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Keywords = rail dampers

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19 pages, 1860 KiB  
Article
Multi-Objective Vibration Control of a Vehicle-Track-Bridge Coupled System Using Tuned Inerter Dampers Based on the FE-SEA Hybrid Method
by Xingxing Hu, Qingsong Feng, Min Yang and Jian Liu
Appl. Sci. 2025, 15(15), 8675; https://doi.org/10.3390/app15158675 (registering DOI) - 5 Aug 2025
Abstract
To address the adverse effects of Tuned Inertia Dampers (TIDs) on track slab vibrations while controlling high-frequency rail vibrations, a hybrid Finite Element-Statistical Energy Analysis (FE-SEA) method is developed for modeling the vehicle-track-bridge coupled system. Short-wavelength track irregularities are introduced as high-frequency excitation, [...] Read more.
To address the adverse effects of Tuned Inertia Dampers (TIDs) on track slab vibrations while controlling high-frequency rail vibrations, a hybrid Finite Element-Statistical Energy Analysis (FE-SEA) method is developed for modeling the vehicle-track-bridge coupled system. Short-wavelength track irregularities are introduced as high-frequency excitation, and the accuracy and efficiency of this method are validated by comparison with the traditional finite element method (FEM). A vibration control model for track-bridge structures incorporating TIDs is designed, and the effects of the TID’s inertance, stiffness, and damping coefficients on the vertical acceleration responses of the rail and track slab are investigated in detail. The study reveals that although TIDs effectively reduce rail vibrations, they may induce adverse effects on track slab vibrations. Using the vibration acceleration amplitudes of both the rail and track slab as dual control objectives, a multi-objective optimization model is established, and the TID’s optimal parameters are determined using a multi-objective genetic algorithm. The results show that the optimized TID parameters reduce rail acceleration amplitudes by 16.43% and improve the control efficiency by 12.45%, while also addressing the negative effects on track slab vibration. The track slab’s vibration acceleration is reduced by 5.47%, and the vertical displacement and acceleration of the vehicle body are reduced by 14.22% and 47.5%, respectively, thereby enhancing passenger comfort. This study provides new insights and theoretical guidance for vibration control analysis in vehicle-track-bridge coupled systems. Full article
21 pages, 6856 KiB  
Article
Reducing Railway Track Vibrations by Applying Particle-Damping Systems
by Felipe Klein Fiorentin, Cristian Piehowiak, Anelize Zomkowski Salvi, Yesid Ernesto Asaff, Andrea Piga Carboni, Abílio Manuel Pinho de Jesus and Thiago Antonio Fiorentin
Appl. Sci. 2025, 15(9), 5014; https://doi.org/10.3390/app15095014 - 30 Apr 2025
Viewed by 648
Abstract
The wheel–rail contact is an intrinsic characteristic of rail transport. This contact is one of the main reasons why rails are so efficient for transportation, mainly due to the very low friction coefficient between them and the wheels. However, this strong argument also [...] Read more.
The wheel–rail contact is an intrinsic characteristic of rail transport. This contact is one of the main reasons why rails are so efficient for transportation, mainly due to the very low friction coefficient between them and the wheels. However, this strong argument also leads to a disadvantage: the wheel contact is also associated with excessive vibration and noise, which have a strong impact on the passengers’ comfort and especially the surrounding community. These noises and vibrations impact the public in several ways, like disturbing sleep, increasing stress and heart-associated diseases. The main objective of the present work is to investigate the rail vibration attenuation by applying particle dampers. Four different particles will be studied, and their effectiveness in reducing the rail vibrations will be analysed. Promising results were found, where under certain conditions, the particle dampers, such as lead and magnetite particles, were able to reduce peak vibration levels by more than an order of magnitude. The application of this system may have a strong impact on the communities using and in the vicinity of rail systems by reducing the noise and vibration, consequently improving people’s health and well-being. Full article
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29 pages, 3075 KiB  
Article
Parametric Study of a Fully Passive Oscillating Foil on a Swinging Arm
by Dominic Cloutier, Mathieu Olivier and Guy Dumas
Energies 2025, 18(5), 1277; https://doi.org/10.3390/en18051277 - 5 Mar 2025
Cited by 1 | Viewed by 763
Abstract
A NACA 0015 airfoil is connected to a swinging arm by springs and dampers and is let loose in an incompressible and viscous flow at a Reynolds number of 3.9×106. The foil operates in a power-extracting regime and is [...] Read more.
