Vibrations of a Cantilevered Thick Plate

It has been for the first time that an analytical solution to the problem of free vibrations of a cantilevered thick orthotropic plate is presented. This problem is quite cumbersome for using the exact methods of the theory of elasticity; therefore, methods based on the variational approach were developed to solve it. The paper suggests using the superposition method to construct a general solution of the vibration equations of a plate in the series form of particular solutions obtained with the help of a variables separation. The particular solutions of one of the coordinates are built in the form of trigonometric functions of a special type (modified trigonometric system). The constructed solution, in contrast to the solutions known in the literature on the basis of the variational approach, accurately satisfies the equations of vibrations. The use of a modified trigonometric system of functions makes it possible to obtain uniform formulas for even and odd vibration shapes and to reduce the quantity of boundary conditions on the plate sides from twelve to nine ones, while five of the nine boundary conditions are also accurately satisfied. The structure of the presented solution on the plate boundary is such that, each of the kinematic or force characteristics of the plate is represented as a sum of two series, i.e. a trigonometric series and a series in hyperbolic functions. Remaining boundary conditions make it possible to obtain an infinite system of linear algebraic equations with respect to the unknown coefficients of the series representing the solution. The convergence of the solution by the reduction method of the infinite system is investigated numerically. Examples of the numerical implementation are given; numerical studies of the spectrum of natural frequencies of the cantilevered thick plate were carried out based on the obtained solution, both with varying elastic characteristics of the material and with varying geometric parameters.

Analysis of chaotic characteristics of trigonometric function system

Chaos, as an important subject of nonlinear science, plays an important role in solving problems in both natural sciences and social sciences such as the fields of secure communications, fluid motion, particle motion and so on. Aiming at this problem, this paper proposes a nonlinear dynamic system composed of product trigonometric functions and studies its chaotic characteristics. Through the mathematical derivation of the system’s period, the analysis of the necessary conditions at the fixed point, the experimental drawing of the Lyapunov exponential graph and the branch graph of the system, it is proved that the system has larger chaotic interval and stronger chaotic characteristics. The parameters of the proposed dynamic system are generated randomly, and then the chaotic sequence can be generated. The chaotic sequence is used to encrypt the digital image, a good encryption effect is obtained, and there is a large key space. At the same time, the motion of the particles in the space magnetic field is simulated, which further proves that the trigonometric system has strong chaotic characteristics.

Cornu Spirals and the Triangular Lacunary Trigonometric System

This work is intended to directly supplement the previous work by Coutsias and Kazarinoff on the foundational understanding of lacunary trigonometric systems and their relation to the Fresnel integrals, specifically the Cornu spirals [Physica 26D (1987) 295]. These systems are intimately related to incomplete Gaussian summations. The current work provides a focused look at the specific system built off of the triangular numbers. The special cyclic character of the triangular numbers modulo m carries through to triangular lacunary trigonometric systems. Specifically, this work characterizes the families of Cornu spirals arising from triangular lacunary trigonometric systems. Special features such as self-similarity, isometry, and symmetry are presented and discussed.

The Problem of Damping the Transverse Oscillations on a Longitudinally Moving String

Introduction. The problem under consideration is relevant to production processes associated with the longitudinal movement of materials, for example, for producing paper webs. For these processes transverse disturbances, which in the vertical section are described by the hyperbolic equation of a longitudinally moving string, are extremely undesirable. That gives the problem of damping these oscillations within a finite time. Materials and Methods. To solve the problem of damping the oscillations, the authors suggest reducing it to the trigonometric problem of the moments at an arbitrary time interval. When considering moving materials, the construction of the basis systems forming the moment problem is a special challenge, since the hyperbolic equation contains a mixed derivative (Coriolis acceleration). Therefore, the classical method of separating variables is not applicable in this case. Instead, a new method is used to find self-similar solutions of non-stationary equations, which makes it possible to find the basis systems explicitly. Results. In the case of paper web, it is necessary to find a minimal in the whole class of admissible perturbations time interval, within which the trigonometric system forming the problem of moments is the Riesz basis, that make it possible through using the system conjugate with it to find the optimal control way in the form of a series and, therefore, to build a so-called optimal damper. Conclusions. As a result of the study, a generalized solution of the problem of transverse oscillations is constructed. For the problem of damping oscillations, the exact damping time is obtained, namely, a time T0 at which the total energy of the system is zero. Optimum control is found in the form of a Fourier series. Keywords: damping oscillations, hyperbolic equation, Coriolis acceleration, trigonometric moment problem, Riesz base For citation: Muravey L. A., Petrov V. M., Romanenkov A. M. The Problem of Damping the Transverse Oscillations on a Longitudinally Moving String. Vestnik Mordovskogo universiteta = Mordovia University Bulletin. 2018; 28(4):472–485. DOI: https://doi.org/10.15507/0236-2910.028.201804.472-485 Acknowledgements: The work was supported by grant No. 16-01-00425 A from the Russian Foundation for Basic Research.