Jiří Náprstek

Experimental Set-Up for Advanced Aeroelastic Tests on Sectional Models

This article describes an original and multipurpose experimental set-up for the analysis of complex linear and non-linear aspects of aero-elastic behaviour of beam cross-sections. The apparatus meets rigorous theoretical assumptions and allows very precise and quick adjustment of stiffness and mass of a cross-section, which is not always possible with the traditional “parallel spring-supported bridge” approach used by many researchers. The principal advantages are described together with key construction details. Examples of the large amplitude non-linear response are presented, to illustrate the capacity and usefulness of the stand.

Finite element method analysis of Fokker–Plank equation in stationary and evolutionary versions

Abstract The problems that often arise in stochastic dynamics can be investigated using the Fokker–Planck (FP) equation. The response of a such systems being subjected to additive and/or multiplicative random noise is represented by probability density function (PDF) that gives the full information about a response random character. Various analytic and semi-analytic solution methods have been developed for various systems to obtain results requested. However numerical approaches offer a powerful alternative. In particular the Finite Element Method (FEM) seems to be very effective. A couple of single dynamic linear/non-linear (Duffing and Van Der Pol type) systems under additive and multiplicative random excitations are discussed using FEM as a solution tool of the FP equation. The resulting PDFs are analyzed and if the analytic results exist mutually compared.

Non-linear Model of a Ball Vibration Absorber

Wind excited vibrations of slender structures such as towers, masts or certain types of bridges can be reduced using passive or active vibration absorbers. If there is available only a limited vertical space to install such a device, a ball type of absorber can be recommended. In general, it is a semi-spherical horizontal dish in which a ball of a smaller diameter is rolling. Ratio of both diameters, mass of the rolling ball, quality of contact surfaces and other parameters should correspond with characteristics of the structure. The ball absorber is modelled as a holonomous system. Using Lagrange equations of the second type, governing non-linear differential system is derived. The solution procedure combines analytical and numerical processes. As the main tool for dynamic stability investigation the 2nd Lyapunov method is used. The function and effectiveness of the absorber identical with those installed at the existing TV towers was examined in the laboratory of the Institute of Theoretical and Applied Mechanics. The response spectrum demonstrates a strongly non-linear character of the absorber. The response amplitudes at the top of a TV tower with ball absorber were reduced to 15–40 % of their original values.

Non-stationary response of structures excited by random seismic processes with time variable frequency content

Seismic random processes are characterized by high non-stationarity and, in most cases, by a marked variability of frequency content. The hypothesis modeling seismic signal as a simple product of a stationary signal and a deterministic modulation function, consequently, is hardly ever applicable. Several mathematical models aimed at expressing the recorded process by means of a system of stationary random processes and deterministic amplitude and frequency modulations are proposed. Models oriented into the frequency domain with subsequent response analysis based on integral spectral resolution and models oriented into the time domain based on the multicomponent resolution are investigated. The resolution into individual components (non-stationary signals) is carried out by three methods. The resolution into intrinsic mode functions seems to possess the best characteristics and yields results almost not differing from the results obtained by stochastic simulation. An example of the seismic response of an existing bridge obtained by two older models and three variants of multicomponent resolution is given.

Theoretical background and implementation of the finite element method for multi-dimensional Fokker–Planck equation analysis

Abstract Fokker–Planck equation is one of the most important tools for investigation of dynamic systems under random excitation. Finite Element Method represents very effective solution possibility particularly when transition processes are investigated or more detailed solution is needed. However, a number of specific problems must be overcome. They follow predominantly from the large multi-dimensionality of the Fokker–Planck equation, shape of the definition domain and usual requirements on the nature of the solution which are out of a conventional practice of the Finite Element employment. Unlike earlier studies it is coming to light that multi-dimensional simplex elements are the most suitable to be deployed. Moreover, new original algorithms for the multi-dimensional mesh generating were developed as well as original procedure of the governing differential and algebraic systems assembling and subsequent analysis. Finally, an illustrative example is presented together with aspects typical for the problem with large multi-dimensionality.

