Two problems in stochastic structural dynamics

This thesis treats two different subjects within the field of random vibrations of structural systems. For this reason the thesis falls in two parts and consequently this abstract is also split into two parts. Part I The first part concerns a fast simulation technique useful for the examination of the stochastic characteristics of the random development of the plastic displacement processes of elasto-plastic oscillators (EPOs) driven by Gaussian white noise. These oscillators are specific examples of the more general class of systems with hysteresis. The results obtained herein are, however, valid for general hysteretic systems. The simulation procedure consists in simulating clumps of plastic displacement increments by use of the so-called Slepian model and simulating, by simple means, the inter-clump waiting times. The Slepian model has earlier been applied to ideal EPOs for which analytical results have been derived. Similar results cannot be derived for non-ideal EPOs, why simulations are required. In the present work the analytical results valid for the ideal EPO forms the basis of a semi-analytical simulation scheme used for non-ideal EPOs with strain hardening and softening. The simulation algorithm, which is basically a time integration algorithm, is fast as it takes time-steps of half a period or larger. Especially when simulating waiting times it may use time steps of the same magnitude as the mean waiting time, which gives a considerable time gain compared to traditional direct numerical time integration. In former works an approximate waiting time distribution based on asymptotic analytical results has been suggested. Here an improved waiting time description accounting for the fact that convergence to the asymptotic result is slow is presented. The problem is that for realistic yield levels clumps arrive in groups. A phenomenon which is not properly accounted for by the asymptotic results. For the waiting time simulation a model for the amplitude process of a linear oscillator driven by white noise is applied. It is derived from two simple Slepian models for the displacement and velocity processes. In order that the Slepian model simulation of the clumps be sufficiently simple to code and the computation of the plastic increments not too time costly, a simplification of the general hysteretic restoring force diagram is suggested and proved useful. Since yield levels are generally asymmetric with respect to the equilibrium point the notion of symmetrized yield levels is introduced as they simplify the waiting time simulation. This implies the so-called clump definition different from what has traditionally been used. It is demonstrated that the simulation scheme proves well using these symmetrized yield levels without limiting the applicability of the algorithm. As the results obtained by the Slepian Model Simulation Method are compared to results obtained by direct time integration a thorough discussion of the applied direct time integration scheme is given. It is proven that the scheme is at the same time a fair and a realistic benchmark for the Slepian Model Simulation Method with respect to time consumption and accuracy. Finally, simulation results are presented. These proves that the Slepian Simulation Method gives good approximate results and that it exhibits good time gain factors. Especially it is proven that the merging of clump simulations and waiting time simulations, due to the suggested waiting time distribution works notably better than results obtained by exponentially distributed waiting times. Part II The second part deals with the random response of a certain class of structures with random properties subjected mainly to deterministic excitation. In investigations of noise radiated from submerged ship hulls it turns out that the vibrations of the ship hull are influenced by the presence of numerous minor devices elastically attached to the hull. Furthermore, information about the entity of the devices is uncertain, why the devices are termed fuzzy substructures. The fuzzies are most realistically modeled by stochastic means. Thus the ship hull, the so-called master, may be considered a structure with random vibration properties. Herein it is investigated whether the dynamical damping effect that the master experiences due to the fuzzies, is obtained if the fuzzies are modeled by a continuous random field. Some researches have claimed that this is not the case. Constructing, step by step, a piecewise continuous one-dimensional Poisson square wave field from a discrete model of the fuzzies, and further proving a certain asymptotic equivalence of the discrete model and the field, it is shown that continuous fields lead to damping effects too. Furthermore, a specific example is given in which quantification of the damping effect is obtained by use of Winterstein approximations to the distribution of the change-in-impedance of the master as caused by the fuzzies. The strength of these results is that, via the Winterstein approximations, a mapping from standard Gaussian variables to the change-in-impedance quantities is obtained thus opening for possible applications in numerical reliability methods. Though the theory of fuzzy substructures has arisen in the field of submarine engineering the authors of [22] see potential use of this theory in structural engineering as well.