The Induced Oscillations of Flexible Prestressed Elements of Structures (symmetrical System)

The results of the investigations of dynamic behaviour of the flexible prestressed structure elements are presented in the paper. The given physical model can be applied to the flexible structures like sloping arches, shells, bending plates, elements of the large space antenna fields (LSAF). The dynamic behaviour of the investigated systems is described by the equations where ϵ is damping coefficient, α,β are coefficients determining the character of non-linear restoring force are parameters of outer effect. The analysis of the “skeleton” curves disclosed the double qualities of system (1). Thus, “large” oscillations possess the peculiarities of the rigid system behaviour, and “small” oscillations possess the qualities of soft systems. The character of the oscillation amplitude changing with the increase or decrease of the excitation frequencies is followed in Fig 1. The establishment of the forced oscillation regimes from one branch to another is accompanied not only by the transition from “large” oscillations to “small”, or vice versa, but also by the development of the combination tones (2ω, 3ω 5ω, …, ω/2, ω/3). The analytical solutions for “large” and “small” forced oscillations are given by harmonic balance method. The solution was found in the form φ = Acosωt for “large” oscillation, and for “small” oscillation, where . The for curves disclosed unstable branches of amplitude-frequency curves and critical value amplitude of “large” oscillations were obtained. The methods and results of the computing experiment are presented in the paper. For working out the software necessary for the given task, the method of numerical integration (Runge-Kutta method of the fourth order), spectral analysis (Hertzel algorithm), computer graphics, etc were used. The results of the numerical integration are well-coordinated with the analytical solution for the “framework” curves and for the amplitude-frequency curves of forced oscillations.

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