Dissertations / Theses on the topic 'Equations of motion'

The work presented in this thesis is devoted to the study of geodesic motion in the context of General Relativity. The motion of a single test particle is governed by the geodesic equations of the given space-time, nevertheless one can be interested in the collective behavior of a family (congruence) of test particles, whose dynamics is controlled by the Raychaudhuri equations. In this thesis, both the aspects have been considered, with great interest in the latter issue. Geometric quantities appear in these evolution equations, therefore, it goes without saying that the features of a given space-time must necessarily arise. In this way, through the study of these quantities, one is able to analyze the given space-time. In the first part of this dissertation, we study the relation between geodesic motion and gravity. In fact, the geodesic equations are a useful tool for detecting a gravitational field. While, in the second part, after the derivation of Raychaudhuri equations, we focus on their applications to cosmology. Using these equations, as we mentioned above, one can show how geometric quantities linked to the given space-time, like expansion, shear and twist parameters govern the focusing or de-focusing of geodesic congruences. Physical requirements on matter stress-energy (i.e., positivity of energy density in any frame of reference), lead to the various energy conditions, which must hold, at least in a classical context. Therefore, under these suitable conditions, the focusing of a geodesics "bundle", in the FLRW metric, bring us to the idea of an initial (big bang) singularity in the model of a homogeneous isotropic universe. The geodesic focusing theorem derived from both, the Raychaudhuri equations and the energy conditions acts as an important tool in understanding the Hawking-Penrose singularity theorems.

This thesis presents an approximate solution of second order relative motion equations. The equations of motion for a Keplerian orbit in spherical coordinates are expanded in Taylor series form using reference conditions consistent with that of a circular orbit. Only terms that are linear or quadratic in state variables are kept in the expansion. A perturbation method is employed to obtain an approximate solution of the resulting nonlinear differential equations. This new solution is compared with the previously known solution of the linear case to show improvement, and with numerical integration of the quadratic differential equation to understand the error incurred by the approximation. In all cases, the comparison is made by computing the difference of the approximate state (analytical or numerical) from numerical integration of the full nonlinear Keplerian equations of motion.
Master of Science

This work is a study of the relationship between Brownian motion and elementary, linear partial differential equations. In the text, I have shown that Brownian motion is a Markov process, and that Brownian motion itself, and certain Stochastic processes involving Brownian motion are also martingales. In particular, Dynkin's formula for Brownian motion was shown. Using Dynkin's formula and Brownian motion, I then constructed solutions for the classical Dirichlet problem and the heat equation, given by Δu=0 and ut= 1/2Δu+g, respectively. I have shown that the bounded solution is unique if Brownian motion will always exit the domain of the function once it has started at a point in the domain. The heat equation also has a unique bounded solution.

Thesis (Ph. D.)--Missouri University of Science and Technology, 2008.
Vita. The entire thesis text is included in file. Title from title screen of thesis/dissertation PDF file (viewed August 21, 2008) Includes bibliographical references (p. 124-131).

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2006.
Includes bibliographical references (p. 183-197).
In this thesis we will explore three approaches to aspects of the fundamental structure of string theory. We first provide a brief review of perturbative string theory, and briefly discuss how each of the three topics to be discussed in the body of this thesis depart from this starting point. We then study the open string one-loop tadpole diagram in Witten cubic open string field theory. We compute this diagram both analytically and numerically and study the divergences arising from the collective behavior of open string fields in the short-distance region of the diagram. We demonstrate that this region of the diagram encodes information about the linearized Einstein equation describing the shift in the closed string fields in reaction to the D-brane supporting the open strings. We also show that the manner in which this information is encoded is somewhat singular, and comment on the implications for the quantum consistency of open bosonic string field theory. We next compute the closed string radiation from a decaying D-brane in type II string theory. The calculation is made possible by noting that the integrals involved in the requisite disk one-point functions reduce to integrals over the group manifold of a product of unitary groups.
(cont.) We find that the total number and energy of strings radiated during the decay process diverges for D-branes of small enough dimension, in precise analogy to the bosonic case. Finally, we investigate a simple class of type II string compactifications which incorporate nongeometric "fluxes" in addition to "geometric flux" and the usual H-field and R-R fluxes. We develop T-duality rules for NS-NS geometric and nongeometric fluxes, which we use to construct a superpotential for the dimensionally reduced four-dimensional theory. The resulting structure is invariant under T-duality, so that the distribution of vacua in the IIA and IIB theories is identical when nongeometric fluxes are included.
by Jessie (Julia) Shelton.
Ph.D.

