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T. Lu
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1
Journal article
(2022)
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Bin Fang, Tao Lü, Liwei Cheng, Dongdong Wang, Yang Ni, Bowen Fan, Jiuling Meng, Thijs J.H. Vlugt, Fulong Ning
Methane hydrate dissociation kinetics can be inhibited in NaCl solutions; however, this effect is reversed by promoting bubble formation that enhances dissociation. The negative and positive effects of inorganic salt injection on gas production from hydrate-bearing sediments are still controversial. Here, molecular dynamics simulations were performed to investigate the characteristics of NaCl solution invasion into hydrate-occupied nanopores and the effects on the confined hydrate dissociation kinetics. Two initial configurations comprising liquid and silica pore phases were studied with a low or high NaCl concentration, respectively. The results show that, under the simulation conditions, salt invasion decelerated hydrate dissociation within the silica pore as NaCl invasion into the pore is stepwise. Initially, few ions can diffuse into the pore phase, and gas nanobubbles form on the solid surface mainly via confinement and surface effects, independent of NaCl solution invasion. Subsequently, gradual salt diffusion immersed the residual hydrate in the salt solution and hindered hydrate decomposition until the dissociation finished. More ions could diffuse into the pore phase at the high NaCl concentrations with a low diffusion efficiency, leading to surface nanobubble growth toward the residual hydrate and somewhat accelerated hydrate dissociation. This severely hinders the escape of released methane from the pore. This study yields molecular-level insight into the origin of the negative effect of salt invasion on hydrate dissociation, which should be avoided during gas production from hydrate reservoirs with low permeabilities via salt injection combined with thermal stimulation.
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Methane hydrate dissociation kinetics can be inhibited in NaCl solutions; however, this effect is reversed by promoting bubble formation that enhances dissociation. The negative and positive effects of inorganic salt injection on gas production from hydrate-bearing sediments are still controversial. Here, molecular dynamics simulations were performed to investigate the characteristics of NaCl solution invasion into hydrate-occupied nanopores and the effects on the confined hydrate dissociation kinetics. Two initial configurations comprising liquid and silica pore phases were studied with a low or high NaCl concentration, respectively. The results show that, under the simulation conditions, salt invasion decelerated hydrate dissociation within the silica pore as NaCl invasion into the pore is stepwise. Initially, few ions can diffuse into the pore phase, and gas nanobubbles form on the solid surface mainly via confinement and surface effects, independent of NaCl solution invasion. Subsequently, gradual salt diffusion immersed the residual hydrate in the salt solution and hindered hydrate decomposition until the dissociation finished. More ions could diffuse into the pore phase at the high NaCl concentrations with a low diffusion efficiency, leading to surface nanobubble growth toward the residual hydrate and somewhat accelerated hydrate dissociation. This severely hinders the escape of released methane from the pore. This study yields molecular-level insight into the origin of the negative effect of salt invasion on hydrate dissociation, which should be avoided during gas production from hydrate reservoirs with low permeabilities via salt injection combined with thermal stimulation.
A periodically supported beam on a visco-elastic half-space is considered to model the vibration of railway tracks. The viscosity of the half-space is assumed to be of the Kelvin-Voigt type. Making use of the concept of equivalent dynamic stiffness, the reaction of the half-space to the sleepers is replaced by a system of identical spring located under each sleeper. The frequency-dependent equivalent stiffness of the springs is a function of the phase shift of vibrations of neighbouring supports. The equivalent stiffness is derived analytically employing the contour integration technique, resulting in a comprehensive expression for different phase velocities of the waves in the beam with respect to the wave speeds of the half-space. Apart from the Rayleigh type surface wave (quasi-elastic wave), an extra visco-elastic surface wave may exist in a visco-elastic half-space depending on the parameters of the half-space and the frequency range. The existence of this second surface wave has not been addressed within the field of train-induced ground vibration. The importance of this wave for the equivalent stiffness is analysed. An effective method to determine the frequency range for the visco-elastic surface wave to exist is proposed.
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A periodically supported beam on a visco-elastic half-space is considered to model the vibration of railway tracks. The viscosity of the half-space is assumed to be of the Kelvin-Voigt type. Making use of the concept of equivalent dynamic stiffness, the reaction of the half-space to the sleepers is replaced by a system of identical spring located under each sleeper. The frequency-dependent equivalent stiffness of the springs is a function of the phase shift of vibrations of neighbouring supports. The equivalent stiffness is derived analytically employing the contour integration technique, resulting in a comprehensive expression for different phase velocities of the waves in the beam with respect to the wave speeds of the half-space. Apart from the Rayleigh type surface wave (quasi-elastic wave), an extra visco-elastic surface wave may exist in a visco-elastic half-space depending on the parameters of the half-space and the frequency range. The existence of this second surface wave has not been addressed within the field of train-induced ground vibration. The importance of this wave for the equivalent stiffness is analysed. An effective method to determine the frequency range for the visco-elastic surface wave to exist is proposed.
