The objective of this dissertation is to develop a mechanical model of a nonlinear piezoelectric energy harvesting system induced by Duffing oscillator and predict the periodic motions of such a nonlinear dynamical system under different excitation frequency. In this dissertation, analytical distributed-parameter electromechanical modeling of a piezoelectric energy harvester will be presented. The electromechanically coupled circuit equation excited by infinitely many vibration modes is derived. The governing electromechanical equations are reduced to ordinary differential equations in modal coordinates and eventually an infinite set of algebraic equations is obtained for the complex modal vibration response and the complex voltage response of the energy harvester beam. One single vibration mode is chosen and discussed. The periodic motions are obtained through an implicit mapping method with high accuracy, stability and bifurcations of periodic motions are determined by the eigenvalue analysis. Frequency-amplitude characteristics of periodic motions are achieved by the Fourier transform

A study is presented of the instability mechanisms of a damped axisymmetric circular disk of uniform thickness rotating about its axis with constant angular velocity and subjected to various transverse space-fixed loading systems. The natural frequencies of spinning floppy disks are obtained for various nodal diameters and nodal circles with a numerical and an approximate method. Exploiting the fact that in most physical applications the thickness of the disk is small compared with its outer radius, we use their ratio to define a small parameter. Because the nonlinearities appearing in the governing partial-differential equations are cubic, we use the Galerkin procedure to reduce the problem into a finite number of coupled weakly nonlinear second-order equations. The coefficients of the nonlinear terms in the reduced equations are calculated for a wide range of the lowest modes and for different rotational speeds. We have studied the primary resonance of a pair of orthogonal modes under a space-fixed constant loading, the principal parametric resonance of a pair of orthogonal modes when the disk is subject to a massive loading system, and the combination parametric resonance of two pairs of orthogonal modes when the excitation is a linear spring. Considering the case of a spring moving periodically along the radius of the disk, we show how its frequency can be coupled to the rotational speed of the disk and lead to a principal parametric resonance. In each of these cases, we have used the method of multiple scales to determine the equations governing the modulation of the amplitudes and phases of the interacting modes. The equilibrium solutions of the modulation equations are determined and their stability is studied. / Master of Science

Islanding occurs when a connected load to the grid is disconnected from the grid and energized solely by local generators. Islanding can result in frequency and voltage instability, changes in current, and overall poor power quality. Poor power quality can interrupt industrial operations, damage sensitive electrical equipment, and induce outages upon the resynchronization of the island with the grid. This study proposes an islanding detection method employing Duffing oscillators to analyze fluctuations at the point of common coupling (PCC) under a high-noise environment, focusing on decreasing detection period, zero power mismatch nondetection zone, and power quality degradation. Unlike existing methods, which overlook the noise effect, this study mitigates noise impact on islanding detection. Power system noise in PCC measurements arises from switching transients, harmonics, grounding issues, voltage sags, voltage swells, electromagnetic interference, and power quality issues that affect islanding detection. Transient events, like lightning-induced traveling waves can also introduce noise levels exceeding the voltage amplitude, disturbing conventional detection techniques~cite{IEEE1313}. The noise interferes with measurements and increases the nondetection zone (NDZ), causing failed or delayed islanding detection. Duffing oscillator nonlinear dynamics enable detection capabilities at a high noise level. The proposed methods are designed to detect the PCC measurement fluctuations based on the IEEE standard 1547 through the Duffing oscillator. The basic idea is that the Duffing oscillator phase trajectory changes from periodic to chaotic mode and sends an islanded operation command to the inverter. The proposed islanding detection method can distinguish switching transients and faults from an islanded operation. / Master of Science / This study introduces a method to detect one of the important power grid issues, called islanding. Islanding occurs when a power grid section becomes isolated and continues to operate independently, leading to power quality problems and safety hazards. The method is developed using a Duffing oscillator that can detect special signals under a high-noise environment. The proposed method monitors electrical characteristics for islanding detection, including frequency, phase angle, and voltage amplitude. The method can quickly and accurately identify when an islanded operation occurs by analyzing these signals. This method combines the advantages of passive and active detection methods while avoiding common drawbacks, such as failed and delayed detection and power quality degradation. The proposed method is tested on a setup and meets IEEE standard 1547 criteria for safety and performance. The method is important because it detects islanded operations in a high-noise environment when the other methods cannot detect islanded operations correctly. The method maintains accuracy with decreased power quality degradation in noise. This method is a cost-effective solution for modern power grids increasingly integrating renewable energy resources.