Thesis (M.S. in Mechanical Engineering) Naval Postgraduate School, Sept. 2001.
Thesis advisor, Papoulias, Fotis A. "September 2001." Includes bibliographical references (p. 47). Also Available in print.

The primary purpose of the work undertaken in this thesis is to investigate soliton scattering in the non linear CP² sigma model. This has two spatial and one temporal dimension. The vector fields used to represent the model have three components and hence there exists a global SU(3) symmetry. The effects of adding an Hopflike term to the basic lagrangian is considered. A review of the model is given in chapter I. The second chapter discusses Noether's theorem which states that each symmetry of the lagrangian has associated with it a conserved charge. In the third chapter, the eight charges relating to the internal symmetry are calculated. Explanations are provided for the results calculated during the numerical simulations. The results for the CP¹ model are also discussed. In the fourth chapter, these charges are used to predict the qualitative behaviour of the solitons. It will provide an explanation for the effect of the coefficient of the hopflike term on the scattering. The single soliton ansatz is also investigated. In the penultimate chapter, an alternative approach is used. This involves looking for the closest static approximation to the evolved solution. It is able to predict the trajectory for pure CP² and some confirmation is provided for the ansatz used in the full lagrangian. The last chapter summarises the results. It also provides some suggestions for further work.

2006/2007
Lo scopo di questa tesi di dottorato è la stima del moto forte del suolo nell’area delle Alpi Sud-Orientali. A tal fine sono state proposte delle relazioni empiriche che stimano i parametri del moto in funzione della magnitudo, della distanza dall’epicentro e della classificazione geologica del suolo; successivamente tali relazioni sono state usate per calibrare il software ShakeMaps con il fine di generare in tempo reale le mappe di scuotimento del terreno per la regione Friuli-Venezia Giulia. Le GMPEs (Ground Motion Predictive Equations) per PGA, PGV e SA sono state calcolate nell’area delle Alpi Sud-Orientali utilizzando registrazioni del moto forte del terreno. Sono state selezionate 900 forme d’onde accelerometriche filtrate tra 0.1 Hz e 30 Hz; la distanza epicentrale varia tra 1 km a 100 km, mentre la magnitudo locale, opportunamente calibrata confrontando diversi cataloghi, varia in un intervallo relativamente ampio (3.0 <= ML <= 6.3). Sono stati testati diversi modelli di attenuazione e il miglior risultato è stato individuato utilizzando specifici criteri di valutazione derivanti da considerazioni di carattere statistico (valore di R2, uso dell’ANOVA test, analisi dei residui). I coefficienti del modello finale sono stati determinati oltre che da ML, dalla distanza epicentrale e dagli effetti dovuti al sito, anche dalla saturazione della magnitudo, dalla correlazione tra magnitudo e distanza e dagli effetti di “near-source”. I coefficienti delle GMPEs sono stati calcolati per le componenti verticali ed orizzontali (rappresentata sia con la componente maggiore sia con la somma vettoriale delle due componenti); la tecnica dell’analisi dei gruppi ha permesso di ridurre l’incertezza finale sulle relazioni empiriche. Il confronto con i risultati ottenuti precedentemente evidenzia come le relazioni ottenute in questa tesi abbiano una maggiore attenuazione a basse magnitudo e a grandi distanze; risultati analoghi sono stati ottenuti per le relazioni ricavate dai dati registrati in tutta l’Italia Settentrionale. L’evoluzione recente delle reti sismiche rende oggi disponibile una grossa mole di dati acquisiti in tempo reale, per cui risulta fattibile stimare velocemente lo scuotimento del terreno tramite mappe; il software “ShakeMap” è stato adattato alle Alpi Sud-Orientali implementato allo scopo di ottenere una stabile interfaccia con il sistema di acquisizione dati “Antelope” che garantisca l’estrazione dei parametri del moto dalle forme d’onda e la creazione delle mappe di scuotimento entro 5 minuti dall’evento sismico. Questa procedura richiede una fitta e uniforme distribuzione spaziale degli strumenti di registrazione sul territorio e una classificazione geologica del suolo fatta usando le velocita’ medie, Vs30, dei primi 30m del mezzo immediatamente sotto gli strumenti. La classificazione geologica del suolo prevede la suddivisione in tre categorie (suolo rigido, suolo addensato e suolo soffice) mentre i coefficienti di amplificazione sono stati calcolati usando le relazioni proposte da Borcherdt (1994). Le relative mappe vanno calcolate usando le GMPEs e le relazioni empiriche che legano il moto del terreno all’intensità macrosismica, basate ambedue su dati registrati nella regione alpina. Le GMPEs discusse in precedenza sono state inserite nel software “ShakeMap” per la produzione delle mappe di scuotimento in tempo reale e quasi-reale nell’Italia Nord-Orientale. Per valutare l’effetto della densità di stazioni sulle mappe di scuotimento sono stati calcolati dei sismogrammi sintetici relativi al terremoto di Bovec 2004 variando il passo di griglia e la geometria dei ricevitori. I risultati ottenuti indicano come una distribuzione fitta e uniforme di strumenti sul territorio e una scelta accurata delle dimensioni della griglia dei ricevitori siano cruciali per calibrare le mappe di scuotimento in una ben determinata area geografica. Le mappe di scuotimento del suolo sono state generate per otto terremoti avvenuti nell’area considerata negli ultimi 30 anni; inoltre