The effect of large-scale variation in the support stiffness on railway track degradation is studied using a frequency-domain approach. The model used can deal with parametric excitation due to both the discrete sleeper spacing and arbitrary large-scale spatial track non-uniformity. Adopted stiffness profiles are based on realistic datasets in the literature. The sensitivity to degradation is assessed by quantifying the energy dissipation in the substructure over the influence zone. Results show that the effect of spatial stiffness variation generally increases with the speed, for any subgrade condition; system resonance however leads to increased degradation at resonance speeds and increases with the mean value of the track stiffness. The speed is shown to have a larger influence in the presence of non-uniformity than it has for uniform track with a mean value of the same non-uniform track stiffness, independent of this mean value. In general, support stiffness non-uniformity and poor track conditions (in terms of a low overall stiffness) may have comparable effects; the combination of both is a worst-case scenario. Predictions are independent from the randomness for measured datasets and have therefore general validity. Further, an excellent correlation is found between the spatial variation of the dynamic track stiffness, the differential energy dissipation in the substructure, and the work performed by the moving contact load with respect to the track, independent of the train speed. This confirms existing empirical evidence of the dynamic track stiffness for non-uniform track as an indicator for degradation.
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The effect of large-scale variation in the support stiffness on railway track degradation is studied using a frequency-domain approach. The model used can deal with parametric excitation due to both the discrete sleeper spacing and arbitrary large-scale spatial track non-uniformity. Adopted stiffness profiles are based on realistic datasets in the literature. The sensitivity to degradation is assessed by quantifying the energy dissipation in the substructure over the influence zone. Results show that the effect of spatial stiffness variation generally increases with the speed, for any subgrade condition; system resonance however leads to increased degradation at resonance speeds and increases with the mean value of the track stiffness. The speed is shown to have a larger influence in the presence of non-uniformity than it has for uniform track with a mean value of the same non-uniform track stiffness, independent of this mean value. In general, support stiffness non-uniformity and poor track conditions (in terms of a low overall stiffness) may have comparable effects; the combination of both is a worst-case scenario. Predictions are independent from the randomness for measured datasets and have therefore general validity. Further, an excellent correlation is found between the spatial variation of the dynamic track stiffness, the differential energy dissipation in the substructure, and the work performed by the moving contact load with respect to the track, independent of the train speed. This confirms existing empirical evidence of the dynamic track stiffness for non-uniform track as an indicator for degradation.
Journal article
(2020)
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Andrei B. Faragau, Traian Mazilu, Andrei V. Metrikine, Tao Lu, Karel N. van Dalen
Transition zones in railway tracks are areas with considerable variation of track properties (e.g., foundation stiffness) which may cause strong amplification of the response, leading to rapid degradation of the track geometry. Two possible indicators of degradation in the supporting structure are identified, namely the wheel-rail contact force and the power/energy input by the vehicle. This paper analyses the influence of accounting for the interaction between the vehicle and the supporting structure on the contact force and on the power/energy input. To this end, a one-dimensional model is formulated, consisting of an infinite Euler-Bernoulli beam resting on a locally inhomogeneous Kelvin foundation, interacting with a moving loaded oscillator that has a nonlinear Hertzian spring. The solution is obtained using the Green's-function method. To obtain the Green's function of the inhomogeneous and infinite beam-foundation sub-system, the finite difference method is used for the spatial discretization and non-reflective boundary conditions are applied. Accounting for the interaction between the moving oscillator and the supporting structure generally leads to stronger wave radiation, caused by the variation of the vertical momentum of the moving mass. Results show that for relatively high velocity and small transition length, the maximum contact force as well as the energy input exhibit a significant increase compared to the moving constant load case. Furthermore, for relatively high velocities, the maximum contact force also increases significantly with increasing stiffness dissimilarity, findings which supplement the existing literature. Finally, the two degradation indicators can be used in the preliminary stages of design to assess the performance of railway track transition zones.
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Transition zones in railway tracks are areas with considerable variation of track properties (e.g., foundation stiffness) which may cause strong amplification of the response, leading to rapid degradation of the track geometry. Two possible indicators of degradation in the supporting structure are identified, namely the wheel-rail contact force and the power/energy input by the vehicle. This paper analyses the influence of accounting for the interaction between the vehicle and the supporting structure on the contact force and on the power/energy input. To this end, a one-dimensional model is formulated, consisting of an infinite Euler-Bernoulli beam resting on a locally inhomogeneous Kelvin foundation, interacting with a moving loaded oscillator that has a nonlinear Hertzian spring. The solution is obtained using the Green's-function method. To obtain the Green's function of the inhomogeneous and infinite beam-foundation sub-system, the finite difference method is used for the spatial discretization and non-reflective boundary conditions are applied. Accounting for the interaction between the moving oscillator and the supporting structure generally leads to stronger wave radiation, caused by the variation of the vertical momentum of the moving mass. Results show that for relatively high velocity and small transition length, the maximum contact force as well as the energy input exhibit a significant increase compared to the moving constant load case. Furthermore, for relatively high velocities, the maximum contact force also increases significantly with increasing stiffness dissimilarity, findings which supplement the existing literature. Finally, the two degradation indicators can be used in the preliminary stages of design to assess the performance of railway track transition zones.
