Andreas Haigermoser a, Bernd Luber b, Jochen Rauh c & Gunnar Gräfe d a Siemens Austria AG, Eggenbergerstraße 31, Graz 8020, Austria

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1 This article was downloaded by: [SIEMENS AG OESTERREICH] On: 27 July 2015, At: 00:13 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: 5 Howick Place, London, SW1P 1WG Vehicle System Dynamics: International Journal of Vehicle Mechanics and Mobility Publication details, including instructions for authors and subscription information: Road and track irregularities: measurement, assessment and simulation Andreas Haigermoser a, Bernd Luber b, Jochen Rauh c & Gunnar Gräfe d a Siemens Austria AG, Eggenbergerstraße 31, Graz 8020, Austria Click for updates b Kompetenzzentrum Das Virtuelle Fahrzeug Forschungsgesellschaft mbh, Austria c Daimler AG, Sindelfingen, Germany d 3D Mapping Solutions GmbH, Holzkirchen, Germany Published online: 21 May To cite this article: Andreas Haigermoser, Bernd Luber, Jochen Rauh & Gunnar Gräfe (2015) Road and track irregularities: measurement, assessment and simulation, Vehicle System Dynamics: International Journal of Vehicle Mechanics and Mobility, 53:7, , DOI: / To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content.

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3 Vehicle System Dynamics, 2015 Vol. 53, No. 7, , Road and track irregularities: measurement, assessment and simulation Andreas Haigermoser a, Bernd Luber b, Jochen Rauh c and Gunnar Gräfe d a Siemens Austria AG, Eggenbergerstraße 31, Graz 8020, Austria; b Kompetenzzentrum Das Virtuelle Fahrzeug Forschungsgesellschaft mbh, Austria; c Daimler AG, Sindelfingen, Germany; d 3D Mapping Solutions GmbH, Holzkirchen, Germany (Received 6 February 2015; accepted 31 March 2015 ) Road and track irregularities have an important influence on the dynamic behaviour of vehicles. Knowledge of their characteristics and magnitude is essential for the design of the vehicle but also for comparable homologation and acceptance tests as well as for the planning and management of track maintenance. Irregularities of tracks and roads are regularly measured using various measurement technologies. All have advantages and weaknesses and require several processing steps. Characterisation of irregularities is done in the distance as well as in the wavelength domain. For rail irregularities, various distance domain description methods have been proposed and are in use. Methods have been analysed and compared with regard to their processing steps. Several methods have been analysed using measured irregularity and vehicle response data. Characterisation in the wavelength domain is done in a similar way for track and road irregularities. Here, an important issue is the estimation of the power spectral densities and the approximation by analytical formulas. For rail irregularities, periodic defects also play an important role. The use of irregularities in simulations requires various processing steps if measured irregularities are used, as well as if synthetic data are utilised. This paper gives a quite complete overview of rail irregularities and points out similarities and differences to the road. Keywords: homologation; rail wheel interaction; vehicle infrastructure interaction; vehicle track interaction; irregularites; irregularities measurement 1. Introduction and background Tracks as well as roads are in general neither straight nor perfectly even. If a vehicle runs over the track or the road surface with its irregularities, movements of the wheels and as a consequence of the vehicle are induced. The dynamic system vehicle road/track is excited to produce vibrations. The induced dynamic movements of the wheels are associated with accelerations of the vehicle bodies and forces inside the vehicle and between wheel and road/rail. Track and road irregularities are not the only, albeit major, source of vehicle oscillations and high wheel forces and accelerations are linked to wear, damage and safety issues such as derailment of rail vehicles or ride discomfort. Therefore, it is necessary to know and understand the crucial characteristics of the track and road irregularities and their influence on the dynamics of the vehicles and track. This relates to several fields and stakeholders in the vehicle track/road system. Three of them shall be dealt with in detail here. *Corresponding author. andreas.haigermoser@siemens.com c 2015 Siemens AG Austria

4 1.1. Track and road irregularities for design of vehicles Vehicle System Dynamics 879 First of all, knowledge of the irregularities is necessary for the design of vehicles. This applies to running safety of the vehicles, ride comfort, dynamic loading and fatigue of components and also structure-borne noise and noise radiation. In the case of rail vehicles, large irregularities may lead to high dynamic forces between wheel and rail. These forces are superimposed to the static wheel/rail forces arising from the mass and the load of the vehicle and some additional components arising from the lateral accelerations when traversing a curve. Above a certain level of track irregularity, the operation of some vehicles will become unsafe and may lead to derailment. For the design of vehicles, it is necessary to know or to estimate the most extreme conditions the vehicle has to operate in with a sufficient remaining safety margin, for example, against derailment or lateral track shift. It is not easy to define this level. It also depends on the vehicle characteristics, operating conditions, combinations of conditions and many other things. Moreover, it requires a method to describe track geometry in a way that results in high correlation between track geometry description and vehicle reaction. For rail vehicles, the ride comfort is determined by the vehicle s response to the track layout, by rigid body vibrations in response to track irregularities in a frequency range typically between 0.2 and 6 Hz, and by the car-body structure in response to track irregularities in a frequency range starting near the first elastic eigenfrequencies of the car body (typically above 7 Hz). For vehicle design, it is necessary to use track irregularities covering the full frequency range with a representative distribution of the level over speed ranges and track classes. Track irregularities giving frequencies above approx. 20 Hz (rail roughness) may lead to noise radiation, especially by the wheels and rails, and to structure-borne noise transmitted from the wheel rail contact to the car body, or as ground-borne vibrations in buildings. For vehicle design, it is necessary to know the typical and required spectral content of the rail roughness. Fatigue loads are determined by some quasi-static effects (loading/unloading, traction/braking and curving) and by track geometry-induced vibrations in a frequency range up to 150 Hz. Here, not only the spectral content (distribution of amplitudes against frequency) is important, but also the frequency distribution of the defects determining the number of load cycles. Road vehicles can be seen to be very similar in many cases. But some differences must be stated: It is the driver s responsibility to reduce speed for a safe ride over severe road irregularities. Sometimes there may be warnings or speed limits set by the responsible road authorities to inform users about potholes and other irregularities, and the driver will adjust his speed or his track in order to avoid severe road excitations, which might even damage his car. Basically, a road vehicle has to master all kinds of roads with all kinds of irregularities. Of course, a sports car might come to its limits where off road vehicles may still be able to continue. The designer and manufacturer of road vehicles will provide products that can master all the road conditions the customers will most probably encounter Track and road irregularities for homologation and acceptance tests For rail vehicles in most countries, the behaviour of newly developed vehicles is assessed by on-track tests. Measured data are analysed, statistically evaluated and maximum expected values (or in some cases measured values) are assessed against defined limit values. This relates to running safety tests and also to ride comfort or noise radiation tests.

5 880 A. Haigermoser et al. As the result depends very much on the track irregularities, it is evident that they have to be defined. The ideal definition would be such that the result of tests on different tracks, all fulfilling the defined track irregularities requirements, is equal. In Europe, the testing of the running characteristics is mostly done according to EN [1] This European Standard is based on the International Union of Railways (UIC) leaflet 518 [2] and is referenced by the new Technical Specification of interoperability (TSI) relating to locomotives and passenger rolling stock (RST TSI).[3] EN describes tests to be carried out under defined (reference) conditions, amongst others, track irregularity conditions. The specification of the track irregularities is intended to be such that tests of a vehicle on two tracks with the same track geometry assessment (TGA) level (and the other test conditions such as speed, curvature, contact geometry and friction) lead to the same results for the vehicle assessment quantities. Another legal performance requirement is the level of noise radiation from the vehicle. In Europe, the requirements are given in a special TSI.[4] Concerning the testing of the vehicle and the conditions for the test track, it refers to [5], where rail roughness in the wavelength range of mm is specified. Other track irregularity specifications for homologation tests exist in several countries. Later in this contribution, we will discuss their characteristics and shortcomings. In many cases, the purchasing contract includes several performance characteristics of the vehicle. One typical example is a number for ride comfort. Comfort assessment is an assessment of the car-body accelerations. The result also depends on the track geometry conditions during the tests. In order to permit the supplier to forecast and guarantee the result of the comfort tests, it is necessary to define the track geometry on the test tracks. Again, this requires a description of track geometry quality giving a good correlation between track geometry and vehicle reaction here for filtered accelerations in the car body. The road vehicle design has to consider all foreseeable customer use-cases. Some vehicles might only be sold in certain regions of the world, but in most cases the road vehicle design will result in vehicles that can be used on any road worldwide. During the development of road vehicles, the durability aspect is covered by manufacturerspecific design rules, which are checked by manufacturer-specific standardised test procedures. The same holds for the design of ride comfort properties. The assessment of durability and ride comfort of a road vehicle is done by the customer. The buying decision might also consider test results published in auto magazines, or from consumer organisations, or make use of subjective impressions on a test drive. A standardised assessment of road irregularities influencing driving safety does not exist. Of course, the design of the vehicle s chassis systems considers these interactions and reduces their impact on driving safety Track and road maintenance As excessively high track irregularities may impact the running safety of the rail vehicle, checks and maintenance of the track irregularities is necessary and done regularly. The relevant maintenance technologies for rail and track irregularities are tamping, grinding and renewal of tracks. As most of the modern railway lines are heavily loaded by many, densely scheduled trains, such maintenance activities often have a huge influence on the operation on the line. Therefore, long-term maintenance planning is essential, which implies that it is necessary to know the actual status of the track irregularities and their expected development (prognosis and deterioration rate). For both fields, running safety and deterioration, the forces between wheel and rail are relevant. Tracks with large irregularities will result in high dynamic forces between wheel

6 Vehicle System Dynamics 881 and rail, which cause more wear and might cause failures in the rail and track components. In general, higher track irregularities lead to a faster deterioration of track geometry. At some level of track irregularity, operation of some vehicles may become unsafe. Track maintenance management has to find a cost-optimal method to maintain the track to the required quality level and to avoid an exceedance of safety limits. This requires knowledge of the actual track geometry and the deterioration rates. If some level of track irregularities is exceeded, the vehicle speed has to be reduced on these parts of the line. As aforementioned in the field of homologation tests, a TGA method is required in which the assessment quantities correlate highly with the resulting vehicle reactions, especially the wheel rail forces connected with running safety and track geometry deterioration. Road maintenance follows similar principles. Highways and main roads in particular are regularly monitored to detect irregularities (see Section for details). Measurements are collected for longitudinal and transversal evenness, skid resistance and surface distress. These measurements are the basis for maintenance plans, so that road security can be preserved and the existing budgets can be used in an economically wise manner. It is important that maintenance is always done before damage has become excessive, because beyond certain damage levels, irregularities will grow exponentially and road maintenance costs will do the same. Longitudinal waves, for example, will influence all passing vehicles, especially trucks. Even if safety is not yet affected, damage may rise exponentially if the passing trucks are all stimulated in the same way. Once ruts have been formed in the transversal profile, cars and trucks will have the tendency to use them more and more, so that surface deterioration again grows exponentially. Longitudinal waves close to vehicle s eigenfrequencies, rut depth and skid resistance are most critical for road safety and have to be monitored closely Organisation of this paper The structure of this paper is derived from the necessary parts of work aimed at providing the information described earlier. It starts with some definitions and classification for frequency and wavelength ranges. It continues with information on measuring systems, their principles and the strength and weaknesses of the different principles. Further on, the processing of measured data is discussed and an overview on methods for characterising irregularities in distance and wavelength domain is given. The paper ends with an overview of tasks and problems in preparing and using irregularities in simulations. This paper contains more information on rail irregularities, because an equal and full treatment of both sides would go beyond the scope of a state-of-the-art paper. Therefore, the authors have attempted to provide practically complete information concerning rail irregularity and point out similarities and differences to the road sector. 2. Definitions and outline 2.1. Track geometry and irregularities Track geometry is essentially the variation of lateral and vertical track position in relation to the longitudinal position [6] or length parameter. On perfect, straight track, its geometry is a straight line. However, the track is neither constantly straight nor perfect, and consists of elements such as straights, curves, transitions, S-shaped features, switches and track irregularities. This is generally referred to as design geometry (track layout, designed

7 882 A. Haigermoser et al. Figure 1. Track geometry with design geometry and irregularities. shape or alignment) and deviations from it (irregularities, roughness and track geometry quality). These parameters are usually handled separately, in that they have different effects on vehicles. If the design geometry is known, it is theoretically possible to separate the two parts and extract track irregularities from track layout. Figure 1 shows the two rails of a track consisting of a straight section and a curve, both in a longitudinal gradient. It also shows deviations from the design geometry in vertical and lateral directions. Additionally, it illustrates the separation of design geometry and irregularities. Generally, design geometry is of long wavelength and defects are of short wavelength. However, it is possible for deviations to be longer than design alignments. In particular, some transition curves are very short, and would, in other circumstances, be considered to be irregularities.[6] However, in practice, the boundary in terms of wavelength between geometry and defects is impossible to define. As track geometry measurement cars always measure (parts of the) full geometry, it is not a trivial task to extract track irregularities from the measurement signals. As the full track geometry is necessary for simulations, it is also not a trivial task to reconstruct it from results of a measurement car. For the design geometry or alignment of the track, projections to a horizontal and a vertical plane are used and may be described by the following elements and their characteristics: Horizontal circular curves with defined radius and cant. Horizontal curves with varying curvature (transition) and super-elevation ramps. Longitudinal track gradient and vertical curves. In Europe, the EN standard describes requirements for the track alignment parameters. Part 1 [7] handles plain track and part 2 [8] covers switches, crossings and comparable alignment design situations with abrupt changes of curvature. Figure 2 shows the coordinate system (as used in Europe), which moves along the centre line of the track s design. The four quantities y L (x), z L (x) for the left rail and y R (x), z R (x) define the position of the rails along the track position x (in reference to the design track alignment). The rail coordinate system used in Europe is orientated with the x-axis in the travelling direction, y parallel to the running surface and z pointing downwards normal to the running surface. In other countries, we find deviating orientations, even not always righthand-oriented systems. For road vehicles, ISO 8855 defines right-handed coordinate systems

8 Vehicle System Dynamics 883 Figure 2. Rail and track coordinate system. with z pointing upwards,[9] an orientation which is also used in some rail publications. The position of rails can also be defined in the so-called track coordinate system, where the spatial position of the rails is related to the track centre line. Often used terms, not always clearly dedicated to one of those coordinate systems, are Longitudinal level, top, profile or surface for z(x) of the track centre line or z L (x), z R (x) of the two rails Alignment (alinement), line or horizontal profile for y(x) of the track centre line or y L (x), y R (x) of the two rails Cross level, cant or super-elevation for cl(x) = b A *δ (x) of the tracks running surface Gauge or gage for g(x) = g 0 + g Both coordinate systems are equivalent and data can easily be transformed. If the track is loaded by the weight of a vehicle, the rails change their position depending on the local track stiffness. As the stiffness changes along the track length, this introduces an additional component of rail irregularity. More precisely, this effect is an excitation by a changing system parameter and should be distinguished from the excitation by the uneven track, acting as excitation due to the forced movement of the rail. In practice, both parts are used together, as most measurements are done in loaded track condition. Track irregularities are a result of fabrication tolerances, installation imperfections and various degradation processes. Examples are rail or sleeper fabrication tolerances, component wear, track settlement and other effects. Design features such as bolted and welded joints of the rails or characteristics of the carrying plates of slab tracks are also an important source of irregularities. Much of the geometric deterioration of track is due to wear and tear in the ballast. Repeated pounding by trains usually causes the ballast to compact and ballast stones to fracture. Also maintenance, for example, tamping may cause damage of the ballast. The rate at which the track deteriorates varies from section to section of the track and therefore generally leads to increasing roughness of ride for trains; which in turn increases the dynamic forces. Ballast deterioration can cause breakage of rail fastenings, rail pads to work loose more readily, sleepers to crack and the ballast to become increasingly ineffective. The remainder of the deterioration of track geometry is primarily dependent upon the magnitude and variations in subgrade stiffness.[10] As with ballast, deterioration of the soil formation affects track irregularities.[11]

9 884 A. Haigermoser et al. The different deterioration mechanisms affect different wavelength ranges. Faults in the 0 2 m band are mainly due to corrugation, production tolerances and wear of rail shape, joints and tolerances in the welds. Periodic effects usually are separated in corrugation (wavelength mm), short waves ( mm) and long waves (up to 2 m) Waves between 3 and 25 m originate from several settlement mechanisms in ballast and subsoil. Resonance frequencies of the vehicles may lead to distinct irregularities. Waves between 25 and 70 m are often caused by track layout irregularities (the shortest transition curves are around 30 m), especially in short transitions or by settlement of the subsoil because of insufficient load carrying capabilities. Waves above 70 m originate from problems in the track layout. Irregularities with wavelengths below 2 3 m are usually denoted as rail roughness or rail irregularities. They are actually variations of the rail profile, caused by production tolerances or wear. Although they may also lead to a deviation in the lateral direction, in most cases only the irregularity in the vertical direction is measured and analysed. Irregularities with wavelength above 2 3 m are usually denoted as track irregularities. Higher track irregularities often occur at locations with discontinuities in support and construction such as level crossings, bridges, tunnels, abutments, culverts, but also switch panels, ballast and foundation stiffness variations. Rail and track irregularities act as an excitation mechanism if the vehicle moves with a speed v along the track. Then, a wheel running over a sinusoidal vertical rail irregularity of wavelength L (m) will perceive an excitation frequency f (Hz), that is, f = v L. (1) A range of vehicle speeds v = m/s in combination with irregularity wavelengths L = m cause excitation frequencies in the frequency interval f = Hz. Longer wavelengths give lower frequencies. Figure 3 illustrates the relationship between irregularity wavelengths, excitation frequency and their relevance to vehicle behaviour. In the middle part of the figure, the relevant frequency ranges are indicated for the different dynamic phenomena. Regarding ride comfort, for example, frequencies between 0.5 and 30 Hz are the most important ones. At 100 km/h this corresponds to a wavelength range between 1 and 50 m. At 350 km/h the corresponding range is m. For other domains, other frequency ranges and therefore wavelength ranges are important and indicated in the figure. The red bars in the figure shift along the lines of constant excitation frequency if the speed changes. In the upper part of the figure, the wavelength ranges of some periodic effects are shown. Corrugations and short waves typically have wavelengths between 30 and 200 mm. Sleeper spacing is usually at m and acts as an excitation mechanism due to the changing track stiffness. Rail manufacturing defects may result in periodic excitations at wavelengths around 2 4 m. An important periodic excitation is the rail length (and its harmonics), especially if bolted joints are built, but it is also visible in the case of welded joints. Rail lengths are between 10 and 120 m. Measurement of track irregularities is done in two domains: Short wave measurements cover the range between 30 mm and 1 m, track-recording cars measure them between 3 and 70 m, some special systems up to 150 m. Comparing this with relevant ranges, it becomes clear that there is a gap between measured and those required for a full assessment or for simulations. This applies to short wavelengths between 1 and 3 m and wavelengths between 70 and 200 m for higher speeds.