per gli eventi del Friuli 1976 e Bovec 1998 è stato utilizzato il modello di faglia finita con i parametri di sorgente stimati in precedenti studi. La validazione del modello è stata fatta calcolando il misfit tra le intensità macrosismiche osservate (catalogo DBMI04) e quelle “strumentali” che sono state ottenute dai sismogrammi sintetici tramite relazioni empiriche tra moto del suolo ed intensità. L’analisi è stata fatta per i terremoti del Cansiglio (1936), del Friuli (1976) e di Bovec (1998). I sismogrammi sintetici sono stati calcolati ad una frequenza massima di 10 Hz applicando il modello della riflettività; i parametri del moto sono stati estratti dai segnali sintetici calcolati nelle attuali stazioni di registrazione e successivamente sono state generate le mappe di scuotimento. L’intensità macrosismica “strumentale” è stata ricavata applicando diverse relazioni; il minor misfit è stato ottenuto usando le relazioni proposte da Kästli and Fäh (2006) per tutti e tre i terremoti considerati, il che sembra validare il nostro modello di Shake Maps.
The aim of this PhD thesis is to estimate ground motions in the South-Eastern Alps area. For this purpose we purposed empirical relationships that estimate the ground motion parameters as function of the magnitude, the epicentral distance and the soil geological characterization. Later on these relationships are used to calibrate the ShakeMaps software to generate ground motion shake maps in real time for the Friuli-Venezia Giulia region. The GMPEs (Ground Motion Predictive Equations) for PGA, PGV and SA are computed in the South-Eastern Alps area using strong motion observations. 900 accelerometric waveforms are selected and filtered between 0.1 Hz and 30 Hz; the epicentral distance varies from 1 km to 100 km, while the local magnitude, calibrated by comparison with various catalogues, varies in a relatively wide range (3.0 <= ML <= 6.3). Various attenuation models are tested and the best result is selected by the adoption of specific evaluation criteria derived from statistical considerations (R2 value, ANOVA test, residuals analysis). The coefficients of the final model are determined from ML, the epicentral distance, the site effects, the magnitude saturation, the correlation between the distance and the magnitude and the near-source effects. The coefficients of the GMPEs are computed from vertical and horizontal components (the latter represented both as the largest horizontal component and the vectorial addiction); the cluster analysis reduces the final uncertainties on the empirical relations. The comparison with the previous results evidences that the obtained relationships are characterized by a strong attenuation at low magnitudes and large distances. Similar results are obtained for the relationships derived from data recorded all over Northern Italy. The recent evolution of the seismic networks provides a large number of data, available in real time, so it is possible to quickly estimate shake maps. The “ShakeMap” software has been adapted to the South-Eastern Alps region and implemented to obtain a stable interface with the “Antelope” acquisition system in order to extract the ground motion parameters from the waveforms and the generation of the shake maps within 5 minutes from the earthquake occurrence. This procedure requires a dense and uniform spatial distribution of the recording instruments in the field and a geological classification of the soil derived from the average velocities of the S waves in the first 30m below the recording instruments (Vs30). In the geological classification the soil is divided into three classes (bedrock, stiff soil and soft soil), and the amplification coefficients are computed using the relationships proposed by Borcherdt (1994). The related maps are generated using the GMPEs and the empirical relations that predict the macroseismic intensity from the ground motion, both derived from data observed in the Alpine region. The GMPEs that are obtained in this thesis are inserted in the ShakeMap software to generate shake maps in real time or quasi real time in North-Eastern Italy. To evaluate the effects of the station coverage on the shake maps, synthetic seismograms are computed for the Bovec 2004 earthquake by varying the grid size and the network geometry. The results indicate that a dense and uniform spatial distribution in the field and a careful choice of the grid size are crucial to calibrate the shake maps in a given geographical area. The shake maps are generated for eight earthquakes occurred in the studied area in the last 30 years. Furthermore, the finite-fault model is utilized for the seismic events of the Friuli 1976 and Bovec 1998 selecting the source parameters proposed in previous studies. The model validation is done computing the misfit value between the observed macroseismic data (DBMI04 catalogue) and the “instrumental” intensities that are obtained from the synthetic seismograms using empirical relationships between the ground motion and intensity. This analysis has been done for the earthquakes of Cansiglio (1936), Friuli (1976) and Bovec (1998). The synthetic seismograms are calculated for an upper cutoff frequency of 10 Hz applying the reflectivity model. The ground motion parameters are extracted from synthetic signals computed at the presently operating seismic stations and the shake maps are generated. The macroseismic intensity is derived from various relationships; the lowest misfit is obtained using the relation proposed by Kästli and Fäh (2006) for all considered seismic events and this seem to validate our Shake Maps model.
XX Ciclo
1978

