Free Vibration Analysis of Conical Helicoidal Rods Having Elliptical Cross Sections Positioned in Different Orientation
In this study, the free vibration analysis of conical helicoidal rods with two different elliptically oriented cross sections is investigated and the results are compared by the circular cross-section keeping the net area for all cases equal to each other. Problems are solved by using the mixed finite element formulation. Element matrices based on Timoshenko beam theory are employed. The finite element matrices are derived by directly inserting the analytical expressions (arc length, curvature, and torsion) defining helix geometry into the formulation. Helicoidal rod domain is discretized by a two-noded curvilinear element. Each node of the element has 12 DOFs, namely, three translations, three rotations, two shear forces, one axial force, two bending moments and one torque. A parametric study is performed to investigate the influence of elliptical cross sectional geometry and its orientation over the natural frequencies of the conical type helicoidal rod.
[1] A. A Lucas and P. Lambin, “Diffraction by DNA, carbon nanotubes and other helical nanostructures”, Rep. Prog. Phys., vol. 68, pp. 1181-1249, 2005.
[2] J. Cherusseri, R. Scharma and K.K. Kar, “Helically coiled carbon nanotube electrodes for flexible supercapacitors”, Carbon, vol. 105, pp.113-125, 2016.
[3] D. Tentori, A. Garcia-Wadner and J.A. Rodriguez-Garcia, “Use of fiber helical coils to obtain polarization insensitive fiber devices”, Opt. Fiber Technol., vol. 31, pp.13-19, 2016.
[4] E.H. Egelman, “Three-dimensional reconstruction of helical polymers”, Arch. Biochem. Biophys., vol. 581, 54-58, 2015.
[5] N. Barbieri, R. Barbieri, R.A. de Silva, M.J. Mannala and L.D.S.V. Barbieri “Nonlinear dynamic analysis of wire-rope isolator and Stockbridge damper”, Nonlinear Dyn., doi: 10.1007/s11071-016-2903-1, 2016.
[6] A. Baratta and I. Corbi, “Equilibrium models for helicoidal laterally supported staircases”, Comput. Struct., vol. 124, pp. 21-28, 2013.
[7] J.E. Mottershead, “Finite elements for dynamical analysis of helical rods”, Int. J. Mech. Sci., vol. 22, pp. 267–283, 1980.
[8] M.H. Omurtag and A.Y. Aköz, “The mixed finite element solution of helical beams with variable cross-section under arbitrary loading”, Comput. Struct., vol. 43, pp. 325–331, 1992.
[9] V. Haktanır and E. Kıral, “Statical analysis of elastically and continuously supported helicoidal structures by the transfer and stiffness matrix methods”, Comput. Struct., vol. 49, pp. 663–677, 1993.
[10] V. Yıldırım, “Investigation of parameters affecting free vibration frequency of helical springs”, Int. J. Numer. Methods Eng., vol. 39, pp. 99–114, 1996.
[11] S.A. Alghamdi, M.A. Mohiuddin and H.N. Al-Ghamedy, “Analysis of free vibrations of helicoidal beams”, Eng. Comput., vol. 15, pp. 89–102, 1998.
[12] W. Busool and M. Eisenberger, “Exact static analysis of helicoidal structures of arbitrary shape and variable cross section”, J. Struct. Eng., vol. 127, pp. 1266–1275, 2001.
[13] J. Lee, “Free vibration analysis of cylindrical helical springs by the pseudospectral method”, J. Sound Vib., vol. 302, pp. 185–196, 2007.
[14] A.M. Yu, C.J. Yang and G.H. Nie, “Analytical formulation and evaluation for free vibration of naturally curved and twisted beams”, J. Sound Vib., vol. 329, pp. 1376–1389, 2010.
[15] N. Eratlı, M. Ermis and M.H. Omurtag, “Free vibration analysis of helicoidal bars with thin-walled circular tube cross-section via mixed finite element method”, Sigma Journal of Engineering and Natural Sciences, vol. 33(2), pp. 200-218, 2015.
[16] N. Eratlı, M. Yılmaz, K. Darılmaz and M.H. Omurtag, “Dynamic analysis of helicoidal bars with non-circular cross-sections via mixed FEM”, Struct. Eng. Mech., vol. 57(2), pp. 221-238, 2016.
[17] M. Ermis, The dynamic analysis of non-cylindrical viscoelastic helical bars using mixed finite element, Msc Thesis, Istanbul Technical University, Istanbul, 2015.
[18] M. Ermis, N. Eratlı, Argeso H., A. Kutlu and M.H. Omurtag, “Parametric Analysis of Viscoelastic Hyperboloidal Helical Rod”, Adv. Struct. Eng., doi: 10.1177/1369433216643584,2016.
[19] J.T. Oden and J.N. Reddy, Variational Method in Theoretical Mechanics, Springer-Verlag, Berlin, 1976.
[20] S. Timoshenko and J.N. Goodier, Theory of elasticity, McGraw-Hill, New York, 1951.
[1] A. A Lucas and P. Lambin, “Diffraction by DNA, carbon nanotubes and other helical nanostructures”, Rep. Prog. Phys., vol. 68, pp. 1181-1249, 2005.