10 Vehicle System Dynamics 885 Figure 3. Wavelength, speed, frequency, measurement ranges and physical features Road geometry and irregularities Road geometry is primarily defined by the shape of the landscape to cross. Since road vehicles possess good manoeuvrability, roads can have steep gradients and small radii. For higher cruising speeds and traffic volume, road geometry becomes straighter through the use of bridges, landfills, tunnels, etc. The same elements as defined for tracks can be used to formally describe the road geometry in applications for vehicle system dynamics, as defined in Section 2.1: the layout is defined by the curvature of a reference line, and is complemented by grade and super-elevation defining the longitudinal and lateral slopes. While the longitudinal inclination of the road is of relevance for all energy-related work, for example, the curvature, super-elevation and all longitudinal and lateral slope changes are relevant for vehicle handling assessment. The ride comfort might also be influenced by road geometry, if lateral forces and particularly their fluctuations are inconvenient for the driver and his passengers at the desired travel speed. High longitudinal and lateral forces acting on the vehicle are also relevant for vehicle durability. The typical frequency range of road geometry-induced fluctuations should always be far below 1 Hz in order to be compensated for by the driver. All irregularities caused by construction faults and deterioration will be relevant for handling and ride comfort, and also for the durability of the vehicle. While higher frequency excitations are responsible for NVH (noise, vibration and harshness) phenomena, the frequency range of the irregularities relevant for vehicle system dynamics applications is Hz. Depending on the travel speed, this is equivalent to wavelengths of m for slow driving (6 m/s) and up to m for fast driving (60 m/s). Depending on the desired cruising speed, the horizontal road shape is mathematically designed using a chain of design elements such as clothoids, circles and straight lines. Clothoids are used to model the road curvature transition section between segments with constant curvature, such as circles or straights. Clothoids are typically chosen as mathematical design elements, because they allow smooth vehicle steering, which is the desired target for ideal road design. Road geometry design is standardised by an official set of regulations, which are used to maintain a general standard for road geometry that can be safely used by all vehicles. The

11 886 A. Haigermoser et al. Figure 4. Example of cross slope transition. road design elements are completed by the gradient in height, which describes the necessary parameters for longitudinal slope, cross slope, super-elevation, line of sight regulations or the minimum height radius in valleys or on summits. In Germany, the regulations for road design are split into three main parts to cover all necessary road types.[12 14] The standards have been reorganised and reviewed between 2007 and 2012 with the aim of defining regulations for a modern set of different road types and subtypes As-built road design control Necessary preconditions for safely drivable roads are, for example, that maximum lateral acceleration is limited, that the necessary line of sight is guaranteed or that the water drainage is always sufficient. The last point is regarded to be one of the critical parameters related to driving safety. Road design has to make sure that transition sections in which the cross slope of the road changes its sign are as short as possible. If part of the transition section cross slope is zero and the section is too long, then water will gather on the road and drivers will be placed in a potentially dangerous situation. Due to sophisticated, precise engineering surveying methods, the 3D design of existing roads can be completely reconstructed from survey data. For new roads, these methods reveal differences between the building result and the planned road design. For existing roads, the survey results permit the identification of potential risks due to road degeneration. An example for a cross slope transition is shown in Figure 4. The left plot shows a high-resolution digital road surface model of a highway intersection. The graphic is colour coded in height, so that the colour changes give the same visual impression as contour lines. The zoomed section on the right-hand side reveals that for the north direction the main lane (3) and the side lane (4) have the same cross slope to the right. For the southern direction, the cross slope for the side lane (1) is also tilted to the right, but the main southern lane (2) changes the sign of the cross slope from north to south and contains a significant cross slope transition zone Road condition determination and road condition rating Naturally, road surface irregularities have a decisive influence on vehicle behaviour and driving itself, because they directly affect safety and comfort for the road users as well as

12 Vehicle System Dynamics 887 Figure 5. Figure 6. Example of colour-coded road surface model. Longitudinal profile of the example data. noise production and potential hazards due to aquaplaning. Vibrations caused by vertical irregularities may also affect the vehicle cargo, because they may reduce load stability or may damage the cargo itself. Road surface grip is decisive for braking distance and lateral driving stability. Image examples and explanations for all kinds of road surface damage can be found in [15]. For road condition measurements, it is irrelevant whether the problems result from construction or are caused by surface or underground degeneration. The precise detection and quantification of damage to prevent hazards is decisive for road safety and cost-effective maintenance. For this reason, road condition monitoring is done continuously (compare Section 3.2). Figure 5 shows the colour-coded road surface model for a section of concrete paving slabs. The contour lines reveal significant vertical shifts at the edges between the different slabs. Figure 6 shows a longitudinal profile through the data set of Figure 5. The vertical irregularities reach a maximum of 3.5 cm, which is almost certain to create hazardous situations, at least for truck loads and cargo.

13 888 A. Haigermoser et al. 3. Measurement of irregularities Measurement of irregularities is a prerequisite for all subsequent analysis steps. Similar principles for roads and tracks can be found, although requirements and details differ Measurement and pre-processing of track geometry irregularities As previously analysed, there are a number of different use cases for track geometry irregularities, often linked with diverging requirements. For vehicle design in terms of ride comfort in the high-speed range, track irregularities up to 200 m wavelength may be relevant or even critical. On the other extreme for simulation of noise radiation, irregularities in a wavelength range around 10 2 m are important. It is evident that these four orders of magnitude in wavelength and the resulting differences in the requirements lead to different measurement technologies. It is not possible to measure this full range of track geometry with one measuring device or even measuring principle. The dividing line of the usual measuring systems is at around 1 2 m wavelength, consistent with the separation in rail irregularities (rail roughness) and track irregularities Measurement of rail roughness Trolley systems are used for the measurement of rail roughness and short local defects, where the track is more or less unloaded, and vehicle-based systems, where the track is loaded by the vehicle. Several surveys for these measurement systems have been published, see [16 19]. Two principles are used in trolley systems: measurement using displacement sensors and measurement using accelerometers.[16] In the case of measurement using displacement sensors, the vertical displacement is measured with respect to a reference beam mounted on the rail (displacement transducers running on a stationary straightedge). This reference beam, with a length of in general about 1 m, can be either fixed at a specific location (discrete measurement) or can be moved along the rail using the trolley for a continuous measurement of rail geometry. In the case of measurement using an accelerometer, the vertical displacement is basically determined by double integration of the acceleration signal that is measured when pushing the trolley along the rail surface. In both cases, the position along the rail is recorded simultaneously, such that the result is a measure of the vertical rail irregularity as a function of the position along the rail.[16] Such systems are available from different suppliers; see e.g. [20] or[21]. Vehicle-mounted systems use either optical [22] or mechanical measurements of the surface irregularities in relation to a versine [20] or accelerometers mounted on axle boxes. The latter method measures the combined roughness of rail and wheel. Using a non-braked wheelset with comparably smooth wheels enables the estimation of the rail roughness.[23] Besides the combined wheel and rail roughness, the axle box accelerometer signal is also influenced by the vibrations from the rail/track and the wheelset. This is difficult to accurately compensate and is a shortcoming of this measurement technique. At Trafikverket (Swedish infrastructure manager) the conversion from acceleration to roughness levels is done by a speed-dependent transfer function.[23] The transfer function is calibrated by comparing the spectrum of the measured acceleration to the spectrum of the measured roughness, measured by a trolley with displacement sensors, which has considerably better accuracy than the axle-box system.

14 Measurement of track geometry and track irregularities Vehicle System Dynamics 889 Track geometry has been measured since railways exist. It is necessary for the acceptance of track construction or maintenance works. It was also soon used to decide on necessary track works. The first systems were mechanical devices measuring track gauge or cant. Track construction was checked by geodetic measurements. In [24], we find a description of possibly one of earliest track-recording cars from 1877 (Eisenbahn Revisions Wagen). This system was able to measure track gauge and cross level. Additionally, the vehicle movements were registered to identify locations on the track with poor track geometry. A line of 250 km length with two tracks could be registered per day. The measuring car of the German Eisenbahn-Zentralamt measured the longitudinal level as early as This was done by mechanical means as chord measurement including the movement of the recording pens.[24] An early form of measuring track quality from the reactions of a moving train was the use of the Hallade machine at the beginning of the last century. This instrument, invented by the French Emile Hallade, was carried in an ordinary carriage and relied on a system of damped pendulums attached to a pen scribe-paper roll that recorded the oscillation of the carriage. These records enabled the engineers to see exactly where joints required packing up or where curves required re-aligning. In the first half of the twentieth century, track-recording cars using chord measurement systems were introduced including systems with gyroscopes producing an artificial horizon. The earliest inertial systems were developed by NS (Nederlandse Sporwegen) and BR (British Rail) in the 1970s Requirements and principles Measurement methods may be divided into Discrete working systems, such as hand measurements and Continuously working systems, such as recording cars.[25] The measurement principles depend on the measured quantity. Track gauge can be measured by wheels or sliders (mechanical) or by optical means. For the measurement of cross level the direction of gravitational forces is used as base. To determine it, gyros are normally used in continuously working systems. Track gauge and cross level have the advantage that their measurements need no (additional) reference system. This is more difficult for alignment and longitudinal level, because the theoretical track layout gives high values in an inertial system. Measurement principles may be divided into Geodetic measurements using some fixed-points, Chord measuring systems, Inertial measuring systems and Vehicle reaction measuring systems (indirect method). The measuring systems are required to preserve amplitude and phase of the irregularities. This means that each irregularity in the required wavelength range should be measured with its real amplitude and no phase shift between real shape and measurements should occur. A limited wavelength range is beneficial, because excessively long wavelengths have very high amplitudes. It is very difficult and often not necessary to measure them together with small wavelengths with small amplitude. As measurements are required that do not disturb

15 890 A. Haigermoser et al. railway operation, it is necessary that they measure at the speed of the in-service trains. The measuring car should have typical mass and stiffness parameters in order to measure the track in the usual deformed status. It is also necessary to acquire the location where the data are measured. In the past, this has been done by measuring the wheel rotation and synchronising with selected track magnets or other transponders. An accuracy of a few meters can be obtained, but this is not good enough if incipient faults are to be tracked offline; sub-metre accuracy is necessary for this as well as positive verification of the line reference. Global positioning system (GPS) location alone is not sufficient. However, by using both, GPS and track magnet data, this accuracy can be achieved. Geodetic measurements are perfect in their amplitude and phase characteristics, but they are expensive, not continuous and not wavelength limited. On modern railway lines, it is even impossible to do such measurements without closing the line for some time. They are used mainly for track construction. Handheld systems and trolley-based systems as well as the devices used for geodetic measurements are not heavy enough to deform the track as during normal operation. As shown in the later text, different track measurement systems have been developed by railways and companies during the last centuries. Based on their systems, different track geometry standards were generated, which were not comparable. In other words, measurements in one country were not comparable with those made in other countries. This is no longer adequate in an interoperable European railway system. Here, it is necessary to define a minimum track geometry quality to ensure safe operation of trains. Therefore, a series of standards EN Track Geometry Quality were created with six parts, which define requirements on the quantities measured by such systems by requirements on the systems themselves. EN [26] defines the principal parameters for characterisation of track geometry and the requirements for them to be respected. EN ,[27] EN [28] and EN [29] define the requirements on the different measuring systems. EN Geometric quality assessment [30] defines the minimum requirements for the quality levels of track geometry, and specifies safety-related limits for each parameter as defined in EN EN [31] characterises the quality of track geometry, gives the different classes and considerations how this classification can be used Chord measuring systems Chord measuring systems rely on the principle that a base line is formed between two points on the rails and at a third position the distance between this base line and the rail at this position is measured. The sensors may be of mechanical type such as wheels or without contact using optical methods. The upper part of Figure 7 illustrates the principle. The drawback of this method is that this system shows a transfer function that varies between two and zero in amplitude. The phase is also distorted.[32] This is illustrated in the lower parts of Figure 7.[32] If the measuring system runs over an isolated defect, as shown in the middle part of the figure, the resulting chord signal shows three peaks as indicated in the lower part of the figure. There are two additional peaks located at the half of the chord length before and after the defect. In the symmetric case, their amplitude is the half of the defect. The behaviour of this measuring system may be analysed more generally by deriving its transfer function.[33] Figure 8 shows the principle of the system and the amplitude transfer functions (ratio of measurement amplitude and irregularity amplitude) of two different systems. For a symmetric chord (here a = b = 5 m, left part of the figure), there are two extreme cases: If the chord length s is an even multiple of the irregularities wavelength, then

16 Vehicle System Dynamics 891 Figure 7. Principle of chord measuring systems, measured signal from single defect. Figure 8. system. Principle of a chord measurement, amplitude transfer function for mid-chord (left) and asymmetric chord all three wheels are at the same height. The measured result is always zero. If the chord length s is an uneven multiple of the irregularities wavelength, then the middle wheel moves in the opposite direction to the outer ones. It shows the double amplitude of the irregularity. This behaviour of the symmetric chord (mid-chord offset (MCO) system) is visible in the transfer

17 892 A. Haigermoser et al. function on the left part of the figure. At long wavelengths it shows low values asymptotically to zero. The transfer function increases and reaches two at the chord length. In the next part, the transfer function becomes lower and reaches zero at half the wavelength. This behaviour is periodically repeated. For the symmetric chord, there is no phase shift. The amplitude behaviour can be enhanced if the chord is asymmetric. In the right part of the figure, the amplitude transfer function for a chord with a = 6 m and b = 2.6 m is plotted. It shows no more zeros, but there are still peaks with values up to two. A drawback of this system is that the phase is no longer zero. If this transfer function is known and does not exhibit zeros, then it is theoretically possible to calculate the signal for the irregularity. This decolouring is described in detail in a later part of this contribution. Chord measuring systems are widely used.[24,34 37] They are not too complicated in their assembly and measuring techniques and unlike inertial measuring systems, there is no requirement of a minimum speed and hence there is no loss when measuring in stations at low speeds. The main disadvantage of chord measuring systems is their transfer function, especially in the case of mid-chord systems. Data from such systems are hard to use in multibody simulations. When using non-contact sensors, chord measurement requires a lot of optical devices, which increases the risk of failure or bad calibration. At least six optical sensors are required; more than six, if compensation of car-body bending is needed. Additionally, the sensitivity to dust, condensation, driving rain and snow is increased. Design of modern trackrecording cars is similar nowadays to those of passenger vehicles with light and flexible car bodies, hence it is more and more necessary to compensate for the influence of a car-body bending Inertial measurement systems Inertial measurement systems rely on the measurement of accelerations by accelerometers and rotation rates (angular velocities) by rate gyroscopes (inertial measurement unit, IMU). The measured accelerations are integrated twice to determine the position of the sensor in an inertial system. Figure 9 shows the principle used in the measuring coaches developed by DBAG ( Oberbaumesswageneinheit OMWE and RAILAB [24]). It consists of an inertial reference system O1, which is mounted on a vibration-isolated table O2 in the car body. This forms the measuring base. It is mounted in this way because the sensors are highly sensible and unwanted effects from vibrations of the car body, bogie or wheelsets are avoided. On both bogies, measuring frames O3 and O4 are mounted on the axle boxes. The spatial displacements between the reference system O1 and the measuring frame O3 is determined by a so-called vector measuring system (Vektor-Abstands-Messung VAM), an optical device. In order to permit better registration of long waves, the second bogie is also equipped with a measuring frame O4. The angles between the two systems O3 and O4 is measured with two lasers and used for the analysis. Both measuring frames contain on both sides a device for the vertical position (VMS) and the lateral position (HMS). These are laser triangulation sensors, able to contactlessly measure the distance between the sensor and the rail. Track gauge is calculated directly from the results of the HMS s. Cross level is calculated from the relative positions of the two VMS s to the inertial reference system. Curvature is calculated from the change of the track axis against the north axis of the inertial reference system. The vertical and horizontal positions are defined as the difference between the measured spatial position and the compensation curves in horizontal and vertical directions. This system is able to measure irregularities between 1 and 150 m with a transfer function

18 Vehicle System Dynamics 893 Figure 9. Measurement principle of Railab from DBAG. close to one. There is some phase shift because of the necessary filtering of long waves (compensation curves). Similar systems are used by other railways [11,38 42] and produced by a number of suppliers, see e.g. [22,43 47]. Advantages of inertial measuring systems are that they are mechanically quite simply and easily mounted on the car body or bogie frame. They are suitable for higher speeds; higher speeds even reduce the uncertainty of the measurements. The transfer function is equal to one, but there may be distortions from the necessary high-pass filtering. Few sensors are necessary, which reduces the risk of sensor failures. On the other hand, inertial systems can only operate above a minimum speed, typically above km/h. As speed reduces, the output from the inertial sensors becomes smaller and smaller, until noise and offsets predominate. Hence, it is not feasible to measure at low speeds as, for example, in sidetracks situated in big stations. Although the transfer function is equal to one in the pass band, isolated defects are distorted due to the characteristics of the filters. It is necessary to achieve good synchronisation between cameras and inertial units, which makes the calibration of the system quite complicated. The required optical sensors are sensitive to dust, condensation, driving rain and snow; this applies also to chord systems using optical sensors. Problems of both systems are that they are mostly equipped in expensive special cars. The cost of operating them with highly skilled employees is very high, original equipment manufacturer replacement parts prices increase yearly, and there is a continual need to upgrade the technology over the life of the vehicle, which is quite long. As railways try to increase the number of trains operating on a track of finite capacity, the difficulties of timetabling a dedicated self-propelled vehicle is complex due to its non-regular schedule and the fact that it takes revenue train capacity away from the route. Due to its infrequent passage, dedicated track geometry recording cars may identify track infrastructure defects quite late, possibly not until the imposition of speed restrictions and unplanned maintenance activities are required. The speed of a dedicated monitoring vehicle is another problem. As average running speeds increase, the ability of the dedicated vehicle to measure at or near normal track running speed is limited by the availability of high-speed vehicles. Surely, the optimum method of measuring track must be done by a vehicle similar

19 894 A. Haigermoser et al. to that used in normal operation that is capable of accessing the track consistently in terms of speed, axleload and dynamic bogie behaviour. This was the motivation for development of Autonomous Track Geometry Measurement Systems (ATGMS) [48] or Unattended Geometry Measurement System. They are intended to improve railway safety by increasing the availability of track geometry data. Routine use of ATGMS by a minimised interference of inspections to revenue operations, increased inspection frequencies and reduced lifecycle cost of inspection operations (maintenance planning). The larger set of data available from the frequent measurements enables more accurate trending for maintenance planning. Inertial measurement systems are mounted on passenger trains in normal service and the measurement results are made available to the infrastructure managers. The systems usually consist of the measuring unit including signal processing and storage, communication module and a land-based server system. Systems are available from [22,49] or[46], one system has been developed by Federal Railroad Administration (FRA).[48] A simple inertial measurement system uses accelerometers on axle boxes to calculate the rails longitudinal level by double integration and filtering.[50 54] Difficulties with such systems are the very wide dynamic range of the signals and the fact that track geometry is determined by rather low frequencies. It is often stated that for inertial systems to operate effectively, the dynamic range has to be restricted by mounting the accelerometer on a sprung platform (best on the car body). But this requires measurement of the displacements between rail and platform. For lateral irregularities, there is another problem: In general, the wheelset does not exactly follow the rail, but moves inside the wheel rail gauge clearance. There have been proposals to correct this by using a second-order dynamic model of the lateral wheelset movement,[55] where an accelerometer and a gyroscope on the bogie frame have been used. A similar approach is to measure accelerations on the bogie frame or in the car body and estimate the track irregularities from the measured accelerations.[56] In[57], estimation is done in the frequency domain. In [58], system identification techniques are used to identify a parametric model or transfer functions between vertical track irregularity and accelerations in the vehicle. Several methods for the identification have been applied. The proposal was to implement them as inverse models in a Digital Signal Processor platform of a measuring system. In [59,60], track irregularities estimation techniques from car-body acceleration and/or pitch rate are proposed with inverse analysis and Kalman filter. In [61], a method is presented where axle box accelerations are measured. An estimation technique for wheel load and lateral force variation is then applied, based on the frequency response method and calculating estimated wheel load and lateral force variations Reaction measurements (indirect assessment of track geometry) Additionally or even instead of geometry measurements, vehicle reactions are sometimes measured as an indicator of the track geometry quality. Such systems include measurements of accelerations on the axle boxes or within the bogie or car body.[62] There are also systems where the forces between wheel and rail are measured using load measuring wheelsets,[63] in Germany at DBAG, there is even an internal standard for applying this method to highspeed lines and lines where tilting trains operate [64] (part 2002). In the USA (see [65,66]), infrastructure managers are obliged to measure accelerations or even wheel rail forces on tracks of defined classes, speeds and cant deficiencies. In Japan, the car-body acceleration of commercial trains on Shinkansen lines has been measured every day since the inauguration of the Tokaido shinkansen line in 1964.[52] Since 2010, four 700 series shinkansen trains of JR Central have been equipped with acceleration measurement devices.

