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'.     

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.     

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’.     

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.     

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.     

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.     

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

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.     

High Frequency Characteristics of a Front Wheel Suspension

This paper presents a modeling method to analyze the frequency characteristics of a suspension system. The method employs the superelement and lumped methods for structural modeling of flexible bodies, and it is incorporated with a multibidy dynamics formulation in which joint nonlinearities can be considered effectively. To verify the accuracy of the results obtained by the modeling method, a quarter car test was made. Test results showed that the computational model provided fairly accurate frequency characteristics for the suspension system. Other useful conclusions were also drawn from this study.     

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.     

High Frequency Characteristics of a Front Wheel Suspension

SUMMARY

This paper presents a modeling method to analyze the frequency characteristics of a suspension system. The method employs the superelement and lumped methods for structural modeling of flexible bodies, and it is incorporated with a multibidy dynamics formulation in which joint nonlinearities can be considered effectively. To verify the accuracy of the results obtained by the modeling method, a quarter car test was made. Test results showed that the computational model provided fairly accurate frequency characteristics for the suspension system. Other useful conclusions were also drawn from this study.     

In-plane Dynamics of Tires on the Road Based on an Experimentally Verified Rolling Ring Model

The dynamic behavior of tires rolling on the road is predicted based on a ring model that is experimentally verified to well represent a real tire. The road contact is kinematically represented by two displacement constraints, one in radial and one in tangential direction. The displacement constraints are enforced via the receptance theory. The receptance matrix developed for the rolling tire is found to be Hermitian. Numerical results show that the natural (or eigen-) frequencies of the rolling tire with road contact are, as expected, higher than those of the tire with no contact. The influence of the rotating speed to the eigen-frquencies of the tire is also shown. Analytic expressions for the mode shapes of a rolling tire with ground contact are derived.     

Almost disturbance decoupling control of mimo nonlinear system subject to feedback linearization and a feedforward neural network: Application to half-car active suspension system

A novel tracking and almost disturbance decoupling problem of multi-input, multi-output (MIMO) nonlinear systems based on feedback linearization and a multi-layered feedforward neural network approach has been proposed. The feedback linearization and neural network controller guarantees exponentially global uniform ultimate bounded stability and almost disturbance decoupling performance without using any learning or adaptive algorithms. The new approach renders the system to be stable with the almost disturbance decoupling property at each step when selecting weights to enhance the performance if the proposed sufficient conditions are maintained. One example, which cannot be solved by the existing approach of the almost disturbance decoupling problem because it requires the sufficient conditions that the nonlinearities that multiply the disturbances satisfy structural triangular conditions, is proposed to exploit the fact that the tracking and the almost disturbance decoupling performances are easily achieved by the proposed approach. In order to demonstrate the practical applicability, a famous half-car active suspension system is investigated.     

In-plane Dynamics of Tires on the Road Based on an Experimentally Verified Rolling Ring Model

SUMMARY

The dynamic behavior of tires rolling on the road is predicted based on a ring model that is experimentally verified to well represent a real tire. The road contact is kinematically represented by two displacement constraints, one in radial and one in tangential direction. The displacement constraints are enforced via the receptance theory. The receptance matrix developed for the rolling tire is found to be Hermitian. Numerical results show that the natural (or eigen-) frequencies of the rolling tire with road contact are, as expected, higher than those of the tire with no contact. The influence of the rotating speed to the eigen-frquencies of the tire is also shown. Analytic expressions for the mode shapes of a rolling tire with ground contact are derived.     

Simulation of vertical dynamic vehicle–track interaction using a two-dimensional slab track model

The vertical dynamic interaction between a railway vehicle and a slab track is simulated in the time domain using an extended state-space vector approach in combination with a complex-valued modal superposition technique for the linear, time-invariant and two-dimensional track model. Wheel–rail contact forces, bending moments in the concrete panel and load distributions on the supporting foundation are evaluated. Two generic slab track models including one or two layers of concrete slabs are presented. The upper layer containing the discrete slab panels is described by decoupled beams of finite length, while the lower layer is a continuous beam. Both the rail and concrete layers are modelled using Rayleigh–Timoshenko beam theory. Rail receptances for the two slab track models are compared with the receptance of a traditional ballasted track. The described procedure is demonstrated by two application examples involving: (i) the periodic response due to the rail seat passing frequency as influenced by the vehicle speed and a foundation stiffness gradient and (ii) the transient response due to a local rail irregularity (dipped welded joint).     

