Application of Sensitivity Methods to Analysis and Synthesis of Vehicle Dynamic Systems

SUMMARY

The development and application of sensitivity methods for determining the effects of parameter changes on the response of vehicle dynamic systems is presented. The procedures shown can be used to enhance the analysis and synthesis processes of virtually any road or rail vehicle system regardless of its complexity. The parametric sensitivity of vehicle models in time domain, steady state models and vehicle models in frequency domain can be investigated using different types of sensitivity functions, both dimensional and dimensionless including first order standard, percentage, logarithmic, second order standard, and logarithmic and percentage sensitivity measures. These sensitivity functions and measures are determined as functions of partial derivatives of system variables taken with respect to system parameters. In the case of sensitivity functions in the frequency domain the variable values are computed as either the magnitude or phase angle of a complex element of the transfer function matrix. The methods presented enable to determine the influence of all system primary (constant) and secondary (non-constant) parameters on system primary and secondary variables. The primary variables are state variables or elements of the transfer function matrix and the secondary variables may be any functions of primary variables and system parameters. Typical secondary system parameters which can be examined include initial conditions, time variant coefficients, natural frequencies, loads, and typical secondary variables are forces, weight transfers, stability factors and energy components. The analysis of sensitivity results obtained for three vehicle handling models in both linear and nonlinear regimes of vehicle performance and utilizing various types of sensitivity functions is also presented.     

Sensitivity Analysis of Vehicle Design Attributes in Frequency Domain

This paper presents a procedure for determining the sensitivity matrices of a vehicle dynamic system in the frequency domain as derivatives of the transfer function matrix with respect to system parameters. First and second order logarithmic sensitivity functions which possess normalized coefficients have been introduced to enhance analysis. The sensitivity analysis of amplitude-frequency characteristics on changes of selected parameters of a 3 degree-of-freedom vehicle handling model has been performed. It has been demonstrated that the general sensitivity measure proposed can be used to determine the combined influence of system parameters divided into groups such as design, environmental, driver, kinematic, etc., on the multi-parameter system response, prior to determining their order of influence within the groups.     

Vehicle Response to Stochastic Roadways

Dynamic response calculations for vehicles traversing irregular surfaces are usually accomplished using frequency domain methods involving spectral densities and transfer functions. Here an alternative procedure is developed which allows direct computation of mean square values and correlations of system variables for both transient and steady-state conditions. The method is based upon the differential equation for the covariance matrix which is directly related to the state equations for the vehicle. Multiple white noise inputs can be incorporated as well as inputs at two wheels which follow the same track at a distance from one another..The method is suitable for computer implementation without the complex algebra associated with finding all necessary transfer functions and the necessity of evaluating integrals in order to find mean square values using the conventional approach. As an illustration, a simple vehicle model is worked out completely and the variation of pitch and heave motion as a function of vehicle speed is plotted.     

Vehicle Response to Stochastic Roadways

SUMMARY

Dynamic response calculations for vehicles traversing irregular surfaces are usually accomplished using frequency domain methods involving spectral densities and transfer functions. Here an alternative procedure is developed which allows direct computation of mean square values and correlations of system variables for both transient and steady-state conditions. The method is based upon the differential equation for the covariance matrix which is directly related to the state equations for the vehicle. Multiple white noise inputs can be incorporated as well as inputs at two wheels which follow the same track at a distance from one another..The method is suitable for computer implementation without the complex algebra associated with finding all necessary transfer functions and the necessity of evaluating integrals in order to find mean square values using the conventional approach. As an illustration, a simple vehicle model is worked out completely and the variation of pitch and heave motion as a function of vehicle speed is plotted.     

Frequency Spectral Analysis of Bridge-Vehicle System due to Road Surface Roughness

The paper presents a new method to study the dynamic properties of the bridge-vehicle system. The transfer function of the system is obtained by iteration in the frequency domain instead of the time domain. The relationship between vehicle speed and the lowest natural frequency of the system is investigated and a parametric study of the system stability is made. The varying parameters concerned are the vehicle speed, the ratio of vehicle mass to bridge mass, the ratio of vehicle eigenfrequency to bridge eigenfrequence, and the relative damping of the vehicle and bridge.     

Frequency Spectral Analysis of Bridge-Vehicle System due to Road Surface Roughness

SUMMARY

The paper presents a new method to study the dynamic properties of the bridge-vehicle system. The transfer function of the system is obtained by iteration in the frequency domain instead of the time domain. The relationship between vehicle speed and the lowest natural frequency of the system is investigated and a parametric study of the system stability is made. The varying parameters concerned are the vehicle speed, the ratio of vehicle mass to bridge mass, the ratio of vehicle eigenfrequency to bridge eigenfrequence, and the relative damping of the vehicle and bridge.     