Ring-like structures are very commonly used in civil, mechanical and aerospace engineering. Typical examples of such structures are components in turbomachinery, compliant gears, conventional pneumatic tires and more recent non-pneumatic tires, to name a few. In this paper, a ring on elastic foundation is considered. The foundation, modelled as distributed springs, connects the inner surface of the ring to an immovable axis. Focus is placed on the in-plane response of the ring subjected to in-plane load only. A high-order ring model, which accounts for the through-thickness variations of displacements is adopted for the study. Two loading situations of a ring structure are of interest in practice: (i) a stationary ring subjected to a circumferentially moving constant load; and (ii) a rotating ring under a stationary constant load. For the first situation, it is well-known that resonances occur when the rotational speeds of the load satisfy certain conditions. In a series of recent investigations, such resonance speeds have been predicted for a rotating ring subjected to a stationary load. In this paper the case of the rotating ring under a stationary constant load and that of a stationary ring subjected to a moving load are compared in terms of their resonance speeds, as well as the steady-state responses for various parameters. It is found that these two cases are distinguishable even for system parameters which result at similar critical speeds.
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Ring-like structures are very commonly used in civil, mechanical and aerospace engineering. Typical examples of such structures are components in turbomachinery, compliant gears, conventional pneumatic tires and more recent non-pneumatic tires, to name a few. In this paper, a ring on elastic foundation is considered. The foundation, modelled as distributed springs, connects the inner surface of the ring to an immovable axis. Focus is placed on the in-plane response of the ring subjected to in-plane load only. A high-order ring model, which accounts for the through-thickness variations of displacements is adopted for the study. Two loading situations of a ring structure are of interest in practice: (i) a stationary ring subjected to a circumferentially moving constant load; and (ii) a rotating ring under a stationary constant load. For the first situation, it is well-known that resonances occur when the rotational speeds of the load satisfy certain conditions. In a series of recent investigations, such resonance speeds have been predicted for a rotating ring subjected to a stationary load. In this paper the case of the rotating ring under a stationary constant load and that of a stationary ring subjected to a moving load are compared in terms of their resonance speeds, as well as the steady-state responses for various parameters. It is found that these two cases are distinguishable even for system parameters which result at similar critical speeds.
In the framework of vehicle-track interaction, this work puts an emphasis on clarifying the influence of sleeper finite element types on system dynamic responses. Three sleeper element types, namely the rigid-body, extensible Euler-Bernoulli beam and solid element are used respectively in the track system modelling with detail mathematical formulations presented. The rails are modelled as Timoshenko beams. The ballasted track system is subject to a moving vehicle with coupled wheel-rail interactions. From aspects of both frequency- and time- domain analysis in the numerical study, the effectiveness of this model has been validated. The influences of the sleeper finite element type, the sleeper support stiffness and damping coefficient on the system responses have been investigated.
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In the framework of vehicle-track interaction, this work puts an emphasis on clarifying the influence of sleeper finite element types on system dynamic responses. Three sleeper element types, namely the rigid-body, extensible Euler-Bernoulli beam and solid element are used respectively in the track system modelling with detail mathematical formulations presented. The rails are modelled as Timoshenko beams. The ballasted track system is subject to a moving vehicle with coupled wheel-rail interactions. From aspects of both frequency- and time- domain analysis in the numerical study, the effectiveness of this model has been validated. The influences of the sleeper finite element type, the sleeper support stiffness and damping coefficient on the system responses have been investigated.
The in-plane steady-state response of a rotating ring on elastic foundation subjected to a stationary load is investigated theoretically using a high-order model in the framework of the plane strain assumption. The adopted high-order model accounts for the through-thickness variation of stresses and displacements, as well as the boundary tractions at the inner and outer surfaces of the ring. Based on the ratio of the foundation stiffness to the stiffness of the ring, two configurations of the ring-on-foundation system are investigated, namely soft foundation (stiff ring) and stiff foundation (soft ring). The analytical “method of the images” is used to obtain the ring response. It is found that the response of a stiff ring to a stationary load of constant magnitude is governed by the translational rigid body-like motion. In contrast, in the case of a soft ring, a wave-like deformation is predicted for the rotational speeds higher than a critical one. It is for the first time that such wave-like displacements are predicted using a rotating ring model with the rotation effects being properly considered. The response of a rotating ring to a stationary harmonic load is studied too. The predicted displacements using the high-order model are compared with those obtained from the classical low-order model in which only the radial and circumferential displacements at the middle surface of the ring are considered. It is concluded that only in the case of a stiff ring, the classical low-order model and the high-order model give similar predictions. When the ring is soft, the predictions of the two models deviate significantly. Resonances of a stationary ring under a moving load and a rotating ring subjected to a stationary load are compared in terms of the resonance speeds and the steady-state responses. It is shown that these two situations can not be treated as equal in many cases.
