A comparison of two nonlinear damping mechanisms in a vibration isolator

The vibration transmissibility characteristics of a single-degree-of- freedom (SDOF) passive vibration isolation system with different nonlinear dampers are investigated in this paper. In one configuration, the damper is assumed to be linear and viscous, and is connected to the mass so that it is perpendicular to the spring (horizontal damper). The vibration is in the direction of the spring. The second configuration is one in which the damper is in parallel with the spring but the damping force is proportional to the cube of the relative velocity across the damper (cubic damping). Both configurations are studied for small amplitudes of excitation, when some analysis can be conducted based on analytical expressions, and for large amplitudes of excitation, where the analysis is based on numerical simulations. It is found that the two nonlinear systems can outperform the linear system when force transmissibility is considered. However, for displacement transmissibility, the system with the horizontal damper exhibits some desirable properties, but the system with cubic damping does not. © 2012 Elsevier Ltd.

Energy relaxation in nonlinear one-dimensional lattices

We study energy relaxation in thermalized one-dimensional nonlinear arrays of the Fermi-Pasta-Ulam type. The ends of the thermalized systems are placed in contact with a zero-temperature reservoir via damping forces. Harmonic arrays relax by sequential phonon decay into the cold reservoir, the lower-frequency modes relaxing first. The relaxation pathway for purely anharmonic arrays involves the degradation of higher-energy nonlinear modes into lower-energy ones. The lowest-energy modes are absorbed by the cold reservoir, but a small amount of energy is persistently left behind in the array in the form of almost stationary low-frequency localized modes. Arrays with interactions that contain both a harmonic and an anharmonic contribution exhibit behavior that involves the interplay of phonon modes and breather modes. At long times relaxation is extremely slow due to the spontaneous appearance and persistence of energetic high-frequency stationary breathers. Breather behavior is further ascertained by explicitly injecting a localized excitation into the thermalized arrays and observing the relaxation behavior.

A simplified Fermi Accelerator Model under quadratic frictional force

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Non-uniform drag force on the Fermi accelerator model

Some dynamical properties of a particle suffering the action of a generic drag force are obtained for a dissipative Fermi Acceleration model. The dissipation is introduced via a viscous drag force, like a gas, and is assumed to be proportional to a power of the velocity: F alpha -nu(gamma). The dynamics is described by a two-dimensional nonlinear area-contracting mapping obtained via the solution of Newton's second law of motion. We prove analytically that the decay of high energy is given by a continued fraction which recovers the following expressions: (i) linear for gamma = 1; (ii) exponential for gamma = 2; and (iii) second-degree polynomial type for gamma = 1.5. Our results are discussed for both the complete version and the simplified version. The procedure used in the present paper can be extended to many different kinds of system, including a class of billiards problems.

In-flight and collisional dissipation as a mechanism to suppress Fermi acceleration in a breathing Lorentz gas

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

STRUCTURAL CONTROL OF A SMALL-SCALE TEST-BED SHAKER STRUCTURE USING A SPONGE-TYPE MAGNETO-RHEOLOGICAL FLUID DAMPER

Semi-active damping devices have been shown to be effective in mitigating unwanted vibrations in civil structures. These devices impart force indirectly through real-time alterations to structural properties. Simulating the complex behavior of these devices for laboratory-scale experiments is a major challenge. Commercial devices for seismic applications typically operate in the 2-10 kN range; this force is too high for small-scale testing applications where requirements typically range from 0-10 N. Several challenges must be overcome to produce damping forces at this level. In this study, a small-scale magneto-rheological (MR) damper utilizing a fluid absorbent metal foam matrix is developed and tested to accomplish this goal. This matrix allows magneto-rheological (MR) fluid to be extracted upon magnetic excitation in order to produce MR-fluid shear stresses and viscosity effects between an electromagnetic piston, the foam, and the damper housing. Dampers for uniaxial seismic excitation are traditionally positioned in the horizontal orientation allowing MR-fluid to gather in the lower part of the damper housing when partially filled. Thus, the absorbent matrix is placed in the bottom of the housing relieving the need to fill the entire device with MR-fluid, a practice that requires seals that add significant unwanted friction to the desired low-force device. The damper, once constructed, can be used in feedback control applications to reduce seismic vibrations and to test structural control algorithms and wireless command devices. To validate this device, a parametric study was performed utilizing force and acceleration measurements to characterize damper performance and controllability for this actuator. A discussion of the results is presented to demonstrate the attainment of the damper design objectives.

