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Strategies for computation of Lyapunov exponents estimates from discrete data
Fischer, Cyril ; Náprstek, Jiří
The Lyapunov exponents (LE) provide a simple numerical measure of the sensitive dependence of the dynamical system on initial conditions. The positive LE in dissipative systems is often regarded as an indicator of the occurrence of deterministic chaos. However, the values of LE can also help to assess stability of particular solution branches of dynamical systems. The contribution brings a short review of two methods for estimation of the largest LE from discrete data series. Two methods are analysed and their freely available Matlab implementations are tested using two sets of discrete data: the sampled series of the Lorenz system and the experimental record of the movement of a heavy ball in a spherical cavity. It appears that the most important factor in LE estimation from discrete data series is quality of the available record.
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Probability density determination by means of Gibbs entropy probability density
Náprstek, Jiří ; Fischer, Cyril
A method of random response investigation of a nonlinear dynam-ical system is discussed. In particular, the solution of the probability density of a single/multi-degree of freedom (SDOF/MDOF) system response is investigated. Multiple stable equilibrium states with possible jumps of the snap-through type among them are considered. The system is Hamiltonian with weak damping excited by a set of non-stationary Gaussian white noises. The solution, which is based on the Gibbs principle of the maximum entropy of probability, can be employed in various branches of engineering. The search for the extreme of the Gibbs entropy functional is formulated as a constrained optimization problem. The secondary constraints follow from the Fokker-Planck equation (FPE) for the system considered or from the system of ordinary di_erential equations for the stochastic moments of the response derived from the relevant FPE
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Engineering mechanics 2018. Book of full texts
Fischer, Cyril ; Náprstek, Jiří
The proceedings contains papers presented at the 24th 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 consists of several topic oriented parts: Biomechanics, Fluid mechanics, Dynamics, Fracture mechanics, Kinematics, Mechatronics, Reliability of structures, Mechanics of solids, Techological processes, Thermodynamics. The volume represents a well-balanced overview of theoretical, numerical and experimental work on fundamental and applied studies.
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Forced movement of a ball in spherical cavity under kinematic excitation
Náprstek, Jiří ; Fischer, Cyril
In the paper the response of a heavy ball rolling inside a semi-spherical cavity under horizontal kinematic excitation is investigated. The system with six degrees of freedom with three non-holonomic constraints is considered. The contact between the ball and the cavity surface is supposed to be perfect without any sliding. The mathematical model using the Appel-Gibbs function of acceleration energy is developed and discussed. The most important post-critical regimes are outlined and qualitatively evaluated on the frequency axis. Numerical experiments have been performed when excitation frequency is slowly swept up and down. Results obtained by means of semi-analytical investigation and numerical simulation are evaluated and physically interpreted. Some applications in civil engineering as a tuned mass damper used on slender structures is outlined. Strengths and weaknesses of solution method are evaluated.
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Asymmetry of the response probability density of a system with parametric random noises
Náprstek, Jiří ; Fischer, Cyril
Recording of time variable processes is accompanied by various internal disturbing effects as a rule. They influence parameters of the measuring facility, transducer-device transmission, etc. These parasitic processes are usually of the random character and, consequently, they exercise as parametric noises. Moreover, the input signal mostly consists of a useful signal which can be taken as composed from deterministic and random additive part. Various combinations of these noises are the origin of random and also systematic measuring errors which can have under certain circumstances a cumulative character, deteriorate the output signal quality and can lead finally to the stochastic stability loss. These effects can be theoretically described using differential systems with stochastic coefficients and stochastic right hand side considering all input and output processes to be of the Markov type.
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Stochastic stability of the generalized van der Pol system under random additive excitation
Náprstek, Jiří ; Fischer, Cyril
The paper is motivated by a series of wind tunnel experiments investigating aeroelastic SDOF and TDOF section models of various shape and aeroelastic properties. It reveals that most of them can be theoretically modeled by the van der Pol – Duffing type equations or their combination adjusting degree of individual nonlinear terms or their coefficients. It should be emphasized that this character of the system response is very stable and is obvious in linear as well as in weakly nonlinear domain when the post-critical effect emerges. Moreover, many special effects identified by an experimental way evoke properties recognized in the pure theory of differential equations.
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Numerical assessment of stability of the ball vibration absorber
Fischer, Cyril ; Náprstek, Jiří
The ball vibration absorber (BVA) is in principle a heavy ball rolling inside a semi-spherical cavity. It is a convenient alternative to the usual pendulum-based dampers. The dynamics of the real ball absorber is significantly more complicated in comparison to the pendulum one. However, the numerical examination of the non-linear model of the BVA indicates that its resonance behaviour is similar to the spherical pendulum. Despite the theoretical differences, the resonance curves in both cases form similar patterns. It seems that detailed study of both separate cases can provide complementary results, whose parallels appear in the other model.
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Remarks on inverse of matrix polynomials
Fischer, Cyril ; Náprstek, Jiří
Analysis of a non-classically damped engineering structure, which is subjected to an external excitation, leads to the solution of a system of second order ordinary differential equations. Although there exists a large variety of powerful numerical methods to accomplish this task, in some cases it is convenient to formulate the explicit inversion of the respective quadratic fundamental system. The presented contribution uses and extends concepts in matrix polynomial theory and proposes an implementation of the inversion problem.
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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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