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The in-plane steady-state response of a rotating ring on elastic foundation subjected to a stationary load is investigated theoretically using a high-order model in the framework of the plane strain assumption. The adopted high-order model accounts for the through-thickness variation of stresses and displacements, as well as the boundary tractions at the inner and outer surfaces of the ring. Based on the ratio of the foundation stiffness to the stiffness of the ring, two configurations of the ring-on-foundation system are investigated, namely soft foundation (stiff ring) and stiff foundation (soft ring). The analytical “method of the images” is used to obtain the ring response. It is found that the response of a stiff ring to a stationary load of constant magnitude is governed by the translational rigid body-like motion. In contrast, in the case of a soft ring, a wave-like deformation is predicted for the rotational speeds higher than a critical one. It is for the first time that such wave-like displacements are predicted using a rotating ring model with the rotation effects being properly considered. The response of a rotating ring to a stationary harmonic load is studied too. The predicted displacements using the high-order model are compared with those obtained from the classical low-order model in which only the radial and circumferential displacements at the middle surface of the ring are considered. It is concluded that only in the case of a stiff ring, the classical low-order model and the high-order model give similar predictions. When the ring is soft, the predictions of the two models deviate significantly. Resonances of a stationary ring under a moving load and a rotating ring subjected to a stationary load are compared in terms of the resonance speeds and the steady-state responses. It is shown that these two situations can not be treated as equal in many cases.
A straightforward and practical method of frequency-domain analysis is developed for coupled vehicle-track system. The influence of the track flexibility and spatial coherence of irregularities on frequency response of vehicle-track systems are systematically studied
accounting for train velocity and irregularity wavelength. Calculations show that the track flexibility cannot be ignored to obtain an accurate response of wheels whereas the resonance frequencies of car body motions remain unchanged. The inclusion of track flexibility enables consideration of wave reflections in rail sections between different wheels. The excitations at different wheels due to irregularity have phase lags determined by the train velocity and distances between wheels. This spatial coherence is important to the system response. The influence of contact spring on the system frequency response is examined. It is found that the system response converges at a certain value of the contact stiffness and the track stiffness governs the wheel-rail interaction after then. ...
accounting for train velocity and irregularity wavelength. Calculations show that the track flexibility cannot be ignored to obtain an accurate response of wheels whereas the resonance frequencies of car body motions remain unchanged. The inclusion of track flexibility enables consideration of wave reflections in rail sections between different wheels. The excitations at different wheels due to irregularity have phase lags determined by the train velocity and distances between wheels. This spatial coherence is important to the system response. The influence of contact spring on the system frequency response is examined. It is found that the system response converges at a certain value of the contact stiffness and the track stiffness governs the wheel-rail interaction after then. ...
A straightforward and practical method of frequency-domain analysis is developed for coupled vehicle-track system. The influence of the track flexibility and spatial coherence of irregularities on frequency response of vehicle-track systems are systematically studied
accounting for train velocity and irregularity wavelength. Calculations show that the track flexibility cannot be ignored to obtain an accurate response of wheels whereas the resonance frequencies of car body motions remain unchanged. The inclusion of track flexibility enables consideration of wave reflections in rail sections between different wheels. The excitations at different wheels due to irregularity have phase lags determined by the train velocity and distances between wheels. This spatial coherence is important to the system response. The influence of contact spring on the system frequency response is examined. It is found that the system response converges at a certain value of the contact stiffness and the track stiffness governs the wheel-rail interaction after then.
accounting for train velocity and irregularity wavelength. Calculations show that the track flexibility cannot be ignored to obtain an accurate response of wheels whereas the resonance frequencies of car body motions remain unchanged. The inclusion of track flexibility enables consideration of wave reflections in rail sections between different wheels. The excitations at different wheels due to irregularity have phase lags determined by the train velocity and distances between wheels. This spatial coherence is important to the system response. The influence of contact spring on the system frequency response is examined. It is found that the system response converges at a certain value of the contact stiffness and the track stiffness governs the wheel-rail interaction after then.
A new high-order model for in-plane vibrations of rotating rings is developed in this paper. The inner surface of the ring is connected to an immovable axis through an elastic foundation (distributed springs), whereas the outer surface is traction free. The developed model enables the dynamic analysis of the rings on stiff elastic foundation that rotate with a high speed. The traction force at the inner surface of such rings is so high that it influences significantly the through-thickness stress distribution. This boundary effect cannot be captured by the classical low order theories while the model proposed in this paper can account for this effect. Nonlinear equations of motion are first derived, considering the geometrical nonlinearity of the system while assuming the linear elastic behaviour of the ring material. The formulation accounts for the stress caused by rotation and the significant normal and tangential traction forces at the inner surface of the ring. The displacement fields are assumed to be polynomials of the through-thickness coordinate in both the radial and circumferential directions. The derivation is generic and can yield ring theories of different order, i.e. of the Timoshenko-type and beyond, with proper consideration of both the internal state of the body and the boundary effects at the surfaces. Two types of critical speeds are investigated, namely the one at which the free vibrations become unstable and the one at which the forced vibration of a rotating ring subjected to a constant stationary point load experiences resonance. A comparison is presented of the predictions of the developed model to those of the lower order theories. It is shown that even for thin rings on elastic foundation, high order corrections, beyond the ones of the Timoshenko theory, need to be considered for an accurate estimation of the critical speeds of rotating rings. The new high-order model is superior to the existing ring models in predicting dynamic behaviour of either stationary or rotating rings. Without loss of generality, the model is applicable to both plane strain and plane stress configurations.