Hysteresis in transverse galloping: the role of the inflection points

Transverse galloping is here considered as a one-degree-of-freedom oscillator subjected to aerodynamic forces, which are described by using the quasi-steady hypothesis. The hysteresis of transverse galloping is also analyzed. Approximate solutions of the model are obtained by assuming that the aerodynamic and damping forces are much smaller than the inertial and stiffness ones. The analysis of the approximate solution, which is obtained by means of the method of Krylov–Bogoliubov, reveals the existing link between the hysteresis phenomenon and the number of inflection points at the aerodynamic force coefficient curve, Cy(α)Cy(α); CyCy and αα being, respectively, the force coefficient normal to the incident flow and the angle of attack. The influence of the position of these inflection points on the range of flow velocities in which hysteresis takes place is also analyzed.

The study and measurements of the behaviour and characteristics of electromagnetic relays

The thesis presents a theoretical and practical study of the dynamic behaviour of electromagnetic relays. After discussing the problem of solving the dynamicc equations analytically and presenting a historical survey of the earlier works in the relay and its dynamics, the simulation of a relay on the analogue computer is discussed. It is shown that the simulation may be used to obtain specific solutions to the dynamic equations. The computer analysis provides the dynamic characteristics for design purposes and may be used in the study of bouncing, rebound oscillations and stability of the armature motion. An approximate analytical solution to the two dynamic equations is given based on the assumption that the dynamic variation of the pull with the position of the armature is linear. The assumption is supported by the Computer-aided analysis and experimental results. The solution is intended to provide a basis for a rational design. A rigorous method of analysing the dynamic performance by using Ahlberg's theory is also presented. This method may be justified to be the extension of Ahlberg's theory by taking the mass and frictional damping forces into account. While calculating the armature motion mathematically, Ahlberg considers the equilibrium of two kinds of forces, namely pull and load, and disregards the mass and friction forces, whereas the present method deals with the equilibrium of all four kinds of forces. It is shown how this can be utilised to calculate the dynamic characteristics for a specific design. The utility of this method also extends to the study of stability, contact bounce and armature rebound. The magnetic circuit and other related topics which are essential to the study of relay dynamics are discussed and some necessary experimental results are given.

The non-ideal problem behavior using a dynamic vibration absorber with nonlinear essential stiffness and timedependent damping properties

The purpose of this study is to develop a dynamic vibration absorber using viscoelastic material with nonlinear essential stiffness and time-dependent damping properties for a non-ideal vibrating system with Sommerfeld effect, resonance capture, and jump phenomenon. The absorber is a mass-bar subsystem that consists of a viscoelastic bar with memory attached to mass, in which the internal dissipative forces depend on current, deformations, and its operational frequency varies with limited temperature. The non-ideal vibrating system consists of a linear (nonlinear) oscillator (plane frame structure) under excitation, via spring connector, of a DC-motor with limited power supply. A viscoelastic dynamic absorber modeled with elastic stiffness essentially nonlinearities was developed to further reduce the Sommerfeld effect and the response of the structure. The numerical results show the performance of the absorber on the non-ideal system response through the resonance curves, time histories, and Poincarésections. Furthermore, the structure responses using the viscoelastic damper with and without memory were studied. © IMechE 2012.

A simplified approach to determine aerodynamic damping of railway overheads

The railway overhead (or catenary) is the system of cables responsible for providing electric current to the train. This system has been reported as wind-sensitive (Scanlon et al., 2000), and particularly to the occurrence of galloping phenomena. Galloping phenomena of the railway overhead consists of undamped cable oscillations triggered by aerodynamic forces acting on the contact wire. As is well known, aerodynamic loads on the contact wire depends on the incident flow mean velocity and the angle of attack. The presence of embankments or hills modifies both vertical velocities profiles and angles of attack of the flow (Paiva et al., 2009). The presence of these cross-wind related oscillations can interfere with the safe operation of the railway service (Johnson, 1996). Therefore a correct modelling of the phenomena is required to avoid these unwanted oscillations.

