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Modern information technologies accompany all of us in our daily life. They not only use to serve to our work and for the entertainment, they even have got a great potential to be used in the education. The submitted thesis deals with the appropriate utilization of the special IT application in the pedagogic process, especially in teaching of maths, the topic of trigonometric functions. The aim of this thesis is to create structures with a high potential of the modern technology handling. To tackle this task the graphical program GeoGebra was chosen due to its qualities for creating dynamic structures and the other advantages presented further in the thesis. To the created constructions the detailed instructions how they can be constructed are added. The thesis is also accompanied by worksheets with solved tasks, focusing on the application of trigonometric functions in real situations.

The dissertation focuses on estimating the uncertainty associated with stress/strain prediction procedures from dynamic test data used in turbine blade analysis. An accurate prediction of the maximum response levels for physical components during in-field operating conditions is essential for evaluating their performance and life characteristics, as well as for investigating how their behavior critically impacts system design and reliability assessment. Currently, stress/strain inference for a dynamic system is based on the combination of experimental data and results from the analytical/numerical model of the component under consideration. Both modeling challenges and testing limitations, however, contribute to the introduction of various sources of uncertainty within the given estimation procedure, and lead ultimately to diminished accuracy and reduced confidence in the predicted response. The objective of this work is to characterize the uncertainties present in the current response estimation process and provide a means to assess them quantitatively. More specifically, proposed in this research is a statistical methodology based on a Bayesian-network representation of the modeling process which allows for a statistically rigorous synthesis of modeling assumptions and information from experimental data. Such a framework addresses the problem of multi-directional uncertainty propagation, where standard techniques for unidirectional propagation from inputs' uncertainty to outputs' variability are not suited. Furthermore, it allows for the inclusion within the analysis of newly available test data that can provide indirect evidence on the parameters of the structure's analytical model, as well as lead to a reduction of the residual uncertainty in the estimated quantities. As part of this work, key uncertainty sources (i.e., material and geometric properties, sensor measurement and placement, as well as noise due data processing limitations) are investigated, and their impact upon the system response estimates is assessed through sensitivity studies. The results are utilized for the identification of the most significant contributors to uncertainty to be modeled within the developed Bayesian inference scheme. Simulated experimentation, statistically equivalent to specified real tests, is also constructed to generate the data necessary to build the appropriate Bayesian network, which is then infused with actual experimental information for the purpose of explaining the uncertainty embedded in the response predictions and quantifying their inherent accuracy.

The analysis of vehicle-structure interaction systems plays a significant role in the design and maintenance of bridges. In recent years, the assessment of the health of existing bridges and the design of new ones has gained significance, in part due to the progress made in the development of faster moving locomotives, the desire for lighter bridges, and the imposition of performance criteria against rare events such as occurrence of earthquakes and fire. A probabilistic analysis would address these issues, and also assist in determination of reliability and in estimating the remaining life of the structure. In this thesis, we aim to develop tools for the probabilistic analysis techniques of state estimation, parameter identification and global response sensitivity analysis of vehicle-structure interaction systems, which are also applicable to the broader class of structural dynamical systems. The thesis is composed of six chapters and three appendices. The contents of these chapters and the appendices are described in brief in the following paragraphs. In chapter 1, we introduce the problem of probabilistic analysis of vehicle-structure interactions. The introduction is organized in three parts, dealing separately with issues of forward problems, inverse problems, and global response sensitivity analysis. We begin with an overview of the modelling and analysis of vehicle-structure interaction systems, including the application of spatial substructuring and mesh partitioning schemes. Following this, we describe Bayesian techniques for state and parameter estimation for the general class of state-space models of dynamical systems, including the application of the Kalman filter and particle filters for state estimation, MCMC sampling based filters for parameter identification, and the extended Kalman filter, the unscented Kalman filter and the ensemble Kalman filter for the problem of combined state and parameter identification. In this context, we present the Rao-Blackwellization method which leads to variance reduction in particle filtering. Finally, we present the techniques of global response sensitivity analysis, including Sobol’s analysis and distance-based measures of sensitivity indices. We provide an outline and a review of literature on each of these topics. In our