Design of an Active Unilateral Brake Control System for Five-Axle Tractor-Semitrailer Based on Sensitivity Analysis

This paper presents the results of a parametric sensitivity analysis of a five-axle tractor-semitrailer vehicle combination using 3-DOF linear yaw/plane model. The first order logarithmic sensitivity functions are derived with respect to several vehicle design parameters. For stabilization of the vehicle's directional behaviour a fairly new control concept called “Active Unilateral Braking Control (AUBC)” acting on the tractor rear wheel's in order to produce a stabilizing yaw torque is investigated. The AUBC system improves not only the directional stability, but also affects the roll dynamics of the vehicle. The sensitivity of the controlled vehicle system with linear quadratic controller (LQR) is also examined, a robust controller design procedure is proposed as a result of the sensitivity analysis. The robustness of this controller in the presence of both internal (including parametric uncertainties, non-linear dynamics) and external disturbances (such as road irregularities and side wind) allows its implementation with confidence with a non-linear vehicle model. The applicability of this control system to a non-linear vehicle model is tested using a 34 DOF, non-linear vehicle model of the tractor-semitrailer combination.     

Parameters optimisation of a vehicle suspension system using a particle swarm optimisation algorithm

The purpose of this paper is to determine the lumped suspension parameters that minimise a multi-objective function in a vehicle model under different standard PSD road profiles. This optimisation tries to meet the rms vertical acceleration weighted limits for human sensitivity curves from ISO 2631 [ISO-2631: guide for evaluation of human exposure to whole-body vibration. Europe; 1997] at the driver's seat, the road holding capability and the suspension working space. The vehicle is modelled in the frequency domain using eight degrees of freedom under a random road profile. The particle swarm optimisation and sequential quadratic programming algorithms are used to obtain the suspension optimal parameters in different road profile and vehicle velocity conditions. A sensitivity analysis is performed using the obtained results and, in Class G road profile, the seat damping has the major influence on the minimisation of the multi-objective function. The influence of vehicle parameters in vibration attenuation is analysed and it is concluded that the front suspension stiffness should be less stiff than the rear ones when the driver's seat relative position is located forward the centre of gravity of the car body. Graphs and tables for the behaviour of suspension parameters related to road classes, used algorithms and velocities are presented to illustrate the results. In Class A road profile it was possible to find optimal parameters within the boundaries of the design variables that resulted in acceptable values for the comfort, road holding and suspension working space.     

Design of an Active Unilateral Brake Control System for Five-Axle Tractor-Semitrailer Based on Sensitivity Analysis

SUMMARY

This paper presents the results of a parametric sensitivity analysis of a five-axle tractor-semitrailer vehicle combination using 3-DOF linear yaw/plane model. The first order logarithmic sensitivity functions are derived with respect to several vehicle design parameters. For stabilization of the vehicle's directional behaviour a fairly new control concept called “Active Unilateral Braking Control (AUBC)” acting on the tractor rear wheel's in order to produce a stabilizing yaw torque is investigated. The AUBC system improves not only the directional stability, but also affects the roll dynamics of the vehicle. The sensitivity of the controlled vehicle system with linear quadratic controller (LQR) is also examined, a robust controller design procedure is proposed as a result of the sensitivity analysis. The robustness of this controller in the presence of both internal (including parametric uncertainties, non-linear dynamics) and external disturbances (such as road irregularities and side wind) allows its implementation with confidence with a non-linear vehicle model. The applicability of this control system to a non-linear vehicle model is tested using a 34 DOF, non-linear vehicle model of the tractor-semitrailer combination.     

Characterization of Dynamic Vehicle Stability Using Two Models of the Human Pilot Behaviour

This article deals with a study of the stability of the vehicle/pilot system for two different models of human operator behaviour. These models, are the outcome of various.approximations of the precision model for single loop compensatory situations. The vehicle is represented with two degrees of freedom and the pilot is assumed to respond to the lateral displacement and to the lateral velocity with a time delay. The properties of these resulting systems are presented and it is observed that, for any given forward visibility, a critical velocity defines a domain of controllability from a domain of uncontrollability. Furthermore this critical velocity is shown independant of the vehicle/ pilot parameters and may be considered as a possible vehicle safety criterion.     

移动车辆非定常气动力时域模拟及分析

为了在时域中考虑复数导纳函数,精细化模拟作用在移动列车上的非定常气动力,提出列车复数导纳函数时频变换方法,并建立同时考虑顺风向、横风向和竖向脉动风的移动列车非定常气动力数学模型.首先,推导出相对于移动列车的瞬时风速,并将瞬时风速代入到移动列车风荷载模型中,通过泰勒级数展开和忽略脉动风速、三角函数高阶项,将移动列车非定常...     

Suspension parameter estimation in the frequency domain using a matrix inversion approach

The dynamic lumped parameter models used to optimise the ride and handling of a vehicle require base values of the suspension parameters. These parameters are generally experimentally identified. The accuracy of identified parameters can depend on the measurement noise and the validity of the model used. The existing publications on suspension parameter identification are generally based on the time domain and use a limited degree of freedom. Further, the data used are either from a simulated ‘experiment’ or from a laboratory test on an idealised quarter or a half-car model. In this paper, a method is developed in the frequency domain which effectively accounts for the measurement noise. Additional dynamic constraining equations are incorporated and the proposed formulation results in a matrix inversion approach. The nonlinearities in damping are estimated, however, using a time-domain approach. Full-scale 4-post rig test data of a vehicle are used. The variations in the results are discussed using the modal resonant behaviour. Further, a method is implemented to show how the results can be improved when the matrix inverted is ill-conditioned. The case study shows a good agreement between the estimates based on the proposed frequency-domain approach and measurable physical parameters.     

