer angle a driver would need to impose in order to accomplish a specific maneuver for various vehicles.

It is shown that there are many possible inverse equation sets and that the eigenvalues of the inverse equations are hard to interpret since they may have little to do with the vehicle parameters. The linear single-input single-output case is studied first to fix ideas using a simple example. For the bicycle model vehicle, it is shown that any vehicle may have unstable inverse equations depending upon the response quantity used. Extensions to nonlinear and multiple-input multiple output systems are discussed.     

Impact of Dynamic Fluid Slosh Loads on the Directional Response of Tank Vehicles

SUMMARY

A comprehensive directional dynamics model of a tractor-tank trailer is developed by integrating a non-linear dynamic fluid slosh model to the three-dimensional vehicle dynamics model. The nonlinear fluid slosh equations are solved in an Eulerian mesh to determine dynamic fluid slosh loads caused by the dynamic motion of the vehicle. The dynamic fluid slosh forces and moments are coupled with the vehicle dynamics model to study the directional response characteristics of tank vehicles. The directional response characteristics of partially filled tank vehicles employing dynamic slosh model are compared to those employing quasi-dynamic vehicle model, for steady as well as transient directional maneuvers. Simulation results reveal that during a steady steer maneuver, the dynamic fluid slosh loads introduce oscillatory directional response about a steady-state value calculated from the quasi-dynamic vehicle model. The directional response characteristics obtained using the quasi-dynamic and dynamic fluid slosh models during transient steer inputs show good correlation. Based on this study, it can be concluded that the quasi-dynamic model can predict the directional response characteristics of tank vehicles quite close to that evaluated using the comprehensive fluid slosh model.     

Roll Plane Analysis of Articulated Tank Vehicles During Steady Turning

SUMMARY

The rollover immunity levels of articulated tank vehicles with partial loads are investigated. A static roll plane model of the articulated vehicle employing partially filled cylindrical tank is developed. The vertical and lateral translation of the liquid cargo due to vehicle roll angle and lateral acceleration, encountered during steady turning, are evaluated. The roll moments arising from vertical and lateral translation of the liquid cargo are determined and incorporated in the roll plane model of the vehicle. The adverse influence of the unique interactions of the liquid within the tank vehicle, on the rollover limit of the articulated vehicle is demonstrated. The influence of compartmenting of the tank on the steady turning roll response of the vehicle is analyzed, and an optimal order of unloading the compartmented tank is discussed.     

Critical review of handling diagram and understeer gradient for vehicles with locked differential

The steady-state cornering behaviour of rear-wheel drive vehicles fitted with locked differential is critically analysed by means of simple, albeit carefully formulated, vehicle models, which allow for a rigorous theoretical analysis. Results obtained for some classical manoeuvres, with either constant forward speed, steer angle or turning radius, clearly show that, in the case of locked differential, the vehicle cornering behaviour is strongly affected by the manoeuvre. As an important consequence, the handling diagram is not unique and the understeer gradient is no longer dependent only upon the lateral acceleration, as in vehicles equipped with an open differential. Accordingly, this study shows that some typical tools and concepts of vehicle dynamics are indeed inadequate in the case of locked differential.     

Stationar- und Ubergangsverhalten von Sattel- und Lasttugen bei der Kreisfahrt: Lineare Berechnungen

Steady and Transient Turning of Tractor-Semitrailer and Truck-Trailer Combinations: A Linear Analysis

A simplified analysis is made of the yaw stability and control of the two types of the commercial vehicle combinations (tractor-semitrailer, truck-trailer) at a constant forward velocity during steady and transient turning. The combined vehicle is treated as a linear dynamic system (Fig. 2). The steer angle at the front wheels of the tractor (or truck) and the steady-state responses if the road verhicle train (yaw rate, articulation angles and sideslip angle) are calculated (Equations 18 to 25). Exploratory calculations are performed to determine the influence of the cornering stiffness of the tires for the two types of the vehicle combinations upon the steady-state responses (Figs. 7 to 10). For a linear simplified model of articulated vehicle the steady-state turning behaviour is stable also under conditions of rather high driving speed (70 km/h). A simplified analysis of the transient turning behaviour of the two types of road trains has shown the tractor-semitrailer to preserve stability even under driving speeds exceeding 70 km/h (Fig. 13), whereas the truck-trailer combinations appear to become oscillatory unstable if the driving speed rises above the 60 km/h margin (Fig. 14).     

On Inverse Equations for Vehicle Dynamics

Instead of writing equations which when solved yield the response of a vehicle to an input such as the front wheel steer angle, one can often invert the equations so that a response quantity is specified as an input and a new set of equations is solved yielding the steer angle required as an output. Using these equations one can discover the input steer angle a driver would need to impose in order to accomplish a specific maneuver for various vehicles.

It is shown that there are many possible inverse equation sets and that the eigenvalues of the inverse equations are hard to interpret since they may have little to do with the vehicle parameters. The linear single-input single-output case is studied first to fix ideas using a simple example. For the bicycle model vehicle, it is shown that any vehicle may have unstable inverse equations depending upon the response quantity used. Extensions to nonlinear and multiple-input multiple output systems are discussed.     

Roll Plane Analysis of Articulated Tank Vehicles During Steady Turning

The rollover immunity levels of articulated tank vehicles with partial loads are investigated. A static roll plane model of the articulated vehicle employing partially filled cylindrical tank is developed. The vertical and lateral translation of the liquid cargo due to vehicle roll angle and lateral acceleration, encountered during steady turning, are evaluated. The roll moments arising from vertical and lateral translation of the liquid cargo are determined and incorporated in the roll plane model of the vehicle. The adverse influence of the unique interactions of the liquid within the tank vehicle, on the rollover limit of the articulated vehicle is demonstrated. The influence of compartmenting of the tank on the steady turning roll response of the vehicle is analyzed, and an optimal order of unloading the compartmented tank is discussed.     

Energy Flow in Active Attitude Control Suspensions: A Bond Graph Analysis

Fully active ground vehicle suspensions which completely replace the passive spring and damper elements with a force generating actuator have required a significant amount of power. Alternative systems which retain compliant elements to handle high frequency isolation but include active elements to control the vehicle body attitude have been developed to reduce the power requirements. These suspensions are called “low bandwidth” or “fast load leveler” systems and they often incorporate semi-active dampers which produce high frequency controllable forces with low power requirements. Here, two contrasting attitude control systems are studied to show that actuator power can be significantly reduced if the actuator is used to vary a lever ratio instead of being used to compress the suspension spring directly. Both types of systems have been successfully implemented in prototype form. Bond graphs for idealized versions of the suspensions show clearly the significant differences in actuator power and energy requirements even though the abstract mathematical structures of the two systems are remarkably similar. Computer simulations confirm the analytical results.     

汽车转向非线性平衡点求解新方法及其应用

针对汽车转向非线性稳定性平衡点确定问题,提出了求解汽车转向非线性平衡点的一种新方法。建立了汽车转向非线性2自由度模型,采用简化魔术公式轮胎模型描述非线性轮胎侧向力。给出了转向非线性平衡点的表示,对粒子群算法进行了简要分析,提出了求解转向平衡点的粒子群算法。设计了前轮转角、路面附着系数和车速分别变化的3种行驶条件,对3种行驶条件下的转向非线性平衡点进行了求解。结果表明,粒子群算法可以准确求得汽车转向非线性平衡点,适应值小于设置的10-4的精度。