Classical mechanics is the branch of physics concerned with describing the motion of bodies. The subject is based on three simple axioms relating forces and movement. These axioms were first postulated by Newton in the 17th century and are known as his three laws of motion. Lagrangian mechanics is a restatement of the Newtonian formulation. It deals with energy quantities and paths-of-motion instead of forces. This often makes it simpler to use when working with non-trivial mechanical systems. In this thesis, we use the Lagrangian method to model two such systems; A rotating torus and a variant of the classical double pendulum. It soon becomes clear that the complexity of these systems make them difficult to attack by hand. For this reason, we take a computer-based approach. We use a software-package called Sophia which is a plug-in to the computer algebra system Maple. Sofia was developed at the Department of Mechanics at KTH for the specific purpose of modeling mechanical problems using Lagrange’s method. We demonstrate that this method can be successfully applied to the analysis of motion of complex mechanical systems. The complete equations of motion are derived in a symbolic form and then integrated numerically. The motion of the system is finally visualized by means of 3D graphics software Blender.

Although ships have been used for thousands of years, modeling the dynamics of marine craft has historically been restricted by the complex nature of the hydrodynamics. The principal challenge is that the vehicle motion is coupled to the ambient fluid motion, effectively requiring one to solve an infinite dimensional set of equations to predict the hydrodynamic forces and moments acting on a marine vehicle. Additional challenges arise in parametric modeling, where one approximates the fluid behavior using reduced-order ordinary differential equations. Parametric models are typically required for model-based state estimation and feedback control design, while also supporting other applications including vehicle design and submarine operator training. In this dissertation, Lagrangian mechanics is used to derive nonlinear, parametric motion models for marine craft operating in the presence of a free surface. In Lagrangian mechanics, one constructs the equations of motion for a dynamic system using a system Lagrangian, a scalar energy-like function canonically defined as the system kinetic energy minus the system potential energies. The Lagrangian functions are identified under ideal flow assumptions and are used to derive two sets of equations. The first set of equations neglects hydrodynamic forces due to exogenous fluid motions and may be interpreted as a nonlinear calm water maneuvering model. The second set of equations incorporates effects due to exogenous fluid motion, and may be interpreted as a nonlinear, unified maneuvering and seakeeping model. Having identified the state- and time-dependent model parameters, one may use these models to rapidly simulate surface-affected marine craft maneuvers, enabling model-based control design and state estimation algorithms. / Ph. D. / Although ships have been used for thousands of years, modeling the dynamics of marine craft has historically been restricted by the complex nature of the hydrodynamics. The principal challenge is that the vehicle motion is coupled to the ambient fluid motion, effectively requiring one to solve an infinite dimensional set of equations to predict the hydrodynamic forces and moments acting on a marine vehicle. Additional challenges arise in parametric modeling, where one approximates the fluid behavior using reduced-order ordinary differential equations. Parametric models are typically required for model-based state estimation and feedback control design, while also supporting other applications including vehicle design and submarine operator training. In this dissertation, Lagrangian mechanics is used to derive nonlinear, parametric motion models for marine craft operating in the presence of a free surface. In Lagrangian mechanics, one constructs the equations of motion for a dynamic system using a system Lagrangian, a scalar energy-like function canonically defined as the system kinetic energy minus the system potential energies. The Lagrangian functions are identified under ideal flow assumptions and are used to derive two sets of equations. The first set of equations neglects hydrodynamic forces due to exogenous fluid motions and may be interpreted as a nonlinear calm water maneuvering model. The second set of equations incorporates effects due to exogenous fluid motion, and may be interpreted as a nonlinear, unified maneuvering and seakeeping model. Having identified the state- and time-dependent model parameters, one may use these models to rapidly simulate surface-affected marine craft maneuvers, enabling model-based control design and state estimation algorithms.