Within the exact framework established recently, which is a successful marriage between the time dependent density functional theory for open electronic system and quantum dissipation theory formulated in the hierarchical equations of motion, an entirely new scheme is proposed in this thesis to simulate the time-dependent quantum transport in nano-devices at both zero and finite temperature equally without relying on the pole structure of the Fermi distribution function. Neither does it depend on any non-unique parametrization of the line-width matrix, hence, this new practical approach can be integrated with the first principles simulations seamlessly. Beyond the exact framework, a reliable method which works under the Wide- Band-Limit approximation at zero temperature is also developed. At the price of loss of some non-Markovian memory effects on the dynamics, a set of equations of motion which terminates at the first tier instead of the second tier is obtained. Benefiting from the latest advancement of numerical analysis, a hybrid fourth-order Runge-Kutta algorithm is proposed to solve this set of equations of motion which comprises stiff ones. Based on this result, an alternative scheme is considered to deal with the same approximation at finite temperature. As an illustration of these new approaches, the transient current of the one dimensional tight-binding periodical chain with and without a single impurity, driven by some time alternating and/or static bias voltages, are investigated. The influence of temperature and switch-on rate of bias voltage is exemplified. Particularly, in the one dimensional tight-binding chain with a single impurity which breaks its perfect periodicity, an asymmetry between the left and right transient current is found. Comparison between the results under the Wide-Band-Limit approximation and those with the exact description is carried out.
published_or_final_version
Chemistry
Doctoral
Doctor of Philosophy

