
Random response of a dynamic system under polynomial of a white noise
Náprstek, Jiří ; Fischer, Cyril
Many types of external additive random excitation of dynamic systems admit to be modelled as a combination of powers of a Gaussian noise. Such a type of excitation produces a nonGaussian response even if the dynamic system is linear and the excitation is additive only. Although the excitation as a whole is nonGaussian, the problem can be transformed into the form of a linear system with an additive and multiplicative white noise excitation which _nally produces a nonGaussian response. The general method of transformation, the respective FPK equation, basic stochastic moments of the response, and a demonstrative example are discussed.


System response with random imperfections in coefficients on the space of realizations
Náprstek, Jiří ; Fischer, Cyril
The contribution is concerned with the analysis of the simultaneous effect of a random perturbation and a white noise in the coefficient of the system. The excitation of the system of the 1st order is described by the sum of a deterministic signal and an additive white noise which is partly correlated with the parametric noise. The random perturbation in the parameter is considered as a statistics in a set of realizations. It reveals that the density of probability of perturbations must be limited in the phase space, otherwise the system would lose the stochastic stability in probability. The width of the permissible zone depends heavily on the intensity of the parametric noise, the extent of correlation with the additive excitation noise and the type of probability density. The general explanation is demonstrated on cases of normal, uniform and truncated normal densities of probability.


Sensitivity of the generalized van der Pol equation to sub or superharmonic resonance
Fischer, Cyril ; Náprstek, Jiří
Vortex induced vibrations in aeroelasticity of slender structures are often described using the generalized van der Pol equation. This equation provides a good analytical model which is able to reproduce the complex nonstationary behaviour of the structure. Due to the nonlinear character of the underlying physical system, the effect of sub or superharmonic synchronization can be encountered. The contribution presents results of a numerical study aimed at detailed identification of sub or superharmonic resonance effects in the model. This way it supplements the previous works by the authors.


Rapid research with computer algebra systems
Fischer, Cyril
Computer algebra systems (CAS) are gaining popularity not only among young students and scholars but also as a tool for serious work. These highly complicated software systems, which used to just be regarded as toys for computer enthusiasts, have reached maturity. Nowadays such systems are available on a variety of computer platforms, starting from freelyavailable online services up to complex and expensive software packages. The aim of this review paper is to show some selected capabilities of CAS and point out some problems with their usage from the point of view of 25 years of experience.


Local stabilization of the quasiperiodic response of the generalized van der Pol oscillator
Fischer, Cyril ; Náprstek, Jiří
The generalized van der Pol equation is often used for description of various effects originating in the aeroelasticity of large slender engineering structures. This applies mainly to the quasiperiodic beatings that can be encountered especially in lockin regimes when the vortex frequency becomes close to the structure eigenfrequency with a small detuning. The current paper presents numerical analysis of influence of the subor superharmonic excitation on the character of the response of a generalized van der Pol oscillator. This way it complements two previous papers of the authors dealing with stability analysis of certain types of the stationary periodic or quasiperiodic response of the system under study.


Stochastic resonance in dynamics and related disciplines
Náprstek, Jiří ; Fischer, Cyril
Stochastic resonance (SR) is a phenomenon which can be observed in some nonlinear dynamic systems under combined excitation including deterministic harmonic force and random noise. This phenomenon was observed the first in the early 1940s when investigating the Brownian motion. Later several disciplines in optics, plasma physics, biomedicine and social sciences encountered effects of this type. However, the actual discovery and start of intensive period of investigation is dated in early 1980s when the idea of SR initiated remarkable inter disciplinary interest including most areas of physics, chemistry and neurophysiology with a significant overlap to engineering and industrial area. Promising opportunities to employ SR in mechanics emerged only recently to model certain postcritical effects in nonlinear dynamics and simultaneously to develop new vibration damping devices, energy harvesting facilities, sophisticated measuring technics and others. The aim of the paper is to present information about a new challenging discipline offering a large field of basic research and possibilities for practical applications.


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.


Probability density determination by means of Gibbs entropy probability density
Náprstek, Jiří ; Fischer, Cyril
A method of random response investigation of a nonlinear dynamical system is discussed. In particular, the solution of the probability density of a single/multidegree of freedom (SDOF/MDOF) system response is investigated. Multiple stable equilibrium states with possible jumps of the snapthrough type among them are considered. The system is Hamiltonian with weak damping excited by a set of nonstationary 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 FokkerPlanck 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


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 wellbalanced overview of theoretical, numerical and experimental work on fundamental and applied studies.


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 semispherical cavity under horizontal kinematic excitation is investigated. The system with six degrees of freedom with three nonholonomic 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 AppelGibbs function of acceleration energy is developed and discussed. The most important postcritical 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 semianalytical 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.
