<p>In this thesis we study multipole moments of axisymmetric spacetimes. Using the recursive definition of the multipole moments of Geroch and Hansen we develop a method for computing all multipole moments of a stationary axisymmetric spacetime without the use of a recursion. This is a generalisation of a method developed by Herberthson for the static case.</p><p>Using Herberthson’s method we also develop a method for finding a static axisymmetric spacetime with arbitrary prescribed multipole moments, subject to a specified convergence criteria. This method has, in general, a step where one has to find an explicit expression for an implicitly defined function. However, if the number of multipole moments are finite we give an explicit expression in terms of power series.</p> / Note: The two articles are also available in the pdf-file. Report code: LiU-TEK-LIC-2006:4.

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In this thesis we study multipole moments of axisymmetric spacetimes. Using the recursive definition of the multipole moments of Geroch and Hansen we develop a method for computing all multipole moments of a stationary axisymmetric spacetime without the use of a recursion. This is a generalisation of a method developed by Herberthson for the static case. Using Herberthson’s method we also develop a method for finding a static axisymmetric spacetime with arbitrary prescribed multipole moments, subject to a specified convergence criteria. This method has, in general, a step where one has to find an explicit expression for an implicitly defined function. However, if the number of multipole moments are finite we give an explicit expression in terms of power series. / <p>Note: The two articles are also available in the pdf-file. Report code: LiU-TEK-LIC-2006:4.</p>

The vibration of complex mechanical systems that include coupled rotating and stationary bodies motivates this work. A semi-analytical model is developed for high-speed, compliant, rotating bodies. Exploiting the axisymmetry of the rotating body, the developed semi-analytical model only discretizes the two-dimensional radial cross-section; Fourier series are used in the circumferential direction. The corresponding formulation for thin-walled, axisymmetric shells is given. Even though the body is axisymmetric, its deflection as well as external forces, constraints, and supports acting on the body are allowed to be asymmetric. These asymmetric elements can be stationary or rotating. The model includes Coriolis and centripetal effects. The prestress (or geometric) stiffness matrix that arises from external forces and constant centripetal acceleration has additional terms compared to the literature, and these terms can significantly change the natural frequencies. Discrete stiffness-damper elements, elastic foundations, and constraint equations are used to couple the rotating body to other rotating and stationary bodies. The model is developed in a stationary reference frame to avoid time-dependent coefficients in the equations of motion when coupled to stationary components. Surface constraints are developed using equivalent force relations between multiple points on the surface and a reference node. Discrete stiffness-dampers, asymmetric elastic foundation, and asymmetric constraints introduce non-axisymmetry in the system. The speed-dependent natural frequencies and complex-valued vibration modes, presence of multiple Fourier harmonics in each mode, changes to critical speeds, divergence and flutter instability phenomena, and eigenvalue veering are investigated for spinning systems with asymmetric features. The developed semi-analytical model is used for rotationally periodic systems, for example, planetary gears. Rotationally periodic systems consist of multiple vibrating, rotating central components and substructures. The model is developed in a reference frame rotating with the central component that supports the substructures. Structured modal properties of the cyclically symmetric systems and diametrically opposed systems are investigated. The modes of the spinning system are categorized into translational-tilting, rotational-axial, and substructure modes. Time-varying coupling elements act as parametric excitation in the system. Large strain energy in the coupling elements lead to large parametric instability regions. The analytical closed-form expression of the parametric instability bandwidth obtained using a perturbation method compares well with numerical results from Floquet theory. / Doctor of Philosophy / Complex mechanical systems, for example, mechanical transmission, consist of coupled rotating and stationary bodies. The vibrations of