Wheel-Rail Contact Forces in High-Speed Simply Supported Bridges at Resonance

The response of high-speed bridges at resonance, particularly under flexural vibrations, constitutes a subject of research for many scientists and engineers at the moment. The topic is of great interest because, as a matter of fact, such kind of behaviour is not unlikely to happen due to the elevated operating speeds of modern rains, which in many cases are equal to or even exceed 300 km/h ( [1,2]). The present paper addresses the subject of the evolution of the wheel-rail contact forces during resonance situations in simply supported bridges. Based on a dimensionless formulation of the equations of motion presented in [4], very similar to the one introduced by Klasztorny and Langer in [3], a parametric study is conducted and the contact forces in realistic situations analysed in detail. The effects of rail and wheel irregularities are not included in the model. The bridge is idealised as an Euler-Bernoulli beam, while the train is simulated by a system consisting of rigid bodies, springs and dampers. The situations such that a severe reduction of the contact force could take place are identified and compared with typical situations in actual bridges. To this end, the simply supported bridge is excited at resonace by means of a theoretical train consisting of 15 equidistant axles. The mechanical characteristics of all axles (unsprung mass, semi-sprung mass, and primary suspension system) are identical. This theoretical train permits the identification of the key parameters having an influence on the wheel-rail contact forces. In addition, a real case of a 17.5 m bridges traversed by the Eurostar train is analysed and checked against the theoretical results. The influence of three fundamental parameters is investigated in great detail: a) the ratio of the fundamental frequency of the bridge and natural frequency of the primary suspension of the vehicle; b) the ratio of the total mass of the bridge and the semi-sprung mass of the vehicle and c) the ratio between the length of the bridge and the characteristic distance between consecutive axles. The main conclusions derived from the investigation are: The wheel-rail contact forces undergo oscillations during the passage of the axles over the bridge. During resonance, these oscillations are more severe for the rear wheels than for the front ones. If denotes the span of a simply supported bridge, and the characteristic distance between consecutive groups of loads, the lower the value of , the greater the oscillations of the contact forces at resonance. For or greater, no likelihood of loss of wheel-rail contact has been detected. The ratio between the frequency of the primary suspension of the vehicle and the fundamental frequency of the bridge is denoted by (frequency ratio), and the ratio of the semi-sprung mass of the vehicle (mass of the bogie) and the total mass of the bridge is denoted by (mass ratio). For any given frequency ratio, the greater the mass ratio, the greater the oscillations of the contact forces at resonance. The oscillations of the contact forces at resonance, and therefore the likelihood of loss of wheel-rail contact, present a minimum for approximately between 0.5 and 1. For lower or higher values of the frequency ratio the oscillations of the contact forces increase. Neglecting the possible effects of torsional vibrations, the metal or composite bridges with a low linear mass have been found to be the ones where the contact forces may suffer the most severe oscillations. If single-track, simply supported, composite or metal bridges were used in high-speed lines, and damping ratios below 1% were expected, the minimum contact forces at resonance could drop to dangerous values. Nevertheless, this kind of structures is very unusual in modern high-speed railway lines.

Hydrodynamic forces on heave plates for offshore systems oscillating close to the seabed or the free surface

