Subharmonic solutions with prescribed minimal period for a class of second order impulsive systems

Based on variational methods and critical point theory, the existence of subharmonic solutions with prescribed minimal period for a class of second-order impulsive systems is derived by estimating the energy of the solution. And an example is presented to illustrate the result

I. Regular and chaotic motions in a wave tank, II, interactions between a free surface and a shed vortex sheet

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Ocean Engineering, 1992.Includes bibliographical references (leaves 142-143).by Wu-ting Tsai.Ph.D

Theoretical and experimental studies of the dynamics and acoustics of forced ordered granular networks

Ordered arrays of granular particles (beads) have attracted considerable attention in recent years due to their rich dynamical behaviors and interesting properties. Depending on the ratio of static to dynamic deformations between particles the dynamics of granular media is highly tunable ranging from being strongly nonlinear and non-smooth in the absence of static pre-compression, to reducing to weakly nonlinear and smooth for large static pre-compression. The nonlinearity in uncompressed granular media arises from two sources: First, nonlinear Hertzian interactions, which can be modeled mathematically, between beads in contact, and second, bead separations in the absence of compressive forces between them leading to collisions between adjacent beads. When no applied pre-compression exists there is complete absence of linear acoustics in ordered granular media, which results in zero speed of sound as defined in the sense of linear acoustics through the classical wave equation; thus, these media have been characterized as “sonic vacua”. However, various nonlinear waves can still propagate in these media with energy tunable properties. The first part of this dissertation aims to study the frequency responses of a single homogenous granular chain. We consider a one-dimensional uncompressed granular chain composed of a finite number of identical spherical elastic beads with Hertzian interactions. The chain is harmonically excited by an amplitude- and frequency-dependent boundary drive at its left end and has a fixed boundary at its right end. We computationally and experimentally detect time-periodic, strongly nonlinear resonances whereby the particles (beads) of the granular chain respond at integer multiples of the excitation period, and which correspond to local peaks of the maximum transmitted force at the chain’s right, fixed end. In between these resonances we detect local minima of the maximum transmitted forces corresponding to anti-resonances, where chimera states (i.e., coexistence of different stationary and nonstationary waveforms) are noted, in the steady-state dynamics. Furthermore, we construct a mathematical model which can completely capture the rich and complex dynamics of the system. The second part of the study is primarily concerned with the propagatory dynamics of geometrically coupled ordered granular media. In particular, we focus on primary pulse transmission in a two-dimensional granular network composed of two ordered chains that are nonlinearly coupled through Hertzian interactions. Impulsive excitation is applied to one of the chains (denoted as “excited chain”), and the resulting transmitted primary pulses in both chains are considered, especially in the non-directly excited chain (denoted as “absorbing chain”). A new type of mixed nonlinear solitary pulses – shear waves is predicted for this system, leading to primary pulse equi-partition between chains, indicating strong energy exchange between two chains through the geometric coupling. Then, an analytical reduced model for primary pulse transmission is derived to study the strongly nonlinear acoustics in the small-amplitude approximation. In contrast to the full equations of motion the simplified model is re-scalable with energy and parameter-free, and is asymptotically solved by extending the one-dimensional nonlinear mapping technique. The nonlinear maps, which are derived for this two-dimensional system and governing the amplitudes of the mixed-type waves, accurately capture the primary pulse propagation in this system and predict the first occurrence of energy or pulse equi-partition in the network. Moreover, to confirm the theoretical results we experimentally test a series of two-dimensional granular networks, and prove the occurrence of strong energy exchanges leading to eventual pulse equi-partition between the excited and absorbing chains, provided that the number of beads is sufficiently large. Then we analyze the dynamics of a granular network composed of two semi-infinite, ordered homogeneous granular chains mounted on linear elastic foundations and coupled by weak linear stiffnesses under periodic excitation. We first review the acoustic filtering properties of linear and nonlinear semi-infinite periodic media containing two attenuation zones (AZs) and one propagation zone (PZ) in the frequency domain. In both linear and nonlinear systems, under suddenly applied, high-frequency harmonic excitations, “dynamic overshoot” phenomena are realized whereby coherent traveling responses are propagating to the far fields of these media despite the fact that the high frequencies of the suddenly applied excitations lie well within the stop bands of these systems. For the case of the linear system we show that the transient dynamic overshoot can be approximately expressed in terms of the Green’s function at its free end. A different type of propagating wave in the form of a “pure” traveling breather, i.e., of a single propagating oscillatory wavepacket with a localized envelope, is realized in the transient responses of a nonlinear granular network. The pure breather is asymptotically studied by a complexification/averaging technique, showing nearly complete but reversible energy exchanges between the excited and absorbing chains as the breather propagates to the far field. We analytically prove that the reason for this dynamic overshoot phenomenon in both linear and nonlinear networks is the high rate of application of the high-frequency harmonic excitation, which, in essence, amounts approximately to an impulsive excitation of the periodic medium. Verification of the analytical approximations with direct numerical simulations is performed. We further study passive pulse redirection and nonlinear targeted energy transfer in the aforementioned weakly coupled granular network. Periodic excitation in the form of repetitive half-sine pulses is applied to the excited chain. The frequency of excitation is within the pass band of the granular system. At the steady state nearly complete but reversible energy exchanges between the two chains are noted. We show that passive pulse redirection and targeted energy transfer from the excited to the absorbing chain can be achieved by macro-scale realization of the spatial analog of the Landau-Zener quantum tunneling effect. This is realized by finite stratification of the elastic foundation of the excited chain, and depends on the system parameters (e.g., the percentage of stratification) and on the parameters of the periodic excitation. We detect the existence of two distinct nonlinear phenomena in the periodically forced network; namely, (i) energy localization in the absorbing chain due to sustained 1:1 resonance capture leading to irreversible pulse redirection from the excited chain, and (ii) continuous energy exchanges in the form of nonlinear beats between the two chains in the absence of resonance capture. Our results demonstrate that steady state passive pulse redirection in these networks can be robustly achieved under periodic excitation. The final part of present work is concerned with propagating breathers in granular networks under impulsive excitation. We apply a complexification-averaging methodology leading to smooth slow flow reduced models of the dynamics to reveal the nature of 1:1 resonance at fundamental steady-state responses of the system. The primary aim of this analytical study is to provide a predictive way to excite the system at its resonance conditions. In addition to the fundamental resonance we numerically verify the occurrences of subharmonic steady-state responses in such granular networks. We experimentally detect the propagating breathers in a single chain mounted on elastic foundations. Our experimental measurements show good correspondence with the computational results which validate our previous theoretical predications. The results of this work contribute to the design of practical nonlinear acoustic metamaterials and provide a new avenue for understanding ofthe complex nonlinear dynamics of granular media