rotating bodies are transmitted to the other bodies through coupling elements. To reduce weight of the system, the rotating bodies are made thin-walled resulting in increased flexibility of the body. The existing lumped parameter/rigid body models do not account for the flexibility of these rotating bodies. Conventional three-dimensional finite element models lead to a large number of degrees of freedom in the system, increasing the computational cost. We aim to develop a computationally efficient model to analyze the dynamics and vibration of complex mechanical systems. Most rotating bodies can be approximated as axisymmetric. The axisymmetric property of the rotating body is harnessed to reduce the three-dimensional model of the body to a two-dimensional radial cross-section using Fourier series in the circumferential direction. This reduces the system degrees of freedom. Coriolis, centripetal, and prestress effects are included in the model. Discrete stiffness-dampers, elastic foundations, and constraint equations couple the rotating body to other rotating and stationary bodies. Non-axisymmetric coupling elements and forces introduce asymmetry in the system. The system model for these asymmetric systems are developed in a stationary reference frame to avoid time-dependent coefficient equations of motion. Flexible stationary bodies alter the natural frequencies and vibration modes of the system. Instabilities, critical speeds, effects of asymmetry on the natural frequencies and vibration modes of the system are investigated. The model is extended for rotationally periodic systems, for examples, planetary gears and bearings. This model is developed in the reference frame that rotates with the central component that supports substructures. Structured modal characteristics are observed for the rotationally periodic systems. Changing contact conditions act as a source of parametric excitation in systems. Parametric resonances occur when natural frequencies of vibration with large strain energy in the coupling elements sum to the excitation frequency. Parametric instability regions obtained using an analytical equation compare well with numerical results.

Tesis por compendio / [ES] El ruido de rodadura de vehículos ferroviarios presenta importantes desafíos en entornos urbanos y suburbanos, siendo precisa una comprensión integral de sus mecanismos subyacentes para una mitigación efectiva. Esta Tesis ofrece una investigación exhaustiva sobre la generación de ruido de rodadura, tanto en vías rectas como curvas, a través del desarrollo de modelos avanzados de alta frecuencia y herramientas de cálculo para una rigurosa predicción de niveles de ruido. El ruido de rodadura surge de la interacción entre las ruedas y los carriles en presencia de imperfecciones en sus superficies de contacto. Los principales componentes ferroviarios que contribuyen a la emisión acústica son las traviesas, los carriles y las ruedas. Las traviesas y los carriles están conectados por medio de las placas de asiento, mientras que las ruedas y los carriles están dinámicamente acoplados a través de su contacto. En consecuencia, cambios en cualquier componente pueden afectar al comportamiento dinámico y acústico del resto, subrayando la necesidad de un modelo integral para abordar eficazmente la radiación por ruido de rodadura. En esta Tesis se exploran y desarrollan diferentes modelos dinámicos de la vía y del eje montado. La vía, que se asume infinita, se describe utilizando la teoría de estructuras periódicas y se caracteriza por sus números y formas de onda. El eje montado se modeliza mediante el Método de Elementos Finitos y se caracteriza por sus frecuencias naturales y modos de vibración. La interacción rueda/carril se describe como una relación linealizada entre el movimiento relativo de ambos componentes y las fuerzas de contacto. Cabe indicar que los modelos explorados en este trabajo están formulados en el dominio de la frecuencia. Asimismo, se realiza un estudio de influencia del diseño de la vía en la radiación por ruido de rodadura, cuantificando los parámetros contribuyentes a la emisión acústica mediante técnicas estadísticas. Los resultados apuntan a que la geometría del carril tiene un impacto limitado en la radiación, mientras que las propiedades viscoelásticas de la vía, en particular la rigidez de la placa de asiento, desempeñan un papel fundamental en la generación de ruido. Es remarcable que, entre los distintos diseños, se han encontrado variaciones de hasta 7,4 dB(A) en la radiación acústica. Durante el desarrollo de esta investigación, se ha prestado atención al modelizado del eje montado. Su rotación se incluye usando coordenadas