The diploma thesis is focused on the use of heuristic and metaheuristic methods to stabilization and controlling the selected systems distinguished by the deterministic chaos behavior. There are discussed parameterization of chosen optimization methods, which are the genetic algorithm, simulated annealing and pattern search. The thesis also introduced the suitable controlling methods and the definition of the objective function. In the theoretical part of the thesis there is a brief introduction to the deterministic chaos theory. The next chapters describes the most common and deployed methods in~the~control theory, especially OGY and Pyragas methods. The practical part of the thesis is divided into two chapters. The first one describes the~stabilization of the artifical chaotic systems with the time delayed Pyragas method - TDAS and its modification ETDAS. The second chapter shows the real chaotic system control. The Duffing oscillator system was chosen to serve this purpose.

Accurately predicting graft failure following kidney transplantation is essential for identifying high-risk patients and tailoring treatment strategies. This thesis aims to forecast kidney graft failure by estimating and predicting the glomerular filtration rate (GFR) using real data related to pre-transplant and post-operative patients status, provided to us by Hannover Medical School. To achieve this, we implement three Bayesian filtering techniques: the Extended Kalman Filter (EKF), the Unscented Kalman Filter (UKF), and the Particle Filter (PF), on a discrete-time state-space stochastic Duffing oscillator model. We also conduct regression analysis between available GFR measurements and the filters' estimated and predicted values, followed by an error analysis using root mean square error. Our results demonstrate that Particle Filter, utilizing 10,000 particles, consistently produced accurate estimates compared to other filters in most patients. Furthermore, we observe that data interpolation yields more accurate results.

In support of society's technological evolution, the study of nonlinear systems in engineering and sciences has become a vital research area. Aiming to contribute in this field, this thesis investigates the behaviour of nonlinear systems using the 'Wiener theories'. As a useful example the Duffing oscillator is investigated in this work. In many real-life applications, nonlinear systems are excited randomly so this work examines systems under white-noise excitation using the Wiener series. Equivalent Linearisation (EL) is a well-known and simple method that approximates a nonlinear system by an equivalent linear system. However, it has deficiencies which this thesis attempts to improve. Initially, the performance of EL for different types of nonlinearities will be assessed and an alternative method to enhance it is suggested. This requires the calculation of the first Wiener kernel of various system defined quantities. The first Wiener kernel, as it will be shown, is the foundation of this research and a central element of the Wiener theory. In this thesis, an analytical proof to explain the interesting behaviour of the first Wiener kernel for a system with nonlinear stiffness is included using an energy transfer approach. Furthermore, the method mentioned above to enhance EL known as the Single-Pole Fit method (SPF) is to be tested for different kinds of systems to prove its robustness and validity. Its direct application to systems with nonlinear stiffness and nonlinear damping is shown as well as its ability to perform for systems with two degrees of freedom where an extension of the SPF method is required to achieve the desired solution. Finally, an investigation to understand and replicate the complex behaviour observed by the first Wiener kernel in the early chapters is carried out. The groundwork for this investigation is done by modelling an isolated nonlinear spring with a series of linear filters and certain nonlinear operations. Subsequently, an attempt is made to relate the principles governing the successful spring model presented to the original nonlinear system. An iterative procedure is used to demonstrate the application of this method, which also enables this new modelling approach to be related to the SPF method.

Studium chování opticky zachycených částic nám umožňuje porozumět základním fyzikálním jevům plynoucím z interakce světla a hmoty. Předkládaná práce podává vysvětlení zesílení tažné síly působící na opticky svázané částice ve strukturovaném světelném poli, tzv. tažném svazku. Ukazujeme, že pohyb dvou opticky svázaných objektů v tažném svazku je silně závislý na jejich vzájemné vzdálenosti a prostorové orientaci, což rozšiřuje možnosti manipulace hmoty pomocí světla. Následně se práce zaměřuje na levitaci opticky zachycených částic ve vakuu. Představujeme novou metodologii na charakterizaci vlastností slabě nelinearního Duffingova oscilátoru reprezentovaného opticky levitující částicí. Metoda je založena na průměrování trajektorií s určitou počáteční pozicí ve fázovém prostoru sestávajícím z polohy a rychlosti částice a poskytuje informaci o parametrech oscilátoru přímo ze zaznamenaného pohybu. Náš inovativní postup je srovnán s běžně užívanou metodou založenou na analýze spektrální hustoty polohy částice a za využití numerických simulací ukazujeme její použitelnost i v nízkých tlacích, kde nelinearita hraje významnou roli.