Calculating the natural frequency of cantilever tapered beam using classical Rayleigh, modified Rayleigh and finite element methods

  • Authors

    • Luay S.Alansari University of Kufa – Faculty of Engineering – Mechanical Engineering Department
    • Ali M.HAl-Hajjar University of Kufa – Faculty of Engineering – Mechanical Engineering Department
    • Husam Jawad A. University of Kufa – Faculty of Engineering – Mechanical Engineering Department
    2019-02-15
    https://doi.org/10.14419/ijet.v7i4.25334
  • Classical Rayleigh Method, Modified Rayleigh Method, Finite Element Method, ANSYS Workbench, Tapered Beam, Frequency.

  • Abstract

    Beam is a structural element and can be used in different shapes according to its applications and the tapered beam is one of these structural elements. The frequency of tapered beam was investigated in this work using three calculation methods. These methods were Classical Rayleigh Method (CRM), Modified Rayleigh Method (MRM) and Finite Element Method (FEM) using ANSYS Workbench (17.2). The basic idea of Classical Rayleigh Method (CRM) and Modified Rayleigh Method (MRM) was changing the tapered beam into stepped beam with N-steps. The results showed that there was a good agreement between the natural frequency which was calculated by ANSYS and Modified Rayleigh Method (MRM) when the number of steps was (6) and the natural frequency increases when the larger width (or height) increases for different values of smaller width (or height).The frequency ratio is constant when the smaller width (or height) increases. Also, the frequency ratio increases when the width ratio (WL/WS) increases and when the number of divisions (N) increases, the slope of frequency ratio increases too.

     

     

  • References

    1. [1] Firouz-Abadi R.D., Haddadpour H., Novinzadeh A.B., An asymptotic solution to transverse free vibrations of variable-section beams, Journal of Sound and Vibration 2007, 304, 530-540. https://doi.org/10.1016/j.jsv.2007.02.030.

      [2] Hsu J.H., Lai H.Y., Chen C.K., Free vibration of non-uniform Euler-Bernoulli beams with general elastically end constraints using Adomian modified decomposition method, Journal of Sound and Vibration 2008, 318, 965-981. https://doi.org/10.1016/j.jsv.2008.05.010.

      [3] Yagci B., Filiz S., Romero L.L., Ozdoganlar O.B., A spectral-Tchebychev technique for solving linear and nonlinear beam equations, Journal of Sound and Vibration 2009, 321, 375-404. https://doi.org/10.1016/j.jsv.2008.09.040.

      [4] Cleghorn W.L., Tabarrok B., Finite element formulation of a tapered Timoshenko beam for free lateral vibration analysis, Journal of Sound and Vibration 1992, 152, 461-470. https://doi.org/10.1016/0022-460X(92)90481-C.

      [5] Eisenberger M., Dynamic stiffness matrix for variable cross-section Timoshenko beams, Communications in Numerical Methods in Engineering 1995, 11, 507-513. https://doi.org/10.1002/cnm.1640110605.

      [6] Tong X., Tabarrok B., Vibration analysis of Timoshenko beams with non-homogeneity and varying cross-section, Journal of Sound and Vibrations 1995, 186(5), 821-835. https://doi.org/10.1006/jsvi.1995.0490.

      [7] Liao M., Zhong H., Nonlinear vibration analysis of tapered Timoshenko beams, Chaos, Solitons and Fractals 2008, 36, 1267-1272. https://doi.org/10.1016/j.chaos.2006.07.055.

      [8] Attarnejad R., Semnani S.J., Shahba A., Basic displacement functions for free vibration analysis of non-prismatic Timoshenko beams, Finite Elements is Analysis and Design 2010, 46, 916-929.

      [9] Mabie H. H. and Rogers C. B., “Transverse vibrations of double-tapered cantilever beams,’’ Journal of the Acoustical Society of America 36, 463-469, 1964. https://doi.org/10.1121/1.1918979.

      [10] Mabie H. H. and Rogers C. B., “Transverse vibrations of tapered cantilever beams with end support," Journal of the Acoustical Society of America, 44, 1739-1741, 1968. https://doi.org/10.1121/1.1911327.

      [11] Naguleswaran S. “A direct solution for the transverse vibration of Euler-Bernoulli wedge and cone beams". Journal of Sound and Vibration 172: 289–304(1994). https://doi.org/10.1006/jsvi.1994.1176.

      [12] Auciello NM and Nole G "Vibrations of a cantilever tapered beam with varying section properties and carrying a mass at the free end". Journal of Sound and Vibration 214: 105–119, 1998. https://doi.org/10.1006/jsvi.1998.1538.

      [13] Caruntu DI "Classical Jacobi polynomials, closed-form solutions for transverse vibrations". Journal of Sound and Vibration 306: 467–494, 2007) https://doi.org/10.1016/j.jsv.2007.05.046.

      [14] Caruntu DI "Dynamic modal characteristics of transverse vibrations of cantilevers of parabolic thickness. Mechanics Research Communications 36: 391–404, 2009. https://doi.org/10.1016/j.mechrescom.2008.07.005.

      [15] Abrate S. "Vibration of non-uniform rods and beams. Journal of Sound and Vibration, Vol. 185, Issue 4, p. 703-716, 1995. https://doi.org/10.1006/jsvi.1995.0410.

