Hans True

Hans True
Technical University of Denmark | DTU · Department of Applied Mathematics and Computer Science

PhD

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89
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Introduction

Publications

Publications (89)
Article
In this note an explanation and the relevance of a couple of singularities in the mathematical models of rolling friction is presented.
Conference Paper
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Article
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In the paper, we investigate numerically the influence of dry friction yaw damping on the dynamics of a railway vehicle with four wheel sets. The speed of the vehicle is the control parameter. The stick/slip mechanism is considered. The first bifurcation points are found. The results demonstrate the influence of the stick/slip on the dynamics of th...
Conference Paper
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Article
An extended Dichotomy method is proposed for an accurate determination of the critical speed of railway vehicles. The high-dimensional Hermite orthogonal polynomial (HOP) based on the independent and normal distribution parameters is derived. Both the Monte Carlo (MC) method based on a Latin hypercube sampling (LHS) and the HOP expansion method usi...
Article
In this paper, we investigate the lateral dynamics of a railway wheelset suspended under a moving car with linear springs and dry friction dampers. Both theoretical and numerical methods are used to complement each other. The car runs on an ideal, straight and perfect track with a constant speed. A nonlinear relation between the creepages and the c...
Article
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Based on the theory of non-linear dynamic systems, the influence of the perturbation of the wheel rotation speed on the quasi-steady curved motions of a two-axle railway bogie system with a realistic wheel/rail contact relation is investigated in this paper. Since the wheel/rail contact relation is non-linear, it is tabulated in a wheel/rail contac...
Article
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In this paper, a comprehensive analysis is presented to investigate a codimension two bifurcation that exists in a nonlinear railway bogie dynamic system combining theoretical analysis with numerical investigation. By using the running velocity V and the primary longitudinal stiffness K1x as bifurcation parameters the first and second Lyapunov coef...
Article
In this article a theoretical investigation of the dynamics of a railway bogie running on a tangent track with a periodic disturbance of the lateral track geometry is presented. The dynamics is computed for two values of the speed of the vehicle in combination with different values of the wavelength and amplitude of the disturbance. Depending on th...
Conference Paper
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Several attempts of measuring the exact location of the rails by the use of ordinary vehicles have been made. While the method works reasonably well in the vertical direction, the results of the lateral measurements made with different vehicles are so widely scattered that it is virtually impossible to draw any conclusions. We may therefore ask: do...
Article
The lateral dynamic features of a railway vehicle are investigated using two similar wheel/rail contact models: the Vermeulen-Johnson and the Shen-Hedrick-Elkins models. The symmetric/asymmetric bifurcation behaviour and chaotic motions of the railway vehicle are investigated in great detail by varying the speed and using the ‘resultant bifurcation...
Conference Paper
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This work addresses the problem of the reliability of simulations for realistic nonlinear systems, by using efficient techniques for the analysis of the propagation of the uncertainties of the model parameters through the dynamics of the system. We present the sensitivity analysis of the critical speed of a railway vehicle with respect to its suspe...
Article
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We present an approach to global sensitivity analysis aiming at the reduction of its computational cost without compromising the results. The method is based on sampling methods, cubature rules, high-dimensional model representation and total sensitivity indices. It is applied to a half car with a two-axle Cooperrider bogie, in order to study the s...
Article
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The paper contains a report of the experiences with numerical analyses of railway vehicle dynamical systems, which all are nonlinear, non-smooth and stiff high-dimensional systems. Some results are shown, but the emphasis is on the numerical methods of solution and lessons learned. But for two examples the dynamical problems are formulated as syste...
Article
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This paper deals with the study of the nonlinear dynamic behaviour of 2-axle freight wagons in curves, considering the case of one single wagon (neglecting inter-car coupling forces) and of multiple wagons interacting through the buffers and the couplers. A multi-body model of a single wagon and of a three-car assembly is introduced, paying particu...
Conference Paper
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This paper describes the results of the application of Uncertainty Quantification methods to a simple railroad vehicle dynamical example. Uncertainty Quantification methods take the probability distribution of the system parameters that stems from the parameter tolerances into account in the result. In this paper the methods are applied to a low-di...
Article
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In recent years, several authors have proposed ‘easier numerical methods’ to find the critical speed in railway dynamical problems. Actually, the methods do function in some cases, but in most cases it is really a gamble. In this article, the methods are discussed and the pros and contras are commented upon. I also address the questions when a line...