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A new high-order model for in-plane vibrations of rotating rings is developed in this paper. The inner surface of the ring is connected to an immovable axis through an elastic foundation (distributed springs), whereas the outer surface is traction free. The developed model enables the dynamic analysis of the rings on stiff elastic foundation that rotate with a high speed. The traction force at the inner surface of such rings is so high that it influences significantly the through-thickness stress distribution. This boundary effect cannot be captured by the classical low order theories while the model proposed in this paper can account for this effect. Nonlinear equations of motion are first derived, considering the geometrical nonlinearity of the system while assuming the linear elastic behaviour of the ring material. The formulation accounts for the stress caused by rotation and the significant normal and tangential traction forces at the inner surface of the ring. The displacement fields are assumed to be polynomials of the through-thickness coordinate in both the radial and circumferential directions. The derivation is generic and can yield ring theories of different order, i.e. of the Timoshenko-type and beyond, with proper consideration of both the internal state of the body and the boundary effects at the surfaces. Two types of critical speeds are investigated, namely the one at which the free vibrations become unstable and the one at which the forced vibration of a rotating ring subjected to a constant stationary point load experiences resonance. A comparison is presented of the predictions of the developed model to those of the lower order theories. It is shown that even for thin rings on elastic foundation, high order corrections, beyond the ones of the Timoshenko theory, need to be considered for an accurate estimation of the critical speeds of rotating rings. The new high-order model is superior to the existing ring models in predicting dynamic behaviour of either stationary or rotating rings. Without loss of generality, the model is applicable to both plane strain and plane stress configurations.
This study addresses the contribution of spatial variance in the railway track support stiffness to the expected long-term track degradation. Hereto, a novel frequency-domain model is developed with a double periodicity ‘layer’, capable of dealing with both sleeper periodicity and arbitrary non-uniformity in track properties. The model application focuses on a locally reduced support stiffness (hanging sleeper) along the track. The resulting susceptibility to degradation is assessed by quantifying the mechanical energy dissipated over the influence length under a moving train axle. Different descriptions of this energy amount are benchmarked with respect to their predictive value. In the presence of a degraded sleeper support, hanging sleepers are found to develop faster with increasing train speed; the speed effect may be estimated as roughly linear. Moreover, degradation increases progressively with an increasing local relative stiffness reduction. Coincidence of the train speed corresponding to the sleeper passing frequency with the first resonance peak of the system leads to severely increased degradation; increased damping however attenuates dissipation peaks at resonant speeds and shifts their position upwards. The effect of a degraded support is most significant on soft subgrades. The effect of multiple degraded sleeper supports increases up to three sleepers, for any train speed. With respect to the system parameters, particularly the railpad stiffness has significant effect; especially for high-speed tracks a high pad stiffness is very unfavorable. Other effective control parameters in the case of degraded sleeper supports are the sleeper spacing and the rail cross-sectional properties; for example replacing a 54E1 with a 60E1 rail profile may reduce energy dissipation with roughly 30% on high-speed track. An increasing unsprung vehicle mass is unfavorable for track degradation, again with the effect increasing with the train speed. The developed methodology is shown to have significant potential with respect to railway track design in terms of multi-parametric optimization for concrete cases with a given input in terms of soil properties and operational regime.
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This study addresses the contribution of spatial variance in the railway track support stiffness to the expected long-term track degradation. Hereto, a novel frequency-domain model is developed with a double periodicity ‘layer’, capable of dealing with both sleeper periodicity and arbitrary non-uniformity in track properties. The model application focuses on a locally reduced support stiffness (hanging sleeper) along the track. The resulting susceptibility to degradation is assessed by quantifying the mechanical energy dissipated over the influence length under a moving train axle. Different descriptions of this energy amount are benchmarked with respect to their predictive value. In the presence of a degraded sleeper support, hanging sleepers are found to develop faster with increasing train speed; the speed effect may be estimated as roughly linear. Moreover, degradation increases progressively with an increasing local relative stiffness reduction. Coincidence of the train speed corresponding to the sleeper passing frequency with the first resonance peak of the system leads to severely increased degradation; increased damping however attenuates dissipation peaks at resonant speeds and shifts their position upwards. The effect of a degraded support is most significant on soft subgrades. The effect of multiple degraded sleeper supports increases up to three sleepers, for any train speed. With respect to the system parameters, particularly the railpad stiffness has significant effect; especially for high-speed tracks a high pad stiffness is very unfavorable. Other effective control parameters in the case of degraded sleeper supports are the sleeper spacing and the rail cross-sectional properties; for example replacing a 54E1 with a 60E1 rail profile may reduce energy dissipation with roughly 30% on high-speed track. An increasing unsprung vehicle mass is unfavorable for track degradation, again with the effect increasing with the train speed. The developed methodology is shown to have significant potential with respect to railway track design in terms of multi-parametric optimization for concrete cases with a given input in terms of soil properties and operational regime.