Tuning Methodology of Nonlinear Vibration Absorbers Coupled to Nonlinear Mechanical Systems.

A large body of literature exists regarding linear and nonlinear dynamic absorbers, but the vast majority of it deals with linear primary structures. However, nonlinearity is a frequency occurrence in engineering applications. Therefore, the present thesis focuses on the mitigation of vibrations of nonlinear primary systems using nonlinear dynamic absorbers. Because most existing contributions about their design rely on optimization and sensitivity analysis procedures, which are computationally demanding, or on analytic methods, which may be limited to small-amplitude motions, this thesis sets the emphasis on a tuning procedure of nonlinear vibration absorbers that can be computationally tractable and treat strongly nonlinear regimes of motion.The proposed methodology is a two-step procedure relying on a frequency-energy based approach followed by a bifurcation analysis. The first step, carried out in the free vibration case, imposes the absorber to possess a qualitatively similar dependence on energy as the primary system. This gives rise to an optimal nonlinear functional form and an initial set of absorber parameters. Based upon these initial results, the second step, carried out in the forced vibration case, exploits the relevant information contained within the nonlinear frequency response functions, namely, the bifurcation points. Their tracking in parameter space enables the adjustment of the design parameter values to reach a suitable tuning of the absorber.The use of the resulting integrated tuning methodology on nonlinear vibration absorbers coupled to systems with nonlinear damping is then investigated. The objective lies in determining an appropriate functional form for the absorber so that the limit cycle oscillation suppression is maximized.Finally, the proposed tuning methodology of nonlinear vibration absorbers may impose the use of complicated nonlinear functional forms whose practical realization, using mechanical elements, may be difficult. In this context, an electro-mechanical nonlinear vibration absorber relying on piezoelectric shunting possesses attractive features as various functional forms for the absorber nonlinearity can be achieved through proper circuit design. The foundation of this new approach are laid down and the perspectives are discussed