Article
Based on the bifurcation and stability theory of dynamical systems, the symmetric/asymmetric bifurcation behaviours and chaotic motions of a railway bogie system under a complex nonlinear wheel–rail contact relation are investigated in detail by the ‘resultant bifurcation diagram’ method with slowly increasing and decreasing speed. It is found that...
Article
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The paper describes the results of the application of "Uncertainty Quantification" methods in railway vehicle dynamics. The system parameters are given by probability distributions. The results of the application of the Monte-Carlo and generalized Polynomial Chaos methods to a simple bogie model will be discussed.
Conference Paper
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We present an approach to global sensitivity analysis aiming at the reduction of its computational cost without compromising the results. The method is based on sampling methods, cubature rules, High-Dimensional Model Representation and Total Sensitivity Indices. The approach has a general applicability in many engineering fields and does not requi...
Article
The paper describes the results of the application of "Uncertainty Quantification" methods in railway vehicle dynamics. The system parameters are given by probability distributions. The results of the application of the Monte-Carlo and generalized Polynomial Chaos methods to a simple bogie model will be discussed.
Conference Paper
Full-text available
This paper describes the results of the application of Uncertainty Quantification methods to a railway vehicle dynamical example. Uncertainty Quantification methods take the probability distribution of the system parameters that stems from the parameter tolerances into account in the result. In this paper the methods are applied to a low-dimensiona...
Article
Full-text available
In recent years several authors have proposed, "easier" numerical methods' to find the critical speed in railway dynamical problems. Actually the methods do function in some cases, but in most cases it is really a gamble. In this presentation the methods will be discussed and the pros and contras commented. We shall also address the questions when...
Chapter
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-u...
Book
The book combines vehicle systems dynamics with the latest theoretical developments in dynamics of non-smooth systems and numerical analysis of differential-algebraic dynamical systems with discontinuities. These two fields are fundamental for the modelling and analysis of vehicle dynamical sytems. The results are also applicable to other non-smoot...
Chapter
In these notes the fundamentals of the mechanics of rail/wheel contact and deterministic vehicle dynamics is explained. Chapter 1 describes the kinematics and dynamics of rail/wheel contact. Chapter 2 explains why vehicle dynamics must be treated as a nonlinear dynamic problem and how the model problem must be formulated. Chapters 3 and 4 deal with...
Conference Paper
Full-text available
Dry friction dampers have been used in car constructions for several hundred years, and are still extensively used by the railways today. The main reason is that they are much cheaper than hydraulic dampers and more rugged. Their disadvantages are that their function is variable and depends on weather conditions and their state of contamination (di...
Article
The dynamics of two-axle railway freight wagons with the UIC standard suspension is investigated theoretically and the dynamic behaviour is explained. Fully nonlinear models are considered. The hysteresis from dry friction and the effect of impacts between elements of the suspension are included. Different wheel–rail geometries are investigated and...
Article
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According to UIC the equivalent conicity has a decisive role in the assessment of the dynamics of a railway vehicle. The validity of ,decisive' will be contested in this article. After an introductory repetition of known properties of nonlinear dynamic systems the lack of mathematical basis for the equivalent conicity is presented. Examples of earl...
Article
We present a part of Mark Hoffmann's Ph.D. thesis [1] on railway vehicle dynamics. The dynamic problem is non-linear and non-smooth. It is formulated as a multi-body system. Due to its high number of degrees of freedom the problem is analysed numerically taking into account the newest experimental and theoretical results of investigations of non-sm...
Article
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Article
The classical theory of non-linear dynamics treats systems of 'sufficiently smooth' non-linear ordinary differential equations on the form: dx/dt - F(x,t;λ) with appropriate initial conditions. Vehicle dynamic problems have constraints, which are expressed as non-linear algebraic and/or transcendental equations. In addition the vehicle models are n...
Article
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The dynamics of two different two-axle railway freight wagons is investigated theoretically and compared. Fully nonlinear models are considered. The hysteresis from dry friction and the effect of impacts between elements of the suspension are included. Bifurcation diagrams are shown in order to describe the eigen-dynamics of the wagons. Finally, th...
Conference Paper
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We examine three aspects of the dynamics of the Cooperrider truck travelling in a curve with constant radius. First the critical speed is found. Second we investigate the existence of multiple steady solutions to the curving problem. Third - and it is related to the second problem - we examine the position of the truck frame and the wheelsets durin...
Chapter
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We consider a simple model of a wheelset that supports one end of a railway freight wagon by springs with linear characteristics and dry friction dampers. We extend our earlier results to more realistic models, so in this presentation the linear kinematic contact relation in an earlier paper [True and Asmund, 2002] is replaced by the nonlinear rail...
