Dissertations / Theses on the topic 'Harmonic Fourier Series coefficient'

Researchers now have unprecedented access to free Landsat data, enabling detailed monitoring of the Earth's land surface and vegetation.  There are gaps in the data, due in part to cloud cover. The gaps are aperiodic and localized, forcing any detailed multitemporal analysis based on Landsat data to compensate.   Harmonic regression approximates Landsat data for any point in time with minimal training images and reduced storage requirements.  In two study areas in North Carolina, USA, harmonic regression approaches were least as good at simulating missing data as STAR-FM for images from 2001.  Harmonic regression had an R^2"0.9 over three quarters of all pixels. It gave the highest R_Predicted^2 values on two thirds of the pixels.  Applying harmonic regression with the same number of harmonics to consecutive years yielded an improved fit, R^2"0.99 for most pixels.   We next demonstrate a change detection method based on exponentially weighted moving average (EWMA) charts of harmonic residuals. In the process, a data-driven cloud filter is created, enabling use of partially clouded data.  The approach is shown capable of detecting thins and subtle forest degradations in Alabama, USA, considerably finer than the Landsat spatial resolution in an on-the-fly fashion, with new images easily incorporated into the algorithm.  EWMA detection accurately showed the location, timing, and magnitude of 85% of known harvests in the study area, verified by aerial imagery.   We use harmonic regression to improve the precision of dynamic forest parameter estimates, generating a robust time series of vegetation index values.  These values are classified into strata maps in Alabama, USA, depicting regions of similar growth potential.  These maps are applied to Forest Service Forest Inventory and Analysis (FIA) plots, generating post-stratified estimates of static and dynamic forest parameters.  Improvements to efficiency for all parameters were such that a comparable random sample would require at least 20% more sampling units, with the improvement for the growth parameter requiring a 50% increase. These applications demonstrate the utility of harmonic regression for Landsat data.  They suggest further applications in environmental monitoring and improved estimation of landscape parameters, critical to improving large-scale models of ecosystems and climate effects.
Ph. D.

Cette thèse étudie quelques problèmes d’analyse harmonique sur les groupes quantiques compacts. Elle consiste en trois parties. La première partie présente la théorie Lp élémentaire des transformées de Fourier, les convolutions et les multiplicateurs sur les groupes quantiques compacts, y compris la théorie de Hausdorff-Young et les inégalités de Young.Dans la seconde partie, nous caractérisons les opérateurs de convolution positifs sur un groupe quantique fini qui envoient Lp dans L2, et donnons aussi quelques constructions sur les groupes quantiques compacts infinis. La méthode pour étudier les états non-dégénérés fournit une formule générale pour calculer les états idempotents associés aux images deHopf, qui généralise un travail de Banica, Franz et Skalski. La troisième partie est consacrée à l’étude des ensembles de Sidon, des ensembles _(p) et des notions associées pour les groupes quantiques compacts. Nous établissons différentes caractérisations des ensembles de Sidon, et en particulier nous démontrons que tout ensemble de Sidon est un ensemble de Sidon fort au sens de Picardello. Nous donnons quelques liens entre les ensembles de Sidon, les ensembles _(p) et les lacunarités pour les multiplicateurs de Fourier sur Lp, généralisant un travail de Blendek et Michali˘cek. Nous démontrons aussi l’existence des ensembles de type _(p) pour les systèmes orthogonaux dans les espaces Lp non commutatifs, et déduisons les propriétés correspondantes pour les groupes quantiques compacts. Nous considérons aussi les ensembles de Sidon centraux, et nous prouvons que les groupes quantiques compacts ayant les mêmes règles de fusion et les mêmes fonctions de dimension ont des ensemble de Sidon centraux identiques. Quelques exemples sont aussi étudiés dans cette thèse. Les travaux présentés dans cette thèse se basent sur deux articles de l’auteur. Le premier s’intitule “Lp-improving convolution operators on finite quantum groups” et a été accepté pour publication dans Indiana University Mathematics Journal, et le deuxième est un travail intitulé “Lacunary Fourier series for compact quantum groups” et a été publié en ligne dans Communications in Mathematical Physics
This thesis studies some problems in the theory of harmonic analysis on compact quantum groups. It consists of three parts. The first part presents some elementary Lp theory of Fourier transforms, convolutions and multipliers on compact quantum groups, including the Hausdorff-Young theory and Young’s inequalities. In the second part, we characterize positive convolution operators on a finite quantum group G which are Lp-improving, and also give some constructions on infinite compact quantum groups. The methods for ondegeneratestates yield a general formula for computing idempotent states associated to Hopf images, which generalizes earlier work of Banica, Franz and Skalski. The third part is devoted to the study of Sidon sets, _(p)-sets and some related notions for compact quantum groups. We establish several different characterizations of Sidon sets, and in particular prove that any Sidon set in a discrete group is a strong Sidon set in the sense of Picardello. We give several relations between Sidon sets, _(p)-sets and lacunarities for Lp-Fourier multipliers, generalizing a previous work by Blendek and Michali˘cek. We also prove the existence of _(p)-sets for orthogonal systems in noncommutative Lp-spaces, and deduce the corresponding properties for compact quantum groups. Central Sidon sets are also discussed, and it turns out that the compact quantum groups with the same fusion rules and the same dimension functions have identical central Sidon sets. Several examples are also included. The thesis is principally based on two works by the author, entitled “Lp-improvingconvolution operators on finite quantum groups” and “Lacunary Fourier series for compact quantum groups”, which have been accepted for publication in Indiana University Mathematics Journal and Communications in Mathematical Physics respectively

