Chaotic Oscillations of a Kinematically Excited Flat Shell During Geometrically Non-linear Deformation
Journal Title: Проблеми машинобудування - Year 2019, Vol 22, Issue 3
Abstract
We study the forced oscillations of a cantilevered flat shell of constant curvature. These movements are excited by a kinematic periodic embedding motion. To describe geometrically non-linear deformation, the non-linear theory of Donel shells is used. To build a non-linear dynamic system with a finite number of degrees of freedom, the method of specified forms is used. Since the eigen frequencies of longitudinal and torsional oscillations are much higher than bending ones, the inertial forces in the longitudinal and torsional directions are not taken into account. Therefore, the generalized coordinates of longitudinal and torsional oscillations are expressed in terms of bending ones. As a result, a non-linear dynamic system with respect to bending generalized coordinates is obtained. To calculate the eigen forms of linear oscillations, by using which the non-linear dynamic problem decomposes, the Rayleigh-Ritz method is used. Then only kinematic boundary conditions are satisfied. When the solution converges, the force boundary conditions are automatically satisfied. To study the convergence of eigen frequencies, calculations were performed with a different number of basis functions, which are B-splines. A comparison is made with the experimental data on the analysis of eigen frequencies, with the data published in authors' previous article. To numerically analyze the non-linear periodic oscillations, a two-point boundary value problem is solved for ordinary differential equations by the shooting method. The stability of periodic motions and their bifurcations are estimated using multipliers. To study the bifurcations of periodic oscillations, the parameter continuation method is applied. In the region of the main resonance, saddle-node bifurcations, period-doubling bifurcations, and Neimark-Sacker bifurcations are found. To study the steady-state almost periodic and chaotic oscillations, Poincaré sections, spectra of Lyapunov characteristic exponents, and spectral densities are calculated, with the stroboscopic phase portrait used as Poincaré sections. The properties of steady-state oscillations are investigated with a quasistatic change in the frequency of the disturbing action.
Authors and Affiliations
Konstantin V. Avramov, Kseniya F. Cheshko, Oleg F. Polishchuk
Keywords
Related Articles
- EP ID EP641632
- DOI 10.15407/pmach2019.03.026
- Views 74
- Downloads 0
Stressed State in a Finite Cylinder with a Circular Crack at Non-stationary Torsion
This paper considers a solution to an axially symmetric dynamic problem of determining the stress-state in the vicinity of a circular crack in a finite cylinder. The cylinder lower base is rigidly fixed, and the upper on...
Experimental Study of the Effects of a Constant and Variable Electric Fields on the Strength Properties of the Blade Steel Surface Layer
It is known that in wet steam turbines the wet steam flow is electrified when passing through the turbine flow path. In this, the mutual electrification of the condensed moisture flow and flow path surfaces occurs. Until...
Stepwise Optimization of I-section Flexible Elements Under a Fuzzy Approach to Taking into Account Corrosion and Protective Properties of Anticorrosive Coating
This paper is a continuation of research in the field of optimal design of structures under a combined approach to the measurement of corrosion and anticorrosion protective properties of coatings. As noted earlier, such...
Influence of the Location of a Convector on the Distribution of the Room Temperature
This article considers the influence of various options of placing heaters in a room on energy efficiency, and creating a comfortable temperature field for people to stay in. For the research, a square room with an incli...
Modernization of an experimental installation and a procedure for investigating the anisotropic viscoelastic properties of composite materials at elevated temperatures
The paper describes the process of modernizing the existing installation designed for performing long-term tests of steel and aluminum cylindrical specimens for high-temperature creep with the purpose of conducting the e...