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Lyapunov exponents – practical computation
Fischer, Cyril ; Náprstek, Jiří
The Lyapunov exponents serve as numerical characteristics of dynamical systems, which measure possible divergence of nearby trajectories of the solution. In this way they express dependence of the dynamical system on initial conditions. However, the value of Lyapunov exponents consists in their ability to characterise deterministic chaos. The limiting process intrinsic in the definition of Lyapunov exponents, unfortunately, seriously complicates their computation. The short paper presents an overview of difficulties in numerical approaches to enumeration of Lyapunov exponents or at least the largest one and shows a promising method based on QR decomposition of the system Jacobian.
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Non-holonomic planar and spatial model of a ball-type tuned mass damping device
Náprstek, Jiří ; Fischer, Cyril
The area of tuned mass dampers is a wide field of inspiration for theoretical studies in non-linear dynamics and dynamic stability. The studies attempt to estimate behaviour of diverse damping devices and their reliability. The current paper deals with the response of a heavy ball rolling inside a spherical cavity under horizontal kinematic excitation. The non-linear system consists of six degrees of freedom with three non-holonomic constraints. The contact between the ball and the cavity surface is supposed to be perfect without any sliding. The mathematical model using the Appell-Gibbs function of acceleration energy is developed and discussed. Comparison with previous planar (SDOF) model which is based on the Lagrangian procedure is given. The system has an auto-parametric character and therefore semi-trivial solutions and their dynamic stability can be analysed. The most important post-critical regimes are outlined and qualitatively evaluated in resonance domain. Numerical experiments were performed when excitation frequency is slowly swept up and down to identify different modes of response. Some applications in civil engineering as a tuned mass damper, which can be used on slender structures, are mentioned. The proposed device is compared with a conventional pendulum damper. Strengths and weaknesses of both absorbers types are discussed.
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Non-linear normal modes in dynamics-continuous systems
Náprstek, Jiří ; Fischer, Cyril
The paper presents a continuation of an effort started last year, when the authors briefly informed about the Non-linear normal modes (NNM) concerning the version dealing with discrete systems. Although many features of the continuous formulation from the mathematical viewpoint are similar to the discrete case, a couple of specifics should be highlighted from the viewpoint of a real applicability of this tool to investigate particular dynamic systems. Three approaches are mentioned in the paper and the Galerkin-Petrov based procedure is outlined in more details. As a particular subject the cantilever prismatic beam is discussed. Non-linear normal modes for several amplitudes are shown to demonstrate the dependence of their shapes on the actual effective amplitude. Comparison with adequate linear counterpart is done.
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Approximation of the movement of the spherical pendulum
Fischer, Cyril ; Náprstek, Jiří
The horizontally driven two-degrees of freedom spherical pendulum is an auto-parametric system which exhibits a wide variety of response types. Depending on the amplitude and frequency of the excitation the response can vary from stationary to chaotic regime. It has been already shown that the response can be described using time variable parameters of an inscribed ellipse. Such a description has a potential to close the gap between description of stationary, quasi-periodic and chaotic types of response. This study presents methodology and results of an analysis of the experimentally measured data providing the time dependent parameters of a twisting ellipse. The obtained results are discussed and some open problems are indicated.
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Engineering mechanics 2015. 21st International conference May 11-14, 2015, Svratka, Czech Republic. Extended abstracts
Náprstek, Jiří ; Fischer, Cyril
The book of extended abstracts contains the papers presented at the 22nd International Conference on Engineering Mechanics, which has been held in Svratka resort in Czech Republic under auspices of the Czech Society of Mechanics and being a part of IFTOMM (The International Federation for the Promotion of Mechanism and Machine Science) activities. As it corresponds with character of the conference, this proceedings book consists of several topic oriented parts: Linear and Nonlinear Dynamics and Stability, Aeroelasticity, Hydroelasticity and Fluid Mechanics, Biomechanics, Fracture Mechanics, Mechatronics, Reliability of Structures, Mechanics of Solids, Thermomechanics. The publication represents a well-balanced overview of theoretical, numerical and experimental work on fundamental and applied studies.
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Post-critical multi-modal vibration of a continuous inverse pendulum type system
Náprstek, Jiří ; Fischer, Cyril
Auto-parametric effects are processes endangering structures during a seismic attack. Tall structures exposed to a strong vertical component of an earthquake excitation nearby the epicenter can collapse due to auto-parametric resonance effect. Vertical and horizontal response components being independent in the linear regime get into a complex interaction due to non-linear terms in post-critical regime. Two generally different types of the post-critical regimes are presented: (i) post-critical state with possible recovery; (ii) exponentially rising horizontal response leading to a collapse. A special attention is paid to processes of transition from semi-trivial to post-critical state in case of time limited excitation period as it concerns the seismic processes. Solution method combining analytical and numerical approaches is developed and used. Its applicability and shortcomings are commented. A few hints for engineering applications are given.
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EACWE 4. The Fourth European and African Conference on Wind Engineering. Book of extended abstracts
Náprstek, Jiří ; Fischer, Cyril
The proceedings show clearly the progress and recent developments of individual areas of the Wind Engineering, particularly in theoretical approaches, numerical analysis, experimental investigations, in situ measurements, wind climate and structures, wind energy, urban wind problems, design practice and in special problems of Wind Engineering encountered in Africa.
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Engineering Mechanics 2009. Book of extended abstacts
Náprstek, Jiří ; Fischer, Cyril
The objective of the conference is to provide a forum for researchers to discuss the state-of-the-art in Mechanics of Solids, Fluids and Thermomechanics, this particularly in connection with problems under investigation in the Czech Republic. Problems of Engineering Mechanics related to the contemplated topics were discussed, and their solutions presented, during the sessions of the conference.
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Engineering Mechanics 2006. Book of extended abstacts
Náprstek, Jiří ; Fischer, Cyril
The objective of the conference is to provide a forum for researchers to discuss the state-of-the-art in Mechanics of Solids, Fluids and Thermomechanics, this particularly in connection with problems under investigation in the Czech Republic. Problems of Engineering Mechanics related to the contemplated topics were discussed, and their solutions presented, during the sessions of the conference.
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