Transformadores de potência, como produtos não seriados, estão sujeitos a uma série de possíveis desvios de processo. As divergências que transformadores podem apresentar em relação ao seu projeto original podem ter consequências em sua capacidade de suportar carregamentos mecânicos que lhe serão aplicados. Seu transporte da fábrica às subestações elétricas é um dos principais eventos durante o qual suas estruturas são carregadas. Este trabalho foca em descrever e analisar um transformador de potência genérico em um modelo bidimensional, sob o ponto de vista da Mecânica Analítica e da análise de variabilidades com o objetivo de identificar parâmetros importantes à integridade estrutural do equipamento, especialmente aqueles relacionados ao modo de falha de destacamento das chapas do núcleo nos step laps. O modelo numérico do transformador consiste na representação de corpos rígidos conectados entre si por elementos de rigidez e amortecimento, tendo o conjunto do núcleo ferromagnético particularmente mais detalhado de modo a descrever o atrito existente em suas junções. A avaliação dos esforços nesses locais é feita por meio do método dos multiplicadores de Lagrange, ao considerar o atrito como uma restrição ao movimento, sendo que as forças e momentos limites para que ocorra o deslizamento planar das chapas do núcleo são determinados utilizando-se Newton-Raphson ou outras técnicas de otimização. / Power transformers, as non serialized products, are subject to a number of process deviations. The divergences that transformers present from their original design may have consequences on their ability to withstand mechanical loads which will be applied on them. Their transportation from workshop to power plants is one of the main events their structures are loaded. This work focuses on describing and analyzing a generic power transformer in a two-dimensional model, under the sight of Analytical Mechanics and Structural Reliability in order to identify important parameters related to their structural integrity, especially those concerning the failure mode of dismantling of the ferromagnetic core in the step-lap joints. The numerical model of the transformer consists of a representation of rigid bodies, connected to each other by stiness and damping elements, having the assembly of the ferromagnetic core more detailed in order to describe the friction present on their step-lap joint. The evaluation of friction in these locations is made using the Lagrange Multipliers Method, by considering it as a movement restriction, and the limit values of forces and moments to which the planar sliding of the core sheets occurs, damaging the behavior of the electric machine, are determined using Newton-Raphson and Optimization Methods.

This report provides an introduction to geometric mechanics, which seeks to model the behavior of physical mechanical systems using differential geometric objects. In addition to its elegance as a method of representation, this formulation also admits the application of powerful analytical techniques from geometry as an aid to understanding these systems. In particular, it reveals the fundamental role that symplectic geometry plays in mechanics (something which is not at all obvious from the traditional Newtonian formulation), and in the case of systems exhibiting symmetry, leads to an elucidation of conservation and reduction laws which can be used to simplify the analysis of these systems. The contribution here is primarily one of exposition. Geometric mechanics was developed as an aid to understanding physics, and we have endeavored throughout to highlight the physical principles at work behind the mathematical formalism. In particular, we show quite explicitly the entire development of mechanics from first principles, beginning with Newton's laws of motion and culminating in the geometric reformulation of Lagrangian and Hamiltonian mechanics. Self-contained presentations of this entire range of material do not appear to be common in either the physics or the mathematics literature, but we feel very strongly that this is essential in order to understand how the more abstract mathematical developments that follow actually relate to the real world. We have also attempted to make many of the proofs contained herein more explicit than they appear in the standard references, both as an aid in understanding and simply to make them easier to follow, and several of them are original where we feel that their presentation in the literature was unacceptably opaque (this occurs primarily in the presentation of the geometric formulation of Lagrangian mechanics and the appendix on symplectic geometry). Finally, we point out that the fields of geometric mechanics and symplectic geometry are vast, and one could not hope to get more than a fragmentary glimpse of them in a single work, which necessiates some parsimony in the presentation of material. The subject matter covered herein was chosen because it is of particular interest from an applied or engineering perspective in addition to its mathematical appeal. / text