Helicopter agility in nap-of-the-earth flight is widely recognised to be of great importance. Despite this. a general method of quantifying agility does not exist. All previous attempts to quantify agility have been restricted either to flight tests or to simple kinematic modelling - both with obvious disadvantages. A method of quantifying helicopter inherent agility. the agility of the configuration independent of the pilot. utilising inverse solutions of the equations of motion has been developed. A value for the inherent agility of a helicopter is given by studying its performance over a series of standard manoeuvres. The manoeuvres used represent typical tasks undertaken by the configuration under study. The combination of these tasks represent the helicopter's operational role. The helicopter's performance over these standard manoeuvres is found by using an inverse solution of the equations of motion - calculation of the control. and resulting state. time histories needed to fly a given flight path. A six degrees of freedom non-linear mathematical model is used to simulate single main and tail rotor helicopter flight dynamics. The helicopter's performance over each manoeuvre is rated by a quadratic performance function of the state and control variables. The performance function is weighted in such a manner as to penalise undesirably large displacements in the state and control variables of particular importance to that manoeuvre (e.g. large nose down attitude changes in accelerated flight are heavily penalised). An Agility Rating is awarded to a helicopter on the basis of its performance over a wide range of similar manoeuvres. a measure of total inherent agility being a function of the agility ratings for all the manoeuvres relevant to the helicopter's role. The method is illustrated by applying it to two agility studies. Firstly. it is used to show how an optimum tailplane area can be calculated for manoeuvres in the longitudinal plane. Then an "Advanced Rotor Helicopter" is compared with a contemporary battlefield helicopter.

This study presents the node annexation method for modeling kinematic joints between rigid and flexible bodies of rigid-flexible multibody systems. Each node of a flexible body is assumed to have lumped mass and three translational degrees of freedom, resulting in a diagonal mass matrix. Based on the node annexation method, both the nodal- and the modal-coordinate formulations for rigid-flexible multibody dynamics are developed. Conventionally rigid-to-flexible-body joints are treated as kinematic constraints using the Lagrange multiplier method. The formulations based on kinematic constraint method yield coupled equations of motion which have the difficulties associated with modal truncation. On the other hand, the node annexation method transfers the inertia and force effect of connected nodes of a flexible body to the connected rigid body. The mass matrix of the resultant equations of motion consists of two different kind of sub-matrices: one is rigid-body sub-system matrix containing the inertia of both rigid bodies and connected nodes of the flexible body and another is flexible-body sub-system matrix containing the inertia of free nodes of the flexible body. Since there is no off-diagonal terms coupling the sub-matrices, the node annexation method allows the division of the equations of motion into smaller sub-system equations. The node annexation method not only provides computational efficiency but also fundamentally eliminates any kinematic error at rigid-to-flexible-body joints. In addition, the node annexation method preserves the uncoupled nature of modal coordinates, allowing a mathematically justified modal truncation. Computer simulations are performed using a vehicle model with a flexible car body. The simulation results show computational advantage over the kinematic constraint method.

We consider a stochastic functional differential equation with infinite memory driven by a fractional Brownian motion with Hurst parameter $H>1/2$. We prove an existence and uniqueness result of the solution to the stochastic differential equation. We investigate the dependence of the solution on the initial condition and the existence of finite moments of the solution. Furthermore we generalize these results to wider classes of stochastic differential equations. The stochastic integral with respect to fractional Brownian motion is defined as a pathwise Riemann-Stieltjes integral.

Thesis (PhD (Mathematical Sciences. Applied Mathematics))--University of Stellenbosch, 2008.
This dissertation contains a detailed report on the results of a research project on the behaviour of a dynamical system consisting of a hoop to which a heavy particle is fixed at the rim. This loaded hoop rolls on a rough surface while remaining in the vertical plane. The motion of the hoop consists of various, possibly alternating, phases consisting of rolling without slipping, spinning or skidding motion and in some cases ends by hopping off the surface. A general mathematical model is developed, consisting of a system of second order ordinary differential equations, one for each of the three degrees of freedom. Analytic solutions are obtained in some cases; otherwise numerical solutions are used. Three specific applications of the general model are dealt with. In the first application the problem of massless hoops is investigated. The main emphasis is on the somewhat controversial question of what happens after the normal reaction becomes zero in a position where the particle is still moving downwards. A new result shows that the hoop can continue to move horizontally in a motion defined as skimming. The second application deals with rigid hoops and a large number of detailed results are presented. Classification schemes for the different types of behaviour are introduced and summarised in the form of phase diagrams. Some emphasis is placed on the rather amazing number of different patterns of motion that can be obtained by varying the parameters. In the third application two elastic models are analysed, with the primary purpose of explaining one aspect of the reported behaviour of experimental hoops, namely hopping while the particle is moving downwards. A chapter on experimental models rounds off the project.