A NACA 0015 airfoil is connected to a swinging arm by springs and dampers and is let loose in an incompressible and viscous flow at a Reynolds number of 3.9×106. The foil operates in a power-extracting regime and is free to pitch about a pivot that is itself swinging on a circular path; this contraption is called a fully passive oscillating-foil turbine on a swinging arm. This study explores the potential of four different foil configurations: with the swinging arm being either upstream or downstream of its pivot, and with or without the use of gears to control the equilibrium position of the foil with respect to the flow. The results show that the swinging arm concept offers similar performances, i.e., efficiency and power coefficient, as the railed turbine. Indeed, with arm lengths from 3 to 10 chords, efficiency values near 55% and power coefficients reaching 1.57 are obtained. Both the railed and the swinging arm turbines can operate under either a stall-flutter or a coupled-flutter instability. However, it is found that the geared models are the only ones suited when the driving mechanism is the coupled-flutter instability while both geared and gearless configurations are effective under the stall-flutter instability. Full article
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18 pages, 3021 KiB  
Article
Parametric Study on the Effect of Rail Dampers on Track Decay Rate
by Dorina Fologea, Traian Mazilu, Marius-Alin Gheți and Ioana-Izabela Apostol
Appl. Sci. 2024, 14(20), 9541; https://doi.org/10.3390/app14209541 - 19 Oct 2024
Cited by 2 | Viewed by 1259
Abstract
Track decay rate (TDR), meaning the rate of attenuation of bending waves through the rail, is the most important indicator of a track’s dynamic characteristic impacting the rail noise emission. TDR depends on various parameters related to the construction of the track, and [...] Read more.
Track decay rate (TDR), meaning the rate of attenuation of bending waves through the rail, is the most important indicator of a track’s dynamic characteristic impacting the rail noise emission. TDR depends on various parameters related to the construction of the track, and it can be increased using rail dampers. These are mechanical devices working on the principle of dynamic absorbers and are attached to the rail. This paper addresses the track with light rails needing improvements to reduce the rail noise emission using a particular rail damper with a mixed damping system (rubber–oil). The bending waves that propagate through the rail, the frequency response function of the rail, and TDR are investigated considering different scenarios regarding the parameters of the track: soft/stiff rail pad, tampered/settled ballast, and sleeper bay. To this end, an analytic model of the track featuring rail dampers consisting of an infinite Timoshenko beam with discrete attached oscillators is used. Numerical results show the possibility to increase TDR of railway track with light rails for both soft/stiff rail pads from 4 to 500 Hz up to 1250–1600 Hz using rail dampers with a mixed damping system. Full article
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15 pages, 7044 KiB  
Article
Fast Detection of the Stick–Slip Phenomenon Associated with Wheel-to-Rail Sliding Using Acceleration Sensors: An Experimental Study
by Gabriel Popa, Mihail Andrei, Emil Tudor, Ionuț Vasile and George Ilie
Technologies 2024, 12(8), 134; https://doi.org/10.3390/technologies12080134 - 13 Aug 2024
Cited by 3 | Viewed by 6058
Abstract
The stick–slip phenomenon, the initial stage when the traction wheel starts sliding on the rail, is a critical operation that needs to be detected quickly to control the traction drive. In this study, we have developed an experimental model that uses acceleration sensors [...] Read more.
The stick–slip phenomenon, the initial stage when the traction wheel starts sliding on the rail, is a critical operation that needs to be detected quickly to control the traction drive. In this study, we have developed an experimental model that uses acceleration sensors mounted on the wheel to evaluate the amplitude of the stick–slip phenomena. These sensors can alert the driver or assist the traction control unit when a stick–slip occurs. We propose a method to reduce the amplitude of the stick–slip phenomenon using special hydraulic dampers and viscous dampers mounted on the tractive axles of the locomotive to prevent slipping during acceleration. This practical solution, validated through numerical simulation, can be readily implemented in railway systems. The paper’s findings can be used to select the necessary sensors and corresponding vibration dampers. By implementing these sliding reducers, a locomotive can significantly improve traction, apply more torque to the wheel, and increase the load of a carrier train, instilling confidence in the efficiency of the proposed solution. Full article
(This article belongs to the Special Issue Advanced Autonomous Systems and Artificial Intelligence Stage)
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29 pages, 8536 KiB  
Article
A Simulation Approach for Analysis of the Regenerative Potential of High-Speed Train Suspensions
by Haihua Wang, Xinjue Zhang, Ruichen Wang and Guosheng Feng
Energies 2024, 17(14), 3496; https://doi.org/10.3390/en17143496 - 16 Jul 2024
Viewed by 1237
Abstract
This study primarily investigates the adaptability and performance of hydraulic–electric regenerative dampers for high-speed trains by substituting conventional primary dampers with hydraulic–electric regenerative dampers. The primary objectives are to develop a detailed model of primary suspension regenerative damper (PSRD) energy conversion that incorporates [...] Read more.
