Linearization Function of a Complex Nonlinear Two-Force-Element

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'.     

Small Vibrations of Nonlinear Dynamic Systems Containing Non-Rigid Bodies Excited by Stationary Stochastic Processes

From a vibrational point of view, real vehicles form dynamic systems containing non-rigid bodies and nonlinear two-force elements, often excited by stationary random processes. Their theoretical investigation in a broader frequency range is practically possible in the frequency domain only. In the treatment of nonlinearities second-order stochastical linearization methods can be used advantageously. The amount of necessary computations can be vastly reduced by using known receptance matrices of all bodies in the system in its equations of motion after Fourier transformation. The article describes the receptance properties of non-rigid bodies, their use for the determination of the dynamic system's transfer functions and the determination of the output processes' spectral density matrix under stationary random external excitation using stochastical second-order linearization of the nonlinearities.     

Random Vehicle Vibrations as Effected by Dry Friction in Wheel Suspensions

SUMMARY

The changes of vibration comfort and stability of vehicles as effected by the value of frictional force generated in laminated springs are discussed and the probability of departure of the tyres from the ground is described.

A partly automated analogue computer simulation being equivalent to a great number of highway experiments was applied in the study.

The computer model was excited by several stochastic roadprofile - analogue signals generated by filters from the wide-band random signal of a digital noise generator.     

An Analytical and Experimental Investigation of the Driver-Seat-Suspension System

SUMMARY

Vertical seat-suspension systems are characterized by a generalized two- degree-of-freedom model incorporating nonlinearities due to shock absorber damping, linkage friction and bump stops. The analytical model is validated using the results obtained from laboratory tests performed under sinusoidal excitations in the 0.5-8.0 Hz frequency range. Human body models of varying complexities, derived from the mechanical impedance data, are discussed and integrated to the nonlinear seat-suspension model to derive a coupled driver- seat-suspension model. Nonlinear analytical models are expressed by their linear equivalent models using a local equivalent linearization technique based on energy similarity. The vibration attenuation performance characteristics of the seat-suspension and driver-seat-suspension models are investigated for deterministic and random cab floor excitations. The results of the study revealed that the seated human body contributes considerably to the overall ride performance.     

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.     

Optimization of Vehicles Elasto-Damping Elements Characteristics from the Aspect of Ride Comfort

SUMMARY

Spatial random vibrations of a vehicle that arise during driving represent an important factor in functioning of a dynamic system: Driver - Vehicle - Environment. They carry certain information for driver and also cause fatigue of driver and passenger.

This is the reason why the tendency is towards the minimization of vibratory loads, what in practice can be achieved by optimization of characteristics of elasto - damping elements of a vehicle.

In this paper for optimization of elasto - damping elements of a vehicle we used a complex nonlinear model of a driver and a vehicle during the straight - line motion of the vehicle on a rough road. Optimization was performed by application of the Hooke - Jeeves method and by use of outside penalty functions as well as the objective function that enabled simultaneous optimization of vertical vibrations of the driver's seat, vibrations of the steering wheel, and normal reactions in the contact surface of the tyre and road. The optimization was performed with application of the computer HP 9000/800 SE on the example of a medium passenger car.     

Damper Models for Heavy Vehicle Ride Dynamics

SUMMARY

A laboratory rig for testing hydraulic dampers using the ‘hardware-in-the-loop’ method is described, and the accuracy of the test method is investigated. A mathematical model of a hydraulic shock absorber is then developed. The model is suitable for vehicle simulations and has seven parameters which can be determined by simple dynamic measurements on a test damper. The shock absorber model is validated under realistic operating conditions using the test rig, and the relative importance of various features of the model on the accuracy of vehicle simulations is investigated.     

Bond Graphs for Vehicle Dynamics

SUMMARY

For the purposes of analyzing the stability and control properties of aircraft, automobiles, and ships, it is convenient to idealize the vehicle as a rigid body and to use a body-centered coordinate frame. A six-port bond graph is first developed which represents the nonlinear dynamics of a rigid body in a coordinate system rotating with the body. It is then shown how the graph simplifies if only lateral or longitudinal dynamics are studied. Finally, bond graphs for the type of stability studies in which the assumption of small perturbations permits linearization are shown for elementary models of automobiles and aircraft.     

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.     

Optimisation of driver actions in RWD race car including tyre thermodynamics

ABSTRACT

The paper presents an innovative method for a lap time minimisation by using genetic algorithms for a multi objective optimisation of a race driver–vehicle model. The decision variables consist of 16 parameters responsible for actions of a professional driver (e.g. time traces for brake, accelerator and steering wheel) on a race track part with RH corner. Purpose-built, high fidelity, multibody vehicle model (called ‘miMa’) is described by 30 generalised coordinates and 440 parameters, crucial in motorsport. Focus is put on modelling of the tyre tread thermodynamics and its influence on race vehicle dynamics. Numerical example considers a Rear Wheel Drive BMW E36 prepared for track day events. In order to improve the section lap time (by 5%) and corner exit velocity (by 4%) a few different driving strategies are found depending on thermal conditions of semi-slick tyres. The process of the race driver adaptation to initially cold or hot tyres is explained.     