Rotating ring-like structures are very commonly used in civil, mechanical and aerospace engineering. Typical examples of such structures are components in turbomachinery, compliant gears, rolling tyres and flexible train wheels. At the micro-scale, rotating ring models find their applications in the field of ring gyroscopes, in which high accuracy of modelling is required. The in-plane vibrations of rotating rings are of particular interest since such structural components are usually subject to in-plane loads. The focus in this thesis is therefore placed on the in-plane dynamics of rotating rings. While the radial and circumferential motions of a stationary ring are coupled due to curvature, a steadily rotating ring, as any gyroscopic system, is subject to two additional fictitious forces induced by the gyroscopic coupling due to rotation, i.e. the Coriolis and centrifugal forces. Among them the centrifugal force associated with the steady rotation of the ring (quasi-static force) introduces an axi-symmetric radial expansion and a hoop stress; the latter has the tendency to stiffen the ring. In contrast, the dynamic part of the centrifugal force has the tendency to soften the system. Next to that, the Coriolis force bifurcates the natural frequencies of the ring. The proper consideration of the rotation effects is essential to determine the dynamic behaviour of rotating rings, such as stability of free vibrations and resonance of rotating rings under stationary loads. Although various models exist, the considerations of rotation effects are not always in agreement, resulting in distinct theoretical predictions of critical speeds associated with instability and resonance of rotating rings. In addition, in all the existing rotating ring models, the equations of motion were derived assuming the inner and outer surfaces of the ring to be traction-free. However, when one considers a ring whose inner surface is elastically restrained by distributed springs, this assumption is violated. The traction at the inner surface can significantly influence the stress distribution along the thickness of the ring and this effect has to be properly accounted for since the internal stresses may show a strong gradient from the inner surface to the outer surface, especially in the case of rings rotating at high speeds or when the latter are supported by stiff foundation. The primary aim of this thesis is to develop a highly accurate rotating ring model that properly accounts for both the rotation and boundary effects with rigorous mathematical derivation to fill the gap regarding the modelling and prediction of the dynamic behaviour of high-speed rotating rings. To achieve this aim, the following four objectives are set: (i) identify the reasons of disagreements between various existing rotating ring models and clarify the mathematically sound derivations of governing equations; (ii) develop a high-order rotating ring model which properly accounts for the rotation effects, as well as the non-zero tractions at boundaries; (iii) close the debate on the prediction of critical speeds associated with instability of free vibrations and resonance of forced vibrations; and (iv) apply the developed high-order model to predict the steadystate response of rotating rings under stationary loads and the stability of rotating ringstationary oscillator system.
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Rotating ring-like structures are very commonly used in civil, mechanical and aerospace engineering. Typical examples of such structures are components in turbomachinery, compliant gears, rolling tyres and flexible train wheels. At the micro-scale, rotating ring models find their applications in the field of ring gyroscopes, in which high accuracy of modelling is required. The in-plane vibrations of rotating rings are of particular interest since such structural components are usually subject to in-plane loads. The focus in this thesis is therefore placed on the in-plane dynamics of rotating rings. While the radial and circumferential motions of a stationary ring are coupled due to curvature, a steadily rotating ring, as any gyroscopic system, is subject to two additional fictitious forces induced by the gyroscopic coupling due to rotation, i.e. the Coriolis and centrifugal forces. Among them the centrifugal force associated with the steady rotation of the ring (quasi-static force) introduces an axi-symmetric radial expansion and a hoop stress; the latter has the tendency to stiffen the ring. In contrast, the dynamic part of the centrifugal force has the tendency to soften the system. Next to that, the Coriolis force bifurcates the natural frequencies of the ring. The proper consideration of the rotation effects is essential to determine the dynamic behaviour of rotating rings, such as stability of free vibrations and resonance of rotating rings under stationary loads. Although various models exist, the considerations of rotation effects are not always in agreement, resulting in distinct theoretical predictions of critical speeds associated with instability and resonance of rotating rings. In addition, in all the existing rotating ring models, the equations of motion were derived assuming the inner and outer surfaces of the ring to be traction-free. However, when one considers a ring whose inner surface is elastically restrained by distributed springs, this assumption is violated. The traction at the inner surface can significantly influence the stress distribution along the thickness of the ring and this effect has to be properly accounted for since the internal stresses may show a strong gradient from the inner surface to the outer surface, especially in the case of rings rotating at high speeds or when the latter are supported by stiff foundation. The primary aim of this thesis is to develop a highly accurate rotating ring model that properly accounts for both the rotation and boundary effects with rigorous mathematical derivation to fill the gap regarding the modelling and prediction of the dynamic behaviour of high-speed rotating rings. To achieve this aim, the following four objectives are set: (i) identify the reasons of disagreements between various existing rotating ring models and clarify the mathematically sound derivations of governing equations; (ii) develop a high-order rotating ring model which properly accounts for the rotation effects, as well as the non-zero tractions at boundaries; (iii) close the debate on the prediction of critical speeds associated with instability of free vibrations and resonance of forced vibrations; and (iv) apply the developed high-order model to predict the steadystate response of rotating rings under stationary loads and the stability of rotating ringstationary oscillator system.
The long-term behaviour of railway track has attracted increasing attention in recent years. Improvements in long-term structural performance reduce demands for maintenance and increase the continuous availability of railway lines. The focus of this paper is on the prediction of the sensitivity of a track design to long-term deterioration in terms of track geometry. According to the state of the art literature, degradation is often investigated using empirical models based on field measurement data. Although a rough maintenance forecast may be made employing empirical models, the predictions are not generic, and the physical processes which govern track degradation under train operation remain unclear. The first aim of this study is to present a mathematical model to elucidate the underlying physics of long-term degradation of railway tracks. The model consists of an infinitely long beam which is periodically supported by equidistantly discrete sleepers and a moving unsprung mass which represents a travelling train. The mechanical energy dissipated in the substructure is proposed to serve as a measure of the track degradation rate. Secondly, parametric studies on energy dissipation are conducted to identify effects of various track design parameters on the susceptibility of the track to degradation, as well as the effect of the train speed. It has been shown that the track/subgrade stiffness is the most influential parameter on degradation whereas other system parameters do influence the degradation rate but at lower magnitudes. The conclusions can be used to optimise the track design in the early stage for better long-term structural performance of railway tracks.