Transformadores de potência, como produtos não seriados, estão sujeitos a uma série de possíveis desvios de processo. As divergências que transformadores podem apresentar em relação ao seu projeto original podem ter consequências em sua capacidade de suportar carregamentos mecânicos que lhe serão aplicados. Seu transporte da fábrica às subestações elétricas é um dos principais eventos durante o qual suas estruturas são carregadas. Este trabalho foca em descrever e analisar um transformador de potência genérico em um modelo bidimensional, sob o ponto de vista da Mecânica Analítica e da análise de variabilidades com o objetivo de identificar parâmetros importantes à integridade estrutural do equipamento, especialmente aqueles relacionados ao modo de falha de destacamento das chapas do núcleo nos step laps. O modelo numérico do transformador consiste na representação de corpos rígidos conectados entre si por elementos de rigidez e amortecimento, tendo o conjunto do núcleo ferromagnético particularmente mais detalhado de modo a descrever o atrito existente em suas junções. A avaliação dos esforços nesses locais é feita por meio do método dos multiplicadores de Lagrange, ao considerar o atrito como uma restrição ao movimento, sendo que as forças e momentos limites para que ocorra o deslizamento planar das chapas do núcleo são determinados utilizando-se Newton-Raphson ou outras técnicas de otimização. / Power transformers, as non serialized products, are subject to a number of process deviations. The divergences that transformers present from their original design may have consequences on their ability to withstand mechanical loads which will be applied on them. Their transportation from workshop to power plants is one of the main events their structures are loaded. This work focuses on describing and analyzing a generic power transformer in a two-dimensional model, under the sight of Analytical Mechanics and Structural Reliability in order to identify important parameters related to their structural integrity, especially those concerning the failure mode of dismantling of the ferromagnetic core in the step-lap joints. The numerical model of the transformer consists of a representation of rigid bodies, connected to each other by stiness and damping elements, having the assembly of the ferromagnetic core more detailed in order to describe the friction present on their step-lap joint. The evaluation of friction in these locations is made using the Lagrange Multipliers Method, by considering it as a movement restriction, and the limit values of forces and moments to which the planar sliding of the core sheets occurs, damaging the behavior of the electric machine, are determined using Newton-Raphson and Optimization Methods.

In this dissertation, an approach for determining parameters for a nonlinear Lagrangian mechanical system model of a submerged vessel maneuvering near waves is presented. The nonlinear model with determined parameters is capable of capturing nonlinear effects neglected by other linear models, and therefore can be applied to improve maneuvering performance and expand the operating envelope for submerged vessels operating in elevated sea states. To begin, a first principles Lagrangian nonlinear maneuvering (LNM) model for a surface-affected submerged vessel derived by using Lagrangian mechanics cite{BattistaPhD2018} is reformulated to allow the application of data from a medium fidelity potential flow code. In the reformulation process, the order of integration and differentiation in the integro-differential parameters are switched and partial derivatives of the Lagrangian function are computed with readily available data from the panel code solution. As a result, all model parameters can be computed individually using the panel code, wherein the need for additional numerical discretization is circumvented in the computation process through use of solutions already performed by the basic panel code, enabling higher accuracy and lower computational cost. Furthermore, incident wave effects are incorporated into the reformulated LNM model to yield a Lagrangian nonlinear maneuvering and seakeeping (LNMS) model. The LNMS model is