Starting with tensor calculus and the variational form of the Hamiltonian functional, a generalized theory is formulated for doubly curved thin shells. The formulation avoids geometric approximations commonly adopted in other formulations. The theory is then specialized for cylindrical and toroidal shells as special cases, both of interest in the modeling of straight and elbow segments of pipeline systems. Since the treatment avoids geometric approximations, the cylindrical shell theory is believed to be more accurate than others reported in the literature. By adopting a set of consistent geometric approximations, the present theory is shown to revert to the well known Flugge shell theory. Another set of consistent geometric approximations is shown to lead to the Donnell-Mushtari-Vlasov (DMV) theory. A general closed form solution of the theory is developed for cylinders under general harmonic loads. The solution is then used to formulate a family of exact shape functions which are subsequently used to formulate a super-convergent finite element. The formulation efficiently and accurately captures ovalization, warping, radial expansion, and other shell behavioural modes under general static or harmonic forces either in-phase or out-of-phase. Comparisons with shell solutions available in Abaqus demonstrate the validity of the formulation and the accuracy of its predictions. The generalized thin shell theory is then specialized for toroidal shells. Consistent sets of approximations lead to three simplified theories for toroidal shells. The first set of approximations has lead to a theory comparable to that of Sanders while the second set of approximation has lead to a theory nearly identical to the DMV theory for toroidal shells. A closed form solution is then obtained for the governing equation. Exact shape functions are then developed and subsequently used to formulate a finite element. Comparisons with Abaqus solutions show the validity of the formulation for short elbow segments under a variety of loading conditions. Because of their efficiency, the finite elements developed are particularly suited for the analysis of long pipeline systems.

Position estimation using only active phase voltage and current is presented to perform high accuracy position sensorless control of a SRM drive. By extracting the amplitude of the first switching harmonic terms of phase voltage and current for a PWM period through Fourier analysis, flux-linkage and position are estimated without external hardware circuitry such as a modulator and demodulator, resulting in increasing cost, as well as large position estimation error produced when the motional back emf is ignored near zero speed. Hence the proposed position estimation scheme covers the entire speed range including the standstill under various loads and it has high resolution information depending on switching frequency. Fourier series and Fast Fourier transform are employed to decompose the phase voltage and current into its first switching harmonic. A two-phase SRM drive system, consisting of an asymmetrical converter and a conventional closed-loop PI current controller, is utilized to validate the performance of the proposed position estimation scheme in comprehensive operating conditions. The estimated values very closely track the actual values in dynamic simulations and experiments. It is shown that the proposed position estimation scheme using Fourier analysis is sufficiently accurate and works satisfactorily at various operating points. This research also proposes an accurate self-inductance measurement method. In general, when applying circulating currents within the body of a ferromagnetic material under conditions of a time varying magnetic flux, the effects of eddy current losses and resistance changes due to heating decrease the magnetic field strength and thereby the reduced magnetic field decreases the magnetic flux-linkage of SRM. These losses make a challenge to the measurement of magnetic characteristics of SRM. These motives lead to propose a measurement methodology based on 60 Hz sinusoidal excitation using a variable AC power supply, which provides an alternative to time domain integration approaches for self-inductance or flux-linkage measurement as well as eliminates error arising from thermal and eddy currents effects. The validation of the proposed method is verified with the correlation between the measurement and FEA results of flux-linkage. Furthermore, this research proposes the solutions for low cost and high efficiency drive systems, consisting of a split AC converter and a two-phase SRM. Its performance is analyzed and verified with experiments at the rated speed under various loads. It is believed that this drive system combined with the proposed position estimation scheme using Fourier analysis is a strong contender to be a low cost motor drive system with single switch per phase having comparable efficiency and acoustic noise level as an asymmetric drive system.
Ph. D.