La energía eólica marina es uno de los recursos energéticos con mayor proyección pudiendo contribuir a reducir el consumo de combustibles fósiles y a cubrir la demanda de energía en todo el mundo. El concepto de aerogenerador marino está basado en estructuras fijas como jackets o en plataformas flotantes, ya sea una semisumergible o una TLP. Se espera que la energía eólica offshore juegue un papel importante en el perfil de producción energética de los próximos años; por tanto, las turbinas eólicas deben hacerse más fables y rentables para ser competitivas frente a otras fuentes de energía. Las estructuras flotantes pueden experimentar movimientos resonantes en estados de la mar con largos períodos de oleaje. Estos movimientos disminuyen su operatividad y pueden causar daños en los componentes eléctricos de las turbinas y en las palas, también en los risers y moorings. La respuesta de la componente vertical del movimiento puede reducirse mediante diferentes actuaciones: (1) aumentando la amortiguación del sistema, (2) manteniendo el período del movimiento vertical fuera del rango de la energía de la ola, y (3) reduciendo las fuerzas de excitación verticales. Un ejemplo típico para llevar a cabo esta reducción son las "Heave Plates". Las heave plates son placas que se utilizan en la industria offshore debido a sus características hidrodinámicas, ya que aumentan la masa añadida y la amortiguación del sistema. En un análisis hidrodinámico convencional, se considera una estructura sometida a un oleaje con determinadas características y se evalúan las cargas lineales usando la teoría potencial. El amortiguamiento viscoso, que juega un papel crucial en la respuesta en resonancia del sistema, es un dato de entrada para el análisis. La tesis se centra principalmente en la predicción del amortiguamiento viscoso y de la masa añadida de las heave plates usadas en las turbinas eólicas flotantes. En los cálculos, las fuerzas hidrodinámicas se han obtenido con el f n de estudiar cómo los coeficientes hidrodinámicos de masa añadida5 y amortiguamiento varían con el número de KC, que caracteriza la amplitud del movimiento respecto al diámetro del disco. Por otra parte, se ha investigado la influencia de la distancia media de la ‘heave plate’ a la superficie libre o al fondo del mar, sobre los coeficientes hidrodinámicos. En este proceso, un nuevo modelo que describe el trabajo realizado por la amortiguación en función de la enstrofía, es descrito en el presente documento. Este nuevo enfoque es capaz de proporcionar una correlación directa entre el desprendimiento local de vorticidad y la fuerza de amortiguación global. El análisis también incluye el estudio de los efectos de la geometría de la heave plate, y examina la sensibilidad de los coeficientes hidrodinámicos al incluir porosidad en ésta. Un diseño novedoso de una heave plate, basado en la teoría fractal, también fue analizado experimentalmente y comparado con datos experimentales obtenidos por otros autores. Para la resolución de las ecuaciones de Navier Stokes se ha usado un solver basado en el método de volúmenes finitos. El solver usa las librerías de OpenFOAM (Open source Field Operation And Manipulation), para resolver un problema multifásico e incompresible, usando la técnica VOF (volume of fluid) que permite capturar el movimiento de la superficie libre. Los resultados numéricos han sido comparados con resultados experimentales llevados a cabo en el Canal del Ensayos Hidrodinámicos (CEHINAV) de la Universidad Politécnica de Madrid y en el Canal de Experiencias Hidrodinámicas (CEHIPAR) en Madrid, al igual que con otros experimentos realizados en la Escuela de Ingeniería Mecánica de la Universidad de Western Australia. Los principales resultados se presentan a continuación: 1. Para pequeños valores de KC, los coeficientes hidrodinámicos de masa añadida y amortiguamiento incrementan su valor a medida que el disco se aproxima al fondo marino. Para los casos cuando el disco oscila cerca de la superficie libre, la dependencia de los coeficientes hidrodinámicos es más fuerte por la influencia del movimiento de la superficie libre. 2. Los casos analizados muestran la existencia de un valor crítico de KC, donde la tendencia de los coeficientes hidrodinámicos se ve alterada. Dicho valor crítico depende de la distancia al fondo marino o a la superficie libre. 3. El comportamiento físico del flujo, para valores de KC cercanos a su valor crítico ha sido estudiado mediante el análisis del campo de vorticidad. 4. Introducir porosidad al disco, reduce la masa añadida para los valores de KC estudiados, pero se ha encontrado que la porosidad incrementa el valor del coeficiente de amortiguamiento cuando se incrementa la amplitud del movimiento, logrando un máximo de damping para un disco con 10% de porosidad. 5. Los resultados numéricos y experimentales para los discos con faldón, muestran que usar este tipo de geometrías incrementa la masa añadida cuando se compara con el disco sólido, pero reduce considerablemente el coeficiente de amortiguamiento. 