20 3.2. Measurement and pre-processing of road irregularities Vehicle System Dynamics 895 Road inspection and road condition monitoring are the tasks of highly specialised kinematic multi-sensor survey systems, described in Section The technology for road inspection has been established since more than 20 years. Parallel developments have led to mobile road mapping systems (MoSES) for road inventory determination with a far more general survey approach. Due to continuous technical developments over the past approx. 10 years, it has become possible for specially equipped mobile mapping systems to fulfil engineering surveying tasks, for example, for precise as-built highway surveying or for detailed tunnel or railway engineering projects. Remarkable improvements in sensor technology enable the additional usage of high-end mapping system survey data to precisely detect road irregularities described in Section Road condition determination Road condition determination is based on specialised, fast-driving survey systems. The road condition rating based upon the acquired data is a standard procedure, which is regularly repeated in Germany on federal roads every 4 years, especially on highways and federal roads. Since 2008, the surveys also include the connection ramps, entrances, exits, etc. The so-called Zustandserfassung auf Bundesfernstraßen (ZEB) is a joint project between the federal states and the Bundesanstalt für Straßenwesen (BASt). A detailed description can be found in [67]. The following information is acquired about the road surface: Longitudinal and transversal evenness Skid resistance Surface distress The stored data sets are referenced to the road network system, so that the following road condition rating leads to specific analysis values and ratings for each road section. The results are used for all kinds of analysis for road maintenance planning, analysis of certain road building methods or lifecycle monitoring of materials, etc. A detailed description of the different road condition tasks can be found under [68]. The official technical descriptions and regulations, such as [69 71] contain all details about road condition determination in Germany. Longitudinal and transversal evenness: A road s evenness is determined by measuring the longitudinal evenness using a sensor assembly in longitudinal alignment with respect to the direction of travel and by measuring the transverse evenness by a sensor assembly in transverse alignment with respect to the direction of travel. Typically, the systems use contactless measurement technology. Figure 10 shows a typical assembly. The bar for transversal evenness inspection is mounted on the front of the vehicle. The bar for longitudinal evenness measurements is visible on the rear right-hand side of the van. Evenness of the transverse profile/transverse evenness: In former times, the transverse road evenness was checked using a 4 m straightedge, which represented the reference line for a road surface profile measurement. Today s mobile systems for transversal evenness measurements use a bar mounted on the front of the vehicle, which is equipped with a number of high-resolution laser distance sensors covering the width of the driven lane with continuous measurements leading to cross profiles with approx. 10 cm transversal resolution. The ARGUS system in Figure 10, for example, uses up to 37 sensors leading to a maximum number of 37 height values per profile. Each sensor provides distance measurements to the road surface with a standard deviation below 0.1 mm. In the lateral direction, a profile is stored

21 896 A. Haigermoser et al. Figure 10. Road inspection system ARGUS of TÜV Rheinland Schniering GmbH. every 10 cm. Transverse evenness is quantified by analysing the cross profiles to derive rut depth and fictitious water depth as measures for the potential traffic hazard. Longitudinal evenness: The road condition survey systems for longitudinal evenness use a longitudinal straight bar with laser sensors. The sensors are of the same type as for the transversal evenness, but in this case four sensors are used, which are distributed along the straight bar with certain calibrated distances to each other. The quantification of longitudinal evenness is much more complicated compared to transversal evenness, because the longitudinal evenness in road condition determination is derived from frequency analysis. At the moment in Germany the following parameters are generated based on the measurements: Degree of unevenness (AUN) Ripple (W) Effective longitudinal evenness index (LWI) International roughness index The first methods of analysing longitudinal profile data were AUN and W. They have been under criticism since the beginning, because of their inability to correctly characterise the unevenness in terms of its random, periodic and transient parts on a consistent basis. The LWI has been introduced into road condition determination to improve consideration of graduated, usually periodic unevenness as well as single irregularities.[72] Skid resistance measurements: The skid resistance of a road surface is especially critical for vehicle safety due to its decisive influence on vehicle s handling and braking performance. The skid resistance measurements are performed by special truck-based survey systems called SKM (sideways force measurement). The SKM uses a measuring wheel, which is tilted to the direction of travel. The road surface under the wheel is always kept wet with a defined water-film with a thickness of 0.5 mm. This principle of skid resistance measurements requires a lot of effort. Compared to road evenness and surface distress measurements, it is not contact-free, which leads to a number of restrictions in hardware and the handling of

22 Vehicle System Dynamics 897 such systems. A sensor-based approach for contact-free skid measurements would be highly appreciated, but has not yet been technically established. Surface distress: High-resolution road surface images of the driven lane are used for crack detection, mapping of surface repair patches, etc. Typically, the camera system is mounted on the same survey system that does the road evenness measurements, so that these two measurement campaigns are always combined. The cameras are often placed close to the road surface and the image system is synchronised with stroboscope lamps to maintain constant illumination conditions (compare Figure 10 surface camera system is slightly visible at the rear of the vehicle). Image resolution allows crack detection up to mm crack width New technologies for road condition determination Specially equipped high-end mapping systems are not only able to fulfil engineering surveying tasks, but they can also precisely detect road irregularities. An example for such a high-end multi-sensor mapping system is the MoSES. MoSES has been designed as a multisensor survey system to capture all relevant road or railway data without obstruction of the traffic flow. Based on the system concept for the Kinematic Survey System KiSS, different MoSES system setups have been established, which consist of three main sensor modules combining complementary characteristics: Trajectory module consisting of a multi-sensor assembly for precise three-dimensional trajectory determination Multi-camera module equipped with photogrammetric cameras Laser-scanner module with two high-definition laser-scanners Detailed descriptions can be found in [73,74]. Figure 11 shows two examples for sensor setup, which are always similar: The trajectory module with the IMU as a core component is placed close to the laser-scanners in the rear of the vehicle. Various photogrammetric cameras for visual documentation and object detection complete the system. The trajectory module performance is the key to the MoSES s capability to serve highprecision applications. The core of the module is an IMU, which consists of three ring-laser gyros of the highest specification (unaided drift < /hr; random walk < / hr) and three high-end accelerometers. The system is completed by high-precision differential GPS and an odometer as support sensors. These sensors are combined to a powerful self-calibrating position and orientation system. The trajectory sensor data fusion is done Figure 11. Road Mapping System (MoSES); Survey van MMZFS (Mobile Mapping ZF System).

23 898 A. Haigermoser et al. in post-mission data processing. MoSES s applications regularly require position and orientation data in environments where GPS satellite reception is poor or only intermittently available. Under these difficult conditions, post-mission processing techniques have to combine forward and backward Kalman filtering with additional smoothing and sophisticated robust estimation algorithms to generate a best estimated dynamic state vector. As a result, the system continuously provides three-dimensional trajectory information as external orientation with all six degrees of freedom to georeference the collected sensor data. The trajectory comprises position, height and vehicle orientation (pitch, roll and azimuth angle). The maximum internal data rate is 2000 Hz, typically 1000 Hz is used for the results, which means that at a speed of 20 m/s a new vehicle sensor orientation is available every 0.02 m. The absolute accuracy of the position varies between a few cm and a few dm, depending on GPS signal reception conditions. Due to the high-end inertial sensors, the relative accuracy leads to a standard deviation of for roll and pitch angles and for heading. In the direction of driving, the drift is below 1 of the travelled distance. The high specification of the IMU allows generating reliable orientation data even for low survey speed starting at 1 m/s. The camera module consists of a various number of photogrammetrically precisely calibrated cameras. For most configurations, four cameras are used. Two high-definition laser scanners are operated in parallel, so that both scanners digitise the driven corridor with different perspectives. The geometric concept minimises shadows, which may be caused by obstacles in the line of sight, and enhances profile density and redundancy. Maximum survey data rate of each scanner is approx. 1 million points per second with a maximum survey distance of approx. 100 m, which means that at a speed of 20 m/s, each metre of road is scanned in 3D with approx points. Each highdefinition laser-scanner covers a viewing angle of 360 with a profile frequency of more than 200 profiles per second. Each profile contains up to points with a polar coordinate distance resolution of 0.1 mm and an average single point standard deviation of 0.5 mm on the (mostly dark) road surface. Point density on the road surface in the driven lane is 2 3 mm. The photogrammetric cameras and laser scanners are mounted together with the IMU on a precisely calibrated measurement platform. The laser-scanner measurements and the IMU data are time-synchronised with an accuracy of < 10 6 s. The target requirement is the complete compensation of any vehicle movements by the IMU-based trajectory, so that any scan point practically receives its individual external sensor orientation. The resulting georeferenced and calibrated scanner data sets have to be free of disturbing vehicle movements. The time synchronisation of the cameras is a little less rigid due to technical camera reasons and can be specified with 0.1 ms. Figure 12 shows two examples for laser-scanner point clouds. The left image shows a section out of a pilot project for the ÖBB in Austria and the right image shows a section of the Lombard street serpentine in San Francisco. It is visible that Figure 12. Example of laser-scanner data.

24 Vehicle System Dynamics 899 scanner resolution allows the detection of the power lines in the upper image as well as the detailed brick stone structure of the road surface in the lower image Comparison between established road condition determination and high-end mobile mapping The well-established road condition determination systems have been specially designed for their survey purpose. A technical comparison with the high-end mapping system shows that the scanner accuracy is a little less precise compared to the laser sensor in the transversal bar, but it is still well within the spec for road condition determination. On the other hand, the profile point density of scanner data is higher and leads to times more points per transversal profile in the driven lane, so raw data of the scanner-based system have a significantly higher resolution. In addition, the road condition determination systems are only equipped with lower grade inertial sensors, so inertial orientation is poor and does not allow the processing of high-resolution point clouds. Another advantage of scanner-based digitalisation is the complete 360 coverage. Data interpretation may be done for the whole digitised road corridor. Passing cars can be automatically removed from the data. For longitudinal data analysis, the scanner data have the advantage that it is not limited to just one longitudinal profile. Inertial sensor stability allows the detailed analysis of any longitudinal profile, so that advanced and variable data analysis techniques may be applied. Examples are given in Section Comparison between rail and road condition determination Naturally, there are significant differences, but also some identical approaches. Road vehicles are of course free to move sideways, so transversal road evenness and skid resistance are of core importance for cars, but not for railways. On the other hand, the road condition survey is technically easier to handle because the road is inflexible during measurement, whereas the railway tracks and underlying ballast are dynamically warped when the train passes. This explains some differences in the survey equipment approaches, which are based on dynamic measurement principles for trains and on contactless measurements for cars. Quite identical approaches can be found to describe and analyse the longitudinal evenness and also the described problems in detecting single irregularities are comparable. For the train case, the range of speed is much higher, because the road situation is limited to a driving speed of km/h. High-end mapping systems such as MoSES can be used to map railways and theoretically could be mounted to a railway vehicle to contribute to railway condition determination. Precise engineering surveys on railways have been done multiple times, but an evaluation of the data for potential usage of IMU measurements or scanner data for ride and comfort analysis has not yet been done with the MoSES system. Measurements of road vehicle reactions as a method for indirect assessment of road geometry and irregularities are unusual. In modern road traffic, it is common to regularly observe accident black spots to detect road-related hazards. 4. Processing of measured data, characterisation and assessment of track irregularities The measuring systems described earlier deliver information on the spatial position of defined points on the two rails. Depending on the type of the system, the wavelength content of the measured signal is limited. The lower limit is given by the Nyquist frequency and the

25 900 A. Haigermoser et al. necessary anti-aliasing filtering. Typical sampling rates are between 15 and 25 cm. Therefore, short wavelength disturbances such as corrugations are not normally included in the results from track geometry measurement cars but measured by separate systems (see e.g. [16,18,19]). The upper limit of the wavelength content is determined by the measurement principle. For chord measuring systems, this is visible in the amplitude transfer function in Figure 8. Depending on the chord length at 100 m wavelength, less than 10% of the real amplitude is measured. Typical inertia-based systems are principally able to measure track irregularities up to m wavelength. All measurement systems measure the complete geometry within some wavelength range. The measured data contain track layout as well as track irregularities. For instance, a chord measurement will have a constant offset in constant radius curves. A separation is necessary for two reasons: First, the assessment values of track irregularities must not be distorted by track layout. Second, if the track geometry data are used for simulations, it must be ensured that track irregularities and track layout are not wrongly mixed. The separation is more relevant in lines with small radius curves, where the track layout elements (e.g. transition curves) are short and their length may be in the wavelength range used for the analysis of track irregularities. The distinction between and separation of long wave irregularities and curve transitions in particular is often not easy. Track geometry measurement cars use specific algorithms for the separation of these two parts, but these are not specified in the relevant standards and often not fully documented in documents available to the user of the measured track geometry. Three approaches are known: A first approach is to subtract the mean over a defined length from the measured signal. This could be twice the longer wavelength of the required filter bandwidth. The frequency response of such a sliding mean is shown in Figure 13 for 50 m averaging length. It shows characteristics of a high-pass filter with the half-power wavelength ( 3 db) at approx. 65 m. Transitions to curves with radii around 300 m and cant of 150 mm are often less than 50 m long and would be included in the so calculated track irregularities. Additionally, waves at lengths below 50 m are distorted by up to 20%. So it has to be stated that this approach is rather inappropriate. Figure 13. Amplitude transfer function of a sliding mean for eliminating track layout components in track geometry measurements.

26 Vehicle System Dynamics 901 In [66], this is done assessing the deviation from uniformity, which is defined as the average of the measured MCO values for nine consecutive points that are spaced with defined values (e.g. for the 31 chord with 7 9 ). Another approach is to use filters to separate long wavelength features resulting from track design and short wavelength irregularities. In [39], this is done by high-pass filters with 35 and 70 m wavelength. A third approach is to identify design track elements such as curves from the measured track geometry and define track irregularities as deviations from the identified design geometry. This computation is rather complicated and needs a lot of fine-tuning, even if a track database is available. In RAILab [24,75], polynomials are fitted to the measured track geometry (inertial trajectory). For vertical data, a polynomial of degree 1 (linear) is used, while horizontal data are approximated by a cubic polynomial. The fitting length is always 500 m. Problems arise if there is more than one track element within the 500 m interval, which can no longer be described by the polynomial fits. This may lead to an unreasonable behaviour of the polynomials and correspondingly unreasonable long wave content in the data. In order to be practically useable for track and vehicle engineers, the track geometry measurements need to include information on the localisation of the measurement. Usually, this is done by linking the data to the railway specific localisation information (mileposts). This may include information on special track features such as switches, crossings, bridges and stations, where sometimes not all measured quantities are available because of some characteristics of the measurement systems (e.g. track gauge in switches). Many infrastructure managers measure track geometry regularly every six months or even at shorter intervals. This produces a considerably large amount of data, which is often stored in database systems,[76] accessible via the company specific intranet. Standard evaluations are then the analysis of measurements in one position at different times, in order to study possible deterioration of the track geometry. These solutions are all specific to the supplier of the measurement system or the database and no common standards on data formats exist. If track geometry is recorded with a chord measurement system, the measured signals of longitudinal level and alignment are distorted by the transfer characteristics of the chord offset system. The process of restoring the real shape of these signals is called recolouring or decolouring, that is, removing the colour due to the chord measurement. Decolouring of a chord measurement is necessary if the track geometry is to be assessed according to EN Moreover, if track data are to be used for simulation purposes, it must not be distorted due to the chord characteristics. The distortion depends on the chord length and on the chord division. In the case of an asymmetric chord division, it also depends on the running direction of the measurement car. There are a number of methods for decolouring. A commonly used approach is to convert the signal to the wavelength domain, then multiply it with the inverse transfer function, and finally convert it back to the distance domain. A more sophisticated method has been proposed in [77], where a compensation filter is deployed. This filter can be used in online applications and is computationally efficient. Moreover, a direct conversion from one chord system into another chord system can be implemented. A similar approach is proposed in [78,79]. In [80], a method is proposed introducing a transformed curve obtained from the algebraic local curvature and a deconvolution procedure using a Wiener approach. In [33], it is proposed to invert the transfer function in the spatial domain and digital filters are defined allowing the deconvolution of short wavelength irregularities and long wavelength track layout. In [81], a method is proposed to calculate the

27 902 A. Haigermoser et al. long wavelength track geometry from chord measurement using recursive non-linear filtering based on the Bayesian approach. In all methods, the problem of zeros or small values of the transfer function has to be tackled. At such points, the measurement contains noise, but no or nearly no data. Hence, the decolouring process will strongly amplify noise, which can be seen as unrealistic peaks in the power spectral density (PSD) of the signal. In particular, the range of small wavelengths where the transfer function goes up and down can hardly be restored correctly. An amplification threshold must be set in order to avoid an overshooting of noise at the zeros of the transfer function. However, it has to be noted that the decoloured signal in such a wavelength range is mostly determined by noise. Depending on the type of the planned application case, several analysis and evaluation steps are necessary. Vehicle design: Selection of representative track irregularities for the design (determination of vehicle parameters, estimation of load cycles for dimensioning of parts, etc.) Selection of track geometry representative for homologation tests Preparation and pre-processing of data for simulations Selection of limiting extreme cases/manoeuvres (combination of singular defects) Selection of PSD for comfort simulation, structure-borne noise, rolling noise Homologation and acceptance tests Verification of track irregularities during the tests against specification Track maintenance Classification of tracks to quality classes Identification of locations where track quality limits are exceeded Inspection whether present track geometry allows safe running behaviour at the planned speed or what speed reduction is necessary Check whether present track irregularities cause a fast deterioration of the track geometry, estimation of available time until the next track geometry maintenance. Common to most of these problems is that performance indicators have to be defined which classifies track irregularities in terms of the vehicles running behaviour, for example, as good, medium, poor, and unsafe. Rules are required for the extraction of features and combining them in a way that they show a high correlation to the vehicle s running behaviour. This also means a huge data reduction, as more or less longer parts of track should be characterised by a few numbers Characterisation of track irregularities according to the standard method in Europe When Railways started to measure the irregularities on their tracks, soon there was the necessity to find indicators to characterise the quality of the measured track geometry. In many countries, methods for characterising track irregularities have been developed and implemented, leading to a wide range of used parameters and methods. In Europe, there is the intention to harmonise this by EN standards (EN series [26,30,31]).[82]