New modelling tools intended to contribute to the development of components for quieter vehicles are developed. The tools are based on continuum models for wave propagation in poro– and visco–elastic media. By using geometric attributes of the studied components, the computational cost may be radically decreased compared to traditional methods. By assigning known analytical functions for one or two of the spatial directions, the spatial dimension of the remaining numerical problem is reduced. This reduction of spatial dimensions is performed in two di↵erent ways. The first one treats wave propagation in infinitely extended homogeneous and hollowed cylindrical rods, or wave guides, consisting of visco–elastic media. The wave solutions obtained are then used to model rubber vibration isolators of finite length by mode–matching the fields to the radial boundary conditions of interest. The second one is a method for modelling rotationally symmetric multilayered structures consisting of poro–elastic, elastic and fluid domains. By using a harmonic expansion for the azimuthal spatial dependence, the original three–dimensional problem is split up into several, much smaller, two– dimensional ones, radically decreasing the computational load. Moreover, using a mixed measurement/modelling approach, the audible frequency range characteristics of a viscous damper from a truck is studied, illustrating the influence of the rubber bushings by which it is attached to surrounding structures. The modelling approaches presented in this thesis are intended as tools aiding the design process of new vehicles, enabling new technology striving for more sustainable vehicle concepts. More specifically, the tools aim to improve the modelling of sound and vibration properties which are often penalised when seeking new, more sustainable vehicle designs.

QC 20120522

Centre for Eco2 Vehicle Design

Ce travail est constitué de deux parties indépendantes traitant chacune d'un problème issu de la théorie du contrôle des équations aux dérivées partielles. La première partie est consacrée à l'étude d'un feedback explicite et déjà connu, s'appliquant à des systèmes linéaires, réversibles en temps et éventuellement munis d'un opérateur de contrôle non-borné. On justifie le caractère bien posé du problème en boucle fermée via la théorie des semi-groupes puis on étudie le taux de décroissance des solutions du système régulé. La seconde partie concerne un problème d'observation pour la corde vibrante : on détermine comment choisir des instants d'observation pour que la position de la corde à ces instants permette de retrouver les conditions initiales tout en préservant une certaine régularité. La méthode, qui repose sur des résultats d'approximation diophantienne, est ensuite étendue à d'autres systèmes. En utilisant une méthode de dualité on démontre aussi un résultat de contrôlabilité exacte.

Neste trabalho definimos o operador não local, raiz quadrada do Laplaciano com condição de fronteira de Neumann, através do método de extensão harmônica. O estudo foi feito com o auxílio das séries de Fourier em domínios limitados, como sendo o intervalo, o quadrado e a bola. Posteriormente, aplicamos nosso estudo, à problemas elípticos não lineares envolvendo o operador não local raiz quadrada do Laplaciano com condição de fronteira de Neumann.
In this work we define the non-local operator, square root of the Laplacian with Neumann boundary condition, using the method of harmonic extension. The study was done with the aid of Fourier series in bounded domains, as the interval, the square and the ball. Subsequently, we apply our study, the nonlinear elliptic problems involving non-local operator square root of the Laplacian with Neumann boundary condition.

It is known that fission yeast Schizosaccharomyces pombe maintains its nuclear envelope during mitosis and it undergoes an interesting shape change during cell division - from a spherical via an ellipsoidal and a peanut-like to a dumb-bell shape. However, the biomechanical system behind this amazing transformation is still not understood. What we know is, that the shape must change due to forces acting on the membrane surrounding the nucleus and the microtubule based mitotic spindle is thought to play a key role. To estimate the locations and directions of the forces, the shape of the nucleus was recorded by confocal light microscopy. But such data is often inhomogeneously labeled with gaps in the boundary, making classical segmentation impractical. In order to accurately determine the shape we developed a global parametric shape description method, based on a Fourier coordinate expansion. The method implicitly assumes a closed and smooth surface. We will calculate the geometrical properties of the 2-dimensional shape and extend it to 3-dimensional properties, assuming rotational symmetry. Using a mechanical model for the lipid bilayer and the so called Helfrich-Canham free energy we want to calculate the minimum energy shape while respecting system-specific constraints to the surface and the enclosed volume. Comparing it with the observed shape leads to the forces. This provides the needed research tools to study forces based on images.