High-frequency random vibration analysis of a high-speed vehicle–track system with the frequency-dependent dynamic properties of rail pads using a hybrid SEM–SM method

A hybrid Spectral Element Method (SEM)–Symplectic Method(SM) method for high-efficiency computation of the high-frequency random vibrations of a high-speed vehicle–track system with the frequency-dependent dynamic properties of rail pads is presented. First, the Williams-Landel-Ferry (WLF) formula and Fractional Derivative Zener (FDZ) model were, respectively, applied for prediction and representation of the frequency-dependent dynamic properties of Vossloh 300 rail pads frequently used in China's high-speed railway. Then, the proposed hybrid SEM–SM method was used to investigate the influence of the frequency-dependent dynamic performance of Vossloh 300 rail pads on the high-frequency random vibrations of high-speed vehicle–track systems at various train speeds or different levels of rail surface roughness. The experimental results indicate that the storage stiffness and loss factors of Vossloh 300 rail pad increase with the decrease in dynamic loads or the increase in preloads within 0.1–10,000?Hz at 20°C, and basically linearly increase with frequency in a logarithmic coordinate system. The results computed by the hybrid SEM–SM method demonstrate that the frequency-dependent viscous damping of Vossloh 300 rail pads, compared with its constant viscous damping and frequency-dependent stiffness, has a much more conspicuous influence on the medium-frequency (i.e. 20–63?Hz) random vibrations of car bodies and rail fasteners, and on the mid- (i.e. 20–63?Hz) and high-frequency (i.e. 630–1250?Hz) random vibrations of bogies, wheels and rails, especially with the increase in train speeds or the deterioration of rail surface roughness. The two sensitive frequency bands can also be validated by frequency response function (FRF) analysis of the proposed infinite rail–fastener model. The mid and high frequencies influenced by the frequency-dependent viscous damping of rail pads are exactly the dominant frequencies of ground vibration acceleration and wheel rolling noise caused by high-speed railways, respectively. Even though the existing time-domain (or frequency-domain) finite track models associated with the time-domain (or frequency-domain) fractional derivative viscoelastic (FDV) models of rail pads can also be used to reach the same conclusions, the hybrid SEM–SM method in which only one element is required to compute the high-order vibration modes of infinite rail is more appropriate for high-efficiency analysis of the high-frequency random vibrations of high-speed vehicle–track systems.     

路面随机激励下的汽车非线性悬架系统振动分析

基于滤波白噪声生成法模拟了路面不平度,建立了四分之一车辆悬架系统非线性模型,在此基础上利用统计线性化方法对汽车非线性随机振动时域响应进行了研究,并利用MATLAB／Simulink软件对悬架振动进行了数字仿真。计算结果表明,用统计线性化方法求解汽车悬架系统的非线性随机振动问题是可行而且可靠的,仿真输出量可作为评价汽车平顺性的依据。     

斜拉索面内随机参数振动分析

为探讨斜拉索面内承受端部轴向零均值高斯白噪声位移激励下的随机振动特性,推导了基于伊藤方程标准形的斜拉桥拉索平面内索端承受轴向随机位移激励下的随机参数振动微分方程。利用Falsone改进随机线性化法、一般随机线性化法及Mont carlo数值模拟法求解了拉索振动状态向量的二阶矩均方响应。研究表明：改进随机线性化法与Mont carlo数值模拟法结果吻合较好,一般随机线性化法降低了拉索的振动响应。拉索横向均方响应随着激励谱密度强度的增大而增大,随着拉索阻尼比及拉索初始索力的增大而减小。     

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.     

Estimating the ride quality characteristics of vehicles with random decrement analysis of on-the-road vibration response data

This paper describes the application of a practical analytical technique based on the random decrement method to estimate the rigid sprung mass dynamic characteristics (frequency response function) of road vehicles using only vibration response data during constant-speed operation. A brief history and development of the random decrement technique is presented, along with a summary of work undertaken on optimal parameter selection to establish the random decrement signature. Two approaches to estimate the dynamic characteristics from the random decrement signature are described and evaluated. A custom, single-wheeled vehicle (physical quarter car) was commissioned to undertake a series of on-the-road experiments at various nominally constant operating speeds. The vehicle, also instrumented as an inertial profilometer, simultaneously measured the longitudinal pavement profile to establish the vehicle's actual dynamic characteristics during operation. The main outcome of the paper is that the random decrement technique can be used to provide accurate estimates of the sprung mass mode of the vehicle's dynamic characteristics for both linear and nonlinear suspension systems of an idealised vehicle.     

A new parameter identification methodology for a modal analysis test of a rail vehicle

The validation of vehicle mathematical models is a key part of the virtual acceptance process since it is essential to ensure a precise representation of the reality. The model validation procedure should include validation of stationary but also dynamic tests. However, parameter identification from on-track tests is a challenging task due to the non-controlled excitation and the great variability of the test results. Thus, an alternative solution by means of a vehicle modal analysis is proposed, developing a parameter identification methodology for dynamic vehicle model parameters. This methodology calculates estimated values of the vehicle model parameters that have an influence on the excited vehicle vibration modes. Moreover, a new criterion for taking into account the effect of the measurement uncertainties on the selection process of the vehicle parameters is developed. Finally, experimental results show that not only estimations of the suspension stiffness parameters can be obtained, but damping values and structural frequencies from the vehicle bodies can also be estimated.