This study primarily investigates the adaptability and performance of hydraulic–electric regenerative dampers for high-speed trains by substituting conventional primary dampers with hydraulic–electric regenerative dampers. The primary objectives are to develop a detailed model of primary suspension regenerative damper (PSRD) energy conversion that incorporates factors such as oil pressure loss, motor efficiency, and overall system efficiency, and to perform a comprehensive comparative analysis of vibration responses, wheel wear, comfort indices, and power generation using an integrated MATLAB and SIMPACK co-simulation platform. The results reveal that at an operational speed of 350 km/h, the dynamic responses of the carbody, bogie, wheelset, and dampers equipped with the proposed PSRD systems closely align with those of conventional primary vertical damper systems. The detailed PSRDs’ hydraulic–mechanical–electrical model effectively captures the subtleties of oil pressure fluctuations and their impacts. The wear distribution and magnitude across the vehicle remain consistent during acceleration, constant, and deceleration speeds, ensuring uniform wear characteristics. Under real-world railway operational conditions, the ride comfort metrics of vehicles fitted with regenerative dampers are comparable to those with conventional primary vertical dampers. Furthermore, each regenerative damper can generate up to 21.72 W of electrical power, achieving a generation efficiency of 45.28%. Finally, a test rig was designed and fabricated to validate the primary suspension regenerative damper (PSRD) model, showing good agreement between predicted and actual damping force and power regeneration, with results indicating a peak damping force of 12.5 kN and approximately 230 W of regenerated power. This research provides a theoretical foundation and experimental validation for implementing power regeneration mechanisms in railway transportation, demonstrating that the hydraulic–mechanical–electrical PSRD model can fulfil the performance criteria of conventional dampers while offering substantial energy harvesting capabilities. This advancement not only enhances energy efficiency but also contributes to the sustainable development of high-speed rail systems. Full article
(This article belongs to the Section F: Electrical Engineering)
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24 pages, 3776 KiB  
Article
Modelling and Dynamic Analysis of Adaptive Neuro-Fuzzy Inference System-Based Intelligent Control Suspension System for Passenger Rail Vehicles Using Magnetorheological Damper for Improving Ride Index
by Sunil Kumar Sharma, Rakesh Chandmal Sharma, Yeongil Choi and Jaesun Lee
Sustainability 2023, 15(16), 12529; https://doi.org/10.3390/su151612529 - 18 Aug 2023
Cited by 15 | Viewed by 1812
Abstract
The ride comfort and safety of passenger rail vehicles depend on the performance of the suspension system in attenuating vibrations induced by track irregularities. This paper investigates the effectiveness of an Adaptive Neuro-Fuzzy Inference System (ANFIS)-based semi-active controlled suspension system using a magnetorheological [...] Read more.