It is known that fission yeast Schizosaccharomyces pombe maintains its nuclear envelope during mitosis and it undergoes an interesting shape change during cell division - from a spherical via an ellipsoidal and a peanut-like to a dumb-bell shape. However, the biomechanical system behind this amazing transformation is still not understood. What we know is, that the shape must change due to forces acting on the membrane surrounding the nucleus and the microtubule based mitotic spindle is thought to play a key role. To estimate the locations and directions of the forces, the shape of the nucleus was recorded by confocal light microscopy. But such data is often inhomogeneously labeled with gaps in the boundary, making classical segmentation impractical. In order to accurately determine the shape we developed a global parametric shape description method, based on a Fourier coordinate expansion. The method implicitly assumes a closed and smooth surface. We will calculate the geometrical properties of the 2-dimensional shape and extend it to 3-dimensional properties, assuming rotational symmetry. Using a mechanical model for the lipid bilayer and the so called Helfrich-Canham free energy we want to calculate the minimum energy shape while respecting system-specific constraints to the surface and the enclosed volume. Comparing it with the observed shape leads to the forces. This provides the needed research tools to study forces based on images.

Dans ce travail de thèse, nous nous sommes intéressés à la modélisation des phénomènes de diffraction d’ondes électromagnétiques par des objets à symétrie de révolution complexes et fortement hétérogènes. La méthode que nous développons ici consiste en un couplage entre équations aux dérivées partielles (EDP) et équations intégrales (EI). Cette idée est essentiellement connue pour avoir deux avantages. Le premier est que les hétérogénéités de l’objet sont prises en compte naturellement dans la formulation du problème. Le deuxième est dû à l’utilisation des équations intégrales qui donnent une représentation exacte des solutions dans le milieu extérieur en fonction des courants surfaciques. Le domaine de simulation peut ainsi être ramené à l’objet lui-même. L’utilisation de développements en séries de Fourier combinés à la propriété d’invariance par rotation de l’objet permet alors la réduction du problème global 3D à un ensemble dénombrable de problème 2D.L’étude de ces problèmes nous a conduit à décomposer notre analyse en plusieurs parties,chacune ayant à traiter une partie du problème complet ou les méthodes d’intégrations numériques. Ces dernières étant difficiles à réaliser dans le cas des équations intégrales.Nous avons tout d’abord étudié un problème de Maxwell intérieur pour lequel nous avons développé une nouvelle méthode d’éléments finis d’ordre élevé dont nous avons montré l’efficacité et la précision sur de multiples exemples. Puis, nous avons étudié le problème de diffraction d’ondes planes pour des objets parfaitement conducteurs. La méthode d’éléments finis de frontière employée est alors construite par extension de la méthode précédente via l’opérateur de trace tangentielle. En combinant ces deux études, nous avons résolu le problème couplé en introduisant la propriété de symétrie de révolution dans une formulation variationnelle bien choisie. Par construction, les éléments finis qui y sont utilisés sont alors naturellement adaptées. L’algorithme de parallélisation de la méthode de couplage est finalement présentée et des comparaisons entre notre code AxiMax et un code 3D sont illustrées. Dans tous les cas, nous montrons que la méthode d’éléments finis d’ordre élevé permet d’obtenir des résultats d’une grande précision en fonction de la qualité des paramètres de simulation
In this thesis, we are interested in modeling diffraction of electromagnetic waves by axisymmetric and highly heterogeneous objects. Our method consists in a coupling between partial differential equations and integral equations. This idea is mainly interesting for two reasons : heterogeneities are handled naturally in the formulation and integral equations give an analytical representation of solutions outside the object based on surface currents.These advantages allow us to limit the domain of simulation to the object itself. In addition,using Fourier series combined with the rotational invariance property of the object, the 3D problem is reduced to a countable set of 2D problems. The study of these problems is split into several parts. Each part has to deal with aspecific problem as for example the numerical integration of singular integrals which is difficult to achieve. As a first step, we study time-harmonic Maxwell’s equations in a bounded domain for which we develop a new high-order finite element method and present its efficiency and accuracy on many examples. Secondly, we consider the diffraction of plane waves by perfect electric conductors to analyse integral equations for these kind of object.The boundary finite element method applied is defined by extension of the previous one via tangential trace operator. Then, we solve the coupled problem using a well chosen formulation based on the previous studies for which our finite element method is naturally adapted by construction. In order to evaluate its efficiency, a comparison is performed between our program « AxiMax » and one based on a purely 3D model. To conclude, in the last two chapters, we present the numerical integration method and the multi-processing algorithm developed in AxiMax. In all cases, we put forward the fact that our finite element method provides accurate results depending on the quality of the simulation parameters

