FREE VIBRATION ANALYSIS OF HELICOIDAL BARS WITH THIN-WALLED CIRCULAR TUBE CROSS-SECTION VIA MIXED FINITE ELEMENT METHOD

Year 2015, Volume: 33 Issue: 2, 200 - 218, 01.06.2015

Nihal Eratlı Merve Ermiş Mehmet H. Omurtag

Abstract

In this study, the free vibration analysis of cylindrical and non-cylindrical helicoidal bars with thin-walled circular tube cross-section is investigated by using the mixed finite element formulation based on Timoshenko beam theory. Frenet triad is adopted as the local coordinate system in the helix geometry. The curved elements involve two nodes, where each node has 12 DOF, namely three translations, three rotations, two shear forces, one axial force, two bending moment and one torque. Numerical solutions are performed to analyze the dynamic behavior of the helix geometries and benchmark results are presented. Parametric studies are carried out to investigate the influence of the section geometry, the helicoidal geometry, the boundary conditions and the density of the material.

Keywords

Timoshenko beam theory, finite element, non-cylindrical helix, thin-walled circular tube section, free vibration.

References

• [1] Michell JH, The small deformation of curves and surfaces with application to the vibrations of a helix and a circular ring, Messenger of Mathematics, 19:68-82, 1890.
• [2] Love AEM, The propagation of waves of elastic displacement along a helical wire, Transactions of the Cambridge Philosophical Society, 18:364-374, 1899.
• [3] Yoshimura Y and Murata Y, On the elastic waves propagated along coil springs, Institute of Science and Technology, Tokyo University, 6(1):27-35, 1952.
• [4] Wittrick WH, On elastic wave propagation in helical spring, International Journal of Mechanical Science, 8:25-47, 1966.
• [5] Lin Y and Pisano AP, General dynamic equations of helical springs with static solution and experimental verification, J. Appl. Mech., 54:910-917, 1987.
• [6] Mottershead JE, Finite elements for dynamical analysis of helical rods, Int. J. Mech. Sci., 2(1):267-283, 1980.
• [7] Omurtag MH and Aköz AY, The mixed finite element solution of helical beams with variable cross-section under arbitrary loading, Computers and Structures, 43(2):325-331, 1992.
• [8] Girgin K, Free vibration analysis of non-cylindrical helices with, variable cross-section by using mixed FEM, Journal of Sound and Vibration, 297:931-945, 2006.
• [9] Pearson D, The transfer matrix method for the vibration of compressed helical springs, Journal Mechanical Engineering Science, 24(4):163-171, 1982.
• [10] Yıldırım V, Investigation of parameters affecting free vibration frequency of helical springs, Int. J. Num. Meth. Engng., 39(1):99-114, 1996.
• [11] Yıldırım V and İnce N, Natural frequencies of helical springs of arbitrary shape, Journal of Sound and Vibration, 204(2):311-329, 1997.
• [12] Yıldırım V, Free vibration analysis of non-cylindrical coil springs by combined use of the transfer matrix and complementary functions methods, Communications in Num. Meth. Engng., 13:487-494, 1997.
• [13] Yıldırım V, A parametric study on the free vibration of non-cylindrical helical springs, Journal of Applied Mechanics, ASME, 65:157-163, 1998.
• [14] Yıldırım V, Expressions for predicting fundamental natural frequencies of non-cylindrical helical springs, Journal of Sound and Vibration, 252(3):479-491, 2002.
• [15] Busool W and Eisenberger M, Free vibration of helicoidal beams of arbitrary shape and variable cross section, Journal of Vibration and Acoustics, 124:397-409, 2002.
• [16] Lee J, Free vibration analysis of cylindrical helical springs by the pseudospectral method, Journal of Sound and Vibration, 302:185-196, 2007.
• [17] Lee J, Free vibration analysis of non-cylindrical helical springs by the pseudospectral method, Journal of Sound and Vibration, 305:543-551, 2007.
• [18] Oden JT and Reddy JN, Variational Method in Theoretical Mechanics, Berlin: Springer-Verlag; 1976.
• [19] Omurtag MH, Strength of Material 2, Istanbul: Birsen Press; 2013.