Thesis (Ph. D.) University of Missouri-Columbia, 2006.
The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file (viewed on August 1, 2007) Includes bibliographical references.

The KiK-net ground motion database is used to develop ground motion prediction equations for Arias Intensity (Ia), 5-95% Significant Duration (Ds5-95), and 5-75% Significant Duration (Ds5-75). Relationships are developed both for shallow crustal earthquakes and subduction zone earthquakes (hypocentral depth less than 45 km). The models developed consider site amplification using VS30 and the depth to a layer with VS=800 m/s (h800). We observe that the site effect for is magnitude dependent. For Ds5-95 and Ds5-75, we also observe strong magnitude dependency in distance attenuation. We compare the results with previous GMPEs for Japanese earthquakes and observe that the relationships are similar. The results of this study also allow a comparison between earthquakes in shallow-crustal regions, and subduction regions. This comparison shows that Arias Intensity has similar magnitude and distance scaling between both regions and generally Arias Intensity of shallow crustal motions are higher than subduction motions. On the other hand, the duration of shallow crustal motions are longer than subduction earthquakes except for records with large distance and small magnitude causative earthquakes. Because small shallow crustal events saturate with distance, ground motions with large distances and small magnitudes have shorter duration for shallow crustal events than subduction earthquakes.

We present a new pathwise approximation method for stochastic differential equations driven by Brownian motion which does not require simulation of the stochastic integrals. The method is developed to give Wasserstein bounds O(h3/2) and O(h2) which are better than the Euler and Milstein strong error rates O(√h) and O(h) respectively, where h is the step-size. It assumes nondegeneracy of the diffusion matrix. We have used the Taylor expansion but generate an approximation to the expansion as a whole rather than generating individual terms. We replace the iterated stochastic integrals in the method by random variables with the same moments conditional on the linear term. We use a version of perturbation method and a technique from optimal transport theory to find a coupling which gives a good approximation in Lp sense. This new method is a Runge-Kutta method or so-called derivative-free method. We have implemented this new method in MATLAB. The performance of the method has been studied for degenerate matrices. We have given the details of proof for order h3/2 and the outline of the proof for order h2.

Thesis (Ph. D.)--University of California, San Diego, 2008.
Title from first page of PDF file (viewed September 22, 2008). Available via ProQuest Digital Dissertations. Vita. Includes bibliographical references (p. 163-169).

The transient dynamic analysis of constrained mechanical systems may require the solution of a mixed set of algebraic and differential equations of motion. The usual formulation of these equations is expressed in terms of the accelerations of the system components. A canonical form of the equations of motion in terms of the system velocities and the time derivative of the system momenta may be used instead. This is a natural form of the equations in which all the state variables are explicitly expressed, and have the same physical importance. The numerical solution obtained from the canonical equations shows more accuracy and stability, specifically for systems with large and fluctuating forces. For the mechanical systems that undergo an impact, the usual numerical solution of the equations of motion is not valid. Two different methods of analysis of impact problems are presented. In one method, the variations of the impulsive force during the contact period are directly added to the vector of forces in the canonical equations of motion. In the second method, based on the assumption of instantaneous nature of impact, a set of momentum balance-impulse equations is derived by explicitly integrating the canonical equations. These equations are solved at the time of impact for the jump in the system momenta right after impact. Necessary parameters are evaluated for the performance of the two methods of analysis. These parameters include the maximum relative indentation, the maximum contact force, and the coefficient of restitution. The parameters are determined for the collision between two bodies in a system with any general geometric or material properties. The influence of friction modeling in the magnitude and the direction of the total force at the contact surfaces is discussed. The dynamics of a vehicle collision is studied in order to illustrate the efficiency of obtaining a solution to the canonical equations, the simplicity of solving the momentum balance-impulse equations.

