Non-smooth Problems in Vehicle Systems Dynamics

The vehicle systems are modelled mathematically as parameter dependent multi-body systems. The connections between the elements are formulated either as dynamical equations or algebraic, or transcendental or tabulated constraint relations. The connections can rarely be modelled by analytic functions, and the missing analyticity can arise from non-uniqueness or discontinuities in the functions themselves or in their derivatives of any order. In vehicle systems the contact between the vehicle and its support (road or rail) is an important source of missing analyticity. The suspension systems of the vehicles consist of passive and active elements such as springs, dampers and actuators, and their characteristics are only analytic functions within certain intervals of operation. Unilateral contacts in the suspension systems may give rise to changes of the degrees of freedom of the system during operation, and cause impacts or sliding contact during the operation.

This paper deals with the use of nonlinear calculations and bifurcation analysis when investigating running stability during vehicle design and development in the rolling stock industry. Typical methods used for stability analysis in industrial applications are introduced, computation of bifurcation diagram presented and the influence of nonlinearities of the vehicle/track system on the type of Hopf bifurcation investigated. The relationship between the bifurcation diagram and the assessment of safety risk and the dynamic behaviour is discussed.

A closed-form, analytical study that evaluates the response characteristics of a two-degree-of freedom quarter-car model using passive and semi-active dampers is provided as an extension to the results published in past studies for active suspensions. The semiactive methods that are considered include skyhook, groundhook, and hybrid control. The relationships between vibration isolation, suspension deflection, and road-holding are studied using the model. The performance indices that are used include vertical acceleration of the sprung mass, deflection of the suspension (rattle space), and deflection of the tire. The results indicate that hybrid semiactive control yields better comfort than passive suspension and the other semiactive control methods that are considered, without reducing the road-holding quality or requiring larger suspension rattle space for typical passenger cars.

Given a railway vehicle and a track, is it possible to define a wheel limit profile that allows the vehicle to run at a given speed without the occurrence of unsafe hunting motion? Nowadays some railway operators and maintenance staff identify the maximum admissible wear on the profiles comparing the rail equivalent conicity with a limit curve obtained on the basis of experimental evidence. The present paper analyses the relationship between equivalent conicity and the dynamic behaviour of the vehicle in tangent track. This can be used to implement condition – based rail reprofiling strategies.

Due to the relevant costs of rail re-profiling in underground metro lines, stake holders are available to consider alternative solutions. In the present research, a low-cost maintenance operation for avoiding hunting instability occurring on a section of a metro line has been proposed and experimentally verified.

In recent years, railway noise has become a high profile issue and the industry has come under pressure to reduce operating noise because legislative rules concerning noise emission will be tightened in the near future. Especially for freight traffic there is a need to improve, i.e. to reduce, the noise generation of rolling stock.

Curve squealing of railway wheels occurs erratically in tight curves with a frequency of about 4 kHz. Squealing is caused by a self-excited stickslip oscillation in the wheel-rail contact. The mechanism which activates squeal is still unexplained and will be analyzed in the paper at hand. By starting with a modal model of the elastic wheel equipped with a threedimensional hard Coulomb contact, a stability analysis of the stationary run through a curve is performed for the four wheels of the investigated bogie. The results show that in particular the front inner wheel tends to squeal. A numerical simulation performed on the unstable states shows the existence of a self-excited stick-slip oscillation with a frequency that compares well with the ones measured at squeal.

The contribution presents four problems connected with the Colloquium topics. They are of theoretical nature in the wider aspect but all refer to the results of practical simulations of rail vehicle motion in a curved track. The first two can be counted to typical non-smooth problems in rail vehicle dynamics. They refer to two-point contact modelling and vehicle stability in curves, respectively. The third problem is closely related to the second one. It refers to railway vehicle dynamics above critical velocity in transition curves. The forth one can also be recognized as the non-smooth problem but not so typical one. It refers to modelling the kinematics in a curved track. All the problems reveal that one needs to be very careful when studying rail vehicle dynamics and trying to simplify its non-linearities at the same time. Non-smooth functions can influence results of numerical analysis strongly, being the features of the physical and modelling nature.

