КОЛИВАННЯ КРУГЛИХ КІЛЬЦЕВИХ ПЛАСТИ З ВІЛЬНИМ КРАЄМ

Abstract

Round circular plates are used in the type of springs, valves, sensitive elements of devices, lids and bottoms of different type of constructions. Often in the process of planning of such elements it is necessary to know them dynamic descriptions: frequencies and forms of eigentones. For circular round plates with an internal hard edge such information can be found in scientific and certificate literature. For circular round plates with a free internal edge such information is absents. Dynamic descriptions of plates are in-process determined with an internal free edge at the joint or hard fastening of external edge. The equation of vibrations of round plate is demonstrated in the polar coordinate system. The method of section of variables is result equalization only on a radial and angular coordinate with two unknown parameters: frequency of vibrations and unknown number n, that determines the number of waves on a angular coordinate. The decision of this equalization is introduced as a sum which has four independent Bessel function order n, each of which is multiplied on an unknown constant. On an internal and external edge written down for two boundary stipulations and having congeneric system turns which consists of four equalizations in relation to unknown constants. From the condition of equality the zero of determinant of this system the frequency equalization is determined. Frequency equalizations are got for two cases: internal the free, external jammed and internal the free external, hinging. Frequencies are determined for these cases at symmetric ( n = 0) and asymmetrical vibrations ( n = 1, 2) vibrations. Three first frequencies are determined in both cases. Every value of frequency parameter is put in the system of four equalizations in relation to constants, which are expressed through one of them. From the condition of equality the zero of function of movement are determined radiuses of a zero movement. Before the calculation of frequency parameter of Bessel function of order n and their derivatives to the third order result in the proper functions of a zero and first order. This algorithm can be used for determination of dynamic descriptions of circular plates at other fixings of edges. Dynamic descriptions of plates are found at the set method of fixing it is possible to use in the process of planning of elements of machines and devices.