Conference Paper
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In the paper we present the three most common European standard freight wagon suspensions. It is characteristic for the European suspensions that they are all primary suspensions without a bolster. The design and the function of their single elements are described. New results on the nature of dry friction and its influence on the damping character...
Article
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In this article the concept 'critical speed' is examined in the mathematical context of nonlinear dynamics. It is argued that the search for the critical speed must be formulated as a problem of existence and uniqueness of solutions of the dynamical problem rather than as a stability problem. If that is done properly, then the calculated and measur...
Conference Paper
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Although the three-piece-freight truck is a simple design its mathematical model is very complicated. The model is definitely a nonlinear dynamical system, where the nonlinearities arise from the nonlinear kinematic and dynamical contact relations between wheels and rails, the suspensions and the nonlinear dry friction damping with hysteresis and s...
Conference Paper
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We investigate the dynamics of a single-axle bogie under a freight wagon running with constant speed on an ideal, straight and horizontal track. Only dry friction dampers are used. Earlier investigations of a simple version showed that the bogie derails at nearly all speeds. In this work more and more realistic features are introduced in the model...
Article
Full-text available
We investigate the dynamics of a simple model of a wheelset that supports one end of a railway freight wagon by springs with linear characteristics and dry friction dampers. The wagon runs on an ideal, straight and level track with constant speed. The lateral dynamics in dependence on the speed is examined. We have included stick-slip and hysteresi...
Technical Report
Full-text available
The mathematical model is based upon Cooperrider's bogie running through a curve in a quasi-stationary state. The stresses are calculated by Hertz's theory with penetration and Shen-Hedrick-Elkin's theory. The effects of curve radius, speed and superelevation on especially the angle of attack of the wheel sets, the critical speed and the Hopf bifur...
Article
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We investigate the vehicle-track dynamics on a section of a Danish railway line that leads up to and over a concrete bridge. On the bridge the track is supported by concrete sleepers, which are surrounded by rubber hoses and glued on to the concrete bridge. We have simulated the dynamics caused by a DSB electric locomotive type EA of 20 t. nominal...
Article
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We investigate Cooperrider's complex bogie, a mathematical model of a railway bogie running on an ideal straight track. The speed of the bogie v is the control parameter. Taking symmetry into account, we find that the generic bifurcations from a symmetric periodic solution of the model are Hopf bifurcations for maps (or Neimark bifurcations), saddl...
Article
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We present a brief outline of nonlinear dynamics and its applications to vehicle systems dynamics problems. The concept of a phase space is introduced in order to illustrate the dynamics of nonlinear systems in a way that is easy to perceive. Various equilibrium states are defined, and the important case of multiple equilibrium states and their dep...
Article
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The article compares the dynamics of bogie models with and without different orders of discontinuity, highlighting the differences in dynamics. This lead to the conclusion that if the modelling of real discontinuities is not correct in every detail, erroneous dynamic figures may result.
Article
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We consider a mathematical model of the lateral motion of the half of the front car of the IC3 trainset. It consists of the half of the car body and a bogie on two wheel sets. The train runs with the constant speed, V, which is the bifurcation parameter. The wheel profile is the DSB 82-1 profile and the rails are UIC 60 rails with different gauges...
Conference Paper
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The dynamics of moving railroad vehicles is a challenging problem of nonlinear dynamical multibody systems. Already the fundamental guidance system with the flanged steel wheelset on steel rails introduces both kinematic and dynamic nonlinearities in the equations of motion of the vehicle, but there exist also other sources of nonlinearities in mov...
Article
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Cooperrider''s mathematical model of a railway bogie running on a straight track has been thoroughly investigated due to its interesting nonlinear dynamics (see True [1] for a survey). In this article a detailed numerical investigation is made of the dynamics in a speed range, where many solutions exist, but only a couple of which are stable. One o...
Article
Cooperrider's mathematical model of a railway bogie running on a straight track has been thoroughly investigated due to its interesting dynamics (see True [1] for a recent survey). In this paper a detailed investigation of the ultimate transition to chaos at a very high (until now unrealistically high) speed is presented. The transition is interest...
Article
Full-text available
Pantograph overhead line system interactions are studied. The importance of ‘small terms’ in the cable models is examined, i.e. bending stiffness and nonlinearities. The methods of study are wave propagation investigations and direct simulations of an interacting pantograph with the overhead line system. The weak nonlinearities are found to be the...
