Lutz Auersch

Resonances of rail bridges due to the passage of trains have been mainly investigated for single-span bridges. When multi-span bridges are considered, it is of interest if stronger resonance amplifications compared to a single-span bridge must be taken into account. Measurements of eight multi-span structures are presented with their natural frequencies and mode shapes: wooden floor beams, a concrete beam, a steel truss foot bridge, three road bridges, and two rail bridges. Typical effects will be shown, such as multiple fundamental frequencies due to parallel beams and multiple spans, close eigenfrequencies of different mode shapes, clusters for simply-supported or weakly-coupled spans, regular separated frequencies for continuous beams and bridges. The mode shapes are generally global, but sometimes more local mode shapes exist, for example, a longer mid span is dominant in the first eigen mode. The consequences of multi-span bridges for train passages will be discussed based on a frequency-domain theory where three spectra determine the resonance excitation., the axle-sequence spectrum of the train, the transfer function of the bridge, and the modal force spectrum. The method is applied to the two railway bridge examples which are treated as single span bridges at first, then these examples are extended to two-, three-, and multi-span bridges. The resonance of a railway bridge occurs when a maximum of the axle-sequence spectrum (the second or third car length frequency in the examples) meets an eigenfrequency of the bridge (the maximum of the transfer function). The resonance amplitude is determined by the mode shape, which results directly in the modal force spectrum and in the modal mass. Some rules are presented for the modal force (mode shape) spectrum, for the modal mass, and in consequence for the resonance amplitude of multi-span bridges. Only the in-phase mode, where all spans are in phase, can have a similar resonance amplitude as the corresponding single-span bridge. Simply-supported multi-span bridges can have the same resonance amplitude, a continuous bridge with several spans can have a somewhat higher resonance amplitude for the in-phase mode compared to the single span bridge. All other fundamental modes have lower resonance amplitudes than the corresponding single-span bridge.

The Federal Institute of Material Research and Testing has performed many impact tests, from very small laboratory tests to very big “free-field” tests with heavy containers on stiff foundations. The first measurements have been done on a big foundation where it should be guaranteed that the foundation is rigid and the container is tested properly. Later, a smaller drop-test facility has been built on the ground inside an existing building. It had to be controlled by prediction and measurements to ensure that the drop test will not damage the building. Tests from different heights on soft, medium, and stiff targets have been done to find out rules which allow to identify acceptable and unacceptable drop tests. Later, the biggest drop test facility has been built for masses up to 200 t. It was necessary for the design of the foundation to estimate the forces which occur during the drop tests. In addition, the acceptable tests should be selected and controlled by measurements where the impact duration is important. Different sensors, accelerometers, accelerometers with mechanical filters, geophones (velocity transducers), strain gauges, and pressure cells have been applied for these tasks. Signal transformations and model calculations have been used to check and understand the dynamic measurements. The simplest law is the conservation of the momentum which is a good approximation if the impact is short. If the soil under the foundation has an influence on the deceleration of the container, the maximum foundation velocity is lower than the simple estimation.

Measurement results of train-induced vibrations are evaluated for characteristic frequencies, amplitudes and spectra, leading to a prediction which is based on transfer functions of the vehicle–track–soil system, the soil, and the building–soil system. The characteristic frequencies of train-induced vibrations are discussed following the propagation of vibrations from the source to the receiver: out-of-roundness frequencies of the wheels, the sleeper passage frequency, the vehicle–track eigenfrequency, the car-length frequency and multiples, axle-distance frequencies, bridge eigenfrequencies, the building–soil eigenfrequency, and floor eigenfrequencies. Amplitudes and spectra are compared for different train and track types, for different train speeds, and for different soft and stiff soils, where high frequencies are typically found for stiff soil and low frequencies for soft soil. The ground vibration is between the cut-on frequency due to the layering and the cut-off frequency due to the material damping of the soil, but the dominant frequency range also changes with distance from the track. The frequency band of the axle impulses due to the passing static loads obtains a signature from the axle sequence. The high amplitudes between the zeros of the axle-sequence spectrum are measured at the track, the bridge, and also in the ground vibrations, which are even dominant in the far field. A prediction software is presented, which includes all three parts: the excitation by the vehicle–track interaction, the wave transmission through the soil, and the transfer into a building.