28 Vehicle System Dynamics 903 As a typical example, EN [26] defines the principal parameters as track gauge, longitudinal level, alignment, cross level and twist. For longitudinal level and alignment, three different wavelength ranges are defined as D1: 3 m < L 25 m D2: 25 m < L 70 m D3: 70 m < L 150 m for longitudinal level D3: 70 m < L 200 m for alignment Reasons for this are that on the one hand irregularities grow in amplitude with longer wavelengths. If, for example, a range between 3 and 70 m had used, the values would be dominated by the long waves. On the other hand, the vehicle reaction depends very much on the wavelength of the defect. EN [30] declares that three indicators can describe track geometric quality : Extreme values of isolated defects Standard deviation over a defined length, typically 200 m Mean value (only for track gauge) For both, extreme values and standard deviations three levels are defined: Immediate action limit (IAL) refers to the value which, if exceeded, requires taking measures to reduce the risk of derailment to an acceptable level. This can be done either by closing the line, reducing speed or by correction of track geometry; Intervention limit (IL) refers to the value which, if exceeded, requires corrective maintenance in order that the IAL shall not be reached before the next inspection; Alert limit (AL) refers to the value which, if exceeded, requires that the track geometry condition is analysed and considered in the regularly planned maintenance operations. The normative part of EN gives IALs for isolated defects and for mean track gauge. The informative part gives ILs and ALs for isolated defects and mean track gauge, and ALs for standard deviations. All values are defined for several speed categories. This method has been taken over into the Technical specification for interoperability relating to the infrastructure sub-system (INS TSI) [82] and into national rules for maintenance of railway tracks. Similar methods are used in several national rules in European countries or overseas. EN [31] also gives a method for classifying tracks with respect to their irregularities. It defines five classes whose numbers have been derived from the distribution of standard deviations from many measurements in Europe. The method of EN (standard deviations and extreme values) has the advantage of being quite simple. It is very easy to determine the standard deviation of the signals. One difficulty is the distortion of discrete defects by the required filtering (see Figure 17). Another problem is that the results depend significantly on the filter characteristics. As the typical wavelength content of the track geometry parameters grows with the wavelength, both standard deviation and maximum values are determined by signal components with frequencies around the upper wavelength of the required band pass. This leads to a relevant dependency of the result on the specific amplitude and phase characteristics of the applied band-pass filters around and above the higher half-power wavelength, which are usually the focus of the filter specification. Additionally, a track geometry characterisation and assessment using standard deviation and peak values shows a number of shortcomings: The reaction of a railway vehicle to a track defect depends not only on its amplitude, but also on its length and shape.it can easily be shown for a simple model that the force reaction

29 904 A. Haigermoser et al. Figure 14. Transfer function between longitudinal level and vertical wheel force. on longitudinal level is much higher for short wavelengths than for long wavelengths, both of the same amplitude see [83] and Figure 14. Longitudinal level, alignment, cross level and track gauge act together. The vehicle reaction depends on all four variables. The reaction may be higher in a situation where more than one quantity show high levels than in a situation where one quantity shows an even higher level. The separation into three wavelength ranges is problematic. The vehicle reacts to the full wavelength range. There may be a higher reaction in a situation where both D1 and D2 show high values than in a situation where only one of them shows even higher values. These problems are well known and have been often discussed by track engineers and vehicle dynamic specialists. A large number of publications on this question exist and different approaches have been used, mainly for track maintenance purposes Characterisation of track irregularities in distance domain versus time domain Track geometry in principle can be described in the distance domain as f (s), where s denotes the distance along the track, or in the wavelength domain as g(1/l), withl as the wavelength. Characterisation in the distance domain gives information about the amplitudes of the track geometry defects and localisation. However, signals in the distance domain cannot provide explicit information on the wavelengths or the shapes of the defects. This is often handled by looking at defined wavelength ranges. Information on the wavelength spectrum of a signal can be obtained in the wavelength domain. To obtain the wavelength spectrum, track geometry signals must be transformed

30 Vehicle System Dynamics 905 into the wavelength domain. However, the wavelength spectrum does not allow any conclusions to be drawn about the spatial location at which particular frequencies occur. The spatial information is lost when transforming into the wavelength domain. Local events can be detected using windowed Fourier transform techniques. However, as the window has a constant width, only certain frequencies can be detected with ease Overview of characterisation methods in distance domain A multitude of methods have been proposed and are in use in order to characterise track irregularities in the distance domain. Common to all are several analysis steps. Starting point is always the measured track geometry data. It may originate from chord offset systems or from inertial systems. Data from one system type can be converted into the output of the other, considering the remarks on decolouring from above. The track geometry data may be combined with some other data as line speed or curve radius. Then always three steps are applied: (1) Preparing the measured signals by filtering and general algorithms. (2) Extracting features characterising the resulting signals. (3) Evaluating statistical properties of the extracted features or combinations of them. For the first step, a number of possibilities exist and are used. We propose to classify them in the following way: (a) Purely geometrical filtering (b) Transformations by applying mathematical operations that calculate new signals (c) Application of system models, describing the dynamic behaviour of track irregularities and vehicle response This classification is somehow arbitrary and clearly some overlap between the categories exists. It is intended to show commonalities and differences between the methods. Figure 15 demonstrates this classification scheme for an assessment with methods from EN 13848, presented earlier. In the first step, the measured signal (row 1) is filtered with bandpass filters to the D1, D2 and D3 domains (only D1 is shown). We recognise that the first step is done by a purely geometrical assessment. Step 2, feature extraction is here the calculation of standard deviations within a defined length of 200 m and determination of extreme values. For the third step, EN and EN give some possibilities such as calculating the mean of the standard deviations, comparing individual section values with AL, IL or IAL or counting the number of exceedances of limit values. In row 4 of the figure, the cumulative distribution function of the standard deviations is shown, in the right figure, the peak amplitude is plotted against the number of peaks. The latter statistic is not defined within the EN series, but it may be useful for estimating the fatigue load on the running gear Preparing the measured signals by filtering and general algorithms This is the first step in the scheme introduced earlier. Filtering is usually understood as removing some parts of a signal. Here, filtering is treated more generally as an algorithm that transforms or maps (one or more) input signals into an output signal Purely geometrical assessments. This category comprises methods where the resulting output signal is still understood as a geometric quantity (mm). Most of the requirements defined in legal documents and standards belong to this category. In many countries

31 906 A. Haigermoser et al. Figure 15. Three-step process for characterising track irregularities (filtering, feature extraction and statistics on extracted features). documents exist, where requirements on track irregularities are given. Usually, this includes the required filtering, but also information and requirements for the two following assessment steps, namely which features to be extracted and how they shall be assessed (safety limits, recommended limits for track maintenance). In Table 1, some characteristics of these specifications (only for alignment and longitudinal level) are summarised. It starts with the name of the regulation or the country. The second column shows the half-power frequencies (as wavelengths) of required band-pass filters. This has to be applied to an undistorted signal either the output of an inertia-based measuring system or to decoloured signals from chord measurement systems. Columns 3 and 4 indicate chord geometries to be used. This may be implemented directly in the measuring system or for converting data from inertia-based systems to chord signals applying the respective transfer function. So, columns 2 4 describe the processing and filtering of the signals. The characteristics of the feature extraction (step 2, see 4.3.2) are shown in the last column. It should be mentioned that the table only gives a small part of the used regulations. Unfortunately, many of them are not publicly available (at least not in English language). For many countries only secondary literature reports on some details but there is no full public knowledge on all details. For the sake of completeness, it is necessary to bear in mind that most regulations also include requirements for twist, gauge and cross level. For these quantities, the same features are usually used (extreme values and sliding mean for gauge). Filtering is not necessary, because of the limited signal range and the different measurement methods. Twist is assessed on different length bases. In the first row of Table 1, the parameters described in EN are summarised. EN [1] describes the methods and requirements for the assessment of the running characteristics of railway vehicles and uses the same filter characteristics as EN The latest

32 Vehicle System Dynamics 907 Table 1. Characteristics of several track irregularity specifications. Band-pass filtering of Chord geometry (measurement or resampling) measured (inertial) or decoloured signal Lateral Vertical Extracted features EN [26] D1: y, z: 3 25 m Extreme values D2: y, z: m Standard deviation D3: y: m D3: z: m EN [1] D1: y, z: 3 25 m Extreme values Standard deviation INS TSI [82] Reference to EN RST TSI [3] Reference to EN RGS 5021 [84] y: m Extreme values z: 1 35 m Standard deviation y: m z: 1 70 m DB RiL [64] 4 + 6m m Extreme values Standard deviation SNCF see [38,85] m Mauzin 8 wheel base Extreme values Standard deviation FRA [65,66] 31ft MCO ( m) Extreme values 62 ft MCO ( m) 124 ft MCO ( m) Russia [86] m m Peak to peak within 20 and 40 m for y,6,12 and 25 m for z Australia [87] 5 + 5m m Extreme values m India [88] m m Extreme values versions of the technical specifications for interoperability came into force at the beginning of 2015.[3,82] In difference to older versions, they directly refer to the standards for infrastructure [30] and vehicle testing.[1] In Europe, we find some additional nationals standards (which have to be transformed to Notified National Rules ) and maintenance rules. Some of them are shown in Table 1, they differ by the wavelength range or by the use of chord offset signals. In the USA, Russia, Australia and India, chord offset signals are also used. All regulations require the measurement of more or less the same parameters. There are a number of regulations where signals from chord offset measurements are directly used or the signals from inertial systems have to be converted using the transfer function of a chord offset system. The respective chord geometries are given in columns 4 and 5 of the table. Some are symmetric MCO systems, some are asymmetric. A special case is the measurement of the vertical longitudinal level of the French Mauzin car: Here, eight wheels form the base of the chord, which gives a transfer function of approx. one between 2.5 and 10 m.[37] This different filter characteristic of the various chords and band-pass filters causes different results in assessing track irregularities. Figure 16 compares the amplitude transfer functions for the lateral assessments in Table 1. We see a huge difference in the function, which can be interpreted as a wavelength-dependent weighting of the irregularities. As they are amplitude transfer functions, they apply directly to standard deviations. It is evident that values as well as limit values of these different systems may not be compared.

33 908 A. Haigermoser et al. Figure 16. Amplitude transfer functions of several chord offset measurements compared with band-pass D1 according EN Even more complex is the situation for the extreme values, that is, isolated defects. Besides the differences in the amplitude transfer function as shown in Figure 16, the phase shift of the filter and the measurement transfer function also play an important role. This has been visualised in Figure 17. A discrete defect of 10 mm has been simulated as a sine wave with a length of 15 m. It is shown as a blue line in the plots. If a band-pass filter (Butterworth fourth order, 3 25 m) is applied, the shape of the defect changes as shown in the upper left diagram. Shape and amplitude of the defect are distorted. The diagram below shows the signal after zero-phase digital filtering by processing the input data in both the forward and reverse directions. The third diagram shows the result if a sliding mean over 50 m is calculated and subtracted from the unfiltered real shape. This is somehow unrealistic, as a high-pass filter always needs to be applied. The upper three diagrams on the right-hand side show the results if D2 (25 70 m) is considered. The lower two rows show results from four chord offset systems, three chords used in the USA (31, 62 and 124 ft, corresponding to 9.4, 18.8 and 37.8m) and the chord used in Germany for the lateral direction. We see long chords are able to measure the correct amplitude, which is one reason why mid-chord systems are used, although they have zeros in the transfer function and additional peaks in the other direction Transformations. Several proposals have been made to transform the measured irregularities in a more specific way than a band-pass filter or chord offset sampling. EN [31] describes the use combinations of basic irregularity parameters. As an example, the alignment y and cross level cl are summed and a new signal s = y + cl

34 Vehicle System Dynamics 909 Figure 17. Distortion of the shape of isolated defect by using different filters and chords. is created from which the required features can be extracted, for example, by calculating standard deviations in 200 m long sections. In [89], a method is presented describing how the second-order derivatives of the track irregularities, instead of the amplitudes themselves, should be used when evaluating the track/vehicle interaction. This is motivated by an analysis of the equations of motion of a simple model such as that sketched at the top of Figure 14. It is derived such that the dynamic part of the wheel load is related to the inertia forces of the system. Even though the suspended mass is normally larger than the mass of the wheelset, the force of the wheelset will be the largest contributor to the dynamic forces. This is because the acceleration of the suspended mass is normally much lower than the acceleration of the wheelset. It is also shown that the acceleration of the wheelset is dependent on the second-order derivative of the longitudinal level of the track rather than on the amplitude. Some results, presented in [89], look very promising. The results have been produced with simulations in GENSYS. In a master thesis,[90] this method was also applied to the data in DYNOTRAIN. The results were quite inconclusive, however, in some case the second derivative gave better results, in other cases the first derivative and in some cases the amplitude itself gave the highest correlations. In Austria, a Track Geometry Index (TGI) called Maschinen-Durcharbeitungs-Ziffer is used. The intention is to give an indication of the priority of works for recovering track geometry quality. The basic idea is that changes in the vehicle s centre of gravity position are relevant for deterioration of track geometry.[91] Track irregularities are interpreted as circular waves leading to centrifugal accelerations. Changes in the radius of the circular wave lead to changes in the (centrifugal) acceleration, and those changes correspond to changes in the chord offset. Therefore, Schubert [91] derives changes in accelerations from changes in the chord offsets in lateral in the vertical direction. A resulting acceleration change vector from

35 910 A. Haigermoser et al. changes of longitudinal level, alignment and cross level in a height of one metre is derived, leading to the following formula: MDZ = c 1 L/ x L v0.65 i=1 z i2 + ( y i + cl i ) 2, (2) where z i is the difference in the vertical level between two successive points (mm), y i is the difference in the horizontal level between two points (mm), cl i is the difference in the cross level or super-elevation irregularity between two points (mm), v is the train speed (km/h), c is the scaling coefficient, L is the measurement distance (m) and x is the sampling interval (m). Although originally derived from chord measurements, this is also applied to signals from inertia-based systems. The point mass acceleration (PMA) method [31] also originated in Austria and uses a similar approach. It calculates the accelerations of a point in h 0 metre height above the track, which is guided by the track irregularities. This method has been included in EN [31] The following equations are given for the accelerations The vector sum leads to a y = cv n (y δz + h 0 (z δ )), (3) a z = cv n (z δy + h 0 (δ 2 z 2 )). (4) a yz = a 2 y + a2 z, (5) with v being the maximum line speed, h 0 the height of centre of gravity, n the exponent, open for scaling, c the coefficient, open for scaling, x the first spatial derivative, x the second spatial derivative, x the third spatial derivative. In [31], the formulas are then simplified to a y = cv n (y + h 0 (z δ )), (6) a z = cv n z. (7) Another example of a transformation is the approximation of the measured irregularities by some simple forms or motifs, also called geometric parameterisation. The aim is to express the complex track geometry by a small number of parameters that represent the most important properties of the track. A simple approach could be to determine the peak values (reversal points) of the signals and use their amplitudes. However, in an approach applied to the DYNOTRAIN data,[92] it is supposed that the vehicle response will not only depend on the amplitude of the track defect but also on its length. This is obtained by approximating the track geometry by triangles of variable length and height. The height represents the amplitude and the length of the triangle represents the length of the track defect. So, the original signal is replaced by a sequence of triangles, characterised by the two values amplitude A and length L. These can be combined, for example, to a new signal A/L or A/L 2. In the DYNOTRAIN project, wavelet analysis has been studied as another type of geometric parameterisation.[92] The parameterisation using the wavelet analysis was based on the continuous wavelet transform using the Mexican Hat wavelet. In order to describe the track geometry by characteristic values indicating the amplitude and the length of the defects, the maximum values of the coefficient matrix are identified. All peaks above a defined limit value are taken into account. Every peak is defined by its energy, scale and position. The energy relates to the amplitude and the scale to the length of the track defect.

36 Vehicle System Dynamics 911 Figure 18. method. Parameterisation of the measured track irregularity signals by amplitudes and gradients in the WGB A rather complex transformation method stands behind the method called Wirkungsbezogene Gleislagebewertung (WGB) (TGA by vehicle reactions) developed by DBAG in Germany (see [31,93]). Again it applies a geometric parameterisation of the measured track geometry by dividing the signals into consecutive single defects, defined as sections between two subsequent local extreme values of the signal. The following track geometry parameters are used together with the cross-level signal cl: ȳ = 1.2 y l + y r 2 and z = 1.2 z l + z r. (8) 2 The additional factor of 1.2 has been introduced to compensate effects from using the mean of the two rails.[31] To avoid assessing too short and irrelevant wavelengths, a minimum distance between the extreme values should be defined (e.g. 1.0 m). Thereafter, for each single defect of ȳ, z and cl signals, the characteristic parameters amplitude (amp) and gradient (gr) can be derived separately according to Figure 18. Additional input parameters in the signal processing are the absolute value of the locally measured curvature of the track (C H ) and the relevant permissible speed (V). These input parameters are combined according to the following formula (assessment function) resulting in a set of new signals as: R jk = a jk + (b jk ȳ gr + c jk z gr + d jk cl gr + e jk ȳ amp + f jk z amp + g jk cl jk )V, + h jk V 2 C H + i jk V + j jk C H (9) R jk may be interpreted as the local utilisation of the limit value of the vehicle response quantity j of the reference vehicle k. The six vehicle response quantities are Sum of lateral guiding forces per wheelset: Y, Quotient of lateral and vertical contact forces per wheel: Y/Q, Maximum vertical wheel force: Q max, Minimum vertical wheel force: Q min, Maximum lateral car-body acceleration (at floor level): a y, Maximum vertical car-body acceleration: a z.

37 912 A. Haigermoser et al. Figure 19. System view of track irregularities and vehicle reactions. The coefficients of the assessment function (a jk, b jk, c jk, d jk, e jk, f jk, g jk, h jk, i jk, j jk ) above have been determined by a number of multibody system simulations. Combinations of sine waveshaped defects of ȳ, z and cl with varying lengths and amplitudes are applied in different situations of curvature and vehicle speed. The coefficients of the assessment functions itself are then calculated by solving the over-determined system of equations formed by the input parameters in combination with the related output values of the multi body system (MBS) simulations. The coefficients have been determined for five different vehicles including a high-speed train, a locomotive and freight wagons. Features such as standard deviation or maximum values may be extracted from the signals R jk and analysed statistically Systems modelling of the vehicle response. The intention of assessing track irregularities is to obtain good correlation between the result of the assessment and the vehicle reactions. It is therefore obvious to understand the interaction of track irregularities and vehicle as a system with the track irregularities as inputs (together with speed and track layout) and the vehicle reactions (forces between wheel and rail, accelerations in the vehicle) as output. The system may then be understood as an object that maps the input signals into the output signals. Figure 19 shows a scheme of the system describing the relationship between track geometry and vehicle reaction. H is the system model or system function. System theory is a well-founded and widely used science for describing and studying these relationships. The majority of the theory is for a special case of system: the linear time-invariant (LTI) systems. If the characteristics of the vehicle track system are known and describable as a system, it can be used to estimate the vehicle reaction, which may be interpreted as assessment quantities of the track irregularities. In principle, most of the state-of-the-art system models (see e.g. [94]) used in other fields can be applied for the task of track irregularity assessment. There have been a number of proposals and approaches to apply system models in order to estimate the effect of track irregularities and assess them. The differences in the methods come from the manner in which the system H is mathematically formulated and how the parameters of the system model H are determined. All systems proposed are discrete (time) systems. Clearly, it is also possible to use an MBS simulation and solve the full non-linear equations of motion and not rely on transfer functions and digital filters. In [95], it was proposed to solve the equation of motions of a simplified vehicle model in real time to estimate the vehicle response and assess the track geometry conditions. This should then be done for different vehicle types and different speeds. Clearly, this is also possible with more complex models and offline computing. ORE C152: Possibly, the first approach in this direction in railways originates from the 1980s. In the ORE (Office for Research and Experiments) specialist committee for Question C152 (Quantitative evaluation of geometric track parameters determining vehicle behaviour), two methods have been studied [96]: Direct analysis of relationship between measured track irregularities and measured vehicle reactions as a multiple input-single output (MISO) system.