Mathematical and computational modelling for the design of pipe bends and compliant systems

This thesis is divided into three parts. In part I some theoretical and numerical processes are considered which arise when modelling the flow of a fluid through a pipe bend or deflector nozzle. These numerical processes include a new form of numerical integration and a finite element formulation which, it is suggested, could readily be extended to handle further realistic problems based on the pseudo three dimensional model chosen here. An introduction to nonlinear dynamics is included in part II leading towards a classification of bifurcational events in the light of recent advances in dynamics research. Most of the dynamical systems considered are dissipative such that the dynamic behaviour of the system decays onto a final steady state motion which may be modelled by a low order system of equations. In this way any resulting instability will adequately be described, qualitatively at least, by the low order bifurcation classified in part II. In part III the application of the geometrical theory of dynamical systems is used to study the wave driven motions of specified compliant offshore facilities with real data provided from structures currently in use in the offshore industry. In particular predictions are sought of any incipient jumps to resonance of the systems which might lead to potentially dangerous loads in the mooring lines or excessive displacements. Throughout the dynamics work stable steady state paths are closely followed and monitored so that any resulting bifurcation, including the possibility of chaotic behaviour, can be analysed with a view to its subsequent prediction

Identification, reduced order modeling and model updating of nonlinear mechanical systems

In this dissertation, we propose a new method for global/local nonlinear system identification, reduced order modeling and nonlinear model updating, applicable to a broad class of dynamical systems. The global aspect of the approach is based on analyzing the free and forced dynamics of the system in the frequency-energy domain through the construction of free decay or steady-state frequency-energy plots (FEPs). The local aspect of the approach considers specific damped transitions and leads to low-dimensional reduced order models that accurately reproduce these transitions. The nonlinear model updating strategy is based on analyzing the system in the frequency-energy domain by constructing Hamiltonian or forced and damped frequency-energy plots (FEPs). These plots depict the steady-state solutions of the systems based on their frequency-energy dependencies. The backbone branches, branches that correspond to 1:1 resonances, are calculated analytically (for fewer DOFs) or numerically (e.g., shooting method). The system parameters are then characterized and updated by matching these backbone branches with the frequency-energy dependence of the given system by using experimental/computational data. The main advantage of our approach is that we do not assume any type of nonlinearity model a priori, and the system model is updated solely based on numerical simulations and/or experimental results. As such, the approach is applicable to a broad class of nonlinear systems, including systems with strong nonlinearities and non-smooth effects, as will be shown in this dissertation. For larger scale systems, model reduction techniques (e.g., Guyan reduction) are applied to construct reduced order models of the system to which the aforementioned methods are applied

Dynamic stability of thin-walled structures : a semi-analytical and experimental approach