numerically validated by confirming the proposed methods and by comparing steady and unsteady hydrodynamic force calculations from the LNMS model against panel code computations for various vessel motions in calm water and in plane progressive waves. Finally, methods for computing physically intuitive components of the model parameters, as well as methods for making approximations of the terms accounting for memory effects are presented, leading to a model formulation amenable to control design. By applying the methods proposed in this dissertation, each and every parameter of the Lagrangian mechanical system model of a submerged vessel maneuvering in waves can be obtained accurately and with computational efficiency by using a potential flow panel code. The resulting nonlinear motion model provides higher model fidelity than existing unified maneuvering and seakeeping models, especially in applications such as nonlinear control design and simulation. / Doctor of Philosophy / A unified maneuvering and seakeeping model for a submerged vessel maneuvering near waves describes mathematically the relationship between input values to the dynamical system, such as thrust from the propulsors, and output values from the system, such as the position and orientation of the vessel. This unified model has a wide range of applications, ranging from vessel hull form optimization in the early design phase to motion controller tuning after the vessel has been constructed. In order for a unified model to make accurate predictions, for instance, for a submerged vessel making a rapid turn near large waves, nonlinear effects have to be included in the model formulation. To that end, a nonlinear motion model for a marine craft affected by a free surface has been developed using Lagrangian mechanics. This dissertation describes an approach for determining the parameters of the nonlinear motion model using a potential flow panel code, which is originally designed to determine flow velocity of the fluid and pressure distribution over marine vessels. The nonlinear motion model is reformulated and the software implementation is modified to support parameter computations. In addition, the methods are numerically validated by comparing computations using the model against solutions output by the panel code. Compared to traditional parameter estimation approaches, the proposed methods allow for a more accurate and efficient determination of parameters of the nonlinear potential flow model for a submerged vessel operating near waves. The resulting Lagrangian nonlinear maneuvering and seakeeping (LNMS) model with determined parameters is able to capture critical nonlinear effects and has applications such as nonlinear control design, rapid design optimization and training simulator development.

The work is motivated by the goal of linking reality and model, and to see if there is an opportunity to develop an inexpensive educational tool for training in cyber-physical systems. This project has investigated the possibilities to build a cheap inverted pendulum with controller and connect this with the modeling language Acumen. Acumen models is used for comparison with the actual prototype. To solve these problems has a 3D printer been used to create hardware, Arduino UNO for control and Raspberry Pi for enable communication with Acumen over WLAN. The result was a cheap inverted pendulum, which can be built for a cost around 750 SEK. Graphs created in Acumen and from data collected from sensors can be analyzed. With a model of the inverted pendulum system, the results show that Acumen can be used in the development of cyber-physical systems. There are differences between model and reality but also similarities. / Arbetet motiveras av målet att knyta samman verklighet och modell, samt att se om det finns möjlighet att utveckla ett billigt utbildningsverktyg för utbildning i cyberfysiska system. Detta projekt har undersökt möjligheter att bygga en billig inverterad pendel med regulator samt koppla samman denna med modelleringsspråket Acumen. I Acumen skapa en modell av systemet och jämföra den med en fysisk prototyp. För att lösa dessa problem har en 3D skrivare använts för att skapa hårdvara. Arduino UNO för styrning och Raspberry Pi för att möjligöra kommunikation med Acumen över WLAN. Resultatet blev en billig inverterad pendel, som kan byggas för en kostnad runt 750 kr. Grafer från Acumen, och från data samlad från sensorer kan analyseras. Med en modell av en inverterad pendel visar resultaten att Acumen kan användas i utveckling av cyberfysiska system. Skillnader finns mellan modell och verklighet men även likheter.

