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.     

Flexible Bodies in Multibody System Codes

SUMMARY

Multibody codes are'efficient tools to simulate nonlinear dynamic behaviour of rigid and flexible multibody systems undergoing large overall motions overlaid by small elastic deformation. This paper gives an overview of common approaches for the equations of motion of flexible body models and presents a general way to prepare the required data including geometric stiffening terms. In particularly, for the nodal approach the data are derived using standard results of a finite element analysis of the body. The computation of coefficient matrices describing the equations of motion is done outside the finite element code by matrix manipulations only. The data are stored in a standardised object-oriented structure. Consequently, the data set is independent of the formulation of the multibody system code.     

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.     

Damping Performance of Power Transmission System with Hydraulic Torque Converter

Abstract

In this work three first-order nonlinear, nonstationary differential equations describing a hydraulic torque converter in a power transmission system are used.

A linear model is obtained on the basis of these equations. This model is verified both theoretically and by an experimental test. On the basis of the calculations the torsional damping was determined. This permitted the damping performance of a power transmission system with a hydraulic torque converter to be defined and established.     

Tracked Air Cushion Vehicle Suspension Models: Analysis and Comparison

ABSTRACT

In this paper several dynamical models of fluid suspensions for tracked air cushion vehicles are analyzed and compared. Specifically, the models considered conform to: a fifth-order linear suspension, a nonlinear configuration with a constant supply pressure, a nonlinear model with compressor characteristics, and a sixth-order nonlinear model with a supply duct connecting the compressor to the cushion chamber. Nonlinearities included in the analysis are the flexible-skirted cushion capacitance, cushion entrance and exit orifice restrictions, feeding system duct fluid capacitance, and compressor pressure-flow characteristics. Temporal responses are obtained for guideway and external force inputs. Back flow conditions under which the supply compressor can stall are examined. It is shown that the general behavior resulting from linear analysis is in many respects similar to that obtained from the nonlinear analysis. Thus, a linear model could provide a good initial basis for a preliminary design and performance specifications.     

基于滑模控制的时滞自动车辆跟随系统数学模型

基于非线性车辆动力学方程和固定车辆间距跟随策略,对具有时间滞后的自动化公路系统车辆纵向跟随控制问题进行了研究。在假定车队中的每个被控制车辆能够接收到车队领头车辆以及该车前面一个车辆的位移、速度和加速度信息的情况下,应用滑模变结构控制方法,通过对滑模运动方程的分析,得到了关于车辆间距误差的车辆纵向跟随系统的数学模型。该模型属于一类具有时间滞后的无限维非线性关联大系统。在具有时间滞后的车辆纵向跟随控制器设计中,利用该类非线性关联大系统的稳定性判定条件来设计控制参数,可确保车辆纵向跟随控制系统的稳定性。     

Symbolic-numerical Solution of Multibody Systems with Closed Loops

SUMMARY

For multibody systems with closed kinematic Loops a set of ordinary differential equations and decoupled algebraic equations is formulated which can be solved with explicit multistep integration algorithms. This is achieved by introducing a minimal set of generalized coordinates being specified during numerical integration. For avoiding restart of the integration algorithm after changing these variables transformation relationships are given. Velocity and acceleration constraints are satisfied exactly, position constraints are fulfilled approximately by a dynamic invariant projection onto the constraint manifold. The method is demonstrated by an application to a five-point wheel suspension.     

Active Control of Non-stationary Response of Vehicles with Nonlinear Suspensions

SUMMARY

Active control of non-stationary response of a single degree of freedom vehicle model with nonlinear passive suspension elements is considered in this paper. The method of equivalent linearization is used to derive the equivalent linear model and the optimal control laws are obtained by using stochastic optimal control theory based on full state information. Velocity squared quadratic damping and hysteresis type of stiffness nonlinearities are considered. The effect of the nonlinearities on the active system performance is studied. The performance of active suspensions with nonlinear passive elements is found to be superior to the corresponding passive suspension systems.     

Tractor-Semitrailer Dynamics: Design of the Fifth Wheel∗

SUMMARY

The nonlinear equations of motion are derived for a tractor-semitrailer truck where both the itractor and the semitrailer yaw, pitch, roll, and translate. Special emphasis is placed on the constraints imposed by the fifth wheel on the vehicle motion. In particular, the effects of two proposed fifth wheel design changes on the jackknifing behavior of a vehicle in a turning, braking maneuver are studied. The results demonstrate that the tendency of the vehicle to jackknife can be reduced with a geometric modification of the fifth wheel.     

Exact Equations for Tractrix Curves Associated with Vehicle Offtracking

Abstract

Vehicle offtracking behavior at low speeds is closely approximated by a geometric entity called a tractrix. This paper presents differential equations for generalized coordinates of a planar multibody vehicle model based on tractrix behavior. The equations are exact, can be used with any type of input path, are valid for forward and backward movements, and are much simpler than previously published formulations used to compute transient offtracking. The differential equations can be integrated using conventional numerical integration algorithms to obtain plots of the low-speed tracking performance of articulated vehicles. The equations were formulated symbolically by a computer program used to analyze the kinematic and dynamic behavior of multibody systems. Example numerical results are plotted.     