The ride comfort and safety of passenger rail vehicles depend on the performance of the suspension system in attenuating vibrations induced by track irregularities. This paper investigates the effectiveness of an Adaptive Neuro-Fuzzy Inference System (ANFIS)-based semi-active controlled suspension system using a magnetorheological fluid damper in reducing nonlinear lateral vibrations of a passenger rail vehicle. A complete rail vehicle model is developed, including the carbody, front and rear bogies, and the passive suspension system’s nonlinear stiffness and damping characteristics are considered from experimental data. The passive suspension model is validated through experiments, and an ANFIS-based controller is incorporated with the secondary vertical suspension system to improve ride behavior. Three semi-active suspension strategies are considered under varying speeds and track irregularities, and their effectiveness is compared to the nonlinear passive suspension system in terms of rms acceleration, rms displacement, ride quality, and comfort. The results shows that the ANFIS-based semi-active suspension system with a magnetorheological fluid damper outperforms the passive suspension system and semi-active strategies in all tested conditions. There is a reduction in rms acceleration by approximately 11.11% to 23.64% and rms displacement by about 5.36% to 32.06%. Moreover, it significantly improves ride quality (9.20% to 31.02%) and comfort (9.96% to 31.50%). The rms acceleration and displacement are reduced, and the Sperling ride index and Percentage Reduction Index values demonstrate that the ANFIS-based semi-active suspension effectively minimizes the impact of rail irregularities and vibrations, resulting in a significant gain in ride quality and passenger comfort. Full article
(This article belongs to the Special Issue Sustainable Railway Infrastructure and Rolling Stock)
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16 pages, 7339 KiB  
Article
Heat Treatment and Dynamic Mechanical Analysis Strain Sweep Effects on the Phase Structure and Morphology of an Fe-28Mn-6Si-5Cr Shape Memory Alloy
by Mihai Popa, Florin Popa, Bogdan Pricop, Nicanor Cimpoeșu, Nicoleta-Monica Lohan, Gabriel Kicsi, Bogdan Istrate and Leandru-Gheorghe Bujoreanu
Nanomaterials 2023, 13(7), 1250; https://doi.org/10.3390/nano13071250 - 1 Apr 2023
Cited by 1 | Viewed by 2099
Abstract
Fe-Mn-Si-based shape memory alloys (SMAs) have been extensively investigated since 1982 for various useful properties that enhance the development of different applications such as anti-seismic dampers for very tall buildings, pipe joints, or rail fasteners. In particular, the Fe-28Mn-6Si-5Cr (mass. %) alloy has [...] Read more.
Fe-Mn-Si-based shape memory alloys (SMAs) have been extensively investigated since 1982 for various useful properties that enhance the development of different applications such as anti-seismic dampers for very tall buildings, pipe joints, or rail fasteners. In particular, the Fe-28Mn-6Si-5Cr (mass. %) alloy has been mainly used in vibration mitigation or self-adjustable axial displacement applications. Dynamic mechanical analysis (DMA), performed by strain sweeps (SS), enables the monitoring of the evolution of storage modulus and internal friction variations with increasing strain amplitudes at different constant frequencies and temperatures. Thus, applying dynamic bending with various frequencies and amplitudes that actually represents an isothermal mechanical treatment. In the present paper, an Fe-28Mn-6 Si-5Cr (mass. %) SMA was cast by ingot metallurgy, hot-rolled, and water quenched in order to obtain thermally induced martensite and avoid the occurrence of cooling cracks. The influence of the holding time, between 2 and 10 h, at 1050 °C and the effects of DMA-SS performed at three different frequencies were analyzed by a differential scanning calorimetry, an X-ray diffraction, and a scanning electron and atomic force microscopy. The effects of the holding time and mechanical treatment on the structure and morphology of martensite plates were corroborated with the results of the thermal analysis. Full article
(This article belongs to the Special Issue Nanostructural Processing Effects in Shape Memory Alloys)
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13 pages, 7344 KiB  
Article
Effect of the Magnetorheological Damper Dynamic Behaviour on the Rail Vehicle Comfort: Hardware-in-the-Loop Simulation
by Filip Jeniš, Michal Kubík, Tomáš Michálek, Zbyněk Strecker, Jiří Žáček and Ivan Mazůrek
Actuators 2023, 12(2), 47; https://doi.org/10.3390/act12020047 - 19 Jan 2023
Cited by 14 | Viewed by 2467
Abstract
Many publications show that the ride comfort of a railway vehicle can be significantly improved using a semi-active damping control of the lateral secondary dampers. However, the control efficiency depends on the selection of the control algorithm and the damper dynamic behaviour, i.e., [...] Read more.
Many publications show that the ride comfort of a railway vehicle can be significantly improved using a semi-active damping control of the lateral secondary dampers. However, the control efficiency depends on the selection of the control algorithm and the damper dynamic behaviour, i.e., its force rise response time, force drop response time and force dynamic range. This paper examines the influence of these parameters of a magnetorheological (MR) damper on the efficiency of S/A control for several control algorithms. One new algorithm has been designed. Hardware-in-the-loop simulation with a real magnetorheological damper has been used to get close to reality. A key finding of this paper is that the highest efficiency of algorithms is not achieved with a minimal damper response time. Furthermore, the force drop response time has been more important than the force rise response time. The Acceleration Driven Damper Linear (ADD-L) algorithm achieves the highest efficiency. A reduction in vibration of 34% was achieved. Full article
(This article belongs to the Special Issue 10th Anniversary of Actuators)
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18 pages, 7141 KiB  
Article
An Efficient Numerical Model to Predict the Mechanical Response of a Railway Track in the Low-Frequency Range
by Maryam El Moueddeb, François Louf, Pierre-Alain Boucard, Franck Dadié, Gilles Saussine and Danilo Sorrentino
Vibration 2022, 5(2), 326-343; https://doi.org/10.3390/vibration5020019 - 24 May 2022
Cited by 6 | Viewed by 3208
Abstract
With railway interoperability, new trains are allowed to move on the French railway network. These trains may present different designs from standard trains. This work aims to complete the current approach for vehicle admission on the railway network, which is defined in technical [...] Read more.