In this work, the possibility of realising object tracking by implementing predictive methods with static object detectors is explored. The static object detectors are obtained as models trained on a machine learning algorithm, or in other words, a deep neural network. Specifically, it is the single shot detector inception v2 network that will be used to train such models. Predictive methods will be incorporated to the end of improving the obtained models’ precision, i.e. their performance with respect to accuracy. Namely, Lagrangian mechanics will be employed to derived equations of motion for three different scenarios in which the object is to be tracked. These equations of motion will be implemented as predictive methods by discretising and combining them with four different iterative formulae. In ch. 1, the fundamentals of supervised machine learning, neural networks, convolutional neural networks as well as the workings of the single shot detector algorithm, approaches to hyperparameter optimisation and other relevant theory is established. This includes derivations of the relevant equations of motion and the iterative formulae with which they were implemented. In ch. 2, the experimental set-up that was utilised during data collection, and the manner by which the acquired data was used to produce training, validation and test datasets is described. This is followed by a description of how the approach of random search was used to train 64 models on 300×300 datasets, and 32 models on 512×512 datasets. Consecutively, these models are evaluated based on their performance with respect to camera-to-object distance and object velocity. In ch. 3, the trained models were verified to possess multi-scale detection capabilities, as is characteristic of models trained on the single shot detector network. While the former is found to be true irrespective of the resolution-setting of the dataset that the model has been trained on, it is found that the performance with respect to varying object velocity is significantly more consistent for the lower resolution models as they operate at a higher detection rate. Ch. 3 continues with that the implemented predictive methods are evaluated. This is done by comparing the resulting deviations when they are let to predict the missing data points from a collected detection pattern, with varying sampling percentages. It is found that the best predictive methods are those that make use of the least amount of previous data points. This followed from that the data upon which evaluations were made contained an unreasonable amount of noise, considering that the iterative formulae implemented do not take noise into account. Moreover, the lower resolution models were found to benefit more than those trained on the higher resolution datasets because of the higher detection frequency they can employ. In ch. 4, it is argued that the concept of combining predictive methods with static object detectors to the end of obtaining an object tracker is promising. Moreover, the models obtained on the single shot detector network are concluded to be good candidates for such applications. However, the predictive methods studied in this thesis should be replaced with some method that can account for noise, or be extended to be able to account for it. A profound finding is that the single shot detector inception v2 models trained on a low-resolution dataset were found to outperform those trained on a high-resolution dataset in certain regards due to the higher detection rate possible on lower resolution frames. Namely, in performance with respect to object velocity and in that predictive methods performed better on the low-resolution models. / I detta arbete undersöks möjligheten att åstadkomma objektefterföljning genom att implementera prediktiva metoder med statiska objektdetektorer. De statiska objektdetektorerna erhålls som modeller tränade på en maskininlärnings-algoritm, det vill säga djupa neurala nätverk. Specifikt så är det en modifierad version av entagningsdetektor-nätverket, så kallat entagningsdetektor inception v2 nätverket, som används för att träna modellerna. Prediktiva metoder inkorporeras sedan för att förbättra modellernas förmåga att kunna finna ett eftersökt objekt. Nämligen används Lagrangiansk mekanik för härleda rörelseekvationer för vissa scenarion i vilka objektet är tänkt att efterföljas. Rörelseekvationerna implementeras genom att låta diskretisera dem och därefter kombinera dem med fyra olika iterationsformler. I kap. 2 behandlas grundläggande teori för övervakad maskininlärning, neurala nätverk, faltande neurala nätverk men också de grundläggande principer för entagningsdetektor-nätverket, närmanden till hyperparameter-optimering och övrig relevant teori. Detta inkluderar härledningar av rörelseekvationerna och de iterationsformler som de skall kombineras med. I kap. 3 så redogörs för den experimentella uppställning som användes vid datainsamling samt hur denna data användes för att producera olika data set. Därefter följer en skildring av hur random search kunde