The ability of a piezoelectric transducer in energy conversion is rapidly expanding in several applications. Some of the industrial applications for which a high power ultrasound transducer can be used are surface cleaning, water treatment, plastic welding and food sterilization. Also, a high power ultrasound transducer plays a great role in biomedical applications such as diagnostic and therapeutic applications. An ultrasound transducer is usually applied to convert electrical energy to mechanical energy and vice versa. In some high power ultrasound system, ultrasound transducers are applied as a transmitter, as a receiver or both. As a transmitter, it converts electrical energy to mechanical energy while a receiver converts mechanical energy to electrical energy as a sensor for control system. Once a piezoelectric transducer is excited by electrical signal, piezoelectric material starts to vibrate and generates ultrasound waves. A portion of the ultrasound waves which passes through the medium will be sensed by the receiver and converted to electrical energy. To drive an ultrasound transducer, an excitation signal should be properly designed otherwise undesired signal (low quality) can deteriorate the performance of the transducer (energy conversion) and increase power consumption in the system. For instance, some portion of generated power may be delivered in unwanted frequency which is not acceptable for some applications especially for biomedical applications. To achieve better performance of the transducer, along with the quality of the excitation signal, the characteristics of the high power ultrasound transducer should be taken into consideration as well. In this regard, several simulation and experimental tests are carried out in this research to model high power ultrasound transducers and systems. During these experiments, high power ultrasound transducers are excited by several excitation signals with different amplitudes and frequencies, using a network analyser, a signal generator, a high power amplifier and a multilevel converter. Also, to analyse the behaviour of the ultrasound system, the voltage ratio of the system is measured in different tests. The voltage across transmitter is measured as an input voltage then divided by the output voltage which is measured across receiver. The results of the transducer characteristics and the ultrasound system behaviour are discussed in chapter 4 and 5 of this thesis. Each piezoelectric transducer has several resonance frequencies in which its impedance has lower magnitude as compared to non-resonance frequencies. Among these resonance frequencies, just at one of those frequencies, the magnitude of the impedance is minimum. This resonance frequency is known as the main resonance frequency of the transducer. To attain higher efficiency and deliver more power to the ultrasound system, the transducer is usually excited at the main resonance frequency. Therefore, it is important to find out this frequency and other resonance frequencies. Hereof, a frequency detection method is proposed in this research which is discussed in chapter 2. An extended electrical model of the ultrasound transducer with multiple resonance frequencies consists of several RLC legs in parallel with a capacitor. Each RLC leg represents one of the resonance frequencies of the ultrasound transducer. At resonance frequency the inductor reactance and capacitor reactance cancel out each other and the resistor of this leg represents power conversion of the system at that frequency. This concept is shown in simulation and test results presented in chapter 4. To excite a high power ultrasound transducer, a high power signal is required. Multilevel converters are usually applied to generate a high power signal but the drawback of this signal is low quality in comparison with a sinusoidal signal. In some applications like ultrasound, it is extensively important to generate a high quality signal. Several control and modulation techniques are introduced in different papers to control the output voltage of the multilevel converters. One of those techniques is harmonic elimination technique. In this technique, switching angles are chosen in such way to reduce harmonic contents in the output side. It is undeniable that increasing the number of the switching angles results in more harmonic reduction. But to have more switching angles, more output voltage levels are required which increase the number of components and cost of the converter. To improve the quality of the output voltage signal with no more components, a new harmonic elimination technique is proposed in this research. Based on this new technique, more variables (DC voltage levels and switching angles) are chosen to eliminate more low order harmonics compared to conventional harmonic elimination techniques. In conventional harmonic elimination method, DC voltage levels are same and only switching angles are calculated to eliminate harmonics. Therefore, the number of eliminated harmonic is limited by the number of switching cycles. In the proposed modulation technique, the switching angles and the DC voltage levels are calculated off-line to eliminate more harmonics. Therefore, the DC voltage levels are not equal and should be regulated. To achieve this aim, a DC/DC converter is applied to adjust the DC link voltages with several capacitors. The effect of the new harmonic elimination technique on the output quality of several single phase multilevel converters is explained in chapter 3 and 6 of this thesis. According to the electrical model of high power ultrasound transducer, this device can be modelled as parallel combinations of RLC legs with a main capacitor. The impedance diagram of the transducer in frequency domain shows it has capacitive characteristics in almost all frequencies. Therefore, using a voltage source converter to drive a high power ultrasound transducer can create significant leakage current through the transducer. It happens due to significant voltage stress (dv/dt) across the transducer. To remedy this problem, LC filters are applied in some applications. For some applications such as ultrasound, using a LC filter can deteriorate the performance of the transducer by changing its characteristics and displacing the resonance frequency of the transducer. For such a case a current source converter could be a suitable choice to overcome this problem. In this regard, a current source converter is implemented and applied to excite the high power ultrasound transducer. To control the output current and voltage, a hysteresis control and unipolar modulation are used respectively. The results of this test are explained in chapter 7.