With railway interoperability, new trains are allowed to move on the French railway network. These trains may present different designs from standard trains. This work aims to complete the current approach for vehicle admission on the railway network, which is defined in technical baselines. Historically, computation rules for traffic conditions are based on simplified analytical works, which are considerably qualitative. They have evolved through feedback and experimental campaigns to comply with the track structure evolution. An efficient methodology based on numerical simulation is needed to evaluate railway vehicle admission to answer this issue. A perspective to update these computation rules is to evaluate the structural fatigue in the rail. That is to say, fatigue is caused by bending and shear stresses. The complexity of the railway system has led to an investigation at first of the vertical response of the railway track and quantifying its contribution to the rail’s stress response. In that sense, this paper investigates the vertical track response to a moving railway vehicle at low frequencies. For this purpose, a lightweight numerical model for the track, a multi-body model for the vehicle, and a random vertical track irregularity are proposed. More explicitly, the track model consists of a two-layer discrete support model in which the rail is considered as a beam and sleepers are point masses. The rail pads and ballast layer are modelled as spring/damper couples. Numerical results show a negligible effect of track inertia forces due to high track stiffness and damping. Nevertheless, this assumption is valid for normal rail stresses but not for ballast loading, especially in the case of sleeper voids or unsupported sleepers. Hence, the prediction of the mechanical stress state in the rail for fatigue issues is achieved through a static track model where the equivalent loading is obtained from a dynamic study of a simplified vehicle model. A statistical analysis shows that the variability of the vertical track irregularity does not influence the output variabilities like the maximum in time and space of the normal and shear stress. Full article
(This article belongs to the Topic Dynamical Systems: Theory and Applications)
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24 pages, 4693 KiB  
Article
Condition Monitoring of the Dampers in the Railway Vehicle Suspension Based on the Vibrations Response Analysis of the Bogie
by Mădălina Dumitriu
Sensors 2022, 22(9), 3290; https://doi.org/10.3390/s22093290 - 25 Apr 2022
Cited by 17 | Viewed by 3452
Abstract
This paper investigates the possibility of developing a new method for fault detection of a damper in the primary suspension of the railway vehicle, based on the analysis of the vertical vibration’s response of the bogie. To this purpose, experimental data are used, [...] Read more.
This paper investigates the possibility of developing a new method for fault detection of a damper in the primary suspension of the railway vehicle, based on the analysis of the vertical vibration’s response of the bogie. To this purpose, experimental data are used, along with results from numerical simulations regarding the Root Mean Square (RMS) accelerations measured/simulated in four reference bogie points—two points on the chassis, against the suspension, and two points located against the axle boxes. The experimental data are utilized to define the normal area of operating and the damper failure area in the bogie primary suspension, as well as a basis for validating the results of numerical simulations. The numerical simulations are developed on the basis of two original models of the vehicle–track system, rigid-flexible coupled type, which take into account the elasticity of the vehicle carbody and the elasticity of the wheel-rail contact: a reference model with 15 degrees of freedom, for simulating the bogie response to vertical vibrations for the normal operating of the primary suspension dampers, and an extended model with 20 degrees of freedom, for simulating the bogie vibration response to the failure damper of a primary suspension. The presented results show that there are clear premises on the possibilities of developing a fault detection method of any of the four dampers of the primary suspension corresponding to a vehicle bogie, based on the RMS accelerations measured only in two reference points of the bogie. Full article
(This article belongs to the Section Vehicular Sensing)
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20 pages, 6387 KiB  
Article
Study on Possible Application of Rubber Granulate from the Recycled Tires as an Elastic Cover of Prototype Rail Dampers, with a Focus on Their Operational Durability
by Cezary Kraśkiewicz, Bogumiła Chmielewska, Artur Zbiciak and Anna Al Sabouni-Zawadzka
Materials 2021, 14(19), 5711; https://doi.org/10.3390/ma14195711 - 30 Sep 2021
Cited by 11 | Viewed by 2580
Abstract
This study is an attempt to investigate possible applications of rubber granulate SBR (styrene-butadiene rubber) produced from recycled waste tires as an elastic cover for prototype rail dampers, which are aimed at reducing the level of railway noise emitted in the environment. The [...] Read more.
