Study of Stress-Strain State of an Aircraft Body at Steering Loads
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y of Stress-Strain State of an Aircraft Body at Steering Loads Yu. I. Vinogradova,* and G. B. Menkovb,** a
Bauman Moscow State Technical University, Moscow, 105005 Russia bClosed Joint Stock Company“Bolid”, Moscow, 127018 Russia *e-mail: [email protected] **e-mail: [email protected]
Received November 8, 2018; revised November 12, 2019; accepted November 19, 2019
Abstract—A design scheme is considered: a cylindrical shell, which is loaded concentrated through the frame. A classical mathematical model is used for the adopted calculation scheme. Based on the mathematical model, the initial problem is analytically solved and an effective algorithm for multiparameter solution of the corresponding boundary value is constructed. The results of a quantitative analysis of the stress-strain state for the aircraft body are presented. Keywords: analytical solution, initial problem, boundary value problem, efficient computational algorithm, parametric analysis DOI: 10.3103/S0025654420030164
1. INTRODUCTION The design scheme for determining the stress-strain state (SSS) of the aircraft body is adopted in the form of a cylindrical shell, which is affected by a concentrated load through the frame. The frame is modeled by an annular line endowed with the rigidity characteristics of the frame, which is located on the edge of the shell or in cross section away from it. We study the sheath and frame SSS depending on their parameters and load in the form of forces and moments. The study is carried out by the analytical method of pairing intervals in a matrix form convenient for computer implementation. The method ensures the stability of computer calculations. The effectiveness of the method is that the analytical solution does not contain integration constants and parametric analysis is limited to solving a system of algebraic equations when changing the parameters of the calculation scheme. The solution of the system of ordinary differential equations (ODEs), obtained by the Fourier method of separation of variables in partial differential equations, is carried out analytically once in a parametric calculation, as an initial, and not a boundary-value problem. The study focuses on the influence of the rigidity characteristics of the frame on its SSS and shell SSS. 2. A MATHEMATICAL MODEL OF THE MECHANICS OF THE DEFORMATION OF A CYLINDRICAL SHELL The studies were performed on the basis of Vlasov [1] mathematical model of the mechanics of the deformation of a cylindrical shell.
d un 1 − ν 2 dv dw n un + 1 + ν n n + ν n = 0, − 2 dξ dξ 2 2 dξ 2
du d 2v 2 − 1 + ν n n + 1 − ν 2n − n v n − nwn = 0, 2 2 dξ dξ 423
(2.1)
424
VINOGRADOV, MENKOV 2
2 du ν n + nv n + c 2 d 2 − n2 wn + wn = 0, dξ dξ un, v n, wn are the functional coefficients of the trigonometric series of the decompositions u, v and w;
c 2 = h2 /(12R 2 ) is the small parameter; R and h are the radius and thickness of the cylindrical shell; ξ = x /R is the dimensionless coordinate along its axis. The system of linear ordinary differential equations (2.1) is re
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