Chapter
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The drive towards energy savings in the railway industry have forced the railway companies and manufacturers to examine unconventional vehicle designs. The Danish State Railways together with Professor Frederich from RWTH in Aachen and the company Linke-Hoffmann-Busch have developed a steered single-axle bogie, which will save weight compared with...
Article
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A few examples of chaotic motion in railway vehicle dynamics are presented. We show examples of chaos in realistic numerical models as well as an example of chaotic motion from a test on a railway line.
Conference Paper
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The author discusses the definition and existence of a critical speed for the onset of hunting of railroad vehicles. First the field test situation is described. It is argued that the important problem is the determination of the forces and accelerations in the vehicle and the rails, which may be large even when the vehicle does not hunt. Next the...
Article
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In this paper we continue a numerical study of the dynamical behavior of a model of a suspended railway wheelset. We investigate the effect of speed and suspension and flange stiffnesses on the dynamics. Numerical bifurcation analysis is applied and one- and two-dimensional bifurcation diagrams are constructed. The onset of chaos as a function of s...
Article
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We investigate the dynamics of Cooperrider's bogie model with realistic wheel and rail profiles. The results are presented mainly as bifurcation diagrams We find speed ranges with asymptotically stable forward motion along the track centre line, coexisting attractors, symmetric and asymmetric oscillations and chaos. In contrast to earlier investiga...
Article
An example is presented of a nonlinear dynamic problem, in which the slope of a bifurcating periodic orbit is finite in the bifurcation point.
Article
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We discuss the kinematics and dynamics of a wheelset rolling on a railway track. The mathematical model of a suspended wheelset rolling with constant speed on a straight track is set up and its dynamics is investigated numerically. The results are presented mainly on bifurcation diagrams. Several kinds of dynamical behavior is identified within the...
Article
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In this paper we present the results of a numerical investigation of the dynamics of a model of a suspended railway wheelset in the speed range between 0 and 180 km h-1. The wheel rolls on a straight and horizontal track unaffected by external torques. A nonlinear relation between the creepage and the creep forces in the ideal wheel rail contact po...
Article
In this paper we present the results of a refined investigation of the dynamical behaviour of Cooperrider's complex bogie. The earlier results were presented in [4] and [7]. It was discovered, that one of the solution branches in [4] and [5] was one of an asymmetric, periodic oscillation - albeit with a very small offset, but it indicates, that the...
Conference Paper
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Chapter
Railway Vehicle Dynamics is a challenging and interesting application of “finite but many-dimensional” nonlinear dynamics. In this article we shall describe only one aspect, namely the investigation of lateral oscillations of railway vehicles moving along a straight, horizontal and “ideal” track. In order to describe the problem adequately the math...
Article
We examine first a simplified model of a railway car on bogies running with constant speed - we describe stationary, periodic, and biperiodic motion. Next we examine the model of Cooperrider's complex bogie running with constant speed - we briefly describe the chaotic motion that develops at high speed. In both cases the dynamical equations governi...
Article
It has been known for some time that an autonomous system of nonlinear ordinary differential equations of order three or higher may have solutions that depend on time in a very complicated way. Apart from the well known stationary or periodic solutions (with sub- or superharmonics), quasi-periodic solutions of varying complexity and chaotic solutio...
Article
The convective boundary layer of a Boussinesq fluid stabilized by a positive vertical temperature gradient near a heated vertical wall is considered. It is shown that traveling plane waves develop above a critical Rayleigh number and are asympotically stable for Prandtl numbers less than 9.9. When the Prandtl number equals unity, the mathematical m...
Article
The secondary wave motion in the free Ekman layer, on which a constant shear stress acts, is examined by an analytic method. The results show that the waves move away from the center of rotation.
Article
The two cases of stationary Ekman boundary layer flow of an incompressible fluid near i) a plane boundary and ii) a free surface with constant shear are considered. It is proven that a stable secondary flow in the form of traveling waves bifurcates from the stationary flow at a certain Reynolds number, and that the stationary flow is unstable above...

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Projects (3)
Project
The systematic quantification of the uncertainties affecting dynamical systems and the characterization of the uncertainty of their outcomes is critical for engineering design and analysis, where risks must be reduced as much as possible. Uncertainties stem naturally from our limitations in measurements, predictions and manufacturing, and we can say that any dynamical system used in engineering is subject to some of these uncertainties.
Project
To find out how realistic track geometry variations may change the theoretical results we know from ideal track conditions