While moment matching can effectively reduce the dimension of a linear, time-invariant system, it can simultaneously fail to improve the stable time-step for the forward Euler scheme. In the context of a semi-discrete heat equation with spatially smooth forcing, the high frequency modes are virtually insignificant. Eliminating such modes dramatically improves the stable time-step without sacrificing output accuracy. This is accomplished by modal filtration, whose computational cost is relatively palatable when applied following an initial reduction stage by moment matching. A bound on the norm of the difference between the transfer functions of the moment-matched system and its modally-filtered counterpart yields an intelligent choice for the mode of truncation. The dual-stage algorithm disappoints in the context of highly nonnormal semi-discrete convection-diffusion equations. There, moment matching can be ineffective in dimension reduction, precluding a cost-effective modal filtering step.

The problem of pile - soil interaction is examined in the Thesis at hand by means of both theoretical analyses and experimental investigations. Pile foundations in seismically prone areas are subjected to both direct loading, such as axial and lateral forces imposed at their heads, resulting from a phenomenon known as inertial interaction, and indirect loading along their body, such as imposed displacements due to the passage of various types of seismic waves, resulting from a phenomenon known as kinematic interaction. Along this vein, a family of analytical models of the Tajimi type are presented in the framework of linear elastodynamic theory to explore the effects of axial and lateral pile - soil interaction in homogeneous and inhomogeneous soil under static and dynamic (kinematic and inertial) loading. Apart from simplified two-dimensional models of the Baranov - Novak type, few analytical solutions are available to tackle these problems in three dimensions, the majority of which are restricted to the analysis of an elastic half space under static conditions. The proposed models are based on a continuum solution pioneered by Japanese investigators (notably Matuso & Ohara and Tajimi) in the 1960’s. In the realm of this approach the soil is modelled as a continuum, while the pile is conveniently modelled as a rod or a beam by strength-of-materials theory. Displacements and stresses are expressed through Fourier series in terms of the natural modes of the soil medium. Fundamental to the analysis presented in this study is that the influence of horizontal soil displacement on axial pile response and vertical displacement on lateral response, respectively, are negligible. However, their effect on stresses is not negligible which differentiates the proposed models from the classical Tajimi solutions in which the aforementioned displacements are set equal to zero. The above approximations are attractive, as they lead to a straightforward uncoupling of the equations of motions, even in inhomogeneous media, unlike the classical elastodynamic theory where the uncoupling is generally impossible in presence of inhomogeneity. Although approximate, the proposed models are advantageous over available analytical models and rigorous numerical schemes, as they require relative simple computations and provide excellent predictions of pile response at the frequency ranges of interest in earthquake engineering and geotechnics. In addition, they are advantageous over existing simplified analytical approaches of the Winkler type, as they are more accurate, self - standing, free of empirical constants and provide more realistic simulation of the problem. The main advantage over numerical methods (finite and boundary elements) lies in the derivation of the solution in closed form and the elucidation of complex mechanisms related to the dynamic interaction phenomenon, such as radiation damping and wave propagation in in homogeneous media. The main goal of the theoretical effort lies in the derivation of solutions in closed - form