6. Un diseño novedoso de heave plate basado en la teoría fractal ha sido experimentalmente estudiado a diferentes calados y comparado con datos experimentales obtenidos por otro autores. Los resultados muestran un comportamiento incierto de los coeficientes y por tanto este diseño debería ser estudiado más a fondo. ABSTRACT Offshore wind energy is one of the promising resources which can reduce the fossil fuel energy consumption and cover worldwide energy demands. Offshore wind turbine concepts are based on either a fixed structure as a jacket or a floating offshore platform like a semisubmersible, spar or tension leg platform. Floating offshore wind turbines have the potential to be an important part of the energy production profile in the coming years. In order to accomplish this wind integration, these wind turbines need to be made more reliable and cost efficient to be competitive with other sources of energy. Floating offshore artifacts, such oil rings and wind turbines, may experience resonant heave motions in sea states with long peak periods. These heave resonances may increase the system downtime and cause damage on the system components and as well as on risers and mooring systems. The heave resonant response may be reduced by different means: (1) increasing the damping of the system, (2) keeping the natural heave period outside the range of the wave energy, and (3) reducing the heave excitation forces. A typical example to accomplish this reduction are “Heave Plates”. Heave plates are used in the offshore industry due to their hydrodynamic characteristics, i.e., increased added mass and damping. Conventional offshore hydrodynamic analysis considers a structure in waves, and evaluates the linear and nonlinear loads using potential theory. Viscous damping, which is expected to play a crucial role in the resonant response, is an empirical input to the analysis, and is not explicitly calculated. The present research has been mainly focused on the prediction of viscous damping and added mass of floating offshore wind turbine heave plates. In the calculations, the hydrodynamic forces have been measured in order to compute how the hydrodynamic coefficients of added mass1 and damping vary with the KC number, which characterises the amplitude of heave motion relative to the diameter of the disc. In addition, the influence on the hydrodynamic coefficients when the heave plate is oscillating close to the free surface or the seabed has been investigated. In this process, a new model describing the work done by damping in terms of the flow enstrophy, is described herein. This new approach is able to provide a direct correlation between the local vortex shedding processes and the global damping force. The analysis also includes the study of different edges geometry, and examines the sensitivity of the damping and added mass coefficients to the porosity of the plate. A novel porous heave plate based on fractal theory has also been proposed, tested experimentally and compared with experimental data obtained by other authors for plates with similar porosity. A numerical solver of Navier Stokes equations, based on the finite volume technique has been applied. It uses the open-source libraries of OpenFOAM (Open source Field Operation And Manipulation), to solve 2 incompressible, isothermal immiscible fluids using a VOF (volume of fluid) phase-fraction based interface capturing approach, with optional mesh motion and mesh topology changes including adaptive re-meshing. Numerical results have been compared with experiments conducted at Technical University of Madrid (CEHINAV) and CEHIPAR model basins in Madrid and with others performed at School of Mechanical Engineering in The University of Western Australia. A brief summary of main results are presented below: 1. At low KC numbers, a systematic increase in added mass and damping, corresponding to an increase in the seabed proximity, is observed. Specifically, for the cases when the heave plate is oscillating closer to the free surface, the dependence of the hydrodynamic coefficients is strongly influenced by the free surface. 2. As seen in experiments, a critical KC, where the linear trend of the hydrodynamic coefficients with KC is disrupted and that depends on the seabed or free surface distance, has been found. 3. The physical behavior of the flow around the critical KC has been explained through an analysis of the flow vorticity field. 4. The porosity of the heave plates reduces the added mass for the studied porosity at all KC numbers, but the porous heave plates are found to increase the damping coefficient with increasing amplitude of oscillation, achieving a maximum damping coefficient for the heave plate with 10% porosity in the entire KC range. 5. Another concept taken into account in this work has been the heave plates with flaps. Numerical and experimental results show that using discs with flaps will increase added mass when compared to the plain plate but may also significantly reduce damping. 6. A novel heave plate design based on fractal theory has tested experimentally for different submergences and compared with experimental data obtained by other authors for porous plates. Results show an unclear behavior in the coefficients and should be studied further. Future work is necessary in order to address a series of open questions focusing on 3D effects, optimization of the heave plates shapes, etc.

