Abstract
An analytical method is presented to determine the vibration characteristics of a rotating Timoshenko beam with variable cross section and intermediate flexible connections using the differential transform method. Based on the application of Timoshenko beam theory on separate beams and the compatibility requirements on each connection point, the correlations between every two adjacent spans are obtained. The formulation is extended to a point where it would be able to evaluate the cases with internal and external flexible connections. The results will be validated against those reported in the literature and compared with the ones from the modal test. A number of parametric studies are conducted to assess the stiffness of elastic connections, rotating speed, hub radius and tapered ratio effects on the beam natural frequencies and mode shapes. It is observed that by changing the stiffness of the intermediate springs, the general formulation developed here can cover a large array of problems such as cracked or intermediately constrained beams.
Similar content being viewed by others
References
Thakkar D., Ganguli R.: Helicopter vibration reduction in forward fight with induced shear based piezoceramic actuation. Smart Mater. Struct 30(3), 599–608 (2004)
Kumar S., Roy N., Ganguli R.: Monitoring low cycle fatigue damage in turbine blades using vibration characteristics. Mech. Syst. Signal Process. 21(1), 480–501 (2007)
Hodges D.H., Rutkowski M.J.: Free vibration analysis of rotating beams by a variable order finite element method. Am. Inst. Aeronaut. Astron. J. 19, 1459–1466 (1981)
Rao S.S., Gupta R.S.: Finite element vibration analysis of rotating Timoshenko beams. J. Sound Vib. 242(1), 103–124 (2001)
Gunda J.B., Ganguli R.: New rational interpolation functions for finite element analysis of rotating beams. Int. J. Mech. Sci. 50, 578–588 (2008)
Gunda J.B., Ganguli R.: Stiff-String basis functions for vibration analysis of high speed rotating beams. J. Appl. Mech. 75, 245021–245025 (2008)
Gunda J.B., Gupta R.K., Ganguli R.: Hybrid stiff-string polynomial basis functions for vibration analysis of high speed rotating beams. Comput. Struct. 87(3–5), 254–265 (2009)
Bazoune A.: Effect of tapering on natural frequencies of rotating beams. Shock Vib. 14, 169–179 (2007)
Vinod K.G., Gopalakrishnan S., Ganguli R.: Free vibration and wave propagation analysis of uniform and tapered rotating beams using spectrally formulated finite elements. Int. J. Solids Struct. 44(18–19), 5875–5893 (2007)
Wang G., Wereley N.M.: Free vibration analysis of rotating blades with Uniform tapers. Am. Inst. J. Aeronaut. Astron. 42(12), 2429–2437 (2004)
Kuang J.H., Huang B.W.: The effect of blade cracks on mode localization in rotating bladed disks. J. Sound Vibr. 227(1), 85–103 (1999)
Rand O.: Experimental study of the natural frequencies of rotating thin-walled composite blades. Thin-Walled Struct. 21, 191–207 (1995)
Surace G., Anghel V., Mares C.: Coupled bending–bending–torsion vibration analysis of rotating pre-twisted blades: an integral formulation and numerical examples. J. Sound Vib. 206, 473–486 (1997)
Young T.H., Lin T.M.: Stability of rotating pre-twisted tapered beams with randomly varying speed. ASME J. Vib. Acoust. 120(2), 784–790 (1998)
Kar R.C., Neogy S.: Stability of a rotating, pre-twisted, non-uniform cantilever beam with tip mass and thermal gradient subjected to a non-conservative force. Comput. Struct. 33(2), 499–507 (1989)
Chondros T.G., Dimarogonas A.D.: Dynamic sensitivity of structures to cracks. J. Vib. Acoust. Stress Reliab. Des. 111, 251–256 (1989)
Chondros T.G.: The continuous crack flexibility method for crack identification. Fatigue Fract. Eng. Mater. Struct. 24, 643–650 (2001)
Panteliou S.D., Chondros T.G., Argyrakis V.C., Dimarogonas A.D.: Damping factor as an indicator of crack severity. J. Sound Vib. 241(2), 235–245 (2001)
Chang C.C., Chen L.W.: Damage detection of cracked thick rotating blades by a spatial wave let based approach. Appl. Acoust. 65(11), 1095–1111 (2004)
Kim S.S., Kim J.H.: Rotating composite beam with a breathing crack. Compos. Struct. 60(1), 83–90 (2003)
Cheng Y., Zhigang Y., Wu X., Yuan Y.: Vibration analysis of a cracked rotating tapered beam using the p-version finite element method. Finite Elem. Anal. Des. 47, 825–834 (2011)
Ariaei A., Ziaei-Rad S., Malekzadeh M.: Dynamic response of a multi-span Timoshenko beam with internal and external flexible constraints subject to a moving mass. Arch. Appl. Mech. 83, 1257–1272 (2013)
Loya J.A., Rubio L., Fernández-Sáez J.: Natural frequencies for bending vibrations of Timoshenko cracked beams. J. Sound Vib. 290, 640–653 (2006)
Bazoune A., Khulief Y.A., Stephen N.G.: Further results for modal characteristics of rotating tapered Timoshenko beams. J. Sound Vib. 219(3), 157–174 (1999)