for: (i) the static stiffness and the dynamic impedances (dynamic stiffness and damping coefficients) at the pile head, (ii) translational and rotational kinematic response factors (pile head displacement or rotation over free-field response), (iii) actual, depth- dependent, Winkler moduli (spring and damping coefficients), (iv) corresponding average, depth- independent, Winkler moduli to match the pile head stiffness. In addition, simple approximate formulae for Winkler moduli to be used in engineering practice are proposed, to improve the predictions of Winkler models. Pile-to-pile interaction is investigated on the basis of the superposition method for axially loaded piles. Closed-form expressions for attenuation functions are derived to be used individually or in conjunction with more elaborate methods providing more accurate predictions for static and dynamic interaction factors to assess the vertical stiffness of pile groups. New dimensionless frequency ratios controlling pile response are introduced. Finally, new solutions are added in the context of analytical Winkler models for investigating the behaviour of piles under kinematic loading due to vertically-propagating S waves. Emphasis is given on the influence of boundary conditions of the pile. With reference to kinematic pile bending, insight into the physics of the problem is gained through a rigorous superposition scheme involving an infinitely-long pile excited kinematically, and a pile of finite length excited by a concentrated force and a moment at the tip. Contrary to the classical elastodynamic theory where pile response is governed by six dimensionless ratios, in the realm of Winkler theory three only ratios suffice to fully describe the interaction problem, from which the mechanical slenderness and the effective dimensionless frequency are introduced for the first time. The selection of an appropriate value for the Winkler modulus in the accuracy of the kinematic Winkler model is demonstrated. The theoretical results are compared to new experimental data obtained from a series of tests on piles carried out on scaled models performed on the shaking table at University of Bristol Laboratory (BLADE) within the framework of the Seismic Engineering Research Infrastructures (SERIES) program, sponsored by FP7, and contribute in the investigation of pile - soil interaction.
Στην παρούσα διατριβή εξετάζεται το πρόβλημα της αλληλεπίδρασης πασσάλου - εδάφους μέσω συνδυασμένης θεωρητικής ανάλυσης και πειραματικής διερεύνησης. Οι πάσσαλοι, ως μέσο θεμελίωσης σε σεισμογόνες περιοχές, υπόκεινται σε άμεση φόρτιση στην κεφαλή, μέσω δυνάμεων και ροπών, ως αποτέλεσμα του φαινομένου της αδρανειακής αλληλεπίδρασης, αλλά και σε έμμεση φόρτιση σε όλο τους το μήκος, μέσω επιβαλλόμενων εδαφικών μετακινήσεων, ως αποτέλεσμα του φαινομένου της κινηματικής αλληλεπίδρασης. Στην κατεύθυνση αυτή παρουσιάζεται η ανάπτυξη οικογένειας αναλυτικών προσομοιωμάτων τύπου Tajimi, στο πλαίσιο της γραμμικής ελαστοδυναμικής θεωρίας, για τη διερεύνηση της αξονικής και πλευρικής αλληλεπίδρασης πασσάλου - εδάφους σε ομοιογενείς και ανομοιογενείς εδαφικούς σχηματισμούς, υπό στατική και δυναμική φόρτιση κινηματικής και αδρανειακής μορφής. Εκτός από απλοποιημένα διδιάστατα προσομοιώματα τύπου Baranov - Novak, ελάχιστες αναλυτικές λύσεις είναι διαθέσιμες σε τρεις διαστάσεις, οι περισσότερες των οποίων περιορίζονται στην ανάλυση ελαστικού ημίχωρου υπό στατικές συνθήκες. Τα προτεινόμενα προσομοιώματα βασίζονται σε μια πρωτοποριακή λύση συνεχούς μέσου (κατά Matsuo & Ohara και Tajimi) η οποία αναπτύχθηκε στη δεκαετία του 1960, αλλά δεν επεκτάθηκε ουσιαστικά μέχρι την παρούσα εργασία. Στο πλαίσιο αυτής της προσέγγισης το έδαφος προσομοιώνεται ως συνεχές μέσο και ο πάσσαλος ως ράβδος ή δοκός σύμφωνα με τη τεχνική θεωρία της κάμψης (παραδοχή επιπεδότητας των διατομών), ενώ οι μετακινήσεις και οι τάσεις εκφράζονται μέσω αναπτυγμάτων Fourier σε όρους των φυσικών ιδιομορφών του εδαφικού μέσου. Θεμελιώδης παραδοχή της προτεινόμενης ανάλυσης είναι ότι η επιρροή της οριζόντιας εδαφικής