Selected topics in the Hydrodynamics of floating offshore wind turbines: heave damping and second order surge motions

La Energía eléctrica producida mediante tecnología eólica flotante es uno de los recursos más prometedores para reducir la dependencia de energía proveniente de combustibles fósiles. Esta tecnología es de especial interés en países como España, donde la plataforma continental es estrecha y existen pocas áreas para el desarrollo de estructuras fijas. Entre los diferentes conceptos flotantes, esta tesis se ha ocupado de la tipología semisumergible. Estas plataformas pueden experimentar movimientos resonantes en largada y arfada. En largada, dado que el periodo de resonancia es largo estos puede ser inducidos por efectos de segundo orden de deriva lenta que pueden tener una influencia muy significativa en las cargas en los fondeos. En arfada las fuerzas de primer orden pueden inducir grandes movimientos y por tanto la correcta determinación del amortiguamiento es esencial para la analizar la operatividad de la plataforma. Esta tesis ha investigado estos dos efectos, para ello se ha usado como caso base el diseño de una plataforma desarrollada en el proyecto Europeo Hiprwind. La plataforma se compone de 3 columnas cilíndricas unidas mediante montantes estructurales horizontales y diagonales, Los cilindros proporcionan flotabilidad y momentos adrizante. A la base de cada columna se le ha añadido un gran “Heave Plate” o placa de cierre. El diseño es similar a otros diseños previos (Windfloat). Se ha fabricado un modelo a escala de una de las columnas para el estudio detallado del amortiguamiento mediante oscilaciones forzadas. Las dimensiones del modelo (1m diámetro en la placa de cierre) lo hacen, de los conocidos por el candidato, el mayor para el que se han publicado datos. El diseño del cilindro se ha realizado de tal manera que permite la fijación de placas de cierre planas o con refuerzo, ambos modelos se han fabricado y analizado. El modelo con refuerzos es una reproducción exacta del diseño a escala real incluyendo detalles distintivos del mismo, siendo el más importante la placa vertical perimetral. Los ensayos de oscilaciones forzadas se han realizado para un rango de frecuencias, tanto para el disco plano como el reforzado. Se han medido las fuerzas durante los ensayos y se han calculado los coeficientes de amortiguamiento y de masa añadida. Estos coeficientes son necesarios para el cálculo del fondeo mediante simulaciones en el dominio del tiempo. Los coeficientes calculados se han comparado con la literatura existente, con cálculos potenciales y por ultimo con cálculos CFD. Para disponer de información relevante para el diseño estructural de la plataforma se han medido y analizado experimentalmente las presiones en la parte superior e inferior de cada placa de cierre. Para la correcta estimación numérica de las fuerzas de deriva lenta en la plataforma se ha realizado una campaña experimental que incluye ensayos con modelo cautivo de la plataforma completa en olas bicromaticas. Pese a que estos experimentos no reproducen un escenario de oleaje realista, los mismos permiten una verificación del modelo numérico mediante la comparación de fuerzas medidas en el modelo físico y el numérico. Como resultados de esta tesis podemos enumerar las siguientes conclusiones. 1. El amortiguamiento y la masa añadida muestran una pequeña dependencia con la frecuencia pero una gran dependencia con la amplitud del movimiento. siendo coherente con investigaciones existentes. 2. Las medidas con la placa de cierre reforzada con cierre vertical en el borde, muestra un amortiguamiento significativamente menor comparada con la placa plana. Esto implica que para ensayos de canal es necesario incluir estos detalles en el modelo. 3. La masa añadida no muestra grandes variaciones comparando placa plana y placa con refuerzos. 4. Un coeficiente de amortiguamiento del 6% del crítico se puede considerar conservador para el cálculo en el dominio de la frecuencia. Este amortiguamiento es equivalente a un coeficiente de “drag” de 4 en elementos de Morison cuadráticos en las placas de cierre usadas en simulaciones en el dominio del tiempo. 5. Se han encontrado discrepancias en algunos valores de masa añadida y amortiguamiento de la placa plana al comparar con datos publicados. Se han propuesto algunas explicaciones basadas en las diferencias en la relación de espesores, en la distancia a la superficie libre y también relacionadas con efectos de escala. 6. La presión en la placa con refuerzos son similares a las de la placa plana, excepto en la zona del borde donde la placa con refuerzo vertical induce una gran diferencias de presiones entre la cara superior e inferior. 7. La máxima diferencia de presión escala coherentemente con la fuerza equivalente a la aceleración de la masa añadida distribuida sobre la placa. 8. Las masas añadidas calculadas con el código potencial (WADAM) no son suficientemente precisas, Este software no contempla el modelado de placas de pequeño espesor con dipolos, la poca precisión de los resultados aumenta la importancia de este tipo de elementos al realizar simulaciones con códigos potenciales para este tipo de plataformas que incluyen elementos de poco espesor. 9. Respecto al código CFD (Ansys CFX) la precisión de los cálculos es razonable para la placa plana, esta precisión disminuye para la placa con refuerzo vertical en el borde, como era de esperar dado la mayor complejidad del flujo. 10. Respecto al segundo orden, los resultados, en general, muestran que, aunque la tendencia en las fuerzas de segundo orden se captura bien con los códigos numéricos, se observan algunas reducciones en comparación con los datos experimentales. Las diferencias entre simulaciones y datos experimentales son mayores al usar la aproximación de Newman, que usa únicamente resultados de primer orden para el cálculo de las fuerzas de deriva media. 