38 Vehicle System Dynamics 913 Mathematical modelling of the dynamic system, deriving transfer functions and the impulse response functions. Rather, simple recursive and non-recursive filters have been derived and applied as weighting functions on the measured track irregularities. The first method is based on an analysis and comparison of the track geometry and vehicle reaction in the wavelength domain. Transfer functions for the system are calculated using statistical signal processing methods. It is mentioned that this approach requires that the system is linear, that the signal-to-noise ratio of both measurement systems (track geometry and vehicle reaction) is sufficient and that both measurements are synchronous. In terms of categories mentioned earlier, the system here is described and applied in the wavelength domain as a transfer function that has been derived from measurements. In the second approach, the theoretical possibility use of infinite impulse response (IIR)- and finite impulse response (FIR)-filters, calculated from the impulse response, of the system is described. FIR filters lead to a high number of filter coefficients with some practical problems. In terms of categories mentioned earlier, the system here is described and applied in the distance domain as a filter, which has been derived from the systems analytical equations of motion. VRA: In[11], a method called VRA (vehicle response analysis) is described as part of the Track-Recording Systems in the Netherlands. It calculates the vehicle reactions in real time during the track geometry measurements. The response components comprise passenger comfort, as well as Q and Y forces for three vehicles at five speeds. This method was introduced in 1988 and is a method that mainly works in the wavelength domain. In a first step, the measured track geometry is transformed into wavelength domain for sections of 600 m length. A linear multiple-input-multiple-output system is assumed, allowing the calculation of the response of the vehicle to each input signal by multiplication with the related transfer functions. This has to be done for each input and output and the different model variants (vehicle types and speeds). The resulting outputs in the wavelength domain are transformed into the time domain and added (superposition). This is possible, because the system is assumed to be LTI system. In terms of categories mentioned earlier, the system here is described and applied in the wavelength domain as a transfer function, and it has been derived from analytical derivations or from simulations. Pupil: Later, also in the Netherlands at Lloyd s Register, another method for characterisation of track irregularities was developed; the method is called Pupil.[97 99] It uses digital (assessment) filters, specific for several types of vehicles, to analyse track geometry quality. The inputs for the assessment filters are the different parameters describing the track geometry, such as cross level, lateral and vertical alignment together with curvature. The assessment filters describe SISO IIR (single input single output infinite impulse response) transfer functions. The order of the transfer functions is rather low (typically sixth order). The output of these filters is a percentage of the allowable limit. Filters are applied for a series of speeds. The assessment filters were created using system identification techniques [94] applying MATLAB s System Identification Toolbox. For this purpose, simulations were performed in ADAMS/Rail with a defined set of vehicle types (passenger and cargo trains). Theoretical track irregularities were used as inputs, containing all relevant frequencies. Using the input and the simulated output, transfer functions of the vehicle models for different types of irregularities were identified.

39 914 A. Haigermoser et al. The application of these vehicle-dependent assessment filters is then done in the distance domain. The assessment filters give utilisation levels of limit values. They estimate the utilisation for a number of different vehicles at different speeds.[99] In terms of categories mentioned earlier, the system here is described and applied in the distance domain and it has been derived from simulations. Work in Sweden: Work was performed at the former Banverket in Sweden (Swedish railway infrastructure manager, today Trafikverket) and at the Royal Institute of Technology (KTH), also based on filters representing the vehicles transfer behaviour.[ ] This work started with investigations on the vertical behaviour using detailed track models and rather simple vehicle models. The work was also expanded into the lateral direction, but again with a simple vehicle model.[100] The authors used simulation results as data for their experiments. Also, system identification methods were applied to estimate discrete filters using the output error method.[ ] In terms of categories mentioned earlier, the system here is described and applied in the distance domain and it has been derived from simulations. ETF: At the Virtual Vehicle Research Center in Austria, a method called ETF (empirical transfer functions) was developed in research projects between the centre and Siemens and voestalpine.[ ] The ETF-method utilises multibody simulations, MBS and system identification to create typical transfer functions between measured track geometry and vehicle response. Vehicle responses are created by the use of multibody simulations with measured track geometry as input data. When the vehicle response for different sets of track geometry data has been produced, system identification is used to create representative transfer functions between the various input parameters (track irregularities) and output parameters to be investigated. In contrast to methods mentioned before, a greater number of simulations on different track irregularities have been carried out for each of the required speeds and curve radii. The typical transfer functions are then calculated by smoothing and averaging the complex transfer functions. Transfer functions are determined for each combination of input and output quantities and each speed and track layout condition. With these ETFs, the vehicle response can be calculated based on a given set of measured track irregularity data, leading to an estimation of the vehicle reaction. It can be normalised by relating it to the limit values. Work in Japan: In[107], system identification theory and methods from [108] are applied to identify the dynamic characteristics of the vehicle. Track irregularities and vehicle reactions have been measured on a shinkansen test train. In [109], models as described in formulas (11) and (12) are used with the input longitudinal level (mean of the left and right rails) and the outputs acceleration in the car body and dynamic wheel load change Q. Details on system identification are described in the following. The authors compared different methods from [94] implemented in [108] and state that the predicted vehicle behaviour has higher correlation with actual responses and will provide a more suitable TGI than amplitude. In [107], this was extended to lateral accelerations and lateral wheel rail forces. It is stated that the relationship here is not so clear because of the non-linearity of wheelset motion and alignment caused by the flangeway clearances, the non-linearity of wheelset or bogie motion and alignment because of its snaking motion, the combination of centrifugal force and forced wheel movements caused by track irregularities and the combination of alignment and cross-level irregularities. Three prediction models have been suggested: low-frequency lateral acceleration due to dynamic centrifugal force (variations of curvature), middle-frequency lateral acceleration due to track irregularities and middle-frequency lateral force due to track irregularities.

40 Vehicle System Dynamics 915 In the first model, the input signal is the non-compensated lateral acceleration calculated from the measured radius and cant, while the output signal is the dynamic non-compensated lateral acceleration. The radius is derived from the 10 m MCO signal. This model is applied for wavelengths longer than 25 m where a high correlation for non-tilting trains has been found. The models for predicting middle-frequency lateral acceleration and laterals force are statespace models based on formula (12). The inputs are the irregularities for alignment, cross level and gauge. Curvature is included in the alignment signal as this is a chord offset signal. A strong correlation between predicted and measured lateral accelerations is shown in [107]. For lateral forces, a sufficient similarity between predicted and measured values is claimed. In terms of categories mentioned earlier, the system here is described and applied in the distance domain, and it has been derived from measured track irregularities and vehicle reactions. Studies in DYNOTRAIN: Several studies on system identification models intended to define assessment filters have been carried out in the EC-funded research project [110]; on the one hand, several methods from the MATLAB System Identification Toolbox (methods from [94]) have been studied; on the other hand, adaptive filters have been used to estimate system behaviour.[111] System identification (estimation of the filter parameters) has been done for both methods using simulation data and measured data. Polynomial models as well as state-space models have been applied. Polynomial models use a generalised notion of transfer functions to express the relationship among the input, u(t), the output y(t), and the noise e(t) using the following equation: A(q)y(t) = n u i=1 B i (q) F i (q) u i(t nk i ) + C(q) D(q) e(t). (10) The functions A, B, C, D and F are polynomials expressed in the time-shift operator q 1. u i is the i-th input, n u is the total number of inputs and nk i is the i-th input delay that characterises the transport delay.[94,108] Not all the polynomials have to be simultaneously active, leading to a number of different configurations of models. The form n u A(q)y(t) = B i (q)u i (t nk i ) + e(t) (11) i=1 is the simplest one and is called AutoRegressive exogenous (ARX). Other types studied were ARMAX (AutoRegressive Moving Average exogenous), BJ (Box Jenkins) and OE (Output Error). State-space models are models that use state variables to describe a system by a set of firstorder differential or difference equations, rather than by one or more nth-order differential or difference equations. The discrete-time state-space model structure can be written as x(kt + T) = Ax(kT) + Bu(kT) + Ke(kT), y(kt) = Cx(kT) + Du(kT) + e(kt), (12) x(0) = x0, where T is the sampling interval, u(kt) is the input at time instant kt and y(kt) is the output at time instant kt. Experience in DYNOTRAIN was that these methods worked very well on data provided by simulations. The best results were achieved when results from single input simulations

41 916 A. Haigermoser et al. Figure 20. System identification by adaptive filter. (y, z or z neglecting the input y) were used to identify three single-input-single-output (SISO) models, whose outputs are then superposed. Good results were also achieved when the simulation was done with all four irregularity inputs applied simultaneously and a MISO model was identified. If the behaviour is non-linear, larger differences can be found between simulation data (measurement) and estimated data (if the identified model is applied on the input data). This could be improved by applying non-linear system identification methods, leading to significantly more complex models. If the described methods are applied to measured data, some additional problems appear. The available data were measured by different systems. Input data (track irregularities) come from a track-recording car and is sampled in the distance domain. Response data are sampled in the time domain. Although both measurements have been done at the same time, these two systems are some 100 m away. The measurements have to be synchronised and resampled to a unitary sample rate. The problem is that the identification methods require perfect synchronisation. Even a varying delay of some samples may lead to poor identification performance. This problem and the fact that other influences not included in the input variables influence the response lead to a result that the differences become larger if measured data are used. This has been studied in detail in [112]. The other approach studied was the use of adaptive filter techniques and describe the system as a FIR filter in the form M 1 y[k] = w i x[k i], (13) i=0 with M as the length of the filter and w = w 0, w 1,..., w M 1 as the filter coefficients, intended to describe the system. Estimation of the coefficients can be done using the adaptive filter technique, in the simplest approach with the LMS (least mean square) algorithm. Figure 20 shows the scheme of this method. In the identification process, the input signal x[k] is filtered with the adaptive filter (FIR filter w[i]) and its output ys[k] is compared with the measured output of the unknown system y[k]. The difference is used to adapt the filter coefficients, which will be applied to the system input in the next time step. This is repeated until the error signal is below some limit combined with other conditions. There are several adaption algorithms, we used the basic LMS-approach. After completion of the identification process, the filter coefficients w[i] may be applied to other measured input signals and the system response will be estimated and may be used as an assessment quantity for the track irregularities. This approach is applicable for SISO and for MISO systems. The identification was done using SISO simulation data, MISO simulation data and measured data, which is always MISO. Again, an important problem concerning measured data is the required perfect synchronisation of input and response data. As in this case FIR filters are used, the length of the filters is rather high.

42 Vehicle System Dynamics 917 All methods described so far assume a LTI system behaviour. If the system identification is based on simulations, these are preferably from non-linear models and the simulation cases (experiments) should be carefully chosen. During the system identification, using linear models the system is implicitly linearised, although the residual will be larger. Methods for identifying non-linear system are available,[94] first studies about the application to measured data from DYNOTRAIN showed no improvement compared to the linear models.[110] Feature extraction After the measured signals have been processed and filtered some features are extracted from these new signals. The appropriate extracted feature is given in the last column of Table 1. In the compilation of this table, all methods use standard deviation and peak values as features. There are some others, but the vast majority uses these two. Standard deviations are calculated over defined track length, in Europe mostly 200 m.[31] EN requires the calculation in its test sections, whose length depends on the speed and is between 70 and 500 m. Some infrastructure managers use sliding windows of 200 m for the calculation of the standard deviations,[113] others successive sections of a defined length. There are many other possibilities for features to be extracted. Within DYNOTRAIN higher order central moments of the distribution such as kurtosis and the skewness of the track irregularity signals have been studied. Just as arbitrary percentiles may also be extracted, for example, 99.85%, 97.5%, 75%, 25%, 2.5% or 0.15% percentile. More complex features have been developed in the USA. In a research project initiated by the FRA Office of Research and Development, a set of track quality indices has been developed based on a unique, length-based methodology. The basic concept is the use of space curve length to represent track quality. Space curves are generated by track geometry measurement systems on a sample-by-sample basis. For a specified track segment length, the rougher the track surface is, and the longer the space curve will be when stretched into a straight line. The length of a space curve is estimated as the sum of the straight distances between two consecutive data points, as shown in the following equation: n L s = y 2 i + x 2 i, (14) i=1 where L s is the traced length of the space curve, x i is the sampling spacing and y i is the difference in two consecutive measurements. A set of quality indices, each for profile, alignment, cross level and gauge, are computed using the equation for track segments of 528 ft (one-tenth of a mile) ( ) Ls TQI = , (15) L 0 with L 0 as the theoretical length of the segment. In [114], TQIs from some 10,000 track segments have been analysed and their distributions fitted against theoretical distributions. It was found that Gamma and Weibull distributions give the best results. In [115], the results of a study are presented that was conducted to explore the use of fractal analysis of track geometry data for the extraction of features as an indication of the track geometry condition. Fractal analysis, that is, the process used to determine the fractal dimension, was presented by Benoit B. Mandelbrot to characterise those patterns within nature that are irregular, chaotic or fragmented and cannot be effectively quantified using classical geometry of whole-number dimensions. There are various techniques for determining the fractal dimension of rough patterns. In [115], the divider method was used. The divider method is

43 918 A. Haigermoser et al. based on the empirical studies of coastlines, and was used by Mandelbrot to quantify curves whose fractal dimensions were greater than one ( > 1.0). The divider method is based on an equation that expresses the length of a rough line by L(λ) = nλ 1 D R, (16) where λ is the unit of measurement, L(λ) is the length of the rough line based on the unit measurement length λ, n is the number of steps and D R the fractal dimension of the rough line. The fractal dimension of a signal (which can be interpreted as a rough line) can be determined by varying the divider or ruler length λ calculating the length L(λ) and fitting the calculated values by a straight line in a log (L(λ)) versus log(λ) diagram. The slope of the straight line gives the fractal dimension. In [115], data from FRA MCO-measurements have been analysed and they often showed a bi-fractal behaviour with two distinct linear portions of the data and two different fractal dimensions. The first-order dimension D R1 is associated with the relatively large step-length values and the second-order dimension D R2 is associated with the smaller scale roughness. It has been suggested to use these two values possibly together with the location of the fractal breakpoint as numerical parameters for the characterisation of the irregularity. These parameters have been calculated for measurement data and compared with the traditional Running Roughness R 2, which is actually the standard deviation of the longitudinal level. Since fractal dimensions are pattern based, whereas Running Roughness is strictly magnitude-based, the relationship between them is not a constant. A specialist committee of the European Rail Research Institute (ERRI) studied the relationship between track geometry and vehicle reaction.[116] Here, a feature named narrow band wide band (NBWB) parameter was studied that has been defined as [116] NBWB = max{ C xx[p] } p p0, (17) C xx [0] where C xx is the autocorrelation function and p 0 a threshold value.[116] This parameter is an indicator for the predictability of a signal. For white noise, this parameter is zero, for a sinusoid it is equal to one. Figure 21 shows examples for results of feature extraction. For three sections, all 600 m long the aforementioned features have been derived. The three signals are plotted in the first line. They show some differences in the characteristics, although the overall level seems to be similar. The features of the three signals are compared starting at the second line. The first is standard deviation, where the differences are rather small. The second shows the maximum values of the absolute signals. Now, the differences are higher, as this parameter is very specific. The third parameter is the kurtosis. Section 2 has a value around three, which corresponds to the kurtosis value of a normal distributed signal. Section 3 with the high value has more high peaks included, Section 1 is more harmonic with a lower kurtosis. The FRA-TQI values of the signals give similar relations as the standard deviations. The differences in fractal dimension are very low. Again Section 3 gives the highest value. Finally, the NBWB values indicate a rather high similarity to white noise, especially for Sections 1 and Feature combination Before statistics are applied to the features and comparisons to limit values are carried out, sometimes features are combined in order to define new ones. In Section , examples

44 Vehicle System Dynamics 919 Figure 21. Example of feature extraction. have been presented where measured signals have been combined by a mathematical operation to a new signal. There are many rules and proposals where extracted features (mostly standard deviations of different irregularities) are combined to a new feature. A combined standard deviation is defined in EN as cosd = w y σy 2 + w gσg 2 + w zσz 2 + w clσcl 2, (18) with σ as the standard deviation and w as the weighting function. Indices give the direction for y as alignment, g for gauge, z for longitudinal level and cl for cross level. In [117], the concept of Track Geometry Interaction Map (TGIMP) has been introduced. The combined influence of surface (longitudinal level) and alignment irregularities has been studied in simulations. Lines of constant vehicle reaction in figures of the two track irregularities have been produced for several vehicle response parameters such as lateral force Y, vertical force Q, Y/Q and accelerations. These iso-lines can be approximately described by a new track geometry parameter, which is called TGIMP [( ) y m ( ) z m ] 1/m TGIMP = +, (19) y 0 z 0 y and z are the peak to peak values of the track irregularities, y 0 and z 0 are respective limits of alignment and surface variations. TGIMP 1.0 represents safe operating conditions and the safe limit is bounded by TGIMP = 1. In the aforementioned formulation, the strength of the lateral/vertical interaction can be adjusted by the constant m, m 1 for weak interaction, m 1 for strong interaction and 1 < m < 2 for moderate interaction (m = 2 giving a circular or elliptic contour and is equal to the combined standard deviation as shown earlier, if standard deviation is used as feature). The approach has been tested by many simulations; a validation by measured results is not given in [117]. Recently, FRA introduced a new parameter called combined Track Alinement and Surface Deviation.[66] It is defined as y + z y, (20) 0 z 0

45 920 A. Haigermoser et al. where y is the measured alignment deviation (outward is positive), y 0 is the limit value according to the track class, z is the measured surface deviation (downward is positive) and z 0 is the limit value according to the track class. This parameter is assessed in curves that are operated at cant deficiencies greater than 5 in. (127 mm) and in all class 9 tracks. It is limited to 3 y 4 + z y 1. (21) 0 [118] propose another combined index called synthetic track quality coefficient J defined as z 0 J = σ y + σ z + σ tw + 0.5σ g, (22) 3.5 where σ tw is the standard deviation of the twist on a base of 5 m (change of cross level cl over 5 m). This method is used in Poland. [119,120] report on proposals in India to use TGI defined as TGI = 2UI + TI + GI + 6ALI, (23) 10 where UI, TI, GI and ALI are the indices for unevenness, twist, gauge and alignment, respectively. The calculations for the different parameters are obtained by UI(Unevenness Index) = 100 e (σ z,mes σ z,n /σ z,maint σ z,n ), (24) TI(Twist Index) = 100 e (σ tw,mes σ tw,n /σ tw,maint σ tw,n ), (25) GI(Gauge Index) = 100 e (σ g,mes σ g,n /σ g,maint σ g,n ), (26) ALI(Alignment Index) = 100 e (σ y,mes σ y,n /σ y,maint σ y,n ). (27) The second index indicates the source of the standard deviation: mes means measured value, n means the value prescribed for newly laid track and maint means the value prescribed for urgent maintenance. Other forms of combining standard deviations are used in Sweden. The Q-value is a weighted index of the standard deviation of the geometric parameters derived from comfort limits for the different track classes. It is calculated as Q = (σ H/σ Hlim ) + 2(σ S /σ Slim ), (28) 3 where σ H is the mean of the standard deviations of the left and right rails longitudinal level, σ Hlim is an allowable value depending on speed and line category, σ S is the mean standard deviation of alignment, gauge and cross level and σ Slim is a limit value. The ERRI-committee C201 studied the relationship between track geometry and vehicle reaction.[116] The proposal was to apply Neural Networks (NN) to describe the non-linear relation between track irregularity parameters and vehicle response parameters. Analysis was divided into sections with stationary behaviour and sections with instationary characteristics. Algorithms for detecting these section types are given in [116]. For these sections, features such as σ 2, max, NBWB of track irregularities and vehicle response are calculated and the relationship of inputs (track irregularity parameters, speed and curvature) and vehicle response parameters have been modelled with neural networks. The NN model gives then, as a result a non-linear combination of the extracted features, the values of features estimating the vehicle responses to the track irregularities.

