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Stochastic version of the arc-length method
Náprstek, Jiří ; Fischer, Cyril
The solution of a nonlinear algebraic system using the incremental method, based on pre-defined loading steps, fails in the vicinity of local extrema as well as around bifurcation points. The solution involved the derivation of the so-called ’Arc-Length’ method. Its essence lies in not incrementing the system parameter or any of the independent variables but rather the length of the response curve. The stochastic variant of this method allows for working with a system where system parameters include random imperfections. This contribution presents a variant that tracks the first two stochastic moments. Even in this simple case, interesting phenomena can be observed, such as the disappearance of the energy barrier against equilibrium jump due to random imperfections in the system.
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Solution methods for an aeroelastic problem with combined harmonic and stochastic excitation
Fischer, Cyril ; Náprstek, Jiří
Assessing responses in slender engineering structures facing both deterministic harmonic and stochastic excitation is often based on an approximation by the single-degree-of-freedom van der Pol-type nonlinear model. Determining the response probability density function involves solving the Fokker-Planck equation, which is generally a challenging task. Hence, semi-analytical and numerical methods come into play. This contribution reviews several possible techniques and spotlights the exponential-polynomial-closure method. The shown results are limited, as the paper reflects an early stage of the relevant research direction.
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Identification of quasiperiodic processes in the vicinity of the resonance
Fischer, Cyril ; Náprstek, Jiří
In nonlinear dynamical systems, strong quasiperiodic beating effects appear due to combination of self-excited and forced vibration. The presence of symmetric or asymmetric beatings indicates an exchange of energy between individual degrees of freedom of the model or by multiple close dominant frequencies. This effect is illustrated by the case of the van der Pol equation in the vicinity of resonance. The approximate analysis of these nonlinear effects uses the harmonic balance method and the multiple scale method.
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Consistency of mathematical and experimental model of the autoparametric system
Fischer, Cyril ; Náprstek, Jiří
This paper presents a long-overdue comparison of data obtained from experimental investigation of a spherical vibration absorber with the results of two mathematical models of the motion of a heavy sphere in a spherical surface. It shows that the danger posed by the potentially unstable self-parametric nature of the mathematical system is not too great in the case of realistic configurations, and that the values of the parameters describing the realistic structures remain within intervals corresponding to the stable behaviour of the absorber.
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Analyzing stochastic stability of a gyroscope through the stochastic Lyapunov function
Náprstek, Jiří ; Fischer, Cyril
The text delves into the application of first integrals in the construction of Lyapunov functions for analyzing the stability of dynamic systems in stochastic domains. It emphasizes the distinct characteristics of first integrals that warrant the introduction of additional constraints to ensure the essential properties required for a Lyapunov function. These constraints possess physical interpretations associated with system stability. The general approach to testing stochastic stability is illustrated using the example of a 3-degrees-of-freedom system representing a gyroscope.
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Adaptation of methods for cyclo-stationary processes for noisy structural health data
Fischer, Cyril ; Bayer, Jan ; Náprstek, Jiří ; Urushadze, Shota
In structural health analysis, various techniques, including indirect measurement via monitoring vehicles, often yield data with significant randomness and insufficient frequency separation. Conversely, the desired attributes under scrutiny are periodic in nature. Thus, methodologies designed to identify cyclo-stationary properties within noisy data can be adapted for such scenarios, assuming an adequate length of the recorded data.
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Nonstationary vibrations of rectangular plate excited by concentrated force with linearly variable frequency
Náprstek, Jiří ; Fischer, Ondřej
Many physical and technical problems can be modelled as a rectangular plate with certain edge-supports, loaded by forces with variable frequency (e.g. machines in industrial buildings, electrodynamic systems). Such systems can be solved numerically using finite element method, i. e. solving the system of ordinary differnetial equations in time. In some practical cases of frequency-variation this problem can be solved analytically - the advantage of such a procedure is the objectivity and the possibility of parametric analyses of the problem.
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Construction of the Lyapunov function reflecting the physical properties of the model
Náprstek, Jiří ; Fischer, Cyril
Practical experience shows that the random excitation component can affect the system response and its dynamic stability not only negatively but also positively. Such mechanisms are usually developed heuristically and are often not sufficiently justified theoretically. The paper presents a possibility of using the properties of first integrals for the construction of a Lyapunov function for the analysis of a dynamic system stability in the stochastic domain. In such case, the Lyapunov function itself contains information on the examined system and, consequently, it is able to provide a more detailed insight into the system stability properties. The procedure is illustrated by a nonlinear SDOF example.
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Analysis of van der Pol equation on slow time scale for combined random and harmonic excitation
Náprstek, Jiří ; Fischer, Cyril
Vortex shedding represents one of the most important processes that constantly attract the attention of experimental and theoretical research. A number of non-linear effects arise from the fluid-structure interaction. The non-stationary response in the vicinity of the lock-in region has a quasi-periodic character, beating frequency of which varies considerably with the distance from the lock-in frequency. This property is significantly affected by the assumption of combined random and harmonic excitation. This paper describes several details that contribute to the probabilistic characteristics of the system on a time-slow scale using partial response amplitudes.
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Application of first integrals in the construction of the Lyapunov function for the random response stability testing
Náprstek, Jiří ; Fischer, Cyril
The paper deals with a possibility of using the properties of first integrals for the construction of Lyapunov function for the analysis of a dynamic system stability in the stochastic domain. It points out certain characteristics of first integrals resulting in the necessity to introduce additional constraints to assure the principal properties of the Lyapunov function. A number of these constraints has their physical interpretation with reference to system stability. The advantage of this method constructing the Lyapunov function consists in the fact that the Lyapunov function itself contains information on the examined system and, consequently, it is not merely a positive definite function without any relation to the actual case concerned. The presented theory finds application in many dynamical systems. The procedure is illustrated by a nonlinear SDOF example.
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