Eulerianas, lo que resulta un enfoque conveniente ya que el punto de contacto de la rueda con el carril permanece en una posición espacial constante. Dada la simetría axial de su geometría, la respuesta dinámica del eje montado se expande a lo largo de la dirección circunferencial mediante series de Fourier, lo que permite formular el comportamiento dinámico y acústico de este cuerpo tridimensional (3D) en un marco bidimensional (2D), resolviendo analíticamente la coordenada circunferencial. Esta metodología ofrece sin pérdida de generalidad una reducción del tiempo de cálculo computacional, lo que hace que el modelo sea idóneo para su integración en algoritmos de optimización. Por último, se realiza una investigación pionera sobre el ruido de rodadura cuando el vehículo negocia una curva. Si bien la curva está generalmente asociada con el ruido por chirridos, esta Tesis explora y confirma la importancia que también tiene el ruido de rodadura en estas condiciones. Para ello, se modelizan los efectos inerciales y giroscópicos que sufre un eje montado al describir una trayectoria curva. Además, diferentes fenómenos complejos que ocurren en el contacto rueda/carril, como por ejemplo el movimiento relativo entre estos elementos, se incorporan en el modelo de interacción. Los resultados indican que la posición del contacto rueda/carril sirve como un buen indicador del impacto que la negociación de una curva tiene en el ruido de rodadura. / [CA] El soroll de rodament de vehicles ferroviaris presenta importants reptes en entorns urbans i suburbans, requerint una comprensió integral dels seus mecanismes subjacents per a una mitigació efectiva. Aquesta Tesi ofereix una investigació exhaustiva sobre la generació de soroll de rodament, tant en vies rectes com corbes, mitjançant el desenvolupament de models avançats d'alta freqüència i eines de càlcul per a la rigorosa predicció dels nivells de soroll radiat. El soroll de rodament sorgeix de la interacció entre les rodes i els carrils en presència d'imperfeccions en les seues superfícies de contacte. Els principals components ferroviaris que contribueixen a l'emissió acústica són les travesses, els carrils i les rodes. Les travesses i els carrils estan connectats a través de les plaques d'assentament, mentre que les rodes i els carrils estan acoblats mitjançant la seua interacció. En conseqüència, canvis en qualsevol component poden afectar al comportament dinàmic i acústic de la resta, subratllant la necessitat d'un model integral. En aquesta Tesi s'exploren i desenvolupen diversos models dinàmics de la via i de l'eix muntat. La via, que es considera infinita, es descriu utilitzant la teoria d'estructures periòdiques i es caracteritza pels seus números i formes d'ona. L'eix muntat es modelitza mitjançant el Mètode d'Elements Finits i es caracteritza per les seues freqüències naturals i modes de vibració. La interacció entre l'eix muntat i la via es descriu com una relació linealitzada entre el moviment relatiu d'ambdós components i les forces de contacte. Cal assenyalar que els models explorats en aquest treball estan formulats en el domini de la freqüència. Així mateix, es realitza un estudi d'influència del disseny de la via en la radiació, quantificant els paràmetres contribuents a l'emissió acústica mitjançant tècniques estadístiques. Els resultats apunten que la geometria del carril té un impacte limitat en la radiació sonora, mentre que les propietats viscoelàstiques de la via, en particular la rigidesa de la placa d'assentament, tenen un paper fonamental en la generació de soroll. És destacable que, entre els diferents dissenys, s'han trobat variacions de fins a 7,4 dB(A) en la radiació per soroll de rodament. Durant el desenvolupament d'aquesta investigació, s'ha prestat atenció al modelatge de l'eix muntat. La seua rotació s'inclou en la formulació utilitzant coordenades Eulerianes, la qual cosa resulta un enfocament convenient ja que el punt de contacte de la roda amb el carril roman en una posició constant. Donada la simetria axial de la seua geometria, la resposta dinàmica de l'eix muntat s'amplia al llarg de la direcció circumferencial mitjançant sèries de Fourier, permetent formular el comportament dinàmic i acústic d'aquest cos tridimensional (3D) en un marc bidimensional (2D), i resolent la coordenada circumferencial analíticament. Aquesta metodologia ofereix sense pèrdua de generalitat una reducció del temps de càlcul computacional, la qual cosa fa que el model siga idoni per a la seua integració en algoritmes d'optimització. Finalment, es realitza una investigació pionera sobre el soroll de rodament quan el vehicle negocia una corba. Si bé la corba està generalment associada