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.     

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.     

Numerical Simulation of Vehicle Behavior during Straight Line Keeping on Undulating Road Surfaces

SUMMARY

Numerical design of vehicles having optimal straight line stability on undulating road surfaces requires an accurate vehicle model based on knowledge of the relevant phenomena. Therefore, vehicle behavior on undulating straight roads has been analyzed and modeled. Measurements on a flat road surface have shown that the dedicated vehicle model yields accurate simulation results of the steering response to medium steering wheel angle inputs. In addition, the model has been validated by measuring two vehicle responses during normal driving on an undulating straight road: viz. the responses to the small steering wheel angle input and to the input by the global inclination of the road surface.     

Remarks on the Theory of Vehicle Vibrational Analysis Based on the On-the-Road Measurements

SUMMARY

Basic relation between input spectral density matrix and output spectral density matrix of a linear stochastically excited dynamic system is indicated. General conclusions regarding the output processes spectral densities, coherences and phase angles in respect to the input processes stochastic properties are drawn. The possibility of the determination of the system's transfer functions when input and output spectral density matrices are known is discussed. Applications of the obtained results in vehicle vibrational analysis when the vehicle is considered as one-input, two-input or multiinput system are shown.     

Influence of Liquid Load Shift on the Dynamic Response of Articulated Tank Vehicles

SUMMARY

The influence of the lateral load shift on the dynamic response characteristics of an articulated tank vehicle is investigated assuming inviscid fluid flow conditions. A quasi-dynamic roll plane model of a partially filled cleanbore tank of circular cross-section is developed and integrated to a three-dimensional model of the articulated vehicle, assuming constant forward speed. The destabilizing effects of liquid load shift are studied by comparing the directional dynamics of the partially filled tank vehicle to that of an equivalent rigid cargo vehicle subject to steady steer input. Dynamic response characteristics demonstrate that the stability of a partially filled tank vehicle is adversely affected by the Liquid load shift The distribution of cornering forces caused by the liquid load shift yield considerable deviation of the path followed by the liquid tank vehicle. The influence of the vehicle speed on the dynamics of the liquid tank vehicle is also investigated for variations in the fill levels and fluid density.     

A Tractor Handling Study

SUMMARY

Agricultural tractors are relatively slow moving vehicles but a good steer response under all working conditions is generally required. This paper evaluates the effect of ground slope, axle load, ground speed, tyres and ground surface on yaw rate response to steer input.     

Simulation of Multibody Vehicles Moving over Elastic Guideways1

SUMMARY

This paper describes the present state of a general purpose computer program for calculating the dynamic response of vehicles travelling over guideways which may be elastic.

The linearized state-equations of motion for general multibody vehicles are constructed automatically by the program, these equations are supplemented by the equations for the active subsystems. Finally, the vehicle system equations are combined with the modal equations for elastic guideways and the complete set of coupled equations is solved simultaneously by numerical integration.     

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).     

Vergleich einiger Rechen- und Messergebnisse zum Fahrverhalten von Sattelzügen

SUMMARY

In this paper some results of theoretical and experimental investigations on the dynamic directional properties of heavy tractor-semitrailer vehicles are presented.

A nonlinear digital computer model was developed on which the theoretical system analysis is based. This model takes account of the nonUnear tire properties and the friction couple of the fifth wheel. A combination of numerical computation methods (Runge-Kutta and Newton-Raphson techniques) is used for the digital computer simulation.

Full scale road tests with articulated vehicles of 38 ton total weight were conducted for experimental validation of the used theoretical model. As input signals to the vehicle, predetermined steering wheel angle functions were used. The system output signals corresponding to these input functions were measured and stored.

A comparison of the obtained theoretical and experimental results shows a very good qualitative agreement and hence leads to the conclusion that the developed theoretical model can give consistent estimates of the basic dynamic vehicle properties.     

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.     

Additional 4WS and Driver Interaction

SUMMARY

This investigation is based on a complex 4-wheel vehicle model of a passenger car that includes steering system and drive train. The tyre properties are described for all possible combined longitudinal and lateral slip values and for arbitrary friction conditions. The active part is an additional steering system of all 4 wheels, additionally to the driver's steering wheel angle input. Three control levels are used for the driver model that thereby can follow a given trajectory or avoid an obstacle.

The feedback control of the additional 4 wheel steering is based on an observer which can also have adaptive characteristics. Moreover a virtual vehicle model in a feedforward scheme can provide desired steering characteristics.

To get information for critical situations a cornering manoeuvre with sudden u-split conditions is simulated. Further a similar manoeuvre is used to evaluate the reentry in a high friction area from low friction conditions. And finally the performance of the controller is shown in a severe lane change manoeuvre.     

