Study of pure transverse motion in free cylinders and plates in flexural vibration by Ritz’s method |
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| Authors: | A Bayón F Gascón R Medina FJ Nieves F Salazar |
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| Institution: | 1. Departamento de Física Aplicada a los Recursos Naturales, ETSI de Minas, Universidad Politécnica de Madrid, Ríos Rosas 21, 28003 Madrid, Spain;2. Departamento de Física Aplicada II, ETS de Arquitectura, Universidad de Sevilla, R. Mercedes 2, 41012 Sevilla, Spain;1. DIEF, Università di Firenze, Viale Morgagni 40-44, 50134 Firenze, Italy;2. DIIES, Università Mediterranea di Reggio Calabria, Via Graziella, 89122 Reggio Calabria, Italy;3. Lehrstuhl für Mathematik mit Schwerpunkt Digitale Bildverarbeitung and FORWISS, Universität Passau, Innstr. 43, 94032 Passau, Germany;1. I2M, IPB (UMR CNRS 5295), Université de Bordeaux, 33607 Pessac, France;2. Sorbonne University, Université de technologie de Compiègne, LMAC Laboratory of Applied Mathematics of Compiègne, CS 60 319 - 60 203 Compiègne cedex, France;3. Departamento EDAN & IMUS, Universidad de Sevilla, C/Tarfia, s/n, 41012 Sevilla, Spain;1. Department of Mathematics, China University of Mining and Technology, Xuzhou 221116, China;2. Department of Mathematics, Ningbo University, Ningbo 315211, China |
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| Abstract: | In this work the flexural vibration of a free cylinder of any aspect ratio is analysed. A general solution by powers series of the coordinates is proposed here to represent the displacements, with restrictions on the powers of the radial coordinates which prevent potential energy and stress singularities at the axis of the cylinder. By means of an analytic method, it is concluded that certain points of the cylinder have no axial motion. As a result of the pure transverse movement and of the fact that the cylinder bends, it is inferred that the axis is extended. Furthermore, in the symmetric modes, the points situated at the centres of the bases are displaced in the same direction and sense, and hence the distance between them does not vary in time. Flexural natural frequencies are numerically calculated by Ritz’s method with the general solution series proposed. Since the series used are more adequate, convergence is better than with classic series. The results are verified by FEM. Some consequences are extended to a rectangular plate, whose points of the middle surface vibrate transversally in the double-symmetric mode. In order to verify the theoretical results, a set of experiments with a laser interferometer is carried out. The experimental frequencies agree with the theoretical values. |
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