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Exact solutions for free vibrations of axially inhomogeneous Timoshenko beams with variable cross section

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Abstract

A novel method is proposed to simplify the governing equations for the free vibration of Timoshenko beams with both geometrical nonuniformity and material inhomogeneity along the beam axis. For a wide class of Timoshenko beams, this method enables us to reduce the coupled governing differential equations with variable coefficients to a pair of uncoupled second-order differential equations of Sturm–Liouville type with respect to the rotation angle due to bending. The reduced equations contain two important parameters, one describing the variations of translational inertia and bending rigidity along the beam axis, and the other reflecting the comprehensive effect of rotatory inertia and shear deformation. A series of exact analytical solutions are derived from the reduced equations for the first time, and several examples are also provided as benchmarks.

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References

  1. Alshorbagy, A.E., Eltaher, M.A., Mahmoud, F.F.: Free vibration characteristics of a functionally graded beam by finite element method. Appl. Math. Model. 35, 412–425 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Lu, C.F., Chen, W.Q., Xu, R.Q., Lim, C.W.: Semi-analytical elasticity solutions for bi-directional functionally graded beams. Int. J. Solids Struct. 45, 258–275 (2008)

    Article  MATH  Google Scholar 

  3. Kambampati, S., Ganguli, R.: Non-uniform beams and stiff strings isospectral to axially loaded uniform beams and piano strings. Acta Mech. 226, 1227–1239 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Jang, T.S.: A general method for analyzing moderately large deflections of a non-uniform beam: an infinite Bernoulli-Euler-von Kármán beam on a nonlinear elastic foundation. Acta Mech. 225, 1967–1984 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cranch, E.T., Adler, A.A.: Bending vibrations of variable section beams. J. Appl. Mech. 23, 103–108 (1956)

    MATH  Google Scholar 

  6. Storti, D., Aboelnaga, Y.: Bending vibrations of a class of rotating beams with hypergeometric solutions. J. Appl. Mech. 54, 311–314 (1987)

    Article  MATH  Google Scholar 

  7. Wright, A.D., Smith, C.E., Thresher, R.W., Wang, J.L.C.: Vibration modes of centrifugally stiffened beams. J. Appl. Mech. 49, 197–202 (1982)

    Article  MATH  Google Scholar 

  8. Wang, H.C.: Generalized hypergeometric function solutions on the transverse vibration of a class of nonuniform beams. J. Appl. Mech. 34E, 702–708 (1967)

    Article  MATH  Google Scholar 

  9. Abrate, S.: Vibration of non-uniform rods and beams. J. Sound Vib. 185(4), 703–716 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  10. Caruntu, D.I.: Dynamic modal characteristics of transverse vibrations of cantilevers of parabolic thickness. Mech. Res. Commun. 36, 391–404 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Caruntu, D.I.: Classical Jacobi polynomials, closed-form solutions for transverse vibrations. J. Sound Vib. 306, 467–494 (2007)

    Article  MathSciNet  Google Scholar 

  12. Elishakoff, I., Candan, S.: Apparently first closed-form solution for vibrating: inhomogeneous beams. Int. J. Solids Struct. 38, 3411–3441 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Conway, H.D., Dubil, J.F.: Vibration frequencies of truncated wedge and cone beam. J. Appl. Mech. 32E, 932–935 (1965)

    Article  Google Scholar 

  14. Goel, R.P.: Transverse vibration of tapered beams. J. Sound Vib. 47(1), 1–7 (1976)

    Article  MATH  Google Scholar 

  15. Mabie, H.H., Rogers, C.B.: Transverse vibrations of double-tapered cantilever beams with end support and with end mass. J. Acoust. Soc. Am. 55(5), 986–991 (1974)

    Article  Google Scholar 

  16. Elishakoff, I.: Eigenvalues of Inhomogeneous Structures: Unusual Closed-Form Solutions. CRC Press, Boca Raton (2005)

    MATH  Google Scholar 

  17. Huang, Y., Yang, L.E., Luo, Q.Z.: Free vibration of axially functionally graded Timoshenko beams with non-uniform cross-section. Compos. Part B 45, 1493–1498 (2013)