Computer-Aided Analysis of Urban Railway Vehicles

SUMMARY

Computer-aided dynamic simulations are usually employed when designing modern urban railway vehicles. Even if the modeling procedure is similar to the one used for trains, specific features have to be taken into account for tramways: they are designed for low speeds (less than 80 km/h) and narrow curves (less than 20 m of radius). Moreover, in order to improve accessibility, low floor designs have been developed (the floor lying at about 300 mm above the rails level). The simulation procedure has therefore to take account of the occurence of multiple wheel/rail contacts or the modelization of independent wheels. A specific software well adapted to the computer-aided design of urban railway vehicles has been developed by the Faculte Polytechnique de Mons. It performs the following classical analyses:

lateral linearization, modal analysis and root locii plots;

vertical linearization and comfort prediction;

non-linear time simulation in straight track (limit cycles) and in curve (derailment study)

parametric analyses

The vehicle model is formed by combination of bodies ( or flexible bodies, rotating bodies like wheelsets or independent wheels) and interconnection elements ( spring and damper elements). Contact between rails and wheels is treated as a part of the rotating bodies. A residual formulation has been preferred. When combined with the use of a complete iteration matrix, this formulation is well adapted to the treatment of stiff differential equations. It is based on a fast determination of the residues of the dynamic equations combined with the calculation of the iteration matrix through a numerical derivation procedure. The advantages of the approach are discussed. The model of a partial low floor vehicle with wheelsets and independent wheels is described.     

Random Response of Tractor-Semitrailer System

SUMMARY

This work describes an analytical study of the dynamic behaviour of a tractor-semitrailer vehicle. A digital computer simulation was used to describe the longitudinal, vertical, and pitching motions of the vehicle travelling over a stationary random road surface. A man-seat model was also incorporated into the simulation. Vehicle response to road irregularities has been studied by assuming two different roads for loaded and unloaded cases.

Numerical results are presented for vehicle, showing system eigenvalues, power spectral densities and root mean square values of the linear and angular accelerations and displacements. Vehicle acceleration response is compared with the ISO riding comfort standard. All results for the loaded and unloaded cases and for smooth and rough roads indicated that an uncomfortable ride would result from vehicle response.     

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.     

An Investigation into Stick-Slip Vibrations on Vehicle/Track Systems

SUMMARY

A model for the numerical simulation of vehicle/track interaction and stick-slip vibration is presented. A finite element model is developed to calculate vertical contact forces. These forces are then coupled through the contact patch into a non-linear time-domain model by which the stick-slip vibration behaviour of a wheel-rail system is analysed. The investigation suggests that stick-slip vibration may occur if a vehicle which has a maligned or an initial ‘wind-up’ wheeiset meets a vertical irregularity or contaminants on the track.     

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.     

Dynamic Response of a Six-axle Locomotive to Random Track Inputs

SUMMARY

Spectral analysis techniques are employed to analyze the dynamic response of a six-axle locomotive on tangent track to vertical and lateral random track irregularities. The locomotive is represented by a thirty-nine (39) degrees of freedom model. A linear model is employed by considering small displacements, linear suspension elements and a linear theory for the wheel-rail interaction. Power spectral densities of displacements, velocities and accelerations and the statistical average frequencies of the system are obtained for each degree of freedom. Comparison of the calculated dominating frequencies with existing experimental values shows good agreement. The technique of spectral analysis is an effective tool for model validation, and for the determination of rail vehicle response to track irregularities. The probability functions for the response can be used as a measure for the ride quality of rail vehicles and for the study of fatigue damage of components.

     

Modified Second Order Linearization Procedure - Problems Encountered and Their Solution

A modified second order linearization method for solving stationary stochastic vibrations of nonlinear dynamic systems in the frequency domain was described by the author in 1986. Some problems with its use for lightly damped systems containing springs with cubic characteristics were encountered in the praxis. This paper deals with their cause and describes the procedure to be used in this case.     

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.     

Linearization of Non-Linear Stochastically Excited Dynamic Systems by a Modified Second-Order Procedure

An enhanced second-order linearization method for linearization of non-linear dynamic systems excited by stationary stochastic input processes is outlined in the paper. Noise force components originating when deflecting any non-linear two-force element are used as internal excitations of the dynamic system considered. They are noncoherent mutually and with all external excitations. This enables a better approximation of the system's responses under given external excitation as well as a reasonably accurate determination of the coherences of all processes taking place in the system.