Методологія аналізу поперечних коливань прямокутних багатопрогонових пластин змінної згинальної жорсткості

Abstract

Викладено загальну методологію аналізу поперечних коливань прямокутних багатопрогонових ізотропних і трансверсально ізотропних пластин з дискретно змінюваною в одному паралельному до якоїсь пари сторін напрямі згинальною жорсткістю та при використанні перпендикулярних до цього напряму пружних щодо вертикальних переміщень і кутів повертання лінійних опор з рівномірним розподілом уздовж них зосереджених мас (включень, накладок). Для побудови частотних характеристичних рівнянь розрахунку власних частот використовується метод початкових параметрів і асоційованих матриць, а також теорія скінченних автоматів з елементами математичної логіки. Розглянуто приклади застосування методу.

The paper briefly describes the general methodology for the analysis of transverse vibrations of rectangular isotropic and transversely isotropic multispan plates with the bending rigidity, which discretely varies in parallel towards to an arbitrary pair of sides, and using linear bearings perpendicular to this direction, which are elastic towards the vertical displacements and rotation angles, with uniform distribution of the concentrated masses along them (inclusions, stringers). Two opposite edges parallel to the direction of change of the properties of the plate are hinge supported or fixed with sliding, and two others satisfy the arbitrary fixing homogeneous conditions. The research starting point is based on the assumption of possibility of division of the plate onto strips, which are perpendicular to the direction of change of properties, such that certain sections coincide with the beginning or the end of each of the spans, and that within each strip the thickness and/or elastic modulus are constant or can be considered constant with sufficient accuracy. Each section is assumed to be a plate or a beam loaded in joint-sections by supporting forces and moments arising from the interaction of the considered strips with neighboring ones and hinge or sliding supported edges perpendicular to these strips. As each section is described by specific values of the component of the linear and angular displacements and components of internal forces and moments that form the state vector S, with introduction into the consideration of the influence matrix the relationship between the state vectors of the edge section of each of the strips can be obtained. As a result the calculation of the plate is reduced to the analysis of the single object loaded by given forces and additional supporting ones. This allows describing in binary form of topological code of the primary (PP) and the end values (KP) of boundary parameters (conditions) of the graph of investigated elements in the case of calculating the transverse vibrations of the plate using the same number (on two) of arbitrary and fixed parameters. To construct the frequency characteristic equations for calculation of eigenfrequencies the methods of initial parameters and associated matrices, and finite-state machine theory with elements of mathematical logic are used. As an example, it is shown that in the case of the analysis of the plate, one end of which is fixed, and the second is hinge supported or two opposite edges are fixed, and two others are hinge supported, obtained equation of frequency are the same as those obtained using other methods. The oscillations of orthotropic plates subject to forces of inertia of rotation, displacement and longitudinal forces are also considered. Based on the solution of the problem of vibration of cross-beams systems the examples of calculation of circular frequencies of orthotropic plates are considered.