användas för att träna 64 modeller på data av upplösning 300×300 och 32 modeller på data av upplösning 512×512. Vidare utvärderades modellerna med avseende på deras prestanda för varierande kamera-till-objekt avstånd och objekthastighet. I kap. 4 så verifieras det att modellerna har en förmåga att detektera på flera skalor, vilket är ett karaktäristiskt drag för modeller tränade på entagninsdetektor-nätverk. Medan detta gällde för de tränade modellerna oavsett vilken upplösning av data de blivit tränade på, så fanns detekteringsprestandan med avseende på objekthastighet vara betydligt mer konsekvent för modellerna som tränats på data av lägre upplösning. Detta resulterade av att dessa modeller kan arbeta med en högre detekteringsfrekvens. Kap. 4 fortsätter med att de prediktiva metoderna utvärderas, vilket de kunde göras genom att jämföra den resulterande avvikelsen de respektive metoderna innebar då de läts arbeta på ett samplat detektionsmönster, sparat från då en tränad modell körts. I och med denna utvärdering så testades modellerna för olika samplingsgrader. Det visade sig att de bästa iterationsformlerna var de som byggde på färre tidigare datapunkter. Anledningen för detta är att den insamlade data, som testerna utfördes på, innehöll en avsevärd mängd brus. Med tanke på att de implementerade iterationsformlerna inte tar hänsyn till brus, så fick detta avgörande konsekvenser. Det fanns även att alla prediktiva metoder förbättrade objektdetekteringsförmågan till en högre utsträckning för modellerna som var tränade på data av lägre upplösning, vilket följer från att de kan arbeta med en högre detekteringsfrekvens. I kap. 5, argumenteras det, bland annat, för att konceptet att kombinera prediktiva metoder med statiska objektdetektorer för att åstadkomma objektefterföljning är lovande. Det slutleds även att modeller som erhålls från entagningsdetektor-nätverket är lovande kandidater för detta applikationsområde, till följd av deras höga detekteringsfrekvenser och förmåga att kunna detektera på flera skalor. Metoderna som användes för att förutsäga det efterföljda föremålets position fanns vara odugliga på grund av deras oförmåga att kunna hantera brus. Det slutleddes därmed att dessa antingen bör utökas till att kunna hantera brus eller ersättas av lämpligare metoder. Den mest väsentliga slutsats detta arbete presenterar är att lågupplösta entagninsdetektormodeller utgör bättre kandidater än de tränade på data av högre upplösning till följd av den ökade detekteringsfrekvens de erbjuder.

Hamilton-Jacobi theory provides a powerful method for extracting the equations of motion out of some given systems in classical mechanics. On occasion it allows some systems to be solved by the method of separation of variables. If a system with n degrees of freedom has 2n - 1 constants of the motion that are polynomial in the momenta, then that system is called superintegrable. Such a system can usually be solved in multiple coordinate systems if the constants of the motion are quadratic in the momenta. All superintegrable two dimensional Hamiltonians of the form H = (p_x)sup2 + (p_y)sup2 + V(x,y), with constants that are quadratic in the momenta were classified by Kalnins et al [5], and the coordinate systems in which they separate were found. We discuss Hamilton-Jacobi theory and its development from a classical viewpoint, as well as superintegrability. We then proceed to use the theory to find equations of motion for some of the superintegrable Hamiltonians from Kalnins et al [5]. We also discuss some of the properties of the Poisson algebra of those systems, and examine the orbits.

Newton's second law, Schrödinger's equation and Maxwell's equations are all theories composed of at most second-time derivatives. Indeed, it is not often we need to take the time derivative of the acceleration. So why are we not seeing more higher-order derivative theories? Although several studies present higher derivatives' usefulness in quadratic gravity and scalar-field theories, one will eventually encounter a problem. In 1850, the physicist Mikhail Ostrogradsky presented a theorem that stated that a non-degenerate Lagrangian composed of finite higher-order time derivatives results in a Hamiltonian unbounded from below. Explicitly, it was shown that the Hamiltonian of such a system includes linearity in physical momenta, often referred to as the ''Ostrogradsky ghost''. This thesis studies how one can avoid the Ostrogradsky ghost by considering degenerate Lagrangians to put constraints on the momenta. The study begins by showing the existence of the ghost and later cover the essential Hamiltonian formalism needed to conduct Hamiltonian constraint analyses of second-order time derivative systems, both single-variable and systems coupled to a regular one. Ultimately, the degenerate second-order Lagrangians successfully eliminate the Ostrogradsky ghost by generating secondary constraints restricting the physical momenta. Moreover, an outline of a Hamiltonian analysis of a general higher-order Lagrangian is presented at the end.