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

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.     

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.     

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.     

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.     

Effect of obstacles on roads with different waviness values on the vehicle response

Abstract

The effect of the increasing number of cosine-form obstacles, which are superposed on a standard homogeneous longitudinal road profile, is studied when the waviness of the basic road profile changes in the range from 1.5 to 3.5. First, the effect of the composed profile on the power spectral density (PSD) is analysed. It appears that the primary PSD in the form of a straight-line in the log–log plot breaks into two straight-line partial segments with quite different waviness values in the long and short wavelength bands. This seems to be a suitable model of a highly deteriorated road. Next, the effect on the response of two typical vehicles, viz. a personal car and a three-axle truck, is studied, and rather different results were obtained. In a summary valuation, it appears that the presence of additional obstacles causes independence of some of the vehicle response quantities of the primary waviness.     

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.     

Roadway Elevation Profile Generation for Vehicle Simulation

SUMMARY

With a simplified approach for creating road surface elevation information for simulation of vehicle vertical response to roadway unevenness, roadways for single and parallel track simulations and averaged roads for variable velocity simulation are developed. Sets of correctly chosen random roadway slopes are averaged appropriately for the variable velocity simulation. The procedure generates approximately “white” slope spectral density roadways in the frequency ranges of interest, and the elevation profiles are representative of average road profiles. The method is simple in practice yet suffices for many parameter studies of suspensions and vehicle dynamics.     

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.     

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.     

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.     

An Active Suspension with Optimal Linear State Feedback

SUMMARY

In this paper modern optimal control theory is applied to the design of an active suspension system for a motor vehicle. The road profile is assumed to be continuous and random with a power spectral density (P.S.D.) which varies inversely with the square of the frequency. The quadratic integral type performance index employed is a weighted sum of the integral squares of body acceleration, dynamic tyre deflection and relative body-to-axle displacement. A solution is obtained for the infinite time case which is both computationally and physically realizable as an active suspension in which the only continuous measurements required are the body absolute velocity and the body displacement relative to the road. The performance is compared with that of a conventional type passive suspension and found to be significantly better in practically all respects.     

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.     

Power Spectral Density of Road Profiles

The road profile is usually considered to be a random process x ( d ), where x is the road height and d is the distance along the road. As the vehicle travels along the road with velocity v , the random process x ( d ) is converted to a random process x ( t ) which is input to the vehicle suspension via the tyre. The random process x ( d ) is usually described in terms of its power spectral density as a function of frequency in either radians or cycles per unit distance. However, there are several different ways of defining power spectral density, and this makes it difficult to compare published data without knowing how the power spectral density has been defined. The proper calculation of RMS values of vehicle response for an assumed road power spectral density is explained by an example.     

Power Spectral Density of Road Profiles

The road profile is usually considered to be a random process x (d), where x is the road height and d is the distance along the road. As the vehicle travels along the road with velocity v, the random process x (d) is converted to a random process x (t) which is input to the vehicle suspension via the tyre. The random process x (d) is usually described in terms of its power spectral density as a function of frequency in either radians or cycles per unit distance. However, there are several different ways of defining power spectral density, and this makes it difficult to compare published data without knowing how the power spectral density has been defined. The proper calculation of RMS values of vehicle response for an assumed road power spectral density is explained by an example.     

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.     

The Random Vibration of a Nonuniform Cantilever Beam with Concentrated Mass

SUMMARY

The random vibration of a nonuniform cantilever beam with a concentrated mass at its end, is studied as a simple model of the suspension spring of a motor vehicle. The spectral response of the displacement of the concentrated mass is obtained when the beam is subjected to the random motion of the support. The conditions for the validity of the single degree approximation is also discussed.     

Comparison of Ride Comfort Criteria for Computer Optimization of Vehicles Travelling on Randomly Profiled Roads

SUMMARY

This paper discusses and compares some ride comfort criteria which may be suitable for randomly vibrating vehicles. The criteria consider single-figure measures based on response mean square spectral densities for plane linear combined vehicle-passenger models. Eight different measures divided into three groups are studied. The two vehicle suspension damper stiffnesses are numerically optimized with respect to the eight measures of ride comfort. The results are compared and discussed. Two optimizations with respect to five vehicle parameters are also reported.     

Road roughness estimation based on discrete Kalman filter with unknown input

ABSTRACT

The road roughness acts as a disturbance input to the vehicle dynamics, and causes undesirable vibrations associated with the ride and handing characteristics. Furthermore, the accurate measurement of road roughness plays a key role in better understanding a vehicle dynamic behaviour and active suspension control systems. However, the direct measurement by laser profilometer or other distance sensors are not trivial due to technical and economic issues. This study proposes a new road roughness estimation method by using the discrete Kalman filter with unknown input (DKF-UI). This algorithm is built on a quarter-car model and uses the measurements of the wheel stroke (suspension deflection), and the acceleration of the sprung mass and unsprung mass. The estimation results are compared to the measurements by laser profilometer in-vehicle test.     

Optimal Preview Control of a Two-dof Vehicle Model Using Stochastic Optimal Control Theory

An optimal preview control algorithm is applied to a two degree of freedom(dof) vehicle model travelling with constant velocity on a randomly profiled road. The road roughness is modelled as a homogeneous random process being the output of a linear first order filter to white noise. The input from the road irregularity is assumed to be measured at some distance in front of the vehicle and this measured infonnation is utilized by the active controller to prepare the system for the ensuing input. The preview control algorithm is obtained by minimizing a quadratic performance index and by describing the average behaviour of the system by the covariance matrix of the vehicle response state vector. Results are presented for full state feedback and significant improvements in sprung mass acceleration, suspension working space and road holding are observed.     

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.