μετακίνησης στην αξονική απόκριση του πασσάλου, αλλά και η επιρροή της κατακόρυφης μετακίνησης στην πλευρική απόκριση θεωρούνται αμελητέες, ωστόσο η επίδρασή τους στις τάσεις είναι μη μηδενική, πράγμα που τις διαφοροποιεί από τις κλασσικές λύσεις τύπου Tajimi στις οποίες οι ανωτέρω μετακινήσεις μηδενίζονται. Οι ανωτέρω προσεγγίσεις κρίνονται ως ιδιαίτερα ελκυστικές καθώς οδηγούν στην άμεση απόζευξη των εξισώσεων της κίνησης, ακόμη και σε ανομοιογενή μέσα, αντίθετα με την κλασσική ελαστοδυναμική θεωρία, η απόζευξη είναι γενικώς αδύνατη παρουσία εδαφικής ανομοιογένειας. Παρά τον προσεγγιστικό τους χαρακτήρα, τα προτεινόμενα αναλυτικά προσομοιώματα πλεονεκτούν ως προς διαθέσιμα αναλυτικά προσομοιώματα και αυστηρά αριθμητικά σχήματα, καθώς απαιτούν σχετικά απλούς υπολογισμούς και παρέχουν εξαιρετικές προβλέψεις της απόκρισης του πασσάλου για το εύρος συχνοτήτων που παρουσιάζει ενδιαφέρον στη σεισμική μηχανική και τα γεωτεχνικά. Επιπρόσθετα, υπερτερούν ως προς υφιστάμενες αναλυτικές προσεγγίσεις τύπου Winkler, καθώς είναι ακριβέστερα, αυτόνομα, απαλλαγμένα από εμπειρικές σταθερές και προσφέρουν ρεαλιστικότερη προσομοίωση του προβλήματος. Το κύριο πλεονέκτημα έναντι των αριθμητικών μεθόδων (πεπερασμένα και συνοριακά στοιχεία) έγκειται στην εξαγωγή της λύσης σε κλειστή μορφή και στη διερεύνηση πολύπλοκων φαινομένων που σχετίζονται με την αλληλεπίδραση πασσάλου - εδάφους, όπως αυτό της απόσβεσης ακτινοβολίας και της διάδοσης κυμάτων στο έδαφος - ειδικά παρουσία ανομοιογένειας. Ο κύριος στόχος της θεωρητικής διερεύνησης υλοποιείται με την εξαγωγή λύσεων σε κλειστή μορφή για: (i) τη στατική και δυναμική στιφρότητα και απόσβεση στην κεφαλή του πασσάλου, (ii) τους συντελεστές κινηματικής απόκρισης σε μετάθεση και στροφή, (iii) τους πραγματικούς, συναρτήσει του βάθους, συντελεστές Winkler (συντελεστής στιφρότητας ελατηρίων και συντελεστής απόσβεσης), (iv) τους αντίστοιχους μέσους, ανεξάρτητους από το βάθος, συντελεστές Winkler. Επιπρόσθετα, παρουσιάζονται ακριβέστερες των διαθέσιμων στη βιβλιογραφία απλές προσεγγιστικές σχέσεις για τον υπολογισμό του συντελεστή Winkler με σκοπό τη βελτίωση της ακρίβειας των προσομοιωμάτων Winkler. Διερευνάται η αλληλεπίδραση πασσάλου προς πάσσαλο στην περίπτωση αξονικά φορτισμένων πασσάλων με βάση την αρχή της επαλληλίας. Εξάγονται λύσεις σε κλειστή μορφή για τις συναρτήσεις εξασθένισης ώστε να χρησιμοποιηθούν αυτόνομα ή σε συνδυασμό με πιο εκλεπτυσμένες λύσεις δίνοντας με στόχο ακριβέστερες προβλέψεις για τους συντελεστές αλληλεπίδρασης, οδηγώντας έτσι σε πιο ρεαλιστικές εκτιμήσεις της κατακόρυφης στιφρότητας ομάδας πασσάλων. Εισάγονται νέοι αδιάστατοι λόγοι συχνοτήτων που καθορίζουν την απόκριση του πασσάλου. Τέλος, παρουσιάζονται νέες λύσεις σε αναλυτικά προσομοιώματα Winkler για τη διερεύνηση της συμπεριφοράς πασσάλων υποκείμενων σε φόρτιση λόγω της κατακόρυφης διάδοσης διατμητικών κυμάτων στο έδαφος, με έμφαση στην επίδραση των οριακών συνθηκών του προβλήματος. Σε αντίθεση με την κλασσική ελαστοδυναμική θεωρία που η απόκριση του πασσάλου καθορίζεται από έξι αδιάστατους λόγους, στο πλαίσιο της θεωρίας Winkler επαρκούν μόνο τρεις για την πλήρη περιγραφή της αλληλεπίδρασης πασσάλου - εδάφους, εκ των οποίων η μηχανική λυγηρότητα και η ενεργός αδιάστατη συχνότητα παρουσιάζονται για πρώτη φορά. Καταδεικνύεται η σημασία επιλογής της κατάλληλης τιμής του συντελεστή Winkler στην ακρίβεια των εν λόγω προσομοιωμάτων. Προτείνεται σύστημα υπέρθεσης που αποτελείται από ένα απειρομήκη πάσσαλο που διεγείρεται κινηματικά και έναν πάσσαλο πεπερασμένου μήκους που υπόκειται σε αδρανειακή φόρτιση για τη διαλεύκανση της λειτουργίας του θεμελιώδους μηχανισμού που καθορίζει την κινηματική κάμψη του πασσάλου. Τα θεωρητικά αποτελέσματα συγκρίνονται με νέα πειραματικά δεδομένα από σειρά δοκιμών σε διάταξη πασσάλων υπό κλίμακα που εκτελέστηκαν στο σεισμικό προσομοιωτή του Πανεπιστήμιου του Bristol στο πλαίσιο του Ευρωπαϊκού Προγράμματος SERIES, το οποίο χρηματοδοτήθηκε από το κοινοτικό πλαίσιο FP7 που συμβάλλουν στην περαιτέρω διερεύνηση του φαινομένου της κινηματικής αλληλεπίδρασης εδάφους - πασσάλου.