11. Es importante remarcar que las tendencias observadas en los resultados con modelo fijo cambiarn cuando el modelo este libre, el impacto que los errores en las estimaciones de fuerzas segundo orden tienen en el sistema de fondeo dependen de las condiciones ambientales que imponen las cargas ultimas en dichas líneas. En cualquier caso los resultados que se han obtenido en esta investigación confirman que es necesaria y deseable una detallada investigación de los métodos usados en la estimación de las fuerzas no lineales en las turbinas flotantes para que pueda servir de guía en futuros diseños de estos sistemas. Finalmente, el candidato espera que esta investigación pueda beneficiar a la industria eólica offshore en mejorar el diseño hidrodinámico del concepto semisumergible. ABSTRACT Electrical power obtained from floating offshore wind turbines is one of the promising resources which can reduce the fossil fuel energy consumption and cover worldwide energy demands. The concept is the most competitive in countries, such as Spain, where the continental shelf is narrow and does not provide space for fixed structures. Among the different floating structures concepts, this thesis has dealt with the semisubmersible one. Platforms of this kind may experience resonant motions both in surge and heave directions. In surge, since the platform natural period is long, such resonance can be excited with second order slow drift forces and may have substantial influence on mooring loads. In heave, first order forces can induce significant motion, whose damping is a crucial factor for the platform downtime. These two topics have been investigated in this thesis. To this aim, a design developed during HiPRWind EU project, has been selected as reference case study. The platform is composed of three cylindrical legs, linked together by a set of structural braces. The cylinders provide buoyancy and restoring forces and moments. Large circular heave plates have been attached to their bases. The design is similar to other documented in literature (e.g. Windfloat), which implies outcomes could have a general value. A large scale model of one of the legs has been built in order to study heave damping through forced oscillations. The final dimensions of the specimen (one meter diameter discs) make it, to the candidate’s knowledge, the largest for which data has been published. The model design allows for the fitting of either a plain solid heave plate or a flapped reinforced one; both have been built. The latter is a model scale reproduction of the prototype heave plate and includes some distinctive features, the most important being the inclusion of a vertical flap on its perimeter. The forced oscillation tests have been conducted for a range of frequencies and amplitudes, with both the solid plain model and the vertical flap one. Forces have been measured, from which added mass and damping coefficients have been obtained. These are necessary to accurately compute time-domain simulations of mooring design. The coefficients have been compared with literature, and potential flow and CFD predictions. In order to provide information for the structural design of the platform, pressure measurements on the top and bottom side of the heave discs have been recorded and pressure differences analyzed. In addition, in order to conduct a detailed investigation on the numerical estimations of the slow-drift forces of the HiPRWind platform, an experimental campaign involving captive (fixed) model tests of a model of the whole platform in bichromatic waves has been carried out. Although not reproducing the more realistic scenario, these tests allowed a preliminary verification of the numerical model based directly on the forces measured on the structure. The following outcomes can be enumerated: 1. Damping and added mass coefficients show, on one hand, a small dependence with frequency and, on the other hand, a large dependence with the motion amplitude, which is coherent with previously published research. 2. Measurements with the prototype plate, equipped with the vertical flap, show that damping drops significantly when comparing this to the plain one. This implies that, for tank tests of the whole floater and turbine, the prototype plate, equipped with the flap, should be incorporated to the model. 3. Added mass values do not suffer large alterations when comparing the plain plate and the one equipped with a vertical flap. 4. A conservative damping coefficient equal to 6% of the critical damping can be considered adequate for the prototype heave plate for frequency domain analysis. A corresponding drag coefficient equal to 4.0 can be used in time domain simulations to define Morison elements. 5. When comparing to published data, some discrepancies in added mass and damping coefficients for the solid plain plate have been found. Explanations have been suggested, focusing mainly on differences in thickness ratio and distance to the free surface, and eventual scale effects. 6. Pressures on the plate equipped with the vertical flap are similar in magnitude to those of the plain plate, even though substantial differences are present close to the edge, where the flap induces a larger pressure difference in the reinforced case. 7. The maximum pressure difference scales coherently with the force equivalent to the acceleration of the added mass, distributed over the disc surface. 8. Added mass coefficient values predicted with the potential solver (WADAM) are not accurate enough. The used solver does not contemplate modeling thin plates with doublets. The relatively low accuracy of the results highlights the importance of these elements when performing potential flow simulations of offshore platforms which include thin plates. 9. For the full CFD solver (Ansys CFX), the accuracy of the computations is found reasonable for the plain plate. Such accuracy diminishes for the disc equipped with a vertical flap, an expected result considering the greater complexity of the flow. 10. In regards to second order effects, in general, the results showed that, although the main trend in the behavior of the second-order forces is well captured by the numerical predictions, some under prediction of the experimental values is visible. The gap between experimental and numerical results is more pronounced when Newman’s approximation is considered, making use exclusively of the mean drift forces calculated in the first-order solution. 11. It should be observed that the trends observed in the fixed model test may change when the body is free to float, and the impact that eventual errors in the estimation of the second-order forces may have on the mooring system depends on the characteristics of the sea conditions that will ultimately impose the maximum loads on the mooring lines. Nevertheless, the preliminary results obtained in this research do confirm that a more detailed investigation of the methods adopted for the estimation of the nonlinear wave forces on the FOWT would be welcome and may provide some further guidance for the design of such systems. As a final remark, the candidate hopes this research can benefit the offshore wind industry in improving the hydrodynamic design of the semi-submersible concept.