Linearization Function of a Complex Nonlinear Two-Force-Element

Abstract

Output force of a nonlinear two-force-element (TFE) excited by a stationary random process is stationary random as well and is described by its autocorrelation function.

Dependence of this autocorrelation function on the autocorrelation functions of the excitation process and of its velocity and on their crosscovanance function is indicated by the linearization (describing) function.

This paper describes the determination of the linearization function for a complex TFE composed of a parallel spring and damper, both having characteristics described by polynosmial functions. Knowledge of the linearization function is necessary for carrying out the second order linearization procedure of nonlinear dynamic systems excited by stationary random processes described in ‘1’, ‘2’, ‘3’.     

Self-Excited Vibrations of Railway Vehicle with Dry Friction Units

SUMMARY

Some results obtained during developing a new approach to analysis of conditions of self-excitation of lateral vibrations of railway vehicles as systems of variable structure are given in this article. Structure variability is conditioned by the state of constraints with dry friction. An idea of the replacement of a nonlinear multibody system by a number of subsystems, which are linearized in a certain way, is worked out. The efficiency of the method suggested for the determination of the characteristics of self-excited vibrations of active multibody systems with dry friction is illustrated by way of an example of the investigation of the dynamic behaviour of a freight car.     

Observer Design for Electronic Suspension Applications*

SUMMARY

This paper proposes a new methodology for designing observers for automotive suspensions. Automotive suspensions are disturbance-affected dynamic systems. Semi-active suspensions are bilinear while active suspensions with hydraulic actuators are nonlinear. The proposed methodology guarantees exponentially convergent state estimation for both these systems. It uses easily accessible and inexpensive measurements. The fact that sprung mass absolute velocity of the suspension cannot be estimated in an exponentially stable manner with such measurements is also demonstrated.

Controllers using estimated states are implemented experimentally on the Berkeley Active Suspension Test Rig. Experimental results for two cases are presented : use of observer states to improve ride quality in an active suspension and use of observer states to reduce dynamic tire loading in a semi-active heavy vehicle suspension.     

Dynamics of Pneumatic Tire Vehicles with Connected Suspension Systems

SUMMARY

A vehicle model, with 10 degrees of freedom is used to investigate the skidding conditions of any wheel of the vehicle in motion. Equations for the load transfer and equations for the pneumatic tire spring and shock absorber are derived. Parameters such as gradual cornering, U-curve cornering, the wavy road surface of different wave lengths and cases of independent and connected suspension systems are inputs to the system. The tire calculated forces and their corresponding maximum resistance forces are the outputs of the systems. A connected suspension system is found to resist skidding better than the independent suspension system. The system is non-linear, and numerical solutions are obtained.     

Stabilization of High-Speed Railway Vehicles

SUMMARY

The problems of critical speeds of railway vehicles with dry friction units determination are discussed. A new approach is used which extends the field of application of dynamic response linear analysis methods to vital nonlinear multibody systems. The special features concerning the influence of dry friction forces in the body supports on the trucks and parameters of horizontal constraints of wheelsets and truck frames on critical speed are indicated. It is shown that a significant rising of railway vehicle critical speeds can be reached by changing the structure of constraints between the body and the trucks.     

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.     

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.     

Effect of Frame Compliance on the Lateral Dynamics of Motorcycles

SUMMARY

The lateral dynamics of an uncontrolled motorcycle, running on a straight, level road surface, is investigated in this paper. The structural compliances in the front and the rear frames of the motorcycle are taken into account by introducing additional degrees of freedom in the analysis. The kinematics of the tires is represented by linear differential equations which are based on the taut-string model of pneumatic tires. The linear differential equations of motion are solved to yield the eigensolutions of the system. Numerical results, obtained for parameters corresponding to a Honda CB750 motorcycle, are presented and discussed.     

A novel method for identifying inertial parameters of electric vehicles based on the dual H infinity filter

ABSTRACT

Accurate identification of vehicle inertial parameters is essential to the design of vehicle dynamics control systems. In this paper, a novel vehicle inertial parameter identification method based on the dual H infinity filter (DHIF) for electric vehicles (EVs) is proposed. The filter algorithm employs a nonlinear longitudinal vehicle model with three vehicle states. A hierarchical framework is engaged by the DHIF to estimate the vehicle states and inertial parameters concurrently. In order to minimise the disturbance of unknown noise, the vehicle states are estimated by using the linear H infinity filter (LHIF), while the nonlinear H infinity filter (NHIF) utilises the observed states to identify the vehicle inertial parameters. Finally, the proposed estimation method is verified and compared through the dSPACE based hardware-in-the-loop (HIL) simulation experiments. The results indicate that the DHIF-based estimation method is effective to identify the vehicle inertial parameters with high precision, remarkable robustness, and quick convergence.