review of literature, we identify the difficulties encountered when adopting these tools to problems involving vehicle-structure interaction systems, and corresponding to these issues, we identify some open problems for research. These problems are addressed in chapters 2, 3, 4 and 5. In chapter 2, we study the application of finite element modelling, combined with numerical solutions of governing stochastic differential equations, to analyse instrumented nonlinear moving vehicle-structure systems. The focus of the chapter is on achieving computational efficiency by deploying, within a single modeling framework, three sub structuring schemes with different methodological moorings. The schemes considered include spatial substructuring schemes (involving free-interface coupling methods), a spatial mesh partitioning scheme for governing stochastic differential equations (involving the use of a predictor corrector method with implicit integration schemes for linear regions and explicit schemes for local nonlinear regions), and application of the Rao-Blackwellization scheme (which permits the use of Kalman’s filtering for linear substructures and Monte Carlo filters for nonlinear substructures). The main effort in this work is expended on combining these schemes with provisions for interfacing of the substructures by taking into account the relative motion of the vehicle and the supporting structure. The problem is formulated with reference to an archetypal beam and multi-degrees of freedom moving oscillator with spatially localized nonlinear characteristics. The study takes into account imperfections in mathematical modelling, guide way unevenness, and measurement noise. The numerical results demonstrate notable reduction in computational effort achieved on account of introduction of the substructuring schemes. In chapter 3, we address the issue of identification of system parameters of structural systems using dynamical measurement data. When Markov chain Monte Carlo (MCMC) samplers are used in problems of system parameter identification, one would face computational difficulties in dealing with large amount of measurement data and (or) low levels of measurement noise. Such exigencies are likely to occur in problems of parameter identification in dynamical systems when amount of vibratory measurement data and number of parameters to be identified could be large. In such cases, the posterior probability density function of the system parameters tends to have regions of narrow supports and a finite length MCMC chain is unlikely to cover pertinent regions. In this chapter, strategies are proposed based on modification of measurement equations and subsequent corrections, to alleviate this difficulty. This involves artificial enhancement of measurement noise, assimilation of transformed packets of measurements, and a global iteration strategy to improve the choice of prior models. Illustrative examples include a laboratory study on a beam-moving trolley system. In chapter 4, we consider the combined estimation of the system states and parameters of vehicle-structure interaction systems. To this end, we formulate a framework which uses MCMC sampling for parameter estimation and particle filtering for state estimation. In chapters 2 and 3, we described the computational issues faced when adopting these techniques individually. When used together, we come across both sets of issues, and find the complexity of the estimation problem is greatly increased. In this chapter, we address the computational issues by adopting the sub structuring techniques proposed in chapter 2, and the parameter identification method based on modified measurement models presented in chapter 3. The proposed method is illustrated on a computational study on a beam-moving oscillator system with localized nonlinearities, as well as on a laboratory study on a beam-moving trolley system. In chapter 5, we present global response sensitivity indices for structural dynamical systems with random system parameters excited by multiple random excitations. Two new procedures for evaluating global response sensitivity measures with respect to the excitation components are proposed. The first procedure is valid for stationary response of linear systems under stationary random excitations and is based on the notion of Hellinger’s metric of distance between two power spectral density functions. The second procedure is more generally valid and is based on the l2 norm based distance measure between two probability density functions. Specific cases which admit exact solutions are presented and solution procedures based on Monte Carlo simulations for more general class of problems are outlined. The applicability of the proposed procedures to the case of random system parameters is demonstrated using suitable illustrations. Illustrations include studies on a parametrically excited linear system and a nonlinear random vibration problem involving moving oscillator-beam system that considers excitations due to random support motions and guide-way unevenness. In chapter 6 we summarize the contributions made in chapters 2, 3, 4, and 5, and on the basis of these studies, present a few problems for future research. In addition to these chapters, three appendices are included in this thesis. Appendices A and B correspond to chapter 3. In appendix A, we study the effect on the nature of the posterior probability density functions of large measurement data set and small measurement noise. Appendix B illustrates the MCMC sampling based parameter estimation procedure of chapter 3 using a laboratory study on a bending–torsion coupled, geometrically non-linear building frame under earthquake support motion. In appendix C, we present Ito-Taylor time discretization schemes for stochastic delay differential equations found in chapter 5.