      [16] De Rosa M. A. and Auciello N. M. "Free vibrations of tapered beams with flexible ends". Journal of Computers and Structures, Vol. 60, Issue 2, p. 197-202, 1996. https://doi.org/10.1016/0045-7949(95)00397-5.

      [17] Izabela Zamorska "Free Transverse vibrations of non-uniform beams". Scientific Research of the Institute of Mathematics and Computer cience, Vol. 9, Issue 2, p. 244-250, 2010.

      [18] Stanisław Kukla and Izabela Zamojska "Application of the Green’s function method in free vibration analysis of non-uniform beams". Scientific Research of the Institute of Mathematics and Computer Science, Vol. 4, Issue 1, p. 87-94, 2005.

      [19] Mahmoud A. A., et al. "Free vibrations of uniform and non-uniform Euler beam using differential transformation method". Asian Journal of Mathematics and Applications, Vol. 2013, p. 1-16, 2013.

      [20] Downs B "Reference frequencies for validation of numerical-solutions of transverse vibrations of non-uniform beams". Journal of Sound and Vibration 61: 71–78, 1978. https://doi.org/10.1016/0022-460X(78)90042-1.

      [21] Dimaragonas DA and Haddad S "Vibration for Engineers". New York: Prentice-Hall, Inc (1992).

      [22] Banerjee JR "Free vibration of centrifugally stiffened uniform and tapered beams using the dynamic stiffness method". Journal of Sound and Vibration 233: 857–875, 2000. https://doi.org/10.1006/jsvi.1999.2855.

      [23] Attarnejad R, Manavi N and Farsad A "Exact Solution for the Free Vibration of a Tapered Beam with Elastic End Rotational Restraints." Netherlands: Springer Netherlands (2006). https://doi.org/10.1007/978-1-4020-3953-9_146.

      [24] Bazoune A “Effect of tapering on natural frequencies of rotating beams". Shock and Vibration 14: 169–179, 2007. https://doi.org/10.1155/2007/865109.

      [25] Ece MC, Aydogdu M and Taskin V "Vibration of a variable cross-section beam". Mechanics Research Communications 34: 78–84, 2007. https://doi.org/10.1016/j.mechrescom.2006.06.005.

      [26] Liu WK, Jun S, Li S, et al. (1995) Reproducing kernel particle methods for structural dynamics. International Journal for Numerical Methods in Engineering 38: 1655–1679. https://doi.org/10.1002/nme.1620381005.

      [27] P.Nagalatha and P.Sreenivas.â€Structural dynamic reanalysis of beam elements using regression method" .IOSR journal of mechanical and civil engineering.e- ISSN: 2278-1684, volume 9, issue 4.

      [28] B. Rama Sanjeevasresta and Dr.Y.V.Mohan Reddy." Dynamic Reanalysis of beams using polynomial regression method" .Research and applications (IJERA) ISSN: 2248-9622.vol 2, issue 4.

      [29] E.Ozkaya “Linear transverse vibrations of a simply supported beam carrying concentrated massesâ€. Mathematical & Computational applications,vol No 2,pp.147-151,2001.

      [30] Dhayai Hassan Jawad “Free vibration and Buckling Behaviourof Tapered Beam by Finite Element Method†Journal of Babylon University/Engineering Sciences.

      [31] Byoung Koo Lee, et al. “Free vibrations of tapered Beams with general boundary condition". Journal of Civil Engineering, Vol. 6, Issue 3, p. 283-288, 2002.

      [32] Firouz-Abadi R. D, et al. "An asymptotic solution to transverse free vibrations of variable-section beams". Journal of Sound and Vibration, Vol. 304, p. 530-540, 2007. https://doi.org/10.1016/j.jsv.2007.02.030.

      [33] Rossi R. E., Laura P. A. A. "Numerical experiments on vibrating linearly tapered Timoshenko beams". Journal of Sound and Vibration, Vol. 168, Issue 1, 1993, p. 179-183. https://doi.org/10.1006/jsvi.1993.1398.

      [34] Dr. Luay S. Al-Ansari, Calculating of Natural Frequency of Stepping Cantilever Beam, International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:1 2 No:05, (2012).

      [35] Dr. Luay S. Al-Ansari, Calculating Static Deflection and Natural Frequency of Stepped Cantilever Beam Using Modified RAYLEIGH Method", International Journal of Mechanical and Production Engineering Research and Development (IJMPERD) ISSN 2249-6890,Vol. 3, Issue 4, Oct 2013, 113-124© TJPRC Pvt. Ltd., 2013.

      [36] Dr. Luay S. Al-Ansari, Muhannad Al-Waily and Ali M. H. Yusif , "Vibration Analysis of Hyper Composite Material Beam Utilizing Shear Deformation and Rotary Inertia Effects", International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:12 No:04, (2012).

      [37] S. S. Rao "Mechanical Vibrations" Addison-Wesley Publishing Company, (2011).

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  • How to Cite

    S.Alansari, L., M.HAl-Hajjar, A., & Jawad A., H. (2019). Calculating the natural frequency of cantilever tapered beam using classical Rayleigh, modified Rayleigh and finite element methods. International Journal of Engineering & Technology, 7(4), 4866-4872. https://doi.org/10.14419/ijet.v7i4.25334

    Received date: 2019-01-03

    Accepted date: 2019-01-23

    Published date: 2019-02-15