Induced Damping on Vibrating Circular Plates Submerged in still Fluid

When a structure vibrates immersed in a fluid it is known that the dynamic properties of the system are modified. The surrounding fluid will, in general, contribute to the inertia, the rigidity and the damping coefficient of the coupled fluid-structure system. For light structures, like spacecraft antennas, even when the fluid is air the contribution to the dynamic properties can be important. For not so light structures the ratio of the equivalent fluid/structure mass and rigidity can be very small and the fluid contribution could be neglected. For the ratio of equivalent fluid/structure damping both terms are of the same order and therefore the fluid contribution must be studied. The working life of the spacecraft structure would be on space and so without any surrounding fluid. The response of a spacecraft structure on its operational life would be attenuated by the structural damping alone but when the structure is dynamically tested on the earth the dynamic modal test is performed with the fluid surrounding it. The results thus are contaminated by the effects of the fluid. If the damping added by the fluid is of the same order as the structural damping the response of the structure in space can be quite different to the response predicted on earth. It is therefore desirable to have a method able to determine the amount of damping induced by the fluid and that should be subtracted of the total damping measured on the modal vibration test. In this work, a method for the determination of the effect of the surrounding fluid on the dynamic characteristics of a circular plate has been developed. The plate is assumed to vibrate harmonically with the vacuum modes and the generalized forces matrix due to the fluid is thus computed. For a compressible fluid this matrix is formed by complex numbers including terms of inertia, rigidity and damping. The matrix due to the fluid loading is determined by a boundary element method (BEM). The BEM used is of circular rings on the plate surface so the number of elements to obtain an accurate result is very low. The natural frequencies of the system are computed by an iteration procedure one by one and also the damping fluid contribution. Comparisons of the present method with various experimental data and other theories show the efficiency and accuracy of the method for any support condition of the plate.

Effects of Variations in Nonlinear Damping Coefficients on the Parametric Vibration of a Cantilever Beam with a Lumped Mass

Uncertainties in damping estimates can significantly affect the dynamic response of a given flexible structure. A common practice in linear structural dynamics is to consider a linear viscous damping model as the major energy dissipation mechanism. However, it is well known that different forms of energy dissipation can affect the structure's dynamic response. The major goal of this paper is to address the effects of the turbulent frictional damping force, also known as drag force on the dynamic behavior of a typical flexible structure composed of a slender cantilever beam carrying a lumped-mass on the tip. First, the system's analytical equation is obtained and solved by employing a perturbation technique. The solution process considers variations of the drag force coefficient and its effects on the system's response. Then, experimental results are presented to demonstrate the effects of the nonlinear quadratic damping due to the turbulent frictional force on the system's dynamic response. In particular, the effects of the quadratic damping on the frequency-response and amplitude-response curves are investigated. Numerically simulated as well as experimental results indicate that variations on the drag force coefficient significantly alter the dynamics of the structure under investigation. Copyright (c) 2008 D. G. Silva and P. S. Varoto.