In many tasks of railway vibration, the structure, that is, the track, a bridge, and a nearby building and its floors, is coupled to the soil, and the soil–structure interaction and the damping by the soil should be included in the analysis to obtain realistic resonance frequencies and amplitudes. The stiffness and damping of a variety of foundations is calculated by an indirect boundary element method which uses fundamental solutions, is meshless, uses collocation points on the boundary, and solves the singularity by an appropriate averaging over a part of the surface. The boundary element method is coupled with the finite element method in the case of flexible foundations such as beams, plates, piles, and railway tracks. The results, the frequency-dependent stiffness and damping of single and groups of rigid foundations on homogeneous and layered soil and the amplitude and phase of the dynamic compliance of flexible foundations, show that the simple constant stiffness and damping values of a rigid footing on homogeneous soil are often misleading and do not represent well the reality. The damping may be higher in some special cases, but, in most cases, the damping is lower than expected from the simple theory. Some applications and measurements demonstrate the importance of the correct damping by the soil.

The mitigation of train-induced ground vibrations by track solutions is investigated by calculations and measurements. The calculation by a wavenumber domain method includes the correct vehicle–track interaction and the correct track–soil interaction. Some theoretical results for elastic elements and an increased bending stiffness of the track are presented where the force transfer of the track and the vehicle–track interaction are calculated for the high-frequency dynamic mitigation, and the force distribution along the track is calculated for the low-frequency mitigation which is due to the smoother impulses from the passing static loads. Measurement results for the ground vibration near isolated and un-isolated tracks are given for several under-sleeper pads, for under-ballast mats, and for several under-ballast plates and ballast troughs. The elastic elements yield a resonance frequency of the vehicle–track–soil system and a high-frequency reduction of the dynamic axle loads which depends mainly on the softness of the pads or mats and which can be improved by a higher sleeper mass. In addition, all troughs and most of the soft elements show a low-frequency reduction which is attributed to the scattered impulses of the static axle loads. Besides this main contribution of the article, the problem of a soft reference section on a different soil is discussed and recommendations for better ground vibration measurements of mitigation effects are given.

The vehicle–track interaction generates forces and consequently vibrations in the environment. The interaction has been analysed by the simultaneous measurements of vehicle, track and ground vibrations during test runs with varied train speeds. The special effects of the passage over a bridge and through a tunnel are studied and compared with the measurements on a conventional ballasted surface line. The maximum amplitudes, narrow band and one-third octave band spectra are presented for the axle-box accelerations and for the track, bridge and ground vibrations. The different frequencies and frequency bands are related to wheel out-of-roundness, track alignment errors, the sleeper passage and the wheelset–track resonance. An axle impulse component has been observed at the track, at the near-field soil and as a scattered version in the far field. Specific results can be found for the bridge track, where clearly speed-dependent bridge resonances occur due to the axle sequence of the train, and for the tunnel track where soft rail pads are responsible for a strong amplification around the wheelset–track resonance. On the other hand, the axle impulses are strongly reduced by the tunnel track, and the scattered axle impulse component is not as relevant as for the surface track. As a consequence, a strong mid-frequency amplitude reduction of the tunnel compared to the surface line has been measured for low and high train speeds by the Federal Institute of Material Research and Testing (BAM) and by other institutes.

Purpose: The purpose of this article is to check if and how hammer experiments can be applied to the prediction of railway vibration. Methods: The propagation of ground vibrations is theoretically analysed with frequency-wavenumber and simplified methods. Experimental methods are presented which can characterise the site-specific ground vibrations by wave velocities, stiffness and damping. Measurements with hammer and train excitation have been performed at several sites. Results: The one-third octave spectra show the stiffness-dependent amplitudes and the low- and high-frequency filter effects due to the layering and the damping of the soil. Specific train effects, an additional high-frequency filter, the sleeper passage frequency, and an amplified mid-frequency component can be clearly found. The attenuation with distance is analysed in detail where the theoretical exponential and the empirical frequency-dependent power law are considered. Hammer and train excitation show the same site-specific effects which are mainly due to the stronger or weaker damping of the soil. The train attenuation is generally weaker than the hammer attenuation. The attenuation exponent of the power law, which is strongly dependent on the site and the frequency, is reduced for the train vibration by 0.3–0.5 in agreement with the theory. Reasons are discussed for the overall power law and for the dominating mid-frequency component. Conclusion: Therefore, it can be concluded that hammer experiments can well be used for the prediction of train vibrations.