The basis for the analysis of friction in brake systems is the brake pad’s tribological interface. An investigation of this interface reveals friction intensive surface structures. These so-called “patches” are extremely hard and carry the main part of the friction power. By complex interaction processes of wear and heat these patches are generated permanently but leave the system after a certain period of time. So there is an equilibrium of flow of contact patches in the brake pad interface, with the outcome being a dynamic macroscopic friction coefficient, whose “inertia” can be well described by differential equations in the form of special balance equations. Systematic expansions of these balance equations even allow, for the first time, a simulation of different test cycles of the AK-Master test for friction materials with high accuracy. These friction force variations are generated by the dynamics of the local surface geometry and can explain physically effects of measurements, which were up to now described by control theoretic approximations [7, 8].

Beside these effects the dynamics of friction is influenced by lateral vibrational dynamics of these patches on a very fast timescale. This timescale is so fast that processes of patch growth and destruction are negligible. Beyond that, the vibration frequencies of the patches, as well as the actual local friction power on each of these surface structures, vary over a wide range of values, which is the result of a great variety of patch sizes and heights in the interface. Generally, one would expect a smoothing of these local and stochastically distributed vibration effects. It can however be shown, that the oscillations of the patches are subject to synchronization processes, with the result being in-phase patch vibrations on macroscopic areas of the brake pad of significant size. Thereby, self-excited vibrations of the patches can lead to lateral oscillations of the pad’s friction force on a macroscopic scale. These are able to excite the whole system of brake pad and disk.

When modeling dry friction in mechanical elements various models are used to deal with the switch between stick and slip condition. When the standard numerical integration methods come to abrupt change in the system’s state, they start cutting down the integration step to locate the event that increases the required processor time. The paper proposes to deal with that by developing the dry friction element model in parallel with modification of numerical integration method.

This chapter presents a mathematical model of a railway hydraulic damper. The objective is to develop a model suitable to be implemented in a railway simulation program. Computational cost should, therefore, be maintained low not to decrease the simulation speed (rate) to unacceptable values.

A model based mainly in physical characteristics (piston section, volume of the chambers, characteristics of the valves, etc.) is developed. In a first part of the chapter, the modelling of each part of the damper is discussed. Afterwards, the model of the complete damper is constructed and finally, model results are compared satisfactorily with experimental tests.

The chapter also includes a discussion on the numerical problems associated with the developed model and a new simplified version is proposed to overcome the majority of the difficulties presented in the original model.

Vehicle manufacturers are constantly pushed to reduce the aerodynamic drag of vehicles, for example by constructing lower vehicles with less road clearance. This, however, reduces the available margin for oscillations within the suspension. If the oscillation amplitude exceeds a critical value, the suspension will impact a bumpstop. Under periodic excitation, the onset of low-velocity impacts is associated with a strong instability in favor of high-velocity impacts. Such impacts reduce comfort and could be damaging to the vehicle. Efforts should therefore be made to limit impact velocities with the bumpstop, for example by suppressing the instability associated with low-velocity impacts. This paper proposes a low-cost feedback-control strategy, based on making small adjustments to the position of the bumpstop, that serve to suppress the transition to high-velocity impacts with the bumpstop in the case of periodic excitation. The control law is derived from the theory of discontinuity maps. The results demonstrate that the feedback strategy works even when wheel-hop is present.

The lateral vibration of towed wheels, the so-called shimmy, is one of the most intricate phenomena of vehicle system dynamics. In this paper, a simple mechanical model of shimmy is constructed from a towed elastic wheel with a perfectly rigid suspension system. The brief presentation of the mathematical model of the system is followed by the detailed experimental analysis. The physical parameters of a carefully designed experimental rig are determined in different experimental setups. The most relevant stability boundary of the straight-line stationary rolling is calculated theoretically and then validated by experiments. The vibration frequencies of the system in different parameter domains, with special attention to the stable regions, are investigated by the modal analysis of the towed tyre during rolling.