46 Vehicle System Dynamics 921 Figure 22. Schematic neural network architecture for relating track irregularities to vehicle response. A similar approach has been developed by the Transportation Technology Center, Inc. (TTCI).[121,122] It is called performance-based track geometry (PBTG) and uses the NN technique to relate real-time track irregularities to vehicle response. Neural networks are composed of simple elements operating in parallel. These elements are inspired by biological nervous systems. As in nature, the connections between elements largely determine the network function. Figure 22 shows a simplified NN architecture used in [122]. The input variables include curvature, super-elevation and cross level, vertical and lateral irregularity of rails, gauge and speed. The output variables are vehicle response parameters, including vertical and lateral wheel loads, and Y/Q ratio. For each track section, a set of input and output parameters are derived, which are statistical values such as maximum, 98% and 95% percentile or minimum, 2% and 5% percentile. The track section should be long enough, to include the full track irregularity and the maximum vehicle reaction in one section,[122] says that 40 m length was found to be too short. The relationship between an output, F(I), and all inputs can be expressed in terms of the following equations: I = w 0 + w 1 X 1 + w 2 X 2 + w i X i +, (29) 1 F(I) =. 1 + e I (30) In PBTG, the log-sigmoid transfer function is used, but various other functions F(I) are possible. Neural networks have to be trained to perform a particular function by adjusting the values of the connections (weights) w between elements. This is done by data where both the input data and the output data are available. The NN-training procedure consists of two passes through the different layers of the network: a forward pass and a backward pass. In the forward pass, an input vector is applied to the nodes of the network. The effect produced by applying that input vector propagates through the network layer by layer. Consequently, an output is produced as an actual response of the network. If the actual response of the network is not close enough to the desired response, a backward pass takes place and the weights are all adjusted iteratively. This process is referred to as the error correction rule to make the actual response move closer to the desired one. So, it is essential that an input/output database is available from test results. Creation of the NN training database is the major part of the NN modelling effort. The major characteristics of the vehicle behaviour have been present in the training data. NN are able to model non-linear characteristics and, due to their statistical nature, also take into account non-quantified track characteristics on their statistical distribution in the training data. In [122], results from validation by comparison of estimated and measured vehicle response quantities are presented. The agreement is claimed to be acceptable, although there

47 922 A. Haigermoser et al. are differences in magnitudes which, however, can be attributed to factors such as difference in speed. In addition, un-quantified track parameters (rather than track geometry) could also affect instrumented wheelset test results. PBTG can be implemented in track geometry measuring cars as an add-on (PBTG black box). It then identifies track segments that may produce vehicle responses above the predetermined limits (PBTG defects) and generate recommended maintenance actions to them Statistics on features The last column of Table 1 shows which features are derived in order to assess the track irregularities via a comparison with defined limits or some more complex assessments. The standards and regulations (e.g. those summarised in Table 1) give limit values for the extracted features. Often different levels of limits are given, for example, AL, IL and IAL in EN [30] Some standards take also the number of exceedances of defined values into account. Additional assessments are applied for twist, gauge and cross level. Because of the different filtering and differences in chord geometry, it is not easy to compare limit values in the different countries. In [123], limit values of extreme values from the USA, EN 13848, China and Japan are compared. These standards and the regulations in Table 1 can only be compared if a shape and a length of the fault are assumed and the full processing sequence is simulated. The numbers themselves may not be directly compared. EN requires a certain distribution of track geometry during vehicle testing for homologation. It also specifies track geometry during the tests together with the other test conditions. The values of track geometry in test sections (length depends on the speed and is between 70 and 500 m, constant curvature and speed) are assessed against quality levels QN. Sections with maximum values above QN3 may be excluded from the analysis. The distribution of the standard deviations should be such that the 50% value is below QN1 and the 90% value is above QN2. For track geometry maintenance, most European infrastructure managers calculate standard deviations over 200 or 250 m length, either sliding or succecutive. On these some statistics are applied as mean for a specific length of track, cumulative frequency distribution, percentage or number of exceedances of some limits. Most infrastructure managers monitor extreme values and compare them to some limits and count exceedances of the limits. Figure 23 shows an example with several types of feature statistics. Track irregularities from three tracks with local permissible speeds of 160 km/h are shown. The length of the tracks varies between 70 and 90 km. The first row shows the measured signals of the right rail. The irregularities seem to be on an equal level, with slightly lower values in track B. As proposed in EN ,[30] the tracks are divided into 200 m long sections and a number of features as standard deviation and maximum value in the wavelength range D1 (3 25 m) are calculated. This has been done for rail irregularities (y R, y L, z R, z L ) and for track irregularities (y, z, y, z). In the second row, the cumulative frequency distributions of track irregularities are shown. For all three quantities, there are some differences in the shape of the distributions and also in the ratios between the three tracks. For cross level, the three tracks are rather similar, for longitudinal level the differences are quite high. Track C has higher values at the upper end of the distribution, in other words, a small number of sections with higher irregularities. The distributions of the rail irregularities may be compared with the class limits from EN [31] in order to sort the sections into track geometry classes. Results are given in the left diagram of the fourth row, which shows the percentage of track sections with standard deviations in class E, the highest class in EN This may be done for lateral

48 Vehicle System Dynamics 923 and vertical irregularities and shows that track A and C have 15 20% of the sections in class E. Track B is better with a smaller amount of sections with large irregularities. The third row shows some statistics on peak values. Local peaks (positive side and negative side) have been determined, where a local peak is defined as a data sample that is larger than its two neighbouring samples. The distribution of the peak values has been calculated. The diagrams in row 3 show the peak amplitudes against the number of peaks per km. This is a relationship that is relevant for fatigue assessment. The presentation is similar to the S N curves (stress amplitude number of cycles) used for fatigue analysis. It shows a similar shape as the S N curves (relation strength S and number of cycles N), namely a mostly linear characteristic if the x-axis is in a logarithmic scale. In the left diagram, again the specific characteristic of track C is visible with a small number of high amplitude cycles. The rightmost diagrams of the last row show some statistics concerning the maximum values in sections. The percentage of sections exceeding the AL of EN (middle diagram) and the percentage of sections exceeding the IL (right diagram) are plotted. This is a different approach, as here only the number of sections exceeding a defined limit is counted and not the value itself or the number of exceedance events Quality of the assessment methods The aforementioned listing shows that a lot of different methods have been proposed, all intending to improve the assessment methods of track irregularities. Some use results from simulations; others use measurements as proof of the proposed method. But there are no general and comparable assessments available. Such a comparison was one of the aims of DYNOTRAIN. In 2009, the EC-funded research project DYNOTRAIN started to study the problem of describing track geometry together with other important tasks in the field of rail vehicle dynamics.[92] Project partners were major European railway operators, infrastructure managers, rolling stock manufacturers and universities. The project was coordinated by UNIFE and ended in In work package 2 of DYNOTRAIN, standard and alternative methods for describing track geometry quality were studied and new ones were developed. The aim was to find out which Figure 23. Comparison of three tracks by statistics on features: Row 1: distance history, Row 2: cumulative distribution of standard deviations, Row 3: peak statistics, Row 4: exceedance statistics.

49 924 A. Haigermoser et al. Figure 24. Left: Maximum of sum of lateral forces in track sections versus standard deviation of alignment (EN evaluation). Right: Maximum of vertical accelerations in car body versus standard deviation of longitudinal level. Both locomotive BR 120, straight track, 190 < v < 210 km/h. method gives the best correlation to the force reaction of typical vehicles. Additionally, the measured track geometry from on-track tests together with track geometry provided by the partners was analysed in detail. A toolset for creating a track geometry database compliant with the requirements of [1] for vehicle testing was developed. Another focus of this work package was on methods for estimating the track geometry on which reference vehicles are able to operate safely, as well as on methods for transforming test results to other track geometry conditions. A public report on these activities is available at UNIFE.[110] We used results from on-track tests conducted in work package 1 [124] for the study on alternative methods for describing track geometry quality. These tests were carried out in Germany, France, Italy and Switzerland with synchronous measurement of track geometry, rail profiles and vehicle reaction. Six vehicles including a locomotive, a passenger coach and four freight wagons were tested with a total of 10 load measuring wheelsets and some 300 measurement channels. In [92], the use of these measurements to study the effectiveness and quality of different TGA methods is reported. A summary is given in the later text Method to compare and evaluate TGA methods. Comparing and evaluating TGA methods are not that straightforward. A simple comparison in scatter diagrams or statistics as shown in Figure 24 (showing track irregularity against vehicle response in track sections) is only possible if the range of all other parameters is small. As it is important to evaluate methods for different conditions (wide speed range, straight track, curves, etc.), a statistical approach was taken. Figure 25 shows the principle of the method. It uses the vehicle assessment methods of EN 14363, where the vehicle behaviour is characterised by maximum, mean or rms values in track sections of m length. A multiple regression model is applied to these statistical values with a defined set of input variables such as speed, curve radius, cant deficiency, equivalent conicity or the radial steering index (see UIC leaflet 518 [125]). For dynamic vehicle assessment values, a TGA result with one or more outputs is included in the regression model as additional input variables. TGA is here understood to be a combination of steps 1 and 2 from 4.3. The TGA method and its parameters are then varied. The TGA that describes the vehicle behaviour in the best way is deemed to be the best one. The evaluation software tools, used by the test centres specialising in tests according to EN 14363, are not normally able to do such an analysis. Therefore, a complex software system

50 Vehicle System Dynamics 925 Figure 25. Method to assess track geometry description methods. had to be developed to do this analysis. It is completely open to new or changed TGAs or to new test results or even simulation results. It includes the full evaluation method from EN including signal processing, handling of track layout, statistical analysis on section values, etc. A difficult problem was the synchronisation of the different measuring systems (three groups of vehicle reaction measurements, track geometry measurement, rail profile measurement and track layout files). The difficulties originated from differences in the sampling method (time based versus distance based), as well as from several distortions of the synchronisation signals. As the data were also used for system identification and the development of filters modelling the vehicle transfer function behaviour, the requirements for good synchronisation are very high. A large amount of test data is available within DYNOTRAIN. In total, more than 1500 km (vehicles with v max = 200 km/h) or 1000 km (freight vehicles) can be used for the analysis. The analysis was undertaken for two test zones straight track and curves of all radii and for each vehicle with the instrumented bogie in leading and trailing position. Therefore, there is not one single result of the comparison of methods, but many results depending on the vehicle, test zone, instrumented bogie position, and other factors. Figure 26 shows an example of this assessment from [92]. The vertical acceleration in the car body measured in sections in straight track is analysed. The upper plots show the assessment without an input variable for track geometry. As straight track sections are analysed, only speed is used as the input variable. The middle red line gives the regression line together with its confidence interval (inner dashed lines). The outer red dashed lines give the confidence intervals for prediction (confidence level 90%). We see some dependency on speed, but the unexplained variation is very high. The upper right figure shows the fitted values from the regression against the residuals (differences between measured values and values on the regression line). The residuals are very high compared to the fitted values. Both diagrams clearly indicate that the regression is not able to describe the behaviour correctly; the goodness of fit is low. For our purposes, we need a metric to measure this visual impression. One possible metric of a regression s significance is the coefficient of determination R 2 which is the ratio between explained variation and total variation. It is defined as R 2 = variation of the fitted values n variation of the measured values = i=1 (Ŷ i Ȳ) 2 n i=1 (Y i Ȳ) 2, (31)

51 926 A. Haigermoser et al. Figure 26. Multiple regression on vertical acceleration in car body, loco, straight track: top no TGA, bottom standard deviation D1 as TGA. with Y i being the measured value of the assessment variable in section i, Ȳ the mean of the measured values of the assessment variable, Ŷ i the fitted value of the assessment variable in section i and n the number of sections. The value of R 2 varies between 0 and 1, where 1 means that the complete variability is explained by the regression. One problem of R 2 is that it automatically and spuriously increases if additional input parameters are included in the model. This is relevant here as we include TGAs with different numbers of output parameters. Therefore, the adjusted R 2 is used in the analysis (see [92] for details). Another metric, used in [92], is the root mean square error (RMSE) of the regression, which measures the spread of the output values around the regression. In Figure 26, the results for the coefficient of determination and for the RMSE are given in the diagram titles. The small slope of the regression together with the high residuals lead to a small coefficient of determination R 2 = and a high RMSE of 0.154, which is in the range of the fitted values. In the lower plots of Figure 26, the standard deviation of the longitudinal level of track geometry (higher value of both rails) is used as an additional input parameter in the regression model. The lower left figure again shows the output variable z max against the input variable speed (the same diagram as aforementioned). The middle lower diagram shows the same output values against the TGA y D1,std. Here, we find a significant dependency, indicated by a R 2 = 0.64 (simple regression). If a multiple regression is applied to both parameters, the adjusted R 2 reaches a value of The residuals are smaller indicated by RMSE = 0.09 and the range of fitted values is much larger. We clearly see that including standard deviation D1 into the assessment improves the regression model significantly. Scatter plots like those in the lower row of Figure 26 can be misleading because the output variable includes the influence of all input variables, not only the plotted one. In the special

52 Vehicle System Dynamics 927 case with only two inputs, the multiple regression can be visualised in a 3D diagram as shown in [92]. One problem in the scatter plots, as in the lower row of Figure 26, is that they may not properly show the marginal effect of an input variable, given the other input variables in the model. To solve this and to visualise the unique effect of adding, a new input variable to a multiple regression model, added variable plots are used. These plots can sometimes be useful for identifying the nature of the marginal relationship for an input variable in a regression model. The added variable plot shows the relationship between the part of the response unexplained by inputs already in the model and the part of the new input unexplained by inputs already in the model. The unexplained parts are measured by the residuals of the respective regressions. A scatter of the residuals from the two regressions forms the added variable plot. For a detailed description, see [126] or its practical application to railway vehicle dynamics in [92] Compared methods. The intention of the project was to apply as many methods and variants as possible to the DYNOTRAIN data. As indicated in Figure 25, the TGA is based on the measured track irregularity input signals (y L, y R, z L, z R, g, cl the latter two without wavelength restriction). Equivalently, transformed signals such as alignment and longitudinal level at track centre, difference left/right, twist at different length bases have been studied. Selected signals have been filtered by band-pass filters. A number of variants have been studied including D1 and D2from[26] and [1], 10 m bands (1 10 m, m, m,...), 5 m bands (1 5 m, 5 10 m, m, m,...) and bands starting with 1 m (1 10 m, 1 15 m, 1 20 m, 1 25 m,...). Part of the analysis was also the resampling with chord geometry in order to simulate chord measuring systems as required in several regulations. Transformation methods include the vector sum of lateral alignment and cross level, first and second derivatives, parameterisation of track geometry with triangles and wavelets, PMA method from [21] and WGB method from [21]. Additionally, a number of system modelling algorithms (filters of MISO type) have been studied, such as transfer functions derived from analytical models, filters derived by system identification from simulation data and from measured data, ETF derived from simulation data,[104] adaptive Filters derived from simulation data and measured data as well as the Pupil method ( Vehicle Filters developed and used in the Netherlands [99]). Extracted features (characteristic values) in the track sections as indicated in Figure 25 have been calculated using different approaches such as standard deviation, maximum, minimum, maximum of absolute value, kurtosis and skewness. Some methods define a specific rule of which rail (left or right) shall be used for the assessment, partly also depending on the direction of a curve Results. As already explained, the analysis was undertaken for two test zones straight track and curves of all radii and for each vehicle with the instrumented bogie in leading and trailing position. Therefore, there is not one single result of the comparison of methods, but many results depending on the vehicle, test zone, instrumented bogie position and other factors. Here, only a small extract of the results can be shown. In [92], the results without considering track geometry are compared with an assessment using standard deviation 3 25 m (D1) and maximum value in the same wavelength range. For vertical quantities (vertical wheel/rail force Q, vertical car-body acceleration z max (azs) ride characteristics filter, vertical car-body acceleration z S,max (azss) running safety filter), the longitudinal level has been used, for the others the alignment. Selection of the rail was undertaken with the EN approach (higher value of both rails, for alignment in curves the outer rail).

53 928 A. Haigermoser et al. Figure 27. Effect of adding TGA z D1,std and z D1,max in the whole model added variable plot of vertical acceleration in car body, loco BR 120, straight track and y D1,std and y D1,max for Y in curves. If no TGA is included in the regression model for the sections in straight tracks, the coefficients of determination R 2 are very small for all quantities and for all five vehicles. Speed alone is not suitable to explain the variability in the results; this has already been shown in Figure 26 for z max (azs) and is confirmed here for all our quantities and vehicles. For sections in curves and without TGA in the regression, we find different characteristics depending on the investigated quantity. Depending on the filter characteristics, some quantities include quasi-static parts, others such as ÿ max (ays), z max (azs) and z S,max (azss) do not include a quasi-static part. The first ones show a high R 2 even without TGA; the latter show a similar behaviour to the straight track zone. If standard deviation D1 (3 25 m) is included, we see much higher R 2 -values in straight track sections and in curves for quantities with no quasi-static parts. Quantities with quasistatic parts in curves also show higher values, but the improvement is much smaller. If maximum values are used instead of standard deviation, results become worse in most cases, but the difference is rather small. This result was not expected, because the vehicle assessment quantities are maximum values and one would expect that maximum values are better correlated than standard deviations. There are special techniques for visualising the results of a multiple regression. As described earlier, the added variable plot may be used to visualise the marginal effect of an input variable, see [126,92]. The whole model added variable [127] is a presentation, where the slope of the fitted line is the coefficient of the linear combination of the specified inputs projected onto the best-fitting direction. In the upper row of Figure 27, there are only two inputs (speed and track irregularity), there it is a projection onto the regression plane. The vertical car-body acceleration z max in straights is shown in the upper row; Y in curves is shown in the lower row. The first column is without TGA. In the second column, standard deviations in wavelength range D1 are used as TGA and in the third column the maximum values in D1 are used as TGA. We see that the differences between standard deviation and maximum values as TGA are not high. Because of the quasi-static characteristics of Y in the curves, the differences in all three cases are small. This gives a good feeling for the interpretation of changes in the coefficient of determination R 2.

54 Vehicle System Dynamics 929 Figure 28. Comparison of R 2 for no TGA, standard deviation D1 (3 25 m), absolute maximum value D1 z1: straight tracks, z R : curves. Vehicle 1: Loco BR 120, vehicle 2: Coach, vehicle 3: empty four axle wagon, vehicle 4: loaded four axle wagon, vehicle 5: empty two-axle wagon. SY sum of guiding forces Y; Q vertical force; YQ quotient Y/Q; ayp lateral bogie acceleration; ays lateral car-body acceleration ÿ max, vibration behaviour; azs vertical car-body acceleration z max, vibration behaviour; ayss lateral car-body acceleration ÿ S,max, running safety; azs vertical car-body acceleration z S,max, running safety. All parameters according to EN An interesting result to observe in Figure 28 is that the behaviour is very similar for all five vehicles. Therefore, all following results are only given for vehicle 1, the loco BR 120. Detailed results of the analysis of the different methods are given in [110], a summary may be found in [92]. Therefore, here only a very short overview is included. Regarding wavelength ranges D2 (25 70 m) explains much less than D1 (3 25 m), at least at speeds up to 200 km/h. Shorter bands such as 5 and 10 m always give smaller R 2 than D1. If a band-pass filter starting at 1 m and varying upper filter wavelength is applied, we see a distinct maximum at some upper filter wavelength. For lateral quantities, this is at a rather longer wavelength such as 35 m. For vertical quantities, the maximum is at lower values such as 10 or 15 m. There is not always an improvement. If there is an improvement compared to D1, it is rather small. If chord offset filters are applied, the results are similar to those with band-pass filters. In some cases, we see a small improvement of the regression model, in other cases not. It is not necessary to restrict the multiple regression to only one single TGA parameter. One may expect that the vehicle reaction depends on the combination of the different directions of track geometry. Therefore, such combinations have also been studied. Combining standard deviation and maximum value gives a small improvement for some quantities. Combining D1 and D2 improves R 2 for many cases, combining all six 10 m bands gives slightly better values than the combination of D1 and D2. The combined standard deviation from EN [31] improves the coefficient of determination for some quantities, but for other quantities results become worse compared to the standard method. Combinations of alignment and longitudinal level (at track centre) and cross level or twist improve the results quite significantly, especially for lateral quantities. If signals are transformed into derivatives and standard deviations of the second-order derivative are used, significantly lower values of R 2 are achieved compared to the standard method (σ D1 ). The first-order derivative gives better R 2 than the second derivative, but again worse than the standard method. PMA method from EN [31] results in significantly lower coefficients of determination R 2 than the standard method.

