Nonlinear oscillations, bifurcations and chaos in ocean mooring systems

Complex nonlinear and chaotic responses have been recently observed in various compliant ocean systems. These systems are characterized by a nonlinear mooring restoring force and a coupled fluid-structure interaction exciting force. A general class of ocean mooring system models is formulated by incorporating a variable mooring configuration and the exact form of the hydrodynamic excitation. The multi-degree of freedom system, subjected to combined parametric and external excitation, is shown to be complex, coupled and strongly nonlinear. Stability analysis by a Liapunov function approach reveals global system attraction which ensures that solutions remain bounded for small excitation. Construction of the system's Poincare map and stability analysis of the map's fixed points correspond to system stability of near resonance periodic orbits. Investigation of nonresonant solutions is done by a local variational approach. Tangent and period doubling bifurcations are identified by both local stability analysis techniques and are further investigated to reveal global bifurcations. Application of Melnikov's method to the perturbed averaged system provides an approximate criterion for the existence of transverse homoclinic orbits resulting in chaotic system dynamics. Further stability analysis of the subharmonic and ultraharmonic solutions reveals a cascade of period doubling which is shown to evolve to a strange attractor. Investigation of the bifurcation criteria obtained reveals a steady state superstructure in the bifurcation set. This superstructure identifies a similar bifurcation pattern of coexisting solutions in the sub, ultra and ultrasubharmonic domains. Within this structure strange attractors appear when a period doubling sequence is infinite and when abrupt changes in the size of an attractor occur near tangent bifurcations. Parametric analysis of system instabilities reveals the influence of the convective inertial force which can not be neglected for large response and the bias induced by the quadratic viscous drag is found to be a controlling mechanism even for moderate sea states. Thus, stability analyses of a nonlinear ocean mooring system by semi-analytical methods reveal the existence of bifurcations identifying complex periodic and aperiodic nonlinear phenomena. The results obtained apply to a variety of nonlinear ocean mooring and towing system configurations. Extensions and applications of this research are discussed

A study of poststenotic shear layer instabilities

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Nonlinear Normal Modes for Vibrating Mechanical Systems. Review of Theoretical Developments

International audienceTwo principal concepts of nonlinear normal vibrations modes (NNMs), namely the Kauderer–Rosenberg and Shaw–Pierre concepts, are analyzed. Properties of the NNMs and methods of their analysis are presented. NNMs stability and bifurcations are discussed. Combined application of the NNMs and the Rauscher method to analyze forced and parametric vibrations is discussed. Generalization of the NNMs to continuous systems dynamics is also described

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

Chaotic response and stability of offshore equipment

This investigation deals with the chaotic rocking response and overturning stability of offshore equipment subjected to base excitations due to wave-induced motions of the supporting compliant offshore structural system. The equipment compliant structural system is modelled as a free-standing rigid object subjected to horizontal and vertical base accelerations. Three of the major goals of this study are: (1) to identify the major sources of nonlinearity and sensitivity which make the response of the objects difficult to predict and experiments with identical set up and excitations unrepeatable; (2) to develop analytical and quantitative measures to characterize the sensitivity of the response using modern geometric methods; and (3) to determine the relative influence of the various nonlinearities and system parameters on the rocking behavior and overturning stability. Four major sources of nonlinearity are examined in detail. The first source is due to the transition of governing equations after impact when the center of rotation changes from one edge to the other. The second is due to the abrupt change in angular velocity at impact and the associated energy dissipation. The third is due to the geometric effect of the finite slenderness ratio of the object. The fourth is due to the coupling of the vertical (parametric) excitation with the rocking response. To isolate the influence of each source of nonlinearity on the chaotic response and overturning stability, several realistic approximate models containing different combinations of the nonlinearities are examined. Analytical and numerical procedures are developed to determine the motion of rigid objects. It is demonstrated that the nonlinearity associated with the transition of governing equations after impact produces responses that cannot be predicted by classical nonlinear stability analysis. Modern geometric methods are needed to delineate these unanticipated responses. Two new types of stable responses are discovered. In addition to the periodic and overturning responses predicted by classical stability analysis, the Melnikov method and numerical results show the existence of quasi-periodic, and chaotic responses. The stability region of each of the four types of responses --periodic, quasi-periodic, chaotic, and overturning -- are found to be sensitively dependent on the damping, geometry, and the parametric excitation nonlinearities as well as initial conditions and excitation amplitude and frequency parameters. It is found that although the individual responses are very sensitive to small changes in the system and excitation parameters, the following general trends are observed: (1) increasing damping decreases the stability region of the chaotic responses; (2) increasing slenderness ratio increases the stability region of the chaotic responses; and (3) increasing relative magnitude of parametric excitation increases the stability region of the chaotic responses

Applicable Solutions in Non-Linear Dynamical Systems

From Preface: The 15th International Conference „Dynamical Systems - Theory and Applications” (DSTA 2019, 2-5 December, 2019, Lodz, Poland) gathered 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 great effort of the staff of the Department of Automation, Biomechanics and Mechatronics of the Lodz University of Technology. 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 event was attended by over 180 researchers from 35 countries all over the world, who decided to share the results of their research and experience in different fields related to dynamical systems. This year, the DSTA Conference Proceedings were split into two volumes entitled „Theoretical Approaches in Non-Linear Dynamical Systems” and „Applicable Solutions in Non-Linear Dynamical Systems”. In addition, DSTA 2019 resulted in three volumes of Springer Proceedings in Mathematics and Statistics entitled „Control and Stability of Dynamical Systems”, „Mathematical and Numerical Approaches in Dynamical Systems” and „Dynamical Systems in Mechatronics and Life Sciences”. Also, many outstanding papers will be recommended to special issues of renowned scientific journals.Cover design: Kaźmierczak, MarekTechnical editor: Kaźmierczak, Mare

Nonlinear aeroelastic analysis of aircraft wing-with-store configurations

The author examines nonlinear aeroelastic responses of air vehicle systems. Herein, the governing equations for a cantilevered configuration are developed and the methods of analysis are explored. Based on the developed nonlinear bending-bending-torsion equations, internal resonance, which is possible in future air vehicles, and the possible cause of limit cycle oscillations of aircraft wings with stores are investigated. The nonlinear equations have three types of nonlinearities caused by wing flexibility, store geometry and aerodynamic stall, and retain up to third-order nonlinear terms. The internal resonance conditions are examined by the Method of Multiple Scales and demonstrated by time simulations. The effect of velocity change for various physical parameters and stiffness ratio is investigated through bifurcation diagrams derived from Poinar´e maps. The dominant factor causing limit cycle oscillations is the stiffness ratio between in-plane and out-of-plane motion

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”

Nonlinear aeroelastic analysis of aircraft wing-with-store configurations

The author examines nonlinear aeroelastic responses of air vehicle systems. Herein, the governing equations for a cantilevered configuration are developed and the methods of analysis are explored. Based on the developed nonlinear bending-bending-torsion equations, internal resonance, which is possible in future air vehicles, and the possible cause of limit cycle oscillations of aircraft wings with stores are investigated. The nonlinear equations have three types of nonlinearities caused by wing flexibility, store geometry and aerodynamic stall, and retain up to third-order nonlinear terms. The internal resonance conditions are examined by the Method of Multiple Scales and demonstrated by time simulations. The effect of velocity change for various physical parameters and stiffness ratio is investigated through bifurcation diagrams derived from Poinar´e maps. The dominant factor causing limit cycle oscillations is the stiffness ratio between in-plane and out-of-plane motion