Instability Regions in Flexural-Torsional Vibrations of Plates
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Instability Regions in Flexural-Torsional Vibrations of Plates A. G. Egorov1* and B. Affane2** (Submitted by A. M. Elizarov) 1
Institute of Mechanics and Engineering, Federal Research Center Kazan Scientic Center, Russian Academy of Sciences, Kazan, Tatarstan, 420111 Russia 2 N. I. Lobachevskii Institute of Mathematics and Mechanics, Kazan (Volga Region) Federal University, Kazan, Tatarstan, 420008 Russia Received February 3, 2020; revised March 8, 2020; accepted March 19, 2020
Abstract—The paper is devoted to study of the parametric resonance of torsional vibrations caused by large-amplitude flexural vibrations of cantilevered plates. In this study we considered the first two resonant modes of flexural vibrations. The analytical study of the original problem which is reduced to Hill equation, is based on the WKB approximation. According to the obtained results, we built a map of instability gaps in the plane of the two process control parameters. The conclusions are made about the possibility of experimentally detection the studied effect in real systems. DOI: 10.1134/S1995080220070094 Keywords and phrases: flexural vibrations, torsional vibrations, parametric resonance, instability gaps, Hill equation.
1. INTRODUCTION The study of mechanical vibrations of plates in recent decades has witnessed a remarkable and increasing interest due to the various and wide fields of their applications such as atomic microscopy [1], sensors and actuators based on micromechanical oscillators [2], underwater robotic devices [3]. Our interest in this research area is connected with the development of the approach [4–6] to determine the damping properties of materials. The method is based on the measurement of the amplitudes of damped flexural vibrations (FV) of the free end of cantilever test-samples in the first resonant mode, where the thickness h, width b and length L of the samples are such that h b L. It was noticeable [7] that at large-amplitudes of forced FVs, high-frequency torsional vibrations (TV) were also arising. A theoretical study of such phenomenon on the basis of the classical theory is impossible due to the independence between the flexural and torsional oscillations. In this paper, we used the simplest generalization of the vibrations classical theory for elongated plates, proposed in [8] and related to the geometric non-linearity of the motion basic equations. According to the refined model, FV and TV are non-linearly coupled. This makes it basically possible to excite TV by large-amplitude FV due to the phenomenon of parametric resonance (PR). In our work, the possibility to provoke PR is theoretically investigated for the first two FV resonant modes. For simplicity, the equations proposed in [8] are analyzed neglecting both the internal damping of the material and the external aerodynamic drag. The problem statement is given in Section 2 of the paper where two control parameters of the process are highlighted: the dimensionless FV amplitude s and the ratio of the main frequency Ω of TV to the frequency of FV. In Secti
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