55 930 A. Haigermoser et al. Figure 29. Effect of adding TGA z D1,std and Pupil azs in the whole model added variable plot of vertical acceleration in car body, loco, straight track. Also, the parameterisation of track irregularities by triangles and by wavelets (continuous transform, Mexican hat wavelet) does not improve the results compared to the standard method. Using the ratios A/L and A/L 2 mostly worsens the results compared with the cases where the height (amplitude) of the triangle or the energy of the wavelet A is used. The results from the WGB method [31,93] have been disappointing and also surprising. In most cases, the regression is worse than using the standard method. In cooperation with ProRail and Lloyds Register Rail, the measured track geometry was also assessed using different vehicle filters implemented in Pupil. The assessed track geometry was then included in the regression analysis of the measured vehicle reactions. Some improvements for some vehicles against the standard method have been found, especially on straight track. This can be analysed in detail by comparing the whole model added variable plots in Figure 29. Here again, the results for the vertical acceleration in car body z max (azs) are compared with those without TGA and with z D1,std as TGA. The coefficient of determination R 2 is significantly higher (0.78 instead of 0.68), which is also visible in the distribution of the section values around the regression. As the Pupil signals are estimations of the vehicle behaviour, the numbers are also directly comparable to the measured vehicle reactions. Here, we see a small overestimation of the vehicle reaction of about 10%. Although this is the best correlation found in the reported studies, there remain significant residuals with a root mean square of 7% of the limit value. The results from Pupil in curves are worse than for the standard TGA (σ D1 ). Digital filters (FIR-type) estimated as adaptive filters on the basis of multibody simulations showed similar results as Pupil in straight lines. There are cases (vehicle assessment quantities and test zones) where the results are better than for the standard method, but other cases were also found with worse results. The last two methods indicate that there are possibilities for improved description of track geometry, but the studied methods are not able to fully exploit them. The results presented in [92] were not expected and differ from several other comparisons by other authors as mentioned in Section 4.3. Reasons for this may be the influence of other parameters such as wheel rail friction, wheel rail contact geometry and possible measurement uncertainties and noise in the two measurement system (track irregularities and, vehicle response). This is possibly indicated by the remaining residuals or errors in the regression, as visible in Figure 29. It is interesting that if simulation is used for the assessment of these methods rather than measurements, we find significantly better correlations for them than for the standard method (σ D1 ). The reasons for this different behaviour are not fully clear, more detailed studies are necessary.

56 4.4. Characterisation in wavelength domain Vehicle System Dynamics 931 A drawback of describing track geometry in the distance domain is that it is not easy to include characteristics of the shapes of defects and their wavelength content. This can partly be resolved by applying filters to focus the analysis on defined critical wavelength ranges but this approach is not very concise and general. The general approach is spectral analysis, which determines the spectral content of a signal. The essence of spectral analysis is to estimate the distribution of the total power over frequency [128] from a finite record of a stationary data sequence. Track irregularities must be understood as random signals, as they are not completely predictable (which is necessary for deterministic signals). Therefore, methods from random signal processing have to be applied to spectral analysis of irregularities. The irregularities are considered as realisations of a stationary random process. The standard tool for spectral analysis is PSD, which describes how average power (or signal variance) is distributed across frequencies. Formally, it is defined as the Fourier transform of the autocorrelation sequence of a stationary stochastic process (see [128]) and shows the distribution of power as a function of frequency. Spectral density functions can be defined in three different equivalent ways: (a) Via correlation functions (b) Via finite Fourier transforms (c) Via filtering squaring averaging operations There are a number of methods for spectral estimation available, generally divided into non-parametric (classic) and parametric (modern) methods.[129] The parametric approaches assume that the underlying stationary stochastic process has a certain structure that can be described using a small number of parameters (e.g. using an autoregressive (AR) or moving average (MA) model). In these approaches, the task is to estimate the parameters of the model that describes the stochastic process. By contrast, non-parametric methods explicitly estimate the spectrum of the process without assuming that the process has any particular structure. The probably most intuitive (non-parametric) method is to pass the signal through a set of band-pass filters and measure the filter output powers. Non-parametric methods normally rely on Fourier transforms. One way of estimating the PSD of a process is to simply find the discrete-time Fourier transform of the samples of the process (usually done on a grid with an FFT) and appropriately scale the squared magnitude of the result. This estimate is called the periodogram. The modified periodogram windows the time-domain signal prior to computing the DFT in order to smooth the edges of the signal. This has the effect of reducing the height of the sidelobes or spectral leakage. This phenomenon gives rise to the interpretation of sidelobes as spurious frequencies introduced into the signal by the abrupt truncation that occurs when a rectangular window is used. For nonrectangular windows, the end points of the truncated signal are attenuated smoothly, and hence the spurious frequencies introduced are much less severe. On the other hand, nonrectangular windows also broaden the mainlobe, which results in a reduction of resolution. Bartlett s method is a periodogram spectral estimate formed by averaging the discrete Fourier transform of multiple segments of the signal to reduce variance of the spectral density estimate. An improved estimator of the PSD is the one proposed by Welch.[130] The method consists of dividing the time series data into possibly overlapping segments, computing a modified periodogram of each segment, and then averaging the PSD estimates. Other non-parametric methods are, for example, Blackman Tukey s method or multitaper method.[129]

57 932 A. Haigermoser et al. Figure 30. Comparison of several PSD estimation methods. Parametric methods use a different approach for the spectral estimation. Instead of trying to estimate the PSD directly from the data, they model the data as the output of a linear system driven by white noise, and then attempt to estimate the parameters of that linear system. The most common form of parametric estimate uses an AR model. This is an all-pole model, a filter with all of its zeroes at the origin in the z-plane. The output of such a filter for white noise input is an AR process. Common AR methods are the Yule Walker AR method (autocorrelation method), Burg method, covariance method and modified covariance method.[129] Alternative parametric methods include fitting to a MA model and to a full AR moving average model (ARMA).[129] In Figure 30 three methods are compared. At the top of the figure, the signal from a measurement of longitudinal level on 65 km length is shown. On the lower left, the result of the periodogram method is shown. We see a very noisy and rough result with an erratic and wildly fluctuating form. It can be theoretically shown that the periodogram is not a good estimator of PSD because, independently of the signal length N, the standard deviation of the estimator is as large as the quantity to be estimated.[129] In the lower middle diagram, the result for Welch s method is shown. For comparison in grey, the result of the periodogram is plotted again. The form of the PSD is now rather smooth with a number of discrete peaks. These peaks are of physical origin as shown later. An example of a parametric method is shown on the lower right diagram. Burg s method is applied to the measured data assuming an order of 18 for the modelled AR process. This method is not able to model the discrete peaks, but the general shape is estimated quite well Interpretation of PSD In Figure 30, we see the usual form of track irregularity PSD diagrams. In a doublelogarithmic plot, the power density increases with increasing wavelength (decreasing spatial

58 Vehicle System Dynamics 933 frequency) with approximately piecewise linear functions. The rather smooth shape (of a well-estimated PSD) is the result of a stationary random process from track segments constructed in a uniform manner and maintained to a similar performance level.[131] In a first approximation, the PSD could be modelled as a straight line,[132] leading to a description such as S( ) = S 0 ( 0 ) w, = 2π L. (32) Here, 0 [rad/m] denotes a standardised spatial circular frequency, S 0 = S( 0 ) [m 2 /(rad/m)] characterises the roughness at 0 and can be considered as the roughness level, and w is called waviness and characterises whether the irregularities contain long wavelengths (w large) or short wavelengths (w small). In Figure 30, w correponds to the slope of an approximating straight line. In measured track irregularities, the parameter w often is between 2 and 4. In [131], the different values of w or slopes of the PSD are explained by different types of random walk, characterising the underlying physical process. A random walk is an evolutionary process by which a new position is determined from previous positions and a random variable against a reference. Type 1 consists of an attempt to place steps; proportional to white noise, around a straight reference line. Type 2 results if random directions are applied that can be interpreted as approaching a very distant point on the horizon. In type 3 random, the only reference is the reference established by the two previous steps. It can easily be derived that type 1 corresponds in the PSD to a horizontal line (w = 0), type 2 to a slope of 2 decades per decade (w = 2), and type 3 to a slope of 4 decades per decade (w = 4). In an analogous fashion, it is possible to define random walks of even higher orders. For example, a Type 4 random walk is produced from random variations in curvature.[131] Looking at the PSD of measured data, we see that random walks of types 2, 3 and 4 produce PSD s whose asymptotic behaviour at long wavelengths is represented by type 1. Besides this rather smooth PSD (in the Welch estimation), we see pronounced peaks at specific wavelengths. These peaks are only visible in some tracks. Figure 31 shows on the left-hand side the spatial shape of irregularities from two tracks. The overall shape is similar; also signal power seems to be similar. On the right-hand side of the figure, the PSD of the two signals are compared. The main difference is that track A shows distinct peaks at the following sequence of wavelengths: L = 36, 18, 12, 9, 7.2, 6, 5.14, 4.5 m. This corresponds to a sequence L = (L 0 /i), i = 1, 2,... Track B does not show these peaks and the general shape of its PSD is similar to the one of track A except for the distinct periodic peaks. Although no additional information on the two tracks is available, it is most likely that L 0 in track A corresponds to the rail length and this track has bolted joints. In [131], this effect is analysed as a periodic deterministic process that models uniform dips at periodically spaced joints. It is obtained by letting the mean be a function of position in the rail length and by letting all higher order moments vanish. Both together, the stationary random process (all moments be constant) and the periodic deterministic process, are called a periodically modulated random process.[131] The periodic deterministic characteristic is hardly visible in the distance domain figure but readily identifiable in the PSD. This is one of the benefits of analysing track irregularities in the wavelength domain. Also, other hidden periodic characteristics come clearly visible. An important characteristic of PSD is caused by the underlying assumption of a (wide sense) stationary random process. This implies an important averaging property and the loss of phase information. One cannot predict peak values from a PSD. Identical PSD s can result from a wide variety of time histories. For example, the PSD calculated from a time history consisting of a series of large random amplitude pulses spaced at discrete intervals

59 934 A. Haigermoser et al. Figure 31. Example of track irregularities from tracks with and without a periodic deterministic process. and the PSD calculated from a time history composed of small amplitude pulses occurring continuously (at overlapping intervals) will be identical. The two processes will also produce identical mean square values. However, other characteristics of the two processes, for example, peak amplitudes, are quite different.[131] There are approaches to solve these limitations by separating anomalies from the signals. In [131], the anomaly-free track geometry is then modelled by a periodically modulated random process, consisting of two subsets: a continuous stationary random process that accounts for random behaviour uniformly distributed throughout the rail length and a periodic deterministic process that accounts for uniform dips at periodically spaced joints.[131] Under the assumption of an underlying Gaussian random process, not only standard deviation and level crossing but also peak statistics may be calculated from the PSD (see [133] for details). Up to now, the spectral content of the four irregularities has been analysed separately. The dependence of the several irregularities may be studied by the coherence function, which is described more in detail in Section (see formula (51)). Figure 32 shows the magnitude squared coherence functions between different parameters for five different tracks. There is a strong coherence between the longitudinal level of the two rails, as well as between the alignment signals. If track irregularity coordinates are used instead, the coherence vanishes and the signals become more or less independent Analytical description of the PSD As indicated earlier, the PSD of the stationary random process can be approximated by analytical expressions. This may also be related to the aforementioned parametric spectral estimation methods described earlier. For example, ARMA models estimate a PSD in the form of a rational PSD as S( ) = σ 2 B( ) 2 A( ), (33)

60 Vehicle System Dynamics 935 Figure 32. Magnitude squared coherence function between different parameters. A( ) = 1 + a 1 e j + +a p e jp, (34) B( ) = 1 + b 1 e j + +b p e jp. (35) The rational spectra can be associated with a signal obtained by filtering white noise of power σ 2 through a rational filter with H( ) = B( ) A( ). (36) For AR models, the b j are zero; for MA models, the a j are zero (except a 1 = 1). Some of the usual analytical descriptions of PSD are consistent to these models. The first simple analytical expression has been given in Section It can be modified by assuming a bilinear form of the PSD as [132] ( ) w1 0 S 0 0 < I 0 S( ) = ( ) w2. (37) 0 S 0 for 0 II < Recently, a proposal for fitting the PSD by three piecewise linear functions have been published together with results for the Beijing-Shanghai High Speed Railway.[134] The linear functions with w > 0 have the disadvantage that in the limit case of 0 they result in infinite values of the PSD S(0). To avoid this unrealistic case and achieve a better approximation of PSD from measured data, extended roughness models have been proposed. In the USA, the proposal from [131] is widely used, which uses the following

61 936 A. Haigermoser et al. expressions for alignment and longitudinal level and for cross-level and gauge S y,c ( ) = A i ( ) and S cl,g ( ) = ka i 2 11 ( )( (38) 22 ). In Europe, the following formulas have been developed in Germany and used by ERRI [135] and many others in rolling stock procurements: A i 2 S y,z ( ) = (39) ( r )( c ), S cl,g ( ) = A j 2 2 ( r )( c ) (40) b( s ). Various spectra of track irregularities (also referred as PSD Standards) were published by the Chinese Academy of Railway Science (CARS). These spectra rely on five different coefficients and have the general form (cited from [136]) S( ) = a 2 + b c 6 + d 4 + e 2 +. (41) Coefficients are given for different track quality classes and speed ranges. In [136], it is reported that SNCF proposed an analytical description of vertical irregularities as S( ) = A (1 + / 0 ) 3. (42) It should be mentioned that the originally published formulas use different symbols and some of them have been formulated as a function of 1/L, others as a function of. For the sake of comparability, the formulas have partly been rearranged and harmonised. In [136], a summary of these expressions is given including all numbers used in the formulas Presentation of track geometry in wavelength domain The measured track irregularities analysed in Figure 23 have also been studied in the wavelength domain. Exemplary results are shown in Figure 33. It compares the PSD of the full track of km for the three irregularities y, z and z. We use a presentation with the wavelength on the x-axis, contrary to the usual one with the wave number = 1/L. The chosen presentation is more descriptive than the usual one. There are some differences in the overall level of the tracks, which correspond to the differences in the standard deviations, as the variance of the signal equals to the area below the PSD. Significant differences are found in a vertical direction in the wavelength range between 10 and 20 m. Here, the PSDs from track B are significantly lower and the change with increasing wavelength is very low, compared to track A. In the cross level, we find a distinct peak at 3 m in track C. Figure 34 gives a more detailed analysis for the alignment signals. The PSD s from the y-signal of the three tracks is plotted in the middle column. The red line gives the analytical spectrum from [135], which is discussed in Section The left column shows windowed periodograms, calculated by short-distance Fourier transforms. Because the amplitudes grow very much with wavelength, it is useful to normalise these spectra with a reference curve. Here, the analytical spectrum from [135] has been used. With this, variations of the PSD over the distance become visible, for example, sections of tracks with periodic defects.

62 Vehicle System Dynamics 937 Figure 33. PSD of y, z and z of the three tracks. Figure 34. Spectral representation of lateral track irregularities of three tracks. Left: short-distance-periodogram, middle: PSD of whole track, right: normality plot of signal. The right column shows the normal probability plots of the signals. The purpose of this plot is to graphically assess whether the data could come from a normal distribution. If the data are normally distributed, the plot will be linear (red line). Other distribution types will introduce curvature in the plot. Additionally, standard deviation, kurtosis and skewness of

63 938 A. Haigermoser et al. the signal are given in the headline of the plot. Kurtosis is a measure of how outlier-prone a distribution is. The kurtosis of the normal distribution is 3. Distributions that are more outlierprone than the normal distribution have a kurtosis value greater than 3; distributions that are less outlier-prone have a kurtosis value less than 3. Skewness is a measure of the asymmetry of the data around the sample mean. If skewness is negative, the data are spread out more to the left of the mean than to the right. If skewness is positive, the data are spread out more to the right. The skewness of the normal distribution (or any perfectly symmetric distribution) is zero. In these examples, we see that all distributions are not ideally normally distributed. Track C is significantly outlier-prone with a kurtosis of 18. Track B is the one most likely to a normal distribution. 5. Processing of measured data, description and assessment of road irregularities 5.1. Road condition rating The measurement values, described in Section 3.2.1, such as those determined during measurements of transverse evenness and traction, are input variables together with other parameters together with their appropriate weight into the overall road condition rating functions. State variables are calculated for sections of 100 m, which prove to be significant for pavement management analysis. To rate the condition, these physical quantities are finally transferred into condition grades ranging from 1 (very good) to 5 (very poor) Digital road surface models based on laser-scanner data The main scanner data characteristics are high point resolution and very low noise, but the scanner data itself are not a usable result. As a prerequisite for road condition evaluation from scanner data, the raw data, as the georeferenced and calibrated scanner point cloud is considered, has to be transformed. It is necessary to remove disturbances, such as cars, debris or leaves on the road, remove gross sensor errors, establish a regular data structure and reduce noise even further, if possible. The solution is the determination of a regular grid-based digital road surface model. The regular grid for the road model is orientated along the road axis. The grid is based on cross-sections at regular distances, with constant spacing between the points within each profile. This method of generating the grid was chosen because it is mathematically adequate and the results can be easily integrated into standard road planning, automobile driving simulator or simulation software data formats, which define their road surface data typically in the same way. The regular grid density is flexible. As a main effect, the regular grid significantly reduces the amount of data necessary to describe the road surface without significant loss of information. Very variable project requirements demand multi-purpose data organisation. Based on the same MoSES sensor setup and with the data acquisition setup based on high-definition scanners and cameras, data collection is always done in a similar way, however, the project specific road model specifications for post processing may be completely different. Road surface models of a length between only a few meters up to almost 100 km with grid point spacing between and 0.50 m may occur. The generation of 3D digital surface models based on laser-scanner measurements with MoSES requires robust model determination and sophisticated quality control. Both laser-scanners survey large overlapping areas, so that the observables to determine the digital road model are the point clouds of at least two laser scanners. The processing of the road surface survey results is based upon modelling algorithms that are specifically adjusted to the track

64 Vehicle System Dynamics 939 Figure 35. Automobile test track in raw data and as resulting high-resolution road surface model. Figure 36. Automobile test track examples (left: Belgian block, right: obstacle course). surface structure. For each grid coordinate, a best-fit, multi-dimensional surface function is determined using robust estimation techniques. The estimation is based on the surrounding scanner measurements. The resulting function is used to calculate the smoothened height for the grid point. During processing, the quality of the road surface is checked and gross error filtering and automatic obstacle detection ensure that no artefacts are contained within the resulting model of the road surface. During model calculation, the quality of the processed sections is checked for any grid point. Due to the inertial quality and low scanner noise (compare Section 3.2.2), within an approximate 5 m 5 m-patch a relative height accuracy of 1 mm can be obtained. Due to this very high accuracy, the results are used for automobile development purposes as well as for detailed road condition analysis. The example in Figure 35 shows an automobile test track as raw data and as a resulting high-resolution road surface model. Figure 36 shows two additional automobile test track examples. The last example shows the analysis of a cross profile extracted from a road model with 10 cm point spacing and 5 m width, which is analysed for rut depth (left) as well as for fictive water depth Road condition determination based on road surface models The road surface models can be used for transverse as well as for longitudinal analysis or applications using the complete model as input information, such as water flow analysis. Figure 37 contains an example for classical transversal road evenness analysis using a simulated straightedge of 4 m length. The rut depth information is gained by plotting the depth information under the straightedge. The fictive water depth analysis does the same, but with a horizontal reference line. The longitudinal profile can be approximated by its power spectral density according to formula (32) in Section 4.4.1, which is represented as a straight line in a double-logarithmic scaled diagram. The amplitude parameter S 0 shows typical values of 1, 4, 16 cm 3 equivalent

65 940 A. Haigermoser et al. Figure 37. Rut depth (left) and fictive water depth (right). to a road roughness classification of A, B, C found in [137] describing very good to mediocre roads. The waviness parameter w is the slope of the line in the double-logarithmic scaled diagram and shows typical values not far from 2.0. The random properties in the lateral direction can be well represented by the coherence of the longitudinal profiles γ(ñ, ) = S lr ( ) Sll ( ) S rr ( ), (43) where r,l indicate the right and left profiles and ρ is the track width. Small track widths will have coherence values close to 1.0 for a wide range of wavelengths, which means that there is no road-induced camber excitation to a single tyre with low-frequency content. With growing track widths, the coherence already drops at longer wavelengths, which results in road-induced roll excitation of a two-track vehicle even at lower frequencies. More details and further references can be found in [ ]. In Figure 38, this can be illustrated by a longitudinal analysis of 85 km of a motorway measurement (see Section 3.2). In this case, the scanner data of the driven lane have been transformed into a 10 cm regular grid. For analysis purposes, four longitudinal profiles have Figure 38. Motorway sample.