A relatively general formulation for the governing equations of motion, applicable to a large class of flexible multibody systems, is developed using a concise matrix format. The model considered consists of a number of arbitrarily connected flexible deployable members forming branched and closed loop configurations. Joints between bodies are permitted up to six degrees of freedom in translation and rotation. To be effective, the matrix-Lagrangian formulation necessitates development of the kinetic energy expression in a quadratic form in terms of the system velocities. The mass matrix associated with such a quadratic form is known for simple systems such as a collection of point masses, a group of connected rigid bodies, and a discretized flexible structure. However, for a multibody system, where the contributing forces arise from system's translation, rotation, elasticity, deployment, and their interactions, such an expression is not available. To fill this gap, multibody kinematics is developed in terms of the elements of the geometry matrix, which uniquely describes the configuration of branched systems. The characteristic dynamical quantities, i.e., elements of the mass matrix, are identified and the formulation is approached in an increasing order of complexity. The concept of specified and generalized coordinates together with established procedures of analytical dynamics lead to characteristic quantities ( Lagrangian, Hamiltonian, etc. ) and finally result in governing equations of motion which are new to the multibody dynamics. To account for flexibility in a consistent manner, a second-degree nonlinear displacement field is permitted. Alternatively, a linear displacement field can be used if the nonlinear terms up to the fourth-degree are preserved in the strain energy. An algorithm for calculating the stiffness matrix of a flexible element is developed, where terms up to the third-degree of nonlinearity in displacement are retained. Application of this versatile formulation is illustrated through a set of examples of contemporary interest. They pertain to a spacecraft comprising of a central rigid body with attached flexible appendages. The configuration corresponds to a large class of present and planned communication satellites. It can also represent the Space Shuttle based deployment of beam and plate type appendages aimed at scientific experiments or construction of the proposed Space Station. The system static equilibrium and stability are discussed. A computer code is developed and specialized to the specific cases in hand. Typical results of an extensive parametric study are presented for two particular situations : (i) the Space Shuttle based deployment of a beam or a plate type structural member; (ii) the configuration similar to the Waves In Space Plasma (WISP) experiment jointly proposed by Canada and the U.S.A. The problems are analyzed systematically, through progressive introduction of complexity, to help appreciate interactions between librational dynamics, flexibility, deployment, inertia parameters, orbit eccentricity, initial conditions, appendage orientation, etc. The information is fundamental to the missions concerned and essential to help develop appropriate control strategies.
Applied Science, Faculty of
Mechanical Engineering, Department of
Graduate

In case of the PDE's the concept of solving by separation of variables

has a well defined meaning. One seeks a solution in a form of a

product or sum and tries to build the general solution out of these

particular solutions. There are also known systems of second order

ODE's describing potential motions and certain rigid bodies that are

considered to be separable. However, in those cases, the concept of

separation of variables is more elusive; no general definition is

given.

In this thesis we study how these systems of equations separate and find that their separation usually can be reduced to sequential separation of single first order ODE´s. However, it appears that other mechanisms of separability are possible.

Mestrado em Mathematical Finance
O objectivo desta dissertação é o de generalizar um resultado sobre a estabilidade exponencial de soluções triviais de equações diferenciais estocásticas com movimento Browniano fraccionário, desenvolvido por Garrido-Atienza et al., para soluções não-triviais. São apresentadas noções de cálculo fraccionário, assim como a definição e principias propriedades do movimento Browniano fraccionário. De seguida, um framework para equações diferenciais estocásticas com movimento Browniano fraccionário é definido juntamente com resultados de existência e unicidade de soluções. O resultado, original desta dissertação, é aplicado a um modelo Vasicek fraccionário de taxas de juro.
This dissertation aims to generalize a result on the exponential stability of trivial solutions of stochastic differential equations driven by the fractional Brownian motion by Garrido-Atienza et al. to non-trivial solutions in the scalar case. Notions on fractional calculus are presented, as well as the definition and main properties of the fractional Brownian motion. Subsequently the framework for SDEs driven by fractional Brownian motion with a pathwise approach is characterized along with some existence and uniqueness results. The result on stability is then applied to the fractional Vasicek model for interest rates.
info:eu-repo/semantics/publishedVersion

A stochastic differential equationdriven by a Brownian motion where the dispersion is determined by a parameter is considered. The parameter undergoes a change at a certain time point. Estimates of the time change point and the parameter, before and after that time, is considered.The estimates were presented in Lacus 2008. Two cases are considered: (1) the drift is known, (2) the drift is unknown and the dispersion space-independent. Applications to Dow-Jones index 1971-1974  and Goldmann-Sachs closings 2005-- May 2009 are given.