Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Dynamic stiffness approach and differential transformation for free vibration analysis of a moving Reddy-Bickford beam

  • Bozyigit, Baran (Department of Civil Engineering, Dokuz Eylul University) ;
  • Yesilce, Yusuf (Department of Civil Engineering, Dokuz Eylul University)
  • Received : 2015.10.25
  • Accepted : 2016.03.23
  • Published : 2016.06.10
⟨ Previous Next ⟩

Abstract

In this study, the free vibration analysis of axially moving beams is investigated according to Reddy-Bickford beam theory (RBT) by using dynamic stiffness method (DSM) and differential transform method (DTM). First of all, the governing differential equations of motion in free vibration are derived by using Hamilton's principle. The nondimensionalised multiplication factors for axial speed and axial tensile force are used to investigate their effects on natural frequencies. The natural frequencies are calculated by solving differential equations using analytical method (ANM). After the ANM solution, the governing equations of motion of axially moving Reddy-Bickford beams are solved by using DTM which is based on Finite Taylor Series. Besides DTM, DSM is used to obtain natural frequencies of moving Reddy-Bickford beams. DSM solution is performed via Wittrick-Williams algorithm. For different boundary conditions, the first three natural frequencies that calculated by using DTM and DSM are tabulated in tables and are compared with the results of ANM where a very good proximity is observed. The first three mode shapes and normalised bending moment diagrams are presented in figures.

Keywords

References

  1. Arikoglu, A. and Ozkol, I. (2010), "Vibration analysis of composite sandwich beams with viscoelastic core by using differential transform method", Compos. Struct., 92, 3031-3039. https://doi.org/10.1016/j.compstruct.2010.05.022
  2. Bagdatli, S.M., Ozkaya, E. and Oz, H.R. (2011), "Dynamics of axially accelerating beams with an intermediate support", J. Vib. Acoust., 133, 1-10.
  3. Banerjee, J.R. (1997), "Dynamic stiffness for structural elements: A general approach", Comput. Struct., 63, 101-103. https://doi.org/10.1016/S0045-7949(96)00326-4
  4. Banerjee, J.R. and Gunawardana, W.D. (2007), "Dynamic stiffness matrix development and free vibration analysis of a moving beam", J. Sound Vib., 303, 135-143. https://doi.org/10.1016/j.jsv.2006.12.020
  5. Banerjee, J.R. (2012), "Free vibration of beams carrying spring-mass systems-A dynamic stiffness approach", Comput. Struct., 104-105, 21-26. https://doi.org/10.1016/j.compstruc.2012.02.020
  6. Banerjee, J.R. and Jackson, D.R. (2013), "Free vibration of a rotating tapered Rayleigh beam: A dynamic stiffness method of solution", Comput. Struct., 124, 11-20. https://doi.org/10.1016/j.compstruc.2012.11.010
  7. Bao-hui, L., Hang-shan, G., Hong-bo, Z., Yong-shou, L. and Zhou-feng, Y. (2011), "Free vibration analysis of multi-span pipe conveying fluid with dynamic stiffness method", Nucl. Eng. Des., 241, 666-671. https://doi.org/10.1016/j.nucengdes.2010.12.002
  8. Bickford, W.B. (1982), "A consistent higher order beam theory", Develop. Theor. Appl. Mech., 11, 137-150.
  9. 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), 246-269.
  10. Catal, S. (2014), "Buckling analysis of semi-rigid connected and partially embedded pile in elastic soil using differential transform method", Struct. Eng. Mech., 52(5), 971-995. https://doi.org/10.12989/sem.2014.52.5.971
  11. Catal, S. (2006), "Analysis of free vibration of beam on elastic soil using differential transform method", Struct. Eng. Mech., 24(1), 51-63. https://doi.org/10.12989/sem.2006.24.1.051
  12. 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
  13. Catal, S. (2012), "Response of forced Euler-Bernoulli beams using differential transform method", Struct. Eng. Mech., 42(1), 95-119. https://doi.org/10.12989/sem.2012.42.1.095
  14. Chen, C.K. and Ho, S. H. (1986), "Application of differential transformation to eigenvalue problems", Appl. Math. Comput., 79, 173-188.
  15. Chen, L.Q., Tang, Y.Q. and Lim, C.W. (2010), "Dynamic stability in parametric resonance of axially accelerating viscoelastic Timoshenko beams", J. Sound Vib., 329, 547-565. https://doi.org/10.1016/j.jsv.2009.09.031
  16. Ebrahimi, F. and Salari, E. (2015), "Size-dependent free flexural vibrational behavior of functionally graded nanobeams using semi-analytical differential transform method", Compos. Part B, 79, 156-169. https://doi.org/10.1016/j.compositesb.2015.04.010
  17. Eisenberger, M. (2003), "An exact high order beam element", Comput. Struct., 81, 147-152. https://doi.org/10.1016/S0045-7949(02)00438-8
  18. Eisenberger, M. (2003), "Dynamic stiffness vibration analysis using a high-order beam model", Int. J. Numer. Meth. Eng., 57, 1603-1614. https://doi.org/10.1002/nme.736
  19. Heyliger, P.R. and Reddy J.N. (1988), "A higher order beam finite element for bending and vibration problems", J. Sound Vib., 126, 309-326. https://doi.org/10.1016/0022-460X(88)90244-1
  20. Ho, S.H. and Chen, C.K. (2006), "Free transverse vibration of an axially loaded non-uniform spinning twisted Timoshenko beam using differential transform", Int. J. Mech. Sci., 48, 1323-1331. https://doi.org/10.1016/j.ijmecsci.2006.05.002
  21. Jun, L., Hongxing, H. and Rongying, H. (2008), "Dynamic stiffness analysis for free vibrations of axially loaded laminated composite beams", Comput. Struct., 84, 87-98. https://doi.org/10.1016/j.compstruct.2007.07.007
  22. Lal, R. and Ahlawat, N. (2015), "Axisymmetric vibrations and buckling analysis of functionally graded circular plates via differential transform method", Eur. J. Mech. A/Solid., 52, 85-94. https://doi.org/10.1016/j.euromechsol.2015.02.004
  23. Lee, U., Kim, J. and Oh, H. (2004), "Spectral analysis for the transverse vibration of an axially moving Timoshenko beam", J. Sound Vib., 271, 685-703. