Structural Engineering and Mechanics

  • 제53권3호
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  • Pages.537-573
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  • 2015
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  • 1225-4568(pISSN)
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  • 1598-6217(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Differential transform method and numerical assembly technique for free vibration analysis of the axial-loaded Timoshenko multiple-step beam carrying a number of intermediate lumped masses and rotary inertias

  • Yesilce, Yusuf (Department of Civil Engineering, Dokuz Eylul University)
  • 투고 : 2014.08.28
  • 심사 : 2014.11.25
  • 발행 : 2015.02.10
⟨ 이전 논문 다음 논문 ⟩

초록

Multiple-step beams carrying intermediate lumped masses with/without rotary inertias are widely used in engineering applications, but in the literature for free vibration analysis of such structural systems; Bernoulli-Euler Beam Theory (BEBT) without axial force effect is used. The literature regarding the free vibration analysis of Bernoulli-Euler single-span beams carrying a number of spring-mass systems, Bernoulli-Euler multiple-step and multi-span beams carrying multiple spring-mass systems and multiple point masses are plenty, but that of Timoshenko multiple-step beams carrying intermediate lumped masses and/or rotary inertias with axial force effect is fewer. The purpose of this paper is to utilize Numerical Assembly Technique (NAT) and Differential Transform Method (DTM) to determine the exact natural frequencies and mode shapes of the axial-loaded Timoshenko multiple-step beam carrying a number of intermediate lumped masses and/or rotary inertias. The model allows analyzing the influence of the shear and axial force effects, intermediate lumped masses and rotary inertias on the free vibration analysis of the multiple-step beams by using Timoshenko Beam Theory (TBT). At first, the coefficient matrices for the intermediate lumped mass with rotary inertia, the step change in cross-section, left-end support and right-end support of the multiple-step Timoshenko beam are derived from the analytical solution. After the derivation of the coefficient matrices, NAT is used to establish the overall coefficient matrix for the whole vibrating system. Finally, equating the overall coefficient matrix to zero one determines the natural frequencies of the vibrating system and substituting the corresponding values of integration constants into the related eigenfunctions one determines the associated mode shapes. After the analytical solution, an efficient and easy mathematical technique called DTM is used to solve the differential equations of the motion. The calculated natural frequencies of Timoshenko multiple-step beam carrying intermediate lumped masses and/or rotary inertias for the different values of axial force are given in tables. The first five mode shapes are presented in graphs. The effects of axial force, intermediate lumped masses and rotary inertias on the free vibration analysis of Timoshenko multiple-step beam are investigated.

키워드

참고문헌

  1. Chen, D.W. and Wu, J.S. (2002), "The exact solutions for the natural frequencies and mode shapes of nonuniform beams with multiple spring-mass system", J. Sound Vib., 255, 299-322. https://doi.org/10.1006/jsvi.2001.4156
  2. Chen, D.W. (2003), "The exact solutions for the natural frequencies and mode shapes of non-uniform beams carrying multiple various concentrated elements", Struct. Eng. Mech., 16, 153-176. https://doi.org/10.12989/sem.2003.16.2.153
  3. C atal, S. (2006), "Analysis of free vibration of beam on elastic soil using differential transform method", Struct. Eng. Mech., 24(1), 51-62. https://doi.org/10.12989/sem.2006.24.1.051
  4. Catal, S. and Catal, H.H. (2006), "Buckling analysis of partially embedded pile in elastic soil using differential transform method", Struct. Eng. Mech., 24(2), 247-268. https://doi.org/10.12989/sem.2006.24.2.247
  5. Catal, S. (2008), "Solution of free vibration equations of beam on elastic soil by using differential transform method", Appl. Math. Model., 32, 1744-1757. https://doi.org/10.1016/j.apm.2007.06.010
  6. C atal, S. (2012), "Response of forced Euler-Bernoulli beams using differential transform method", Struct. Eng. Mech., 42, 95-119. https://doi.org/10.12989/sem.2012.42.1.095
  7. Gurgoze, M. (1984), "A note on the vibrations of restrained beams and rods with point masses", J. Sound Vib., 96, 461-468. https://doi.org/10.1016/0022-460X(84)90633-3
  8. Gurgoze, M. (1985), "On the vibration of restrained beams and rods with heavy masses", J. Sound Vib., 100, 588-589. https://doi.org/10.1016/S0022-460X(85)80009-2
  9. Gurgoze, M. and Batan, H. (1996), "On the effect of an attached spring-mass system on the frequency spectrum of a cantilever beam", J. Sound Vib., 195, 163-168. https://doi.org/10.1006/jsvi.1996.0413
  10. Gurgoze, M. (1996), "On the eigenfrequencies of a cantilever beam with attached tip mass and a springmass system", J. Sound Vib., 190, 149-162. https://doi.org/10.1006/jsvi.1996.0053
  11. Gurgoze, M. (1998), "On the alternative formulation of the frequency equation of a Bernoulli-Euler beam to which several spring-mass systems are attached in-span", J. Sound Vib., 217, 585-595. https://doi.org/10.1006/jsvi.1998.1796
  12. Gurgoze, M. and Erol, H. (2001), "Determination of the frequency response function of a cantilever beam simply supported in-span", J. Sound Vib., 247, 372-378. https://doi.org/10.1006/jsvi.2000.3618
  13. Gurgoze, M. and Erol, H. (2002), "On the frequency response function of a damped cantilever beam simply supported in-span and carrying a tip mass", J. Sound Vib., 255, 489-500. https://doi.org/10.1006/jsvi.2001.4118
  14. Hamdan, M.N. and Jubran, B.A. (1991), "Free and forced vibrations of a restrained uniform beam carrying an intermediate lumped mass and a rotary inertia", J. Sound Vib., 150, 203-216. https://doi.org/10.1016/0022-460X(91)90616-R
  15. Jaworski, J.W. and Dowell, E.H. (2008), "Free vibration of a cantilevered beam with multiple steps: Comparison of several theoretical methods with experiment", J. Sound Vib., 312, 713-725. https://doi.org/10.1016/j.jsv.2007.11.010
  16. Ju, F., Lee, H. P. and Lee, K. H. (1994), "On the free vibration of stepped beams", Int. J. Solid. Struct., 31, 3125-3137. https://doi.org/10.1016/0020-7683(94)90045-0
  17. Kaya, M.O. and Ozgumus, O.O. (2007), "Flexural-torsional-coupled vibration analysis of axially loaded closed-section composite Timoshenko beam by using DTM", J. Sound Vib., 306, 495-506. https://doi.org/10.1016/j.jsv.2007.05.049
  18. Koplow, M.A., Bhattacharyya, A. and Mann, B.P. (2006), "Closed form solutions for the dynamic response of Euler-Bernoulli beams with step changes in cross-section", J. Sound Vib., 295, 214-225. https://doi.org/10.1016/j.jsv.2006.01.008
  19. Lin, H.P. and Chang, S.C. (2005), "Free vibration analysis of multi-span beams with intermediate flexible constraints", J. Sound Vib., 281, 155-169. https://doi.org/10.1016/j.jsv.2004.01.010
  20. Lin, H.Y. and Tsai, Y.C. (2005), "On the natural frequencies and mode shapes of a uniform multi-span beam carrying multiple point masses", Struct. Eng. Mech., 21, 351-367. https://doi.org/10.12989/sem.2005.21.3.351
  21. Lin, H.Y. and Tsai, Y.C. (2006), "On the natural frequencies and mode shapes of a multiple-step beam carrying a number of intermediate lumped masses and rotary inertias", Struct. Eng. Mech., 22, 701-717. https://doi.org/10.12989/sem.2006.22.6.701
  22. Lin, H.Y. and Tsai, Y.C. (2007), "Free vibration analysis of a uniform multi-span beam carrying multiple spring-mass systems", J. Sound Vib., 302, 442-456. https://doi.org/10.1016/j.jsv.2006.06.080
  23. Lin, H.Y. (2008), "Dynamic analysis of a multi-span uniform beam carrying a number of various concentrated elements", J. Sound Vib., 309, 262-275. https://doi.org/10.1016/j.jsv.2007.07.015
