International Journal of Structural Stability and Dynamics

The paper presents an application of the finite strip method to modeling of vibrations of the collecting electrodes, which are shells with large length (up to 16 m), width of 0.5 m and thickness of 0.002 m. The models and computer programs have been worked out and validated. Comparison of results obtained from numerical simulations and experimental measurements are presented and discussed. The equations of motion have been solved using methods for solution of sparse algebraic equations and Newmark method. The strip method has proved to be numerically effective. The programs enable us to carry out calculations for a system with several hundred thousands of degrees of freedom with time of analysis requiring thousand integration steps during less than 90 min on a PC computer. High numerical efficiency enables the geometrical parameters of the collecting electrodes to be selected in order to ensure large accelerations caused by a beater to be spread evenly over the surface of the electrodes. Conclusions concerning the influence of length of the collecting electrodes on the normal and tangentz accelerations are formulated.

The problem of steady convective mass transfer between a spherical drop and a flow with distributed chemical reaction is considered. It is investigated in case where both Peclet number and the rate constant of the chemical reaction tend to infinity. The quantity of rate constant of the chemical reaction and Peclet number is assumed to have a constant value. It is a boundary value problem for a quasilinear partial elliptical equation with a small parameter multiplying in higher derivatives. In the neighborhood of the saddle point the additional boundary layer arises. The asymptotics of solution is constructed in the neighborhood of the saddle point. The leading term of the asymptotic expansion of solution is constructed in the boundary layer near the rear stagnation point of the drop as the solution for the quasilinear ordinary differential equation.

Calculations of nonlinear displacements and vibrations of inhomogeneous loaded shells with developable principal surface by means of different analytical methods are represented. It is shown that solutions to these methods are the expansions of exact solution in the Taylor series for an independent variable, and in the particular case — for the powers of a natural parameter. A method that provides a polynomial asymptotic approximation of the exact solution of the general form and its meromorphic continuation based on 1D and 2D Padé approximations is proposed. Calculations of nonlinear deformation and stability of elastic flexible circular cylindrical shell under uniform external pressures and of free oscillations of simply supported stringer shell demonstrate the efficiency and accuracy of the proposed method.

Thermal processes occurring in the mechanical clutch or brake systems have a great influence on the strength of elements of these systems as well as on their dynamics. The contact problems exhibited by such systems include heat generated by dry friction contact surfaces. The contact dynamics in general depends on many system parameters, and it attracted attention of many researches focused on analysis of the mentioned phenomena in different kinds of mechanical systems like clutches, brakes, and others. In this work a mathematical model describing the processes of heat generation and its propagation in the mechanical friction clutch is presented. The presented model takes into account the unequal distribution of flux density of produced heat in the clutch, the thermal conductivity of materials of friction linings, and the heat transfer between the friction linings of clutch and its environments. The analyzed object is described by a set of algebraic linear homo- and heterogeneous equations, and it is derived using a computer numerical method. Many interesting numerical and experimental results are obtained, illustrated and discussed. Presented numerical results coincide with experimental data.

In this paper chaotic vibrations of flexible plates of infinite length are studied. The Kirchhoff–Love hypotheses are used to derive the nondimensional partial differential equations governing the plate dynamics. The finite difference method (FDM) and finite element method (FEM) are applied to validate the numerical results. The numerical analysis includes both standard (time histories, fast Fourier Transform, phase portraits, Poincaré sections, Lyapunov exponents) as well as wavelet-based approaches. The latter one includes the so called Gauss 1, Gauss 8, Mexican Hat and Morlet wavelets. In particular, various plate dynamical regimes including the periodic, quasi-periodic, sub-harmonic, chaotic vibrations as well as bifurcations of the plate are illustrated and studied. In addition, the convergence of numerical results obtained via different wavelets is analyzed.

The inverted elastic beam is proposed as an energy harvester. The beam has a tip mass and piezoelectric layers which transduce the bending strains induced by the stochastic horizontal displacement into electrical charge. The efficiency of this nonlinear device is analyzed, focusing on the region of stochastic resonance where the beam motion has a large amplitude. Increasing the noise level allows the motion of the beam system to escape from single well oscillations and thus generate more power.