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The long-term behaviour of railway track has attracted increasing attention in recent years. Improvements in long-term structural performance reduce demands for maintenance and increase the continuous availability of railway lines. The focus of this paper is on the prediction of the sensitivity of a track design to long-term deterioration in terms of track geometry. According to the state of the art literature, degradation is often investigated using empirical models based on field measurement data. Although a rough maintenance forecast may be made employing empirical models, the predictions are not generic, and the physical processes which govern track degradation under train operation remain unclear. The first aim of this study is to present a mathematical model to elucidate the underlying physics of long-term degradation of railway tracks. The model consists of an infinitely long beam which is periodically supported by equidistantly discrete sleepers and a moving unsprung mass which represents a travelling train. The mechanical energy dissipated in the substructure is proposed to serve as a measure of the track degradation rate. Secondly, parametric studies on energy dissipation are conducted to identify effects of various track design parameters on the susceptibility of the track to degradation, as well as the effect of the train speed. It has been shown that the track/subgrade stiffness is the most influential parameter on degradation whereas other system parameters do influence the degradation rate but at lower magnitudes. The conclusions can be used to optimise the track design in the early stage for better long-term structural performance of railway tracks.
In-plane dynamics of rotating rings on elastic foundation is a topic of continuous research, especially in the field of tire dynamics. When the inner surface of a ring is connected to a stiff foundation, the through-thickness variation of radial and shear stress needs to be accounted for. This effect is often overlooked in the ring models proposed in the literature. In this paper, a new high order theory is developed for the in-plane vibration of rotating rings whose inner surface is connected to an immovable hub by distributed springs while the outer surface is stress-free. The high-order terms are chosen such that the boundary conditions at the inner and outer surfaces are satisfied at all times. Instability, which is usually overlooked in the literature, is predicted using the present model. Resonant speeds are investigated, at which modes appear as a stationary displacement pattern to a space-fixed observer. The exact satisfaction of boundary conditions at the inner and outer ring surfaces together with the through-thickness variation of the radial and shear stresses are shown to be of significant importance when the ring rotates at high speeds or is supported by relatively stiff foundation.
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In-plane dynamics of rotating rings on elastic foundation is a topic of continuous research, especially in the field of tire dynamics. When the inner surface of a ring is connected to a stiff foundation, the through-thickness variation of radial and shear stress needs to be accounted for. This effect is often overlooked in the ring models proposed in the literature. In this paper, a new high order theory is developed for the in-plane vibration of rotating rings whose inner surface is connected to an immovable hub by distributed springs while the outer surface is stress-free. The high-order terms are chosen such that the boundary conditions at the inner and outer surfaces are satisfied at all times. Instability, which is usually overlooked in the literature, is predicted using the present model. Resonant speeds are investigated, at which modes appear as a stationary displacement pattern to a space-fixed observer. The exact satisfaction of boundary conditions at the inner and outer ring surfaces together with the through-thickness variation of the radial and shear stresses are shown to be of significant importance when the ring rotates at high speeds or is supported by relatively stiff foundation.
Transition zones in railway lines are areas between different track structures such as the transition from conventional track (ballasted track) to slab track, to a tunnel or a viaduct. The main feature of a transition zone is that it exhibits a dramatic change in structural behaviour to bridge the difference in the adjacent track parts. This change causes high dynamic loads which contribute to quality deterioration of the track. Two main factors influence the magnitude of the interaction forces between trains and track in transition zones. Firstly, the abrupt change in track stiffness. This stiffness is determined by the mechanical features of the entire track structure; the conventional track is a compliant structure, while slab track, tunnels and viaducts are relatively stiff. A train passing a stiffness change induces a variation of track
deflection under the moving dead loads and, consequently, also a variation in the wheelset’s vertical momentum leading to higher (dynamic) loads. Secondly, settlements of the backfill and its foundation are typically larger than those of stiff structures, leading to unevenness of the track. This abstract deals with the issue of the dynamic analysis of an infinite Euler-Bernoulli beam on elastic foundation with transition in foundation stiffness, subjected to a moving oscillator. This model is one of the simplest ones for a vehicle passing a transition zone. The equations of motion are solved by means of the time-domain Green’s function method using convolution integrals in terms of the unknown contact force. Considering the
track as an aperiodic structure, the Green’s functions (receptances) are calculated in a stationary reference frame (i.e., non-moving sources). Two methods of solution are investigated. The first one is based on the Laplace Transform, where the response consists of a contribution from the initial conditions and one from the moving contact force. By choosing the initial conditions in accordance with the response of a beam with homogenous foundation subjected to a moving load, the free vibrations and waves due to oscillator entrance are suppressed and steady-state behaviour is achieved before the oscillator reaches the transition zone. The second method is based on the Fourier Transform, which automatically ensures this
steady-state behaviour. Both methods are exemplified in the paper. The influence of the length of the transition zone and the speed of the moving oscillator on the contact force are analysed; both sub-critical and super-critical speeds are considered.