This study is an attempt to investigate possible applications of rubber granulate SBR (styrene-butadiene rubber) produced from recycled waste tires as an elastic cover for prototype rail dampers, which are aimed at reducing the level of railway noise emitted in the environment. The authors present laboratory procedures and discuss the results of several experimental tests performed on seven different SBR materials with the following densities: 1100, 1050, 1000, 850, 750, 700 and 650 kg/m3. It is proven that rubber granulate SBR produced from recycled waste tires, can be used as an elastic cover in steel inserts in rail dampers, provided that the material density is not lower than 1000 kg/m3. In the conducted tests, samples of the materials with high densities exhibited good static and dynamic elastic characteristics and had sufficient operational durability. Full article
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21 pages, 3690 KiB  
Case Report
Prediction of Theoretical Derailments Caused by Cross-Winds with Frequency
by JunWoo Kim, InHo Song and JeongSeo Koo
Appl. Sci. 2021, 11(7), 2947; https://doi.org/10.3390/app11072947 - 25 Mar 2021
Cited by 6 | Viewed by 2877
Abstract
In this paper, theoretical derailment equations for cross-wind with frequency were derived to assess running safety. For a KTX (Korean high-speed train) unit, the wheel unloading ratios, which are the criteria for evaluating derailments in UIC (International union of railways) and TSI (Technical [...] Read more.
In this paper, theoretical derailment equations for cross-wind with frequency were derived to assess running safety. For a KTX (Korean high-speed train) unit, the wheel unloading ratios, which are the criteria for evaluating derailments in UIC (International union of railways) and TSI (Technical Specification for Interoperability) regulations, were calculated through the formula under the driving regulations according to cross-wind speeds, and the theoretical results were compared and evaluated through a multibody dynamics (MBD) simulation. In addition, the wheel unloading ratios were calculated for various frequencies of cross-winds. As a result of the formula and MBD, the wheel unloading ratios were shown to increase rapidly regardless of the dampers in suspension when the cross-wind frequency and the natural frequency of a vehicle were in agreement. Finally, we calculated the changes of wheel unloading ratio for different track gauges and found that these theoretical equations could calculate more accurate results than the existing Kunieda’s formula. The formula derived in this study has the advantage of considering various variables, such as fluctuant cross-winds, rail irregularities, and derailment behaviors, which were not considered in previous studies or Kunieda’s formula. It could be used for setting suspensions or railway vehicle specifications in the initial design stage. Full article
(This article belongs to the Section Mechanical Engineering)
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17 pages, 1491 KiB  
Article
A Novel Solution to Find the Dynamic Response of an Euler–Bernoulli Beam Fitted with Intraspan TMDs under Poisson Type Loading
by Alberto Di Matteo, Iain Peter Dunn, Giuseppe Failla and Antonina Pirrotta
Infrastructures 2020, 5(5), 40; https://doi.org/10.3390/infrastructures5050040 - 7 May 2020
Cited by 1 | Viewed by 3589
Abstract
This contribution considers a virtual experiment on the vibrational response of rail and road bridges equipped with smart devices in the form of damping elements to mitigate vibrations. The internal damping of the bridge is considered a discontinuity that contain a dashpot. Exact [...] Read more.
This contribution considers a virtual experiment on the vibrational response of rail and road bridges equipped with smart devices in the form of damping elements to mitigate vibrations. The internal damping of the bridge is considered a discontinuity that contain a dashpot. Exact complex eigenvalues and eigenfunctions are derived from a characteristic equation built as the determinant of a 4 × 4 matrix; this is accomplished through the use of the theory of generalized functions to find the response variables at the positions of the damping elements. To relate this to real world applications, the response of a bridge under Poisson type white noise is evaluated; this is similar to traffic loading that would be seen in a bridge’s service life. The contribution also discusses the importance of smart damping and dampers to sustainability efforts through the reduction of required materials, and it discusses the role played by robust mathematical modelling in the design phase. Full article
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