The first part of the paper contains a short review of a series of published papers in the area of system dynamics with coupled rotations as well as of a series of author's various published research results in the area of vector method based on the mass inertia moment vectors and corresponding deviational vector components and vector rotators for the pole and oriented axis, introduced and defined by K. Hedrih in 1991. The vector with principal importance is vector of the rigid body mass inertia moment at the point and for the axis oriented by the unit vector, and with a corresponding component vector of the rigid body mass deviational moment for the rotation axis through the same pole. Second part presents the vector method based on mass moment vectors and vector rotators coupled for pole and oriented nonintersecting axes in application for obtaining vector expressions for kinetic parameters of a rigid body dynamics with multi-coupled rotations around nonintersecting axes. A complete analysis of obtained vector expressions for linear momentum and angular momentum and their corresponding derivatives gives us a series of the kinematical vector rotators around the corresponding directions determined by axes of the rigid body coupled multi-rotations around nonintersecting axes. Series of the theorems are defined.

Recurrence of states, in the meaning that states are arbitrary close after some time, is a fundamental property of deterministic dynamical systems and is typical of nonlinear or chaotic systems. The main idea of the recurrence plots (RPs) method consists in revealing all the times when the phase space trajectory of the dynamical system visits roughly the same area in the phase space. Such recurrence of states is marked within a two-dimensional matrix with ones and zeros. Graphical representation of such a matrix is known in the literature as a recurrence plot. The paper concerns results of research into dynamic properties and determinism of selected mechanical systems. In order to analyze measured responses (time histories) of the considered systems with the use of the recurrence plots, the phase space of these systems was reconstructed by delay embedding. The smallest sufficient embedding dimensions were estimated with the use of the false nearest neighbors algorithm. For the purposes of time delay determining the method of mutual information function was used. All the necessary computations were performed with the use of the CRP Toolbox for MATLAB.

A high speed camera was used for interferometry visualization (in different phases of the motion) of the fluttering NACA0015 profile supported in a translational and rotational manner. First, the simplified mathematical model of the support of investigated profile was identified from minimum least squares differences between modeled and measured system responses. A special graphical Matlab procedure was proposed for evaluation of interferograms. Kinematic analysis defining motion of the profile as a function of time was obtained by a regression using the least squares method. Numerical integration of pressure functions around the airfoil surface allows for calculation of the resulting aerodynamic forces and moments.

Dynamics of an aerodynamic pendulum in steady airflow is studied using different modeling approaches: phenomenological model, where the unsteady effects are simulated using an oscillator attached to the pendulum, and modified method of discrete vortices (MMDV). It is shown that behavior of the pendulum according to the attached oscillator model agrees with data given by MMDV for a wide enough range of parameter values, and the applicability domain of the attached oscillator model is larger than that of the traditional quasi-steady approach. Stability of the "along the flow" equilibrium is studied depending on system parameters. It is shown, in particular, that this equilibrium loses its stability when the moment of inertia of the pendulum increases, if the rotation point is located in a certain range of distances upstream from the center of pressure.

A vast number of papers are devoted to studying the complete integrability of equations of four-dimensional rigid-body motion. Although in studying low-dimensional equations of motion of quite concrete (two- and three-dimensional) rigid bodies in a nonconservative force field, the author arrived at the idea of generalizing the equations to the case of a four-dimensional rigid body in an analogous nonconservative force field. As a result of such a generalization, the author obtained the variety of cases of integrability in the problem of body motion in a resisting medium that fills the four-dimensional space in the presence of a certain tracing force that allows one to reduce the order of the general system of dynamical equations of motion in a methodical way.

The boundary value problem concerning the free vibrations of a slender system subjected to a specific load has been formulated and solved in this work. Heads with parabolic contour have been used to realize the specific load for the first time. The boundary value problem has been formulated using Hamilton's principle. The critical load and the characteristic curves in the plane load–natural frequency have been determined on the basis of the kinetic criterion of stability. Numerical calculations have been assigned to different values of the parameters of the considered system for which the parabolic parameter and the parameter of the roller radius are ranked. The roller is the head of the receiving load. The accuracy of the mathematical model was confirmed on the basis of experimental research based on frequency and modal analysis.

In this paper, the theoretical and calculation model of the forest crane and the lifted load is presented. The model enables the simulation of the motion of the load carried by a forest crane while taking into account the elastic deformations of the boom. The lifted load has been modeled as a 3D rigid body. The application of such a load model enables the simplification and enhancement of the calculation algorithm. The equations describing the coupled motion of the system load and machine elements are presented. The kinematic model enables the analysis of the basic motion of the load as a response of the system to operational control of the forest crane. In order to determine the motion parameters it was assumed that all system elements are rigid. The lifted load is also treated as a rigid body and its motion is the result of the movement of the load suspension point and the dynamic interactions generated during the motion of the system. The presented sample results from the numerical calculations were obtained from the Matlab system. The initial problem has been solved by the Runge–Kutta method. The numerical program has been developed on the basis of the presented model which allows analyzing the motion of load arising from crane operating mechanisms.