[2] J. Cherusseri, R. Scharma and K.K. Kar, “Helically coiled carbon nanotube electrodes for flexible supercapacitors”, Carbon, vol. 105, pp.113-125, 2016.
[3] D. Tentori, A. Garcia-Wadner and J.A. Rodriguez-Garcia, “Use of fiber helical coils to obtain polarization insensitive fiber devices”, Opt. Fiber Technol., vol. 31, pp.13-19, 2016.
[4] E.H. Egelman, “Three-dimensional reconstruction of helical polymers”, Arch. Biochem. Biophys., vol. 581, 54-58, 2015.
[5] N. Barbieri, R. Barbieri, R.A. de Silva, M.J. Mannala and L.D.S.V. Barbieri “Nonlinear dynamic analysis of wire-rope isolator and Stockbridge damper”, Nonlinear Dyn., doi: 10.1007/s11071-016-2903-1, 2016.
[6] A. Baratta and I. Corbi, “Equilibrium models for helicoidal laterally supported staircases”, Comput. Struct., vol. 124, pp. 21-28, 2013.
[7] J.E. Mottershead, “Finite elements for dynamical analysis of helical rods”, Int. J. Mech. Sci., vol. 22, pp. 267–283, 1980.
[8] M.H. Omurtag and A.Y. Aköz, “The mixed finite element solution of helical beams with variable cross-section under arbitrary loading”, Comput. Struct., vol. 43, pp. 325–331, 1992.
[9] V. Haktanır and E. Kıral, “Statical analysis of elastically and continuously supported helicoidal structures by the transfer and stiffness matrix methods”, Comput. Struct., vol. 49, pp. 663–677, 1993.
[10] V. Yıldırım, “Investigation of parameters affecting free vibration frequency of helical springs”, Int. J. Numer. Methods Eng., vol. 39, pp. 99–114, 1996.
[11] S.A. Alghamdi, M.A. Mohiuddin and H.N. Al-Ghamedy, “Analysis of free vibrations of helicoidal beams”, Eng. Comput., vol. 15, pp. 89–102, 1998.
[12] W. Busool and M. Eisenberger, “Exact static analysis of helicoidal structures of arbitrary shape and variable cross section”, J. Struct. Eng., vol. 127, pp. 1266–1275, 2001.
[13] J. Lee, “Free vibration analysis of cylindrical helical springs by the pseudospectral method”, J. Sound Vib., vol. 302, pp. 185–196, 2007.
[14] A.M. Yu, C.J. Yang and G.H. Nie, “Analytical formulation and evaluation for free vibration of naturally curved and twisted beams”, J. Sound Vib., vol. 329, pp. 1376–1389, 2010.
[15] N. Eratlı, M. Ermis and M.H. Omurtag, “Free vibration analysis of helicoidal bars with thin-walled circular tube cross-section via mixed finite element method”, Sigma Journal of Engineering and Natural Sciences, vol. 33(2), pp. 200-218, 2015.
[16] N. Eratlı, M. Yılmaz, K. Darılmaz and M.H. Omurtag, “Dynamic analysis of helicoidal bars with non-circular cross-sections via mixed FEM”, Struct. Eng. Mech., vol. 57(2), pp. 221-238, 2016.
[17] M. Ermis, The dynamic analysis of non-cylindrical viscoelastic helical bars using mixed finite element, Msc Thesis, Istanbul Technical University, Istanbul, 2015.
[18] M. Ermis, N. Eratlı, Argeso H., A. Kutlu and M.H. Omurtag, “Parametric Analysis of Viscoelastic Hyperboloidal Helical Rod”, Adv. Struct. Eng., doi: 10.1177/1369433216643584,2016.
[19] J.T. Oden and J.N. Reddy, Variational Method in Theoretical Mechanics, Springer-Verlag, Berlin, 1976.
[20] S. Timoshenko and J.N. Goodier, Theory of elasticity, McGraw-Hill, New York, 1951.
@article{"International Journal of Architectural, Civil and Construction Sciences:73794", author = "Merve Ermis and Akif Kutlu and Nihal Eratlı and Mehmet H. Omurtag", title = "Free Vibration Analysis of Conical Helicoidal Rods Having Elliptical Cross Sections Positioned in Different Orientation", abstract = "In this study, the free vibration analysis of conical helicoidal rods with two different elliptically oriented cross sections is investigated and the results are compared by the circular cross-section keeping the net area for all cases equal to each other. Problems are solved by using the mixed finite element formulation. Element matrices based on Timoshenko beam theory are employed. The finite element matrices are derived by directly inserting the analytical expressions (arc length, curvature, and torsion) defining helix geometry into the formulation. Helicoidal rod domain is discretized by a two-noded curvilinear element. Each node of the element has 12 DOFs, namely, three translations, three rotations, two shear forces, one axial force, two bending moments and one torque. A parametric study is performed to investigate the influence of elliptical cross sectional geometry and its orientation over the natural frequencies of the conical type helicoidal rod.", keywords = "Conical helix, elliptical cross section, finite element, free vibration.", volume = "10", number = "9", pages = "1144-5", }