Reliability-based sensitivity analysis of vehicle components with non-normal distribution parameters

Techniques from the perturbation method, fourth moment method, reliability-based design theory, and sensitivity analysis approach are employed to present a practical and efficient method for testing the reliability sensitivity of vehicle components with non-normal distribution parameters. With the condition that the first four moments of original random variables are known, the reliability sensitivity theory and cases are researched using the presented numerical method. The variation regularities of reliability sensitivity are obtained and the effects of design parameters on reliability of the vehicle components are studied. The sophisticated formulation provided in this paper is easily amenable to computational procedures. The respective program can be used to obtain the reliability sensitivity of vehicle components with non-normal distribution parameters accurately and quickly. The results obtained are perfect and the solutions compared very well with those from Monte Carlo simulation. The method presents a theoretic basis for the reliability design of the vehicle components.     

A design methodology for mechatronic vehicles: application of multidisciplinary optimization, multibody dynamics and genetic algorithms

A design methodology for mechatronic vehicles is presented. With multidisciplinary optimization (MDO) methods, strongly coupled mechanical, control and other subsystems are integrated as a synergistic vehicle system. With genetic algorithms (GAs) at the system level, the mechanical, control and other relevant parameters can be optimized simultaneously. To demonstrate the feasibility and efficacy of the proposed design methodology for mechatronic vehicles, it is used to resolve the conflicting requirements for ride comfort, suspension working spaces and unsprung mass dynamic loads in the optimization of half-vehicle models with active suspensions. Both deterministic and random road excitations, both rigid and flexible vehicle bodies and both perfect measurement of full state variables and estimated limited state variables are considered. Numerical results show that the optimized vehicle systems based on the methodology have better overall performance than those using the linear quadratic Gaussian (LQG) controller. It is shown that the methodology is suitable for complex design optimization problems where: (1) there is interaction between different disciplines or subsystems; (2) there are multiple design criteria; (3) there are multiple local optima; (4) there is no need for sensitivity analysis for the optimizer at the system level; and (5) there are multiple design variables.     

A design methodology for mechatronic vehicles: application of multidisciplinary optimization,multibody dynamics and genetic algorithms

A design methodology for mechatronic vehicles is presented. With multidisciplinary optimization (MDO) methods, strongly coupled mechanical, control and other subsystems are integrated as a synergistic vehicle system. With genetic algorithms (GAs) at the system level, the mechanical, control and other relevant parameters can be optimized simultaneously. To demonstrate the feasibility and efficacy of the proposed design methodology for mechatronic vehicles, it is used to resolve the conflicting requirements for ride comfort, suspension working spaces and unsprung mass dynamic loads in the optimization of half-vehicle models with active suspensions. Both deterministic and random road excitations, both rigid and flexible vehicle bodies and both perfect measurement of full state variables and estimated limited state variables are considered. Numerical results show that the optimized vehicle systems based on the methodology have better overall performance than those using the linear quadratic Gaussian (LQG) controller. It is shown that the methodology is suitable for complex design optimization problems where: (1) there is interaction between different disciplines or subsystems; (2) there are multiple design criteria; (3) there are multiple local optima; (4) there is no need for sensitivity analysis for the optimizer at the system level; and (5) there are multiple design variables.     

车辆动力传动系固有特性灵敏度分析及动力学修改

分析了车辆动力传动系无阻尼自由振动固有特性的灵敏度，特别给出了固有频率和振型对轴系刚度及惯量等物理参数的灵敏度。基于固有特性灵敏度分析，给出了动力学修改的计算方法。并以某重型车辆动力传动系为实例计算并分析了其固有频率和振型的灵敏度问题，在此基础上进行了动力学修改。     

A Procedure for Optimization of Truck Suspensions

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.     

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.     

不同转向模式的多轴转向车辆性能分析

为解决重型车辆转向时的低速机动性和高速稳定性的问题,提出了多轴动态转向技术,并以三轴车辆为研究对象进行分析。首先建立多轴转向的二自由度车辆模型以及运动微分方程,为提高车辆的稳定性,以零质心侧偏角为目标,推导各轴间的转角比例系数及有关的状态空间矩阵、传递函数,使用MATLAB软件对不同转向模式下的操纵稳定性进行了稳态响应、瞬态响应以及频域响应的仿真。通过分析比较,说明采用多轴动态转向技术,车辆在转向时具有低速机动性高、高速稳定性好的特点。     

基于灵敏度分析的动力总成悬置系统优化

针对某车动力总成悬置系统固有频率偏高和振动耦合严重的问题,建立了该车动力总成悬置系统6自由度模型,并对悬置元件的主刚度参数进行了灵敏度分析.以分析找出的少量敏感参数为设计变量,结合振动解耦和固有频率匹配理论,对该悬置系统参数进行了优化,并对优化前后的动力总成悬置系统进行了台架试验.结果表明,优化后的悬置系统隔振能力有了...