Chaotic Behaviour of Active Magnetic Bearing System by Time Series Analysis

Chaotic behaviour is one of the hardest problems that can happen in nonlinear dynamical systems with severe nonlinearities. It makes the system's responses unpredictable. It makes the system's responses to behave similar to noise. In some applications it should be avoided. One of the approaches to detect the chaotic behaviour is nding the Lyapunov exponent through examining the dynamical equation of the system. It needs a model of the system. The goal of this study is the diagnosis of chaotic behaviour by just exploring the data (signal) without using any dynamical model of the system. In this work two methods are tested on the time series data collected from AMB (Active Magnetic Bearing) system sensors. The rst method is used to nd the largest Lyapunov exponent by Rosenstein method. The second method is a 0-1 test for identifying chaotic behaviour. These two methods are used to detect if the data is chaotic. By using Rosenstein method it is needed to nd the minimum embedding dimension. To nd the minimum embedding dimension Cao method is used. Cao method does not give just the minimum embedding dimension, it also gives the order of the nonlinear dynamical equation of the system and also it shows how the system's signals are corrupted with noise. At the end of this research a test called runs test is introduced to show that the data is not excessively noisy.

Conventional and nonconventional Kramers-Kronig analysis in optical spectroscopy

This Master’s Thesis is dedicated to the investigation and testing conventional and nonconventional Kramers-Kronig relations on simulated and experimentally measured spectra. It is done for both linear and nonlinear optical spectral data. Big part of attention is paid to the new method of obtaining complex refractive index from a transmittance spectrum without direct information of the sample thickness. The latter method is coupled with terahertz tome-domain spectroscopy and Kramers-Kronig analysis applied for testing the validity of complex refractive index. In this research precision of data inversion is evaluated by root-mean square error. Testing of methods is made over different spectral range and implementation of this methods in future is considered.

Dynamic Analysis Model of Spherical Roller Bearings with Defects

Rolling element bearings are essential components of rotating machinery. The spherical roller bearing (SRB) is one variant seeing increasing use, because it is self-aligning and can support high loads. It is becoming increasingly important to understand how the SRB responds dynamically under a variety of conditions. This doctoral dissertation introduces a computationally efficient, three-degree-of-freedom, SRB model that was developed to predict the transient dynamic behaviors of a rotor-SRB system. In the model, bearing forces and deflections were calculated as a function of contact deformation and bearing geometry parameters according to nonlinear Hertzian contact theory. The results reveal how some of the more important parameters; such as diametral clearance, the number of rollers, and osculation number; influence ultimate bearing performance. Distributed defects, such as the waviness of the inner and outer ring, and localized defects, such as inner and outer ring defects, are taken into consideration in the proposed model. Simulation results were verified with results obtained by applying the formula for the spherical roller bearing radial deflection and the commercial bearing analysis software. Following model verification, a numerical simulation was carried out successfully for a full rotor-bearing system to demonstrate the application of this newly developed SRB model in a typical real world analysis. Accuracy of the model was verified by comparing measured to predicted behaviors for equivalent systems.

Esikiristetyn pulttiliitoksen vaikutus hitsausmuodonmuutoksiin

Tässä työssä tutkitaan propulsioyksikön kiinnitysrenkaan pulttiliitosten vaikutusta asen-nushitsauksesta aiheutuviin hitsausmuodonmuutoksiin. Hitsausmuodonmuutoksissa tutki-taan tärkeimpinä kohtina asennuslohkossa laakerin rajapintaa sekä kääntömoottorin kiinni-tyspintaa. Tutkimuksessa asennuslohkon hitsaaminen ja muodonmuutosten arvioiminen toteutettiin käyttämällä epälineaarista elementtimenetelmää. Ensisijaisena tavoitteena työssä on tutkia esikiristettyjen pulttiliitoksien vaikutusta raken-teen muodonmuutoksiin ja pohtia aiheutuvien siirtymien perusteella pulttien tarpeellisuutta rakenteessa. Tämän lisäksi vertaillaan pultillisen ja pultittoman kiinnitystavan eroavaisuuk-sia tuloksia analysoimalla. Saatujen tuloksien perusteella radiaaliset ja aksiaaliset siirtymät eivät olleet riittävän suuria aiheuttamaan haittoja rakenteen toimivuudelle kummassakaan mallissa. Lisäanalyysejä tar-kemmalla lämmöntuonnilla voidaan pitää tarvittavana pulttiliitoksien tarpeellisuuden tar-kemman testaamisen vuoksi.

Bayesian analysis of SEIR epidemic models

This thesis concerns the analysis of epidemic models. We adopt the Bayesian paradigm and develop suitable Markov Chain Monte Carlo (MCMC) algorithms. This is done by considering an Ebola outbreak in the Democratic Republic of Congo, former Zaïre, 1995 as a case of SEIR epidemic models. We model the Ebola epidemic deterministically using ODEs and stochastically through SDEs to take into account a possible bias in each compartment. Since the model has unknown parameters, we use different methods to estimate them such as least squares, maximum likelihood and MCMC. The motivation behind choosing MCMC over other existing methods in this thesis is that it has the ability to tackle complicated nonlinear problems with large number of parameters. First, in a deterministic Ebola model, we compute the likelihood function by sum of square of residuals method and estimate parameters using the LSQ and MCMC methods. We sample parameters and then use them to calculate the basic reproduction number and to study the disease-free equilibrium. From the sampled chain from the posterior, we test the convergence diagnostic and confirm the viability of the model. The results show that the Ebola model fits the observed onset data with high precision, and all the unknown model parameters are well identified. Second, we convert the ODE model into a SDE Ebola model. We compute the likelihood function using extended Kalman filter (EKF) and estimate parameters again. The motivation of using the SDE formulation here is to consider the impact of modelling errors. Moreover, the EKF approach allows us to formulate a filtered likelihood for the parameters of such a stochastic model. We use the MCMC procedure to attain the posterior distributions of the parameters of the SDE Ebola model drift and diffusion parts. In this thesis, we analyse two cases: (1) the model error covariance matrix of the dynamic noise is close to zero , i.e. only small stochasticity added into the model. The results are then similar to the ones got from deterministic Ebola model, even if methods of computing the likelihood function are different (2) the model error covariance matrix is different from zero, i.e. a considerable stochasticity is introduced into the Ebola model. This accounts for the situation where we would know that the model is not exact. As a results, we obtain parameter posteriors with larger variances. Consequently, the model predictions then show larger uncertainties, in accordance with the assumption of an incomplete model.