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Analytical approach of slender structure vibration due to random component of the wind velocity
Náprstek, Jiří ; Hračov, Stanislav
Along wind random vibration of slender structures represents one of the most important aeroelastic effects resulting from wind - structure interaction. The theoretical model being based on one-dimensional elements with continuously distributed mass and stiffness has been introduced in this paper. The system has been considered to be linear self-adjoint with strongly non-proportional linear damping due to both material of the structure and presence of vibration dampers. The additive random excitation continuously distributed in time and space is Gaussian, therefore the response is Gaussian as well. Consequently, mathematical mean value and correlation function are satisfactory for the full description of the generalized solution of the respective PDE in the stochastic meaning. The general results have been obtained mostly in the form of analytical formulae for important cases of input spectral densities. A numerical example dealing with real structure is presented.
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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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Vibration analysis of visco-elastically coupled beams
Hračov, Stanislav ; Náprstek, Jiří
The paper presents a detail analysis of the forced vibration. The dynamic behaviour of the system is investigated parametrically to cover all possible effects of individual subsystems. Damping effectiveness is also analysed for several values of viscosity of the interlayer and damping of the particular beams. The recommendations and limitations for the application of the system are presented and discussed.
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Health monitoring of building structures: preliminary considerations on a case study
Bayer, Jan ; Urushadze, Shota ; Zíma, Pavel
Ageing, erosion, local damage or increase of the operating loads can decrease the reliability and load bearing capacity of building structures. Vibration monitoring may provide a promising solution of this problem, however robust health monitoring techniques are needed before wider practical applications. The presented analytical and experimental case study on a simple supported steel beam is an introductory practical investigation into this topic. It shows the sensitivity of chosen parameters to artificially induced damage and outlines possible directions or further research in conclusions. Special attention is paid to simultaneous monitoring of deformations and strains which can provide us with a useful information about the stress redistribution in monitored or tested structures.
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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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