Buckling refers to a sudden large increase in the deformation of a structure due to a small increase of some external load. If this external load has a dynamic nature, (e.g. a harmonic load, shock load, a step load and/or a random load), such a sudden increase in deformations is denoted as dynamic buckling. Thinwalled structures are often met in engineering practice due to their favourable mass-to-stiffness ratio. Such structures are very susceptible to buckling and are often subjected to dynamic loading. However, fast (pre-) design tools for obtaining detailed insight in the dynamic response and the stability of thinwalled structures subjected to dynamic loading are still lacking. One of the research objectives of this thesis is, therefore, to develop (fast) modelling and analysis tools which give insight in the behaviour of dynamically loaded thinwalled structures. To illustrate and to test the abilities of the developed tools, a number of case studies are examined. The tools are developed for structures with a relatively simple geometry. The geometric simplicity of the structures allows to derive models with a relative low number of degrees of freedom which are, therefore, very suitable for extensive parameter studies (as essential during the design process of thin-walled structures). These models are symbolically derived using a Ritz method in combination with assumptions regarding geometric nonlinear (strain-displacement) relations and the effects of (in-plane) inertia. The resulting models, obtained from energy expressions, are sets of coupled ordinary differential equations which include stiffness nonlinearities and (sometimes) inertia and damping nonlinearities. The modelling approach is implemented in a generic manner in a symbolic manipulation software package, so that model variations can be easily performed. Furthermore, a set of designated numerical tools is combined (e.g. continuation tools for equilibria, periodic solutions and bifurcations, and numerical integration routines) to solve the analytically derived models in a computationally efficient manner. Using this semi-analytical (i.e. analytical-numerical) approach four case studies are performed which include the dynamic buckling of an arch type of structure due to shock loading, snap-through behaviour of a transversally, harmonically excited pre-buckled beam, and the dynamic buckling of a beam and a cylindrical shell structure, both with top mass, which are harmonically loaded in axial direction at their base. For all cases, the effects of several parameter variations are illustrated, including the effect of small deviations from the nominal geometry (i.e. geometric imperfections). For validation, the semi-analytical results are compared with results obtained using the computationally much more demanding finite element modelling technique. However, more important, for two cases (i.e. the axially excited beam and cylindrical shell structures carrying a top mass), the semi-analytical results are also compared with experimentally obtained results. For this purpose, a dedicated experimental set-up has been realized. For the beam structure, the experimental results are in good agreement with the semianalytical results whereas for the cylindrical shell structure, a qualitative match is obtained. It has been illustrated that the differences between the experimental results and the semi-analytical results for the cylindrical shell may be due to the strong dependency of the results with respect to the geometrical imperfections present in the shell. Next to the specific new insights obtained for each case considered, the major result of the thesis is the illustrated power of the semi-analytical approach to obtain practical relevant insights in the phenomena of dynamic buckling of thin-walled structures. In conclusion it can be stated that the semi-analytical approach is a valuable tool in the (pre-) design process of thin-walled structures under dynamic loading

Dynamical systems : mechatronics and life sciences

Proceedings of the 13th Conference „Dynamical Systems - Theory and Applications" summarize 164 and the Springer Proceedings summarize 60 best papers of university teachers and students, researchers and engineers from whole the world. The papers were chosen by the International Scientific Committee from 315 papers submitted to the conference. The reader thus obtains an overview of the recent developments of dynamical systems and can study the most progressive tendencies in this field of science

Vibration, Control and Stability of Dynamical Systems

From Preface: This is the fourteenth time when the conference “Dynamical Systems: Theory and Applications” gathers a numerous group of outstanding scientists and engineers, who deal with widely understood problems of theoretical and applied dynamics. Organization of the conference would not have been possible without a great effort of the staff of the Department of Automation, Biomechanics and Mechatronics. The patronage over the conference has been taken by the Committee of Mechanics of the Polish Academy of Sciences and Ministry of Science and Higher Education of Poland. It is a great pleasure that our invitation has been accepted by recording in the history of our conference number of people, including good colleagues and friends as well as a large group of researchers and scientists, who decided to participate in the conference for the first time. With proud and satisfaction we welcomed over 180 persons from 31 countries all over the world. They decided to share the results of their research and many years experiences in a discipline of dynamical systems by submitting many very interesting papers. This year, the DSTA Conference Proceedings were split into three volumes entitled “Dynamical Systems” with respective subtitles: Vibration, Control and Stability of Dynamical Systems; Mathematical and Numerical Aspects of Dynamical System Analysis and Engineering Dynamics and Life Sciences. Additionally, there will be also published two volumes of Springer Proceedings in Mathematics and Statistics entitled “Dynamical Systems in Theoretical Perspective” and “Dynamical Systems in Applications”