  24. Lin, H.Y. (2010), "An exact solution for free vibrations of a non-uniform beam carrying multiple elasticsupported rigid bars", Struct. Eng. Mech., 34, 399-416. https://doi.org/10.12989/sem.2010.34.4.399
  25. Liu, W.H., Wu, J.R. and Huang, C.C. (1998), "Free vibration of beams with elastically restrained edges and intermediate concentrated masses", J. Sound Vib., 122, 193-207.
  26. Naguleswaran, S. (2002a), "Natural frequencies, sensitivity and mode shape details of an Euler-Bernoulli beam with one-step change in cross-section and with ends on classical supports", J. Sound Vib., 252, 751-767. https://doi.org/10.1006/jsvi.2001.3743
  27. Naguleswaran, S. (2002b), "Vibration of an Euler-Bernoulli beam on elastic end supports and with up to three-step changes in cross-section", Int. J. Mech. Sci., 44, 2541-2555. https://doi.org/10.1016/S0020-7403(02)00190-X
  28. Naguleswaran, S. (2002c), "Transverse vibrations of an Euler-Bernoulli uniform beam carrying several particles", Int. J. Mech. Sci., 44, 2463-2478. https://doi.org/10.1016/S0020-7403(02)00182-0
  29. Naguleswaran, S. (2003a), "Transverse vibration of an Euler-Bernoulli uniform beam on up o five resilient supports including ends", J. Sound Vib., 261, 372-384. https://doi.org/10.1016/S0022-460X(02)01238-5
  30. Naguleswaran, S. (2003b), "Vibration and stability of an Euler-Bernoulli beam with up to three-step changes in cross-section and in axial force", Int. J. Mech. Sci., 45, 1563-1579. https://doi.org/10.1016/j.ijmecsci.2003.09.001
  31. Naguleswaran, S. (2004a), "Transverse vibration and stability of an Euler-Bernoulli beam with step change in cross-section and in axial force", J. Sound Vib., 270, 1045-1055. https://doi.org/10.1016/S0022-460X(03)00505-4
  32. Naguleswaran, S. (2004b), "Vibration of an Euler-Bernoulli stepped beam carrying a non-symmetrical rigid body at the step", J. Sound Vib., 271, 1121-1132. https://doi.org/10.1016/S0022-460X(03)00574-1
  33. Ozgumus, O.O. and Kaya, M.O. (2007), "Energy expressions and free vibration analysis of a rotating double tapered Timoshenko beam featuring bending-torsion coupling", Int. J. Eng. Sci., 45, 562-586. https://doi.org/10.1016/j.ijengsci.2007.04.005
  34. Ozgumus, O.O. and Kaya, M.O. (2006), "Flapwise bending vibration analysis of double tapered rotating Euler-Bernoulli beam by using the differential transform method", Meccanica, 41, 661-670. https://doi.org/10.1007/s11012-006-9012-z
  35. Ozdemir, O. and Kaya, M.O. (2006), "Flapwise bending vibration analysis of a rotating tapered cantilever Bernoulli-Euler beam by differential transform method", J. Sound Vib., 289, 413-420. https://doi.org/10.1016/j.jsv.2005.01.055
  36. Wang, J.R., Liu, T.L. and Chen, D.W. (2007), "Free vibration analysis of a Timoshenko beam carrying multiple spring-mass systems with the effects of shear deformation and rotatory inertia", Struct. Eng. Mech., 26, 1-14. https://doi.org/10.12989/sem.2007.26.1.001
  37. Wu, J.S. and Chou, H.M. (1999), "A new approach for determining the natural frequencies and mode shape of a uniform beam carrying any number of spring masses", J. Sound Vib., 220, 451-468. https://doi.org/10.1006/jsvi.1998.1958
  38. Wu, J.S. and Chen, D.W. (2001), "Free vibration analysis of a Timoshenko beam carrying multiple springmass systems by using the numerical assembly technique", Int. J. Numer. Meth. Eng., 50, 1039-1058. https://doi.org/10.1002/1097-0207(20010220)50:5<1039::AID-NME60>3.0.CO;2-D
  39. Wu, J.S. and Chang, B.H. (2013), "Free vibration of axial-loaded multi-step Timoshenko beam carrying arbitrary concentrated elements using continuous-mass transfer matrix method", Euro. J. Mech. A/Solid., 38, 20-37. https://doi.org/10.1016/j.euromechsol.2012.08.003
  40. Yesilce, Y., Demirdag, O. and Catal, S. (2008), "Free vibrations of a multi-span Timoshenko beam carrying multiple spring-mass systems", Sadhana, 33, 385-401. https://doi.org/10.1007/s12046-008-0026-1
  41. Yesilce, Y. and Demirdag, O. (2008), "Effect of axial force on free vibration of Timoshenko multi-span beam carrying multiple spring-mass systems", Int. J. Mech. Sci., 50, 995-1003. https://doi.org/10.1016/j.ijmecsci.2008.03.001
  42. Yesilce, Y. and Catal, S. (2009), "Free vibration of axially loaded Reddy-Bickford beam on elastic soil using the differential transform method", Struct. Eng. Mech., 31(4), 453-476. https://doi.org/10.12989/sem.2009.31.4.453
  43. Yesilce, Y. (2010), "Effect of axial force on the free vibration of Reddy-Bickford multi-span beam carrying multiple spring-mass systems", J. Vib. Control, 16, 11-32. https://doi.org/10.1177/1077546309102673
  44. Zhou, J.K. (1986), Differential transformation and its applications for electrical circuits, Huazhong University Press, Wuhan China.

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