Properties of Non-Linear Two-Force Elements Used in Vehicle Dynamic Systems under Stationary Stochastic Excitation. Part 1 - Ideal Elements

SUMMARY

Non-linear two-force elements (springs, dampers, etc.) are commonly found in vehicle dynamic systems. Their mathematical idealization by expressing their force-excitation relation by a real function is called to be an ideal element. Excitation of these elements can be often considered as being stationary stochastic. Their general properties under such excitation are discussed and computation of the stochastic properties of the output force for known excitation are shown in this paper. Transfer functions of the elements, i.e. output force - input relations in the frequency domain, under the given excitational conditions form the base for the evaluation.     

Optimal Preview Control of Rear Suspension Using Nonlinear Neural Networks

SUMMARY

The performance of neural networks to be used for identification and optimal control of nonlinear vehicle suspensions is analyzed. It is shown that neuro-vehicle models can be efficiently trained to identify the dynamical characteristics of actual vehicle suspensions. After trained, this neuro-vehicle is used to train both front and rear suspension neuro-controllers under a nonlinear rear preview control scheme. To do that, a neuro-observer is trained to identify the inverse dynamics of the front suspension so that front road disturbances can be identified and used to improve the response of the rear suspension. The performance of the vehicle with neuro-control and with LQ control are compared.     

A Full-Car Roll Model of a Vehicle with Controlled Suspension

SUMMARY

The full-car roll model of a vehicle suspension with static and dynamic control (using wheel, body and seat) is described by means of vertical and lateral input for both static and dynamic states. It is shown that the control deteriorates the static performance of the vertical response and improves the performance of the lateral response.     

A Procedure for Optimization of Truck Suspensions

SUMMARY

The purpose of this paper is to develop a procedure based on covariance analysis and nonlinear programming techniques which can be used for the parameter selection of optimum truck suspensions. The procedure is applied to explore the differences in parameter selection caused by the changes in the frequency content of the road input or by changes in the criteria for optimization. The equations of motion for a tractor-semitrailer truck are cast in state space form. The road excitations are represented by the output of a white noise excited shaping filter taking into consideration the time delays between the different vehicle axles. Shape filters to represent human perception of vibration in both the vertical and longitudinal directions in the time domain are presented and realized in terms of state variables. The suspension parameters of the road-vehicle-human body system are optimized using a direct search algorithm.     

Comparison of All-Wheel Steerings in the System Driver-Vehicle

SUMMARY

Different load or tires and a drive on an ice-coated road can overcharge a driver to such an extend, that the result may be an accident. Therefore the aim of development is a self-acting compensation of the vehicle to different vehicle transfer behaviour (invariant vehicle behaviour).

The calculation of so called optimal characteristics shows, that only rear-wheel steering cannot realize this aim of development. Therefore an additional front-wheel angle, which is not influenced by the driver, is necessary. A transfer function can be calculated in order to get controlled steering of the rear wheels without the influence of load.

It is not possible to realize optimal characteristics, because the parameters of the vehicle are difficult to measure. Only an optimal diagnosis and control of driving condition realize a relief for the driver in every driving situation in order to avoid most of the accidents.

The often demanded sideslip angle compensation only worsens driving conditions on ice-coated roads. Therefore systems which identify the driving condition themselves have to be favoured in any case.     

Properties of Non-Linear Two-Force Elements Used in Vehicle Dynamic Systems under Stationary Stochastic Excitation. Part 2 - Realistic Complex Elements

SUMMARY

Realistic two-force elements used in vehicle dynamic systems have very often complex output force -inputs relations. Determination of their properties under stationary stochastic excitation must be therefore carried out experimentally for every excitational case. Evaluation of the measurements of one type of a passenger car telescopic hydraulic suspension damper and of one type of a heavy duty truck leaf spring is shown in this part of the paper. Transfers of the discussed elements, representing the frequency responses of the elements to the input deflectional process and defined in the first part of this paper, form the base for the evaluation.     

Implementation and validation of a three degrees of freedom steering-system model in a full vehicle model

This paper describes the coupling between a three degrees of freedom steering-system model and a multi-body truck model. The steering-system model includes the king-pin geometry to provide the correct feedback torque from the road to the steering-system. The steering-system model is combined with a validated tractor semi-trailer model. An instrumented tractor semi-trailer has been tested on a proving ground and the steering-wheel torque, pitman-arm angle, king-pin angles and drag-link force have been measured during steady-state cornering, a step steer input and a sinusoidal steering input. It is shown that the steering-system model is able to accurately predict the steering-wheel torque for all tests and the vehicle model is accurate for vehicle motions up to a frequency where the lateral acceleration gain is minimum. Even though the vehicle response is not accurate above this frequency, the steering-wheel torque is still represented accurately.