Masoud A.A., Al-Said S.: A new algorithm for crack localization in a rotating Timoshenko beams. J. Vib. Control 15, 1541–1561 (2009)
Yokoyama T.: Free vibration characteristics of rotating Timoshenko beam. Int. J. Mech. Sci. 30, 743–755 (1988)
Khulief Y.A., Bazoune A.: Frequencies of rotating tapered Timoshenko beams with different boundary conditions. Comput. Struct. 42, 781–795 (1992)
Lee S.Y., Lin S.M.: Bending vibrations of rotating non-uniform Timoshenko beams with an elastically restrained root. J. Appl. Soft Comput. 61, 949–955 (1994)
Du H., Lim M.K., Liew K.K.: A power series solution for vibration of a rotating Timoshenko beams. J. Sound Vib. 175, 505–523 (1994)
Nagaraj V.T.: Approximate formula for the frequencies of a rotating Timoshenko beam. J. Aircr. 33, 637–639 (1996)
Lin S.C., Hsiao K.M.: Vibration analysis of a rotating Timoshenko beam. J. Sound Vib. 240, 303–322 (2001)
Kaya M.O.: Free vibration analysis of a rotating Timoshenko beam by Differential Transformation Method. Aircr. Eng. Aerosp. Technol. 78, 194–203 (2006)
Choi S.T., Chou Y.T.: Vibration analysis of elastically supported turbo machinery blades by the modified differential quadrature methods. J. Sound Vib. 240, 937–953 (2001)
Downs B.: Transverse vibration of cantilever beam having unequal breadth and depth tapers. J. Appl. Soft Comput. 44, 737–742 (1977)
Irie T., Yamada G., Takahashi I.: Determination of the steady state response of a Timoshenko beam of varying section by the use of the spline interpolation technique. J. Sound Vibr. 63, 287–295 (1979)
Irie T., Yamada G., Takahashi I.: Vibration and stability of a non-uniform Timoshenko beam subjected to follower force. J. Sound Vibr. 70, 503–512 (1980)
Lee S.Y., Lin S.M.: Exact vibration solutions for non-uniform Timoshenko beams with attachments. AIAA J. 30, 2930–2934 (1992)
Yaradimoglu B.: Vibration analysis of rotating tapered Timoshenko beam by a new finite element model. Shock Vib. 13, 117–126 (2006)
Thomas J., Abbas B.A.H.: Finite element model for dynamic analysis of Timoshenko beam. J. Sound Vibr. 41, 291–299 (1975)
Friedman Z., Kosmatka J.B.: An improved two-node Timoshenko beam finite element. Comput. Struct. 47(3), 473–481 (1993)
Laura P.A.A., Gutierrez R.H.: Analysis of vibrating Timoshenko beams using the method of differential quadrature. Shock Vib. 1(1), 89–93 (1993)
Stafford R.O., Giurgiutiu V.: Semi-analytic method for rotating Timoshenko beams. Int. J. Mech. Sci. 17, 719–727 (1975)
Ozgumus O.O., Kaya M.O.: Flap wise bending vibration analysis of a rotating tapered cantilever Bernoulli–Euler beam by differential transform method. J. Sound Vib. 289, 413–420 (2006)
Zhou J.K.: Differential Transformation and Its Application for Electrical Circuits. Wuhan, Huazhong University Press, Wuhan (1986)
Mei C.: Application of differential transformation technique to free vibration analysis of a centrifugally stiffened beam. Comput. Struct. 86, 1280–1284 (2008)
Chen C.K., Ju S.P.: Application of differential transformation to transient advective–dispersive transport equation. J. Appl. Math. Comput. 155, 25–38 (2004)
Arikoglu A., Ozkol I.: Solution of boundary value problems for integro-differential equations by using differential transformation method. J. Appl. Math. Comput. 168, 1145–1158 (2004)
Bert C.W., Zeng H.: Analysis of axial vibration of compound bars by differential transformation method. J. Sound Vib. 275, 641–647 (2004)
Ayaz F.: Application of differential transforms method to differential–algebraic equations. J. Appl. Math. Comput. 152, 648–657 (2004)
Ho S.H., Chen C.K.: Free transverse vibration of an axially loaded non-uniform sinning twisted Timoshenko beam using differential transform. Int. J. Mech. Sci. 48, 1323–1331 (2006)
Ertürk V.S., Momani S.: Comparing numerical methods for solving fourth-order boundary value problems. J. Appl. Math. Comput. 188, 1963–1968 (2007)
Kosmatka J.B.: An improved two-node finite element for stability and natural frequencies of axially loaded Timoshenko beams. Comput. Struct. 57(1), 141–149 (1995)
Shames I.H., Dym C.L.: Energy and Finite Element Methods in Structural Mechanics, Chap 4. McGraw-Hill, New York (1985)
Thomson W.T., Dahleh M.D.: Theory of Vibration with Applications, 5th edn. Prentice Hall, Englewood Cliffs (1997)
Lee S.Y., Lin S.M.: Bending vibrations of rotating non-uniform Timoshenko beams with an elastically restrained root. J. Appl. Mech. 61, 949–955 (1994)
Khaji N., Shafiei M., Jalalpour M.: Closed-form solutions for crack detection problem of Timoshenko beams with various boundary conditions. Int. J. Mech. Sci. 51, 667–681 (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Talebi, S., Ariaei, A. Vibration analysis of a rotating Timoshenko beam with internal and external flexible connections. Arch Appl Mech 85, 555–572 (2015). https://doi.org/10.1007/s00419-014-0930-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00419-014-0930-2