COMPARISON OF NUMERICAL AND SEMI-ANALYTICAL SOLUTION OF A SIMPLE NON-LINEAR SYSTEM IN STATE OF THE STOCHASTIC RESONANCE

Stochastic resonance (SR) is a phenomenon which can be observed at some nonlinear dynamic systems under combined excitation including deterministic harmonic force and random noise. The Duffing single degree of freedom oscillator is treated and the Gaussian white noise as the random excitation component is considered. Mathematical basis of this phenomenon follows from properties of the Duffing system with negative linear part of the stiffness. Under certain combinations of the system and excitation parameters the SR can emerge. It manifests by stable periodic hopping between two nearly constant limits perturbed by random noises. SR is observed and practically used in a number of disciplines in physics, biophysics, chemistry, etc. However it seems to be promising as a theoretical model of several aeroelastic post-critical effects arising at a prismatic beam in a cross flow. Three independent theoretical solution methods have been addressed and tested in order to compare results of the system response. The first kind is semi-analytic dealing with the relevant Fokker-Planck Equation (FP). It is solved by means of the stochastic moment procedure. The multiharmonic non-stationary solution of the Probability Density Function (PDF) is expected. The Galerkin approach is adopted. The second way is based on the FEM solution of the FP equation. It is analyzed in an original evolutionary form which enables an analysis of transition effects starting the Dirac type initial conditions. The last procedure represents simulation regarding the original Duffing or relevant Ito stochastic system. Comparison of results provided by the above three methods has revealed appropriate domains of their application to particular problems regarding a preliminary analysis or careful detailed inspection in specific small domains in final stage of an engineering system design.

Maximum Entropy Probability Density Principle in Probabilistic Investigations of Dynamic Systems

In this study, we consider a method for investigating the stochastic response of a nonlinear dynamical system affected by a random seismic process. We present the solution of the probability density of a single/multiple-degree of freedom (SDOF/MDOF) system with several statically stable equilibrium states and with possible jumps of the snap-through type. The system is a Hamiltonian system with weak damping excited by a system of non-stationary Gaussian white noise. The solution based on the Gibbs principle of the maximum entropy of probability could potentially be implemented 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 differential equations for the stochastic moments of the response derived from the relevant FPE. In terms of the application type, this strategy is most suitable for SDOF/MDOF systems containing polynomial type nonlinearities. Thus, the solution links up with the customary formulation of the finite elements discretization for strongly nonlinear continuous systems.

ANALYTICAL APPROACH OF SLENDER STRUCTURE VIBRATION DUE TO RANDOM COMPONENT OF THE WIND VELOCITY

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. Jiřı́ Náprstek and Stanislav Hračov

Multi-dimensional Fokker-Planck equation analysis using the modified finite element method

The Fokker-Planck equation (FPE) is a frequently used tool for the solution of cross probability density function (PDF) of a dynamic system response excited by a vector of random processes. FEM represents a very effective solution possibility, particularly when transition processes are investigated or a more detailed solution is needed. Actual papers deal with single degree of freedom (SDOF) systems only. So the respective FPE includes two independent space variables only. Stepping over this limit into MDOF systems a number of specific problems related to a true multi-dimensionality must be overcome. Unlike earlier studies, multi-dimensional simplex elements in any arbitrary dimension should be deployed and rectangular (multi-brick) elements abandoned. Simple closed formulae of integration in multi-dimension domain have been derived. Another specific problem represents the generation of multi-dimensional finite element mesh. Assembling of system global matrices should be subjected to newly composed algorithms due to multi-dimensionality. The system matrices are quite full and no advantages following from their sparse character can be profited from, as is commonly used in conventional FEM applications in 2D/3D problems. After verification of partial algorithms, an illustrative example dealing with a 2DOF non-linear aeroelastic system in combination with random and deterministic excitations is discussed.

Beating Effects in a Single Nonlinear Dynamical System in a Neighborhood of the Resonance

The exact coincidence of external excitation and basic eigen-frequency of a single degree of freedom (SDOF) nonlinear system produces stationary response with constant amplitude and phase shift. When the excitation frequency differs from the system eigen-frequency, various types of quasi-periodic response occur having a character of a beating process. The period of beating changes from infinity in the resonance point until a couple of excitation periods outside the resonance area. Theabove mentioned phenomena have been identified in many papers including authors’ contributions. Nevertheless, investigation of internal structure of a quasi-period and its dependence on the difference of excitation and eigen-frequency is still missing. Combinations of harmonic balance and small parameter methods are used for qualitative analysis of the system in mono- and multi-harmonic versions. They lead to nonlinear differential and algebraic equations serving as a basis for qualitativeanalytic estimation or numerical description of characteristics of the quasi-periodic system response. Zero, first and second level perturbation techniques are used. Appearance, stability and neighborhood of limit cycles is evaluated. Numerical phases are based on simulation processes and numerical continuation tools. Parametric evaluation and illustrating examples are presented.