66 Vehicle System Dynamics 941 been chosen from the road surface model to demonstrate the usability of scanner-based data for road condition determination. On the left, we can see the altitude profile of the road and the (unfiltered) cross slope (super-elevation) showing in straight sections the typical 2.5% values to ensure sufficient drainage (de-watering) of the road surface. On the right, we see the power spectral density of four longitudinal tracks at + 0.8, + 0.7, 0.7, 0.8 m distance of the lane centre line, and the coherences between each combination of them. We can see a rather good road surface condition, but some irregularity peaks at 2, 5 and 10 m wavelength might result in unexpected vibrations depending on the travel speed and the vehicle s eigenfrequencies. 6. Irregularities and simulation 6.1. Railway vehicle simulation issues Simulation plays an important role for the fields mentioned in Section 1. Regarding the design of railway vehicles, track irregularities have to be taken into account in the issues of running safety, ride comfort, dynamic loading and fatigue of components, and also NVH. The simulation can also be applied to the assessment of track quality for test tracks. In operation, simulation scenarios can provide important insights regarding vehicle track interaction. This information can be used to support decision-making and maintenance strategies. There are different requirements regarding the information of the irregularities depending on the issues. The main criterion is the frequency range of interest of the vehicle reactions. In combination with the vehicle speed, this leads to the required wavelength range content of the measured irregularities. The necessary wavelength information ranges from short wavelengths of centimetres, for example, for NVH issues, up to large wavelengths of 200 m, for example, for comfort issues of high-speed trains, see also Figure 3. Ideally, measured and pre-processed irregularities could be directly used in the simulation. However, due to the limitations of the measurement systems (see Section 3) as well as for reasons of cost, only data with limited wavelength range are available in most cases. To fill the gap, the missing information must be synthetically generated and included in the simulation depending on the issues. Further track information is needed for an appropriate model of the track: Track layout Rail inclination Rail profile Track irregularities Track dynamic parameters This track information forms the input to the so-called track pre-processor that is described in the next section in detail Railway track pre-processing for the simulation The goal of the pre-processing procedure is the calculation of the 3D-position and orientation of the space curve of the right and the left rail as well as the track centre line, given as a function of the arc length. The space curves can be described by the three Euler angles ψ, θ and ϕ (see Figure 1) or by the tangent, normal and binormal vectors t, n and b, which

67 942 A. Haigermoser et al. Figure 39. Block diagram of the track pre-processing procedure. is known as Frenet frame. These quantities are expressed as functions of either the approximated arc length s or the horizontal projected arc length s projected. The choice of the description as well as the calculation of additional information within the pre-processing procedure such as derivatives of the quantities depends on the MBS-formalism.[142] The pre-processing procedure is shown in Figure 39. The input data for the pre-processor are described in the previous section such as track layout, track irregularities, rail inclination and rail profiles Pre-processing of the track layout The track layout is generally described by means of cartographic elements such as straight track, curved track and transition curve. These track layout elements contain information such as Segment length L Horizontal layout (horizontal radius R H or curvature C H ) Vertical layout (vertical slope p, vertical radius R V, curvature C V or grade) Super-elevation (roll/cant angle δ of the track line, super-elevation u in combination with the reference base length b A As another alternative, the space curve can also be described directly by given data points (x, y, z) in Cartesian coordinates. In additional to the description of the track centre line, information about the super-elevation or camber is necessary to define the space curves of the right and left rail Pre-processing of the track irregularities The description of track irregularities is explained in detail in Section 2.1. Based on this description, the track irregularities can be superimposed on the space curves of the right and left rail resulting from the track layout. Care should be taken that the description of the track layout and the track irregularities contain no redundant data information (see Section 4 separation of track layout and track irregularities).

68 Vehicle System Dynamics 943 The description of the track irregularities for the simulation can be based on the following data: Measured track irregularities Synthesis of stochastic track irregularities Synthesis of deterministic track irregularities (isolated defects, periodic irregularities) In the case of measured track irregularities, it should be ensured that the data are adequately pre-processed (see Section 4). This means, for example, that no discontinuities are present, the amplitude as well as the phase of the data is consistent and the data are converted to the coordinate system of the simulation environment (e.g. UIC coordinate system with downward-pointing z-axis). As already mentioned, missing data information can be synthetically generated and superimposed on data of measured track irregularities. The synthesis of stochastic track irregularities is based on the specification of the desired statistical properties. These statistical properties are generally described by polynomial parameters of the desired PSD-shape (see Section 4.4.3). The choice of the synthesis method depends on the needed requirements. For example, test tracks with a long distance require non-repeating sections. Furthermore, when generating synthetic data with random signals, reproducibility of the data for the simulation must be ensured. The generation of synthetic data described in the rail coordinate system also requires the consideration of the wavelengthspecific correlation of the left and the right rail (e.g. by means of coherence functions, see [143]). Otherwise, high amplitudes can occur at long wavelengths. In the literature, different synthesis methods are available for these requirements.[144] As examples, the following methods are mentioned: Superposition of cosine functions with random phase and amplitude Orthogonal basis functions (e.g. wavelets) Shape filters (linear dynamic systems) A detailed description of these methods can be found in Section The synthesis method based on shape filters is commonly used in MBS software packages (e.g. see [145]). The filtering of white noise signals allows the generation of long distance tracks. The reproducibility of the white noise signals has to be ensured by the same initial values of the random number generator, for example. By specifying the desired wavelength range, synthetic track irregularities can be superimposed with measured track irregularities. It is recommended that the resulting signal is analysed regarding realistic maximum values as well as standard deviation values in the distance domain and regarding the shape of the power spectral density function in the wavelength and frequency domains. Stability investigations of railway vehicles in the distance domain and studies on the reactions on single defects may also require synthetically generated isolated defects. Analytical descriptions of different types of defects with their corresponding parameters and their possible occurrences in measured data have been analysed in [146], results are given, for example, in [147]. These defects can be superimposed as singular defects or periodic defects on a synthetically generated basis signal. Redundant data information as well as discontinuities should be avoided as mentioned before. Another example for the use of synthetically generated deterministic signals is the investigation of the dynamic behaviour of railway vehicles. Therefore, a linear chirp signal z(s) can

69 944 A. Haigermoser et al. be used: ( z(s) =ẑsin 2π (f L,0 s + k2 )) s2, f L (s) = f L,0 + ks, (44) k = f L,end f L,0 s end. The instantaneous wavelength frequency f L (s) startsfromf L,0 and varies linearly with distance s. The chirp rate k is the rate of wavelength frequency increase from the starting point to the end point s end. Additionally, the amplitude ẑ can be varied as well to calculate the frequency and amplitude depending transfer behaviour of railway vehicles.[148] There are even some standards that require the simulation of analytically described irregularities. EN [1] requires one of the methods for assessing safety against derailment the use of defined dip in the twisted track. A new FRA regulation [66] requires the carrying out of simulations on a Minimal Compliant Analytical Track MCAT for the qualification of vehicles to operate at high speeds and cant deficiencies. This contains different types of track irregularities with varied amplitudes and wavelengths. Segments are defined such as hunting perturbation, gauge narrowing, gauge widening, repeated vertical defects, repeated lateral defects, single vertical defect, single lateral defect, short twist and combined perturbation Final pre-processing steps The implementation of the track layout and the track irregularities requires a parameterised description as a function of its respective arc length s. Therefore, an analytical description between the nodal points of the curve is introduced using an interpolation scheme (e.g. with splines). The result of the interpolation process should be visually checked because unrealistic amplitudes can occur, depending on the method. Finally, the track layout and the track irregularities are superimposed. The final output of the track pre-processor is the analytical curve description for the track centre line as well as for the right and the left rail. This analytical description can be used by the MBS solver for online evaluation or can be evaluated offline and stored in tabular form. The latter method is used to achieve computational efficiency, whereas a linear interpolation is performed during the integration process.[149] The curve description of the right and the left rail, based on the track layout and the track irregularities, provides a theoretical guidance and not a mechanical constraint. This provides the possibility of a derailment of railway vehicles within the simulation. An adequate superstructure model is also required depending on the desired frequency range.[150] Therefore, the appropriate parameters of the superstructure model have to be known. In practical applications, the determination of these parameters causes high costs. For this reason, a compromise between model depth and cost is usually made Automotive vehicle simulation issues Many similarities exist between railway and road vehicle simulation, but there are also some important differences to mention: Road vehicles can use the total road surface between the left and the right roadside. This brings up the need to provide more than just a longitudinal profile for each track of the vehicle, because the irregularities may be different in the lateral direction.

70 Vehicle System Dynamics 945 Tyres have a significant width and generate forces not only depending on slip and the longitudinal inclination of their contact patch, but also side forces due to the relative camber angle between tyre and road. This becomes obviously clear when lane grooves influence the vehicle motion. This motivates the representation of a road by its 3D surface description. Only in some applications might this be replaced by simplified descriptions based on only two longitudinal profiles, such as two rails as seen in the railway sections aforementioned. This simplified 2-profile approach might also be useful to define test tracks for vehicle testing: due to the missing lateral profiling test, results will become less sensitive to small lateral deviations of the trajectory in repeated test drives Road pre-processing for the simulation Simulation tools for road vehicles often provide road pre-processing features similar to the approaches for railway simulation presented in Section 6.2. There is a significant variety of implementations on the market, mostly using proprietary description languages, data formats and user interfaces. Since tyre models of differing complexity have different requirements on road representations, many of them bring their own road surface representation and pre-processing outside of the vehicle simulation environment, for example, the high-end tyre simulation model FTire.[151] This type of road description represents a full 3D surface, which can be scanned at arbitrary contact points needed for the tyre simulation. Hence, the interface between tyre and road is mainly a function call returning the local height of the road at arbitrary positions. Simpler tyre simulation models sometimes ask for local gradients of the road at a single contact point under the tyre. It is not advisable to provide this information via the road simulation model, because the relevant local gradients depend on tyre properties such as size and orientation. The solution is to introduce an intermediate contact processing layer between the tyre and the road model, which calculates the gradient information by scanning the road at some points under the tyre contact patch, using the relevant tyre properties.[152] Depending on the application, not only the tyre simulation models need the road height information. In current vehicles, assistance systems use camera information to keep the car on the road, and even the preview system Mercedes Magic Body Control uses the road profile seen by a camera to compensate for irregularities.[153] Synthetic road profiles Concerning track irregularities, several methods exist for the generation of road irregularities. The following methods are one-dimensional and therefore used for tracks as well as for a road surface lane Superposition of cosine functions with random phase Orthogonal basis functions (e.g. wavelets) Linear dynamic systems These methods permit the lanes to be generated and assembled in a logical way by using the coherence of the lanes. On the other hand, a two-dimensional profile can be directly generated using other methods: Randomly distributed obstacles Einstein Hopf approach to road surface modelling

71 946 A. Haigermoser et al. Figure 40. Realisation of a road track with the method of Shinozuka and Rice. An ergodic and Gaussian process is completely characterised by its power spectral density spectrum. As already mentioned, it can be described as a piecewise linear function in a double-logarithmic scale. For the following methods, target spectrum Sx v (ω) is given. Using a combination of more than one synthesis approach can also be successful. Kropác and Múcka [154] discusses the combination of homogenous longitudinal road profiles that are superposed by cosine-form obstacles on the spectra properties and on the vehicle reaction. In [155], we find a discussion of various combinations of process models and randomly placed and shaped irregularities with respect to fatigue loads in damage simulations Superposition of cosine functions with random phase. Shinozuka [156,157] and Rice [158] suggest generation using a stationary Gaussian process. This can be realised by the superposition of cosine functions with random phase. This method is based on the inverse Fourier analysis. The process is received by the summation 2 N X s = σ cos( k s + k ), (45) N with the variance k=1 σ = Sx v (ω) dω. (46) The frequencies k and the phases k are random sequences, the phases are uniformly distributed in the interval [0, 2π] and Sx v (ω) is the target power spectral density. This power spectral density can be interpreted as a density function for the random sequences k. That way, the process achieves the demanded power spectral density. It is also possible to fix the frequencies and instead of that use the amplitudes to accommodate to the power spectral density: X s = M a k cos(ω k s + k ), a k = k=1 2 π (ω k+1 ω k 1 )S v x (ω k). (47) This reduces the number of needed terms in comparison to formula (45). The simple and universal applicability is the advantage of this method. The numerical effort, however, is very high. That is the disadvantage of this method. Figure 40 gives an example of a realisation of a road lane with this method. The lane is 400 m long with an increment of 0.01 m. The target power spectral density is a piecewise linear function in a double-logarithmic scale, see Figure 41. The basis for it was the power spectral density of a rough road, which is used for durability testing.

72 Vehicle System Dynamics 947 Figure 41. Target power spectral density. The green line in Figure 41 is the power spectral density of the realisation in Figure 40, the blue line is the power spectral density of the measured data of the rough road and the red line is the target function. As we can see, the generated track matches quite well with the target spectrum, but the realisation of a longer track matches much better (black line). Due to the fact that the rough road is just 250 m long, its power spectral density is not straight and therefore it is not easy to find the right approximation as a piecewise linear function in a double-logarithmic scale. For this reason, it is also important to choose a long track for determination of the target spectrum Linear dynamic systems. Another possibility to generate Gaussian processes is the method of linear dynamic filter systems, see [159,160]. This method uses a dynamic filter under white noise as a model system. By choosing suitable system parameters, the process can be adapted to the target process. The system of functions consists of the eigensolutions of the filter differential equations. Therefore, the computation is more efficient than when using the Shinozuka and Rice method. A linear complete filter system of order n is defined by n n 1 a i Y s (i) = ξ s, X s = b i Y s (i), (48) i=0 where a n 0 and Y s (i) = (d i /dt i )Y t and ξ s is a normalised stationary white noise excitation process. The power spectral density appears to be a fractional rational function of ω i=0 S x (ω) = b2 0 + (b2 1 2b 0b 2 )ω 2 + +b 2 n 1 ω2n 2 a (a2 1 2a 0a 2 ) + +a 2 n ω2n (49)

73 948 A. Haigermoser et al. depending on the homogeneous parameters a i and the inhomogeneous parameters b i of the system in a non-linear way. Because of that non-linear relation, a direct approximation of the spectrum is a highly non-linear optimisation problem. In general, it is hardly solvable for higher order filters if there is no further information about the solution properties or if no approximate solution exists. Hence, a slightly modified objective function is introduced enabling the derivation of a linear strategy to find a suitable set of parameters: The correlation differential operator of the system is compared with the target expression to define the local error function. Due to the fact that the operator is a linear function of the system parameter a i,al 2 -approximation yields to a linear system of equations, which can be solved. The parameters b i can be determined without any further approximation. As already mentioned, this method needs less computing time than the superposition method. In addition, convergence of the method is better. A non-linear approximation procedure to generate non-gaussian processes is also presented in [160] Orthogonal basis functions (e.g. wavelets). The theory of wavelets provides an integrated way of presenting the irregularities: both the spectral and the spatial distribution of the frequency can be shown in a position-frequency level, see [144]. Wavelets form an orthogonal base of the method. They have compact support. For this reason, they can be located regarding frequency and spatial position. A mother wavelet ψ is defined and by shifting and scaling this mother wavelet, new wavelets can be obtained ( x bwt ), (50) 1 ψ a,b : R C, ψ a,b (t) = a WT 1/2 ψ a WT where a WT R\{0} denotes the parameter of scaling and b WT R is the parameter of shifting. A lot of different mother wavelets exist for the different demanded processes. The mother wavelets are the basis functions of the transformation. By using the discrete wavelet transformation (DWT) and defining the wavelet coefficient as random sequences, we obtain a stationary random process. Similar to the method of superposition of cosine functions above, the demanded power spectral density Sx v ( ) determines the density function for the frequency. The wavelet coefficients are almost uncorrelated if the power spectral density of the process declines hyperbolically. In this case, there is no statistical dependence of the coefficients. This is a very good property of this method. The wavelet transform is often compared with the Fourier transform. In fact, the Fourier transform can be viewed as a special case of the wavelet transform with the choice of the mother wavelet ψ(t) = e 2πix. The main difference in general is that wavelets are localised in both time and frequency, whereas the Fourier transform is only localised in frequency. Furthermore, the DWT has a smaller amount of computation time for the synthesis of a signal of length N: the number of computation operations is O(N). The fast Fourier transform FFT needs O(N log 2 (N)) operations (Chapter in [144]) Using coherence to generate two-dimensional profiles. With the methods, superposition of cosine functions with random phase, orthogonal basis functions (e.g. wavelets) and linear dynamic systems, one lane of the road, for example, the left lane, can be generated. To obtain the right lane, it is reasonable to use the coherence function S l,r ( ) γ(, ρ) = Sl,l ( )S r,r ( ), (51) where S l,r ( ) denotes the cross-spectral density of the right and the left lane, and S l,l ( ), S r,r ( ) the power spectral density of the left, or the right lane, respectively. In[161], Ammon

74 Vehicle System Dynamics 949 Figure 42. Example for the coherence using formula (52). and Bormann suggest the following approximation function for this coherence: γ(, ρ) = ( 1 + ( ρ a ) ω ) p. (52) For a = 1 the parameter p and p define basically the position and the gradient in the point of inflection. The parameter a determines how compact the coherence functions are for different lanes. Figure 42 shows an example for p = 0.65, ω = p = 3.5, α = 0.7. This coherence can now be used for modification of the phase in formula (45), so that the left and the right lane satisfy the desired statistical dependence p r = l + (1 γ(, ρ)), (53) with being a uniformly distributed random sequence in [ π, π]. The other lanes between the left and the right lane can also be computed by using the coherence function for the right distance ρ. The result is a two-dimensional, stochastic description of a road surface. Figure 42 shows the result of this method for the coherence. As in Figure 41, it can be seen that the resulting coherence is much better for longer roads: The black line, which is the coherence of the left and the right track of a 10 km long road (generated by superposition of cosine functions, see formula (44)) fits much better than the one of a 270 m long road Synthesis of two track excitations. Two track road excitations with coherence characteristics close to reality are generated through a straightforward mixing function. Two independent excitations ξ 1 and ξ 2 are used as input signals which are already adjusted to the desired longitudinal direction spectrum. From these signals, two coupling signals η 1 and η 2