amb el soroll per grinyols, aquesta Tesi explora i confirma la importància que també té el soroll de rodament en aquestes condicions. Per a això, es modelitzen els efectes inercials i giroscòpics que pateix un eix muntat al descriure una trajectòria corba. A més, diferents fenòmens complexes que ocorren en el contacte roda/carril, com el moviment relatiu entre aquests elements, s'incorporen en el model d'interacció. Els resultats indiquen que la posició del contacte roda/carril serveix com un bon indicador de l'impacte que la negociació d'una corba té en el soroll de rodament. / [EN] Rolling noise emission in railway systems presents significant challenges in urban and suburban environments, requiring a comprehensive understanding of its underlying mechanisms for effective mitigation. This Thesis offers a thorough investigation into rolling noise generation, considering both tangent and curved tracks, through the development of advanced high-frequency models and calculation tools to predict noise levels accurately. Rolling noise arises from the interaction between railway wheels and tracks in the presence of roughness on their contact surfaces. The principal components contributing to the acoustic emission are the sleepers, rails, and wheels. Sleepers and rails are interconnected through rail pads, while wheels and rails are dynamically coupled due to their contact. Consequently, changes in any component might impact the dynamic and acoustic behaviour of all of them, outlining the necessity of a comprehensive model to address rolling noise radiation effectively. Various models for the track and wheelset are explored and developed in this Thesis. The track, assumed to be infinite, is described using periodic structure theory and it is characterised by its wavenumbers and waveshapes. The wheelset is modelled through the Finite Element Method (FEM) and it is characterised by its natural frequencies and vibration modes. The wheelset and track interaction is described as a linearised relationship between the relative motion of both components and the contact forces. The models examined in this work are formulated in the frequency domain. Identified track properties influencing rolling noise radiation are quantified using statistical techniques. While the rail geometry is found to have a limited impact on the total radiation, the viscoelastic properties of the track, particularly the rail pad stiffness, play a crucial role in noise generation. Variations up to 7.4 dB(A) were observed for different track designs. Special attention is devoted to the wheelset modelling in this Thesis. Its rotation is considered using Eulerian coordinates, a convenient approach as the wheel contact point with the rail remains at a constant spatial position. Given the axial symmetry of its geometry, the wheelset dynamic response is expanded around the circumferential direction using Fourier series, which yields a two-dimensional (2D) formulation of the dynamic and acoustic behaviour of this three-dimensional (3D) system, with the circumferential coordinate being solved analytically. This methodology, denoted as axisymmetric approach, offers a significant reduction in the associated computational calculation time while preserving accuracy, making the model well suited for its integration into optimisation algorithms. Lastly, a novel investigation into rolling noise when the vehicle negotiates a curve is conducted. While curved tracks are generally associated with squeal noise, this Thesis offers valuable insights into the importance of rolling noise as well. To achieve this, the inertial and gyroscopic effects associated with a wheelset running on a curve are modelled. Complex phenomena occurring at the wheel/rail contact, such as the relative motion between these two elements, are incorporated into the interaction model. The results indicate that the wheel/rail contact position serves as a reliable indicator of the impact of a curve on the rolling noise. / This Thesis has been supported by Ministerio de Ciencia, Innovación y Universidades in the framework of the FPU grant program. Grants FPU18/03999, EST21/00213, and EST22/00353 as well as projects TRA2017-84701-R, PID2020-112886RA-I00, and PID2023-148483OB-I00 funded by MCIN/AEI/10.13039/501100011033, “ESF Investing in your future”, and “ERDF A way of making Europe” are acknowledged. The author also expresses his gratitude for the support provided by Programa PROMETEO/2021/046 of Generalitat Valenciana. / Andrés Ruiz, VT. (2024). Comprehensive dynamic models of railway wheelsets and tracks for the prediction of rolling noise [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/206074 / Compendio