    Article  Google Scholar 

  18. Huang, Y., Li, X.F.: A new approach for free vibration of axially functionally graded beams with non-uniform cross-section. J. Sound Vib. 329, 2291–2303 (2010)

    Article  Google Scholar 

  19. Shahba, A., Attarnejad, R., Marvi, M.T., Hajilar, S.: Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams with classical and non-classical boundary conditions. Compos. Part B 42, 801–808 (2011)

    Article  Google Scholar 

  20. Tong, X., Tabarrok, B.: Vibration analysis of Timoshenko beams with non-homogeneity and varying cross-section. J. Sound Vib. 186(5), 821–835 (1995)

    Article  MATH  Google Scholar 

  21. Lee, S.Y., Lin, S.M.: Exact vibration solutions for nonuniform Timoshenko beams with attachments. Am. Inst. Aero. Astro. J. 30(12), 2930–2934 (1992)

    MATH  Google Scholar 

  22. Elishakoff, I., Kaplunov, J., Nolde, E.: Celebrating the centenary of Timoshenko’s study of effects of shear deformation and rotary inertia. Appl. Mech. Rev. 67(6), 060802 (2015)

    Article  Google Scholar 

  23. Elishakoff, I.: An equation both more consistent and simper than the Bresse–Timoshenko equation. In: Gilat, R., Banks-Sills, L. (eds.) Advances in Mathematical Modeling and Experimental Methods for Materials and Structures: The Jacob Aboudi Volume, pp. 249–254. Springer, Berlin (2009)

    Chapter  Google Scholar 

  24. Han, S.M., Benaroya, H., Wei, T.: Dynamics of transversely vibrating beams using four engineering theories. J. Sound Vib. 225(5), 935–988 (1999)

    Article  MATH  Google Scholar 

  25. Wang, Z.X., Guo, D.R.: Introduction to Special Functions (in Chinese). Peking University Press, Beijing (2000)

    Google Scholar 

  26. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 7th edn. Elsevier Pte Ltd., Singapore (2007)

    MATH  Google Scholar 

  27. Huang, T.C.: The effect of rotatory inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions. J. Appl. Mech. 28, 579–584 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kaneko, T.: On Timoshenko’s correction for shear in vibrating beams. J. Phys. D Appl. Phys. 8, 1927–1936 (1975)

    Article  Google Scholar 

  29. von Kármán, Th, Biot, M.A.: Mathematical Methods in Engineering. McGraw-Hill Book Company Inc, New York (1940)

    MATH  Google Scholar 

  30. Boiangiu, M., Ceausu, V., Untaroiu, C.D.: A transfer matrix method for free vibration analysis of Euler-Bernoulli beams with variable cross section. J. Vib. Control (2014). doi:10.1177/1077546314550699

    MathSciNet  Google Scholar 

  31. Mindlin, R.D., Deresiewicz, H.: Timoshenko shear coefficient for flexural vibrations of beams. J. Appl. Mech. Trans. ASME 21(3), 286–286 (1954)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Center for Mechanics and Materials and AML, Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, China

    Jianghong Yuan

  2. Department of Civil Engineering, Zhejiang University, Hangzhou, 310058, China

    Yih-Hsing Pao

  3. Department of Engineering Mechanics, Zhejiang University, Hangzhou, 310027, China

    Weiqiu Chen

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  1. Jianghong Yuan
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  2. Yih-Hsing Pao
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  3. Weiqiu Chen
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Correspondence to Jianghong Yuan.

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Yuan, J., Pao, YH. & Chen, W. Exact solutions for free vibrations of axially inhomogeneous Timoshenko beams with variable cross section. Acta Mech 227, 2625–2643 (2016). https://doi.org/10.1007/s00707-016-1658-6

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  • DOI: https://doi.org/10.1007/s00707-016-1658-6

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