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Transition zones in railway lines are areas between different track structures such as the transition from conventional track (ballasted track) to slab track, to a tunnel or a viaduct. The main feature of a transition zone is that it exhibits a dramatic change in structural behaviour to bridge the difference in the adjacent track parts. This change causes high dynamic loads which contribute to quality deterioration of the track. Two main factors influence the magnitude of the interaction forces between trains and track in transition zones. Firstly, the abrupt change in track stiffness. This stiffness is determined by the mechanical features of the entire track structure; the conventional track is a compliant structure, while slab track, tunnels and viaducts are relatively stiff. A train passing a stiffness change induces a variation of track
deflection under the moving dead loads and, consequently, also a variation in the wheelset’s vertical momentum leading to higher (dynamic) loads. Secondly, settlements of the backfill and its foundation are typically larger than those of stiff structures, leading to unevenness of the track. This abstract deals with the issue of the dynamic analysis of an infinite Euler-Bernoulli beam on elastic foundation with transition in foundation stiffness, subjected to a moving oscillator. This model is one of the simplest ones for a vehicle passing a transition zone. The equations of motion are solved by means of the time-domain Green’s function method using convolution integrals in terms of the unknown contact force. Considering the
track as an aperiodic structure, the Green’s functions (receptances) are calculated in a stationary reference frame (i.e., non-moving sources). Two methods of solution are investigated. The first one is based on the Laplace Transform, where the response consists of a contribution from the initial conditions and one from the moving contact force. By choosing the initial conditions in accordance with the response of a beam with homogenous foundation subjected to a moving load, the free vibrations and waves due to oscillator entrance are suppressed and steady-state behaviour is achieved before the oscillator reaches the transition zone. The second method is based on the Fourier Transform, which automatically ensures this
steady-state behaviour. Both methods are exemplified in the paper. The influence of the length of the transition zone and the speed of the moving oscillator on the contact force are analysed; both sub-critical and super-critical speeds are considered.
The in-plane free vibration of a rotating thin ring is revisited in this paper. A new model is proposed which accounts for the elastic foundation and the through-thickness variation of the radial stress. The emphasis is placed on a proper consideration of the geometrical nonlinearity, which is essential for the consistent modelling of the ring stiffening resulting from the radial expansion caused by rotation. The in-plane stability of a thin ring rotating at relatively high speeds is analysed thoroughly. It is shown that the ring can become unstable should the rotational speed exceed a critical value. This result is new as in most known to the authors previous studies the stability problem is either not considered or it is stated that the in-plane vibration of a rotating ring is stable. In the studies which did address the instability, the conclusions and the employed models are prone to criticism. A parametric study is conducted to illustrate the effects of the ring properties on the in-plane stability. Finally, modes, which appear as stationary displacement patterns of the ring to an observer in the space-fixed reference system, are investigated. It is shown that the stationary patterns can occur prior to the onset of the instability for certain ring parameters.
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The in-plane free vibration of a rotating thin ring is revisited in this paper. A new model is proposed which accounts for the elastic foundation and the through-thickness variation of the radial stress. The emphasis is placed on a proper consideration of the geometrical nonlinearity, which is essential for the consistent modelling of the ring stiffening resulting from the radial expansion caused by rotation. The in-plane stability of a thin ring rotating at relatively high speeds is analysed thoroughly. It is shown that the ring can become unstable should the rotational speed exceed a critical value. This result is new as in most known to the authors previous studies the stability problem is either not considered or it is stated that the in-plane vibration of a rotating ring is stable. In the studies which did address the instability, the conclusions and the employed models are prone to criticism. A parametric study is conducted to illustrate the effects of the ring properties on the in-plane stability. Finally, modes, which appear as stationary displacement patterns of the ring to an observer in the space-fixed reference system, are investigated. It is shown that the stationary patterns can occur prior to the onset of the instability for certain ring parameters.
The stability of an oscillator uniformly moving along a thin ring that is connected to an immovable axis by a distributed viscoelastic foundation has been studied. The dynamic reaction of the ring to the oscillator is represented by a frequency and velocity dependent equivalent stiffness. The characteristic equation for the vibration of the oscillator is obtained. It is shown that this equation can have roots with a positive real part, which imply the exponential increase of the amplitude of the oscillator’s vibration in time, i.e. instability. The critical velocity after which instability can occur is determined. With the help of the D-decomposition method, the instability domains are found in the space of the system parameters. Parametric study of the stability domains is carried out.
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The stability of an oscillator uniformly moving along a thin ring that is connected to an immovable axis by a distributed viscoelastic foundation has been studied. The dynamic reaction of the ring to the oscillator is represented by a frequency and velocity dependent equivalent stiffness. The characteristic equation for the vibration of the oscillator is obtained. It is shown that this equation can have roots with a positive real part, which imply the exponential increase of the amplitude of the oscillator’s vibration in time, i.e. instability. The critical velocity after which instability can occur is determined. With the help of the D-decomposition method, the instability domains are found in the space of the system parameters. Parametric study of the stability domains is carried out.