A mechanical model of moderate complexity is presented for evaluating the normal and longitudinal contact force between a tyre of a vehicle and the road. The model consists of a stretched beam on an elastic foundation with elastic tread elements that can touch the road, through which the forces are transmitted. Three friction models are considered, with increasing complexity: the Coulomb model with different static and kinetic friction coefficient, a speed-dependent friction model, and a dynamic model with an internal state. Realistic steady-state tyre characteristics are found. For the more complex friction models, an alternation of regions with low sliding velocity and transition regions with high sliding velocity may be present.

Non–smooth models are often used for mechanical systems, for example to model contact or friction. When system parameters are varied, changes in dynamical behaviour may occur that are different from the standard bifurcations found in smooth systems. Obtaining a unified picture of such non-smooth bifurcations is difficult, because of the wide range of different types of discontinuities commonly encountered. Here we will present theorems concerning boundary induced bifurcations of equilibria and limit cycles in generic Filippov systems, and apply the theory to a model of a friction oscillator.

The phenomenon of vibration displacement means an occurrence of directed on average “slow” changing (particularly motion) due to undirected on average “fast” (vibration) effects.

According to results of carried-out experiments and simulations, dry friction exposed to dither results in viscous–like, frequently linear damping. This means that dither smoothes dry friction as far as damping is concerned. One dithered system of technical importance is a railway freight wagon with friction dampers in the primary suspension developing two–dimensional dry friction. The dither exciting dampers are generated in rolling contact of wheel and rail. Employing proposed rheological model of 2D friction in simulations it has been shown that dither significantly influences ride dynamics of freight wagons.

In large Diesel engines often geartrains are used to drive the camshafts. As the average transmitted load is small, dynamic loads, e.g. gas forces, are typically dominant. This can result in a rattling motion of teeth within the backlash, called gear hammering. We will show that for these impact-like contacts rigid body models are not sufficient, instead, elastic models are necessary to precisely simulate contact forces. The model proposed here is a modally reduced elastic multibody model including a contact algorithm. Due to the size of the modal transformation matrices, additional ways to reduce the computational effort are necessary, e.g. a dynamic reloading scheme. The model is robust and fast enough to allow simulations of many revolutions and many contacts. Furthermore, basic experiments have been carried out to validate the model. Experimental results are presented and agree very well with the simulations.

The occurrence of discontinuous right hand sides in ODE-systems often appears in technical applications. Such applications may be characterised by the cases where the system changes between several states. Each state is defined by a system of ODEs and the transition between states is defined by an algebraic condition. The numerical solution that is done in order to simulate the behaviour of the system will be possible by using standard numerical software but this approach is very ineffecient. We present an alternative approach based upon the tracking of state-changes and accurate numerical determination of transition points. Real applications from railway dynamcs are used to illustrate the approach.

The classical convergence analysis of higher order ODE time integration methods is based on rather strong smoothness assumptions on the right hand side that are typically not satisfied in technical applications since spline approximations of input functions and look-up tables result in frequent discontinuities in derivatives of the right hand side. Practical experience shows, however, that nevertheless the resulting non-smooth model equations may often be solved efficiently by higher order ODE time integration methods. For one typical problem class, the present paper gives a theoretical explanation of this behaviour. The results of the theoretical analysis are illustrated by a series of numerical tests for the simplified model of an agricultural device that moves along a track being defined by the spline approximation of a periodic smooth input function.

Discontinuous system modeling is a present topic when working with practical models of technical systems. Numerical algorithms can only handle models with a given structure of the discontinuity effects. In the paper we show some examples that motivate the investigation of extended problem classes. The sensitivity analysis of all the systems gives important information about the dependency of the model solution on model parameters like controller parameters. We discuss models with nonsmooth switching functions and models with several switching functions influencing the model dynamics at the same time.

Hardware-In-the-Loop (HIL) test rigs are more and more used to develop and/or improve modern control units. Due to the increasing complexity of these systems detailed vehicle models running in real-time are required. The main task in real time applications is to achieve a stable but still sufficiently accurate numerical solution under all operating conditions. Complex vehicles modeled by multi body systems result in stiff differential equations. In particular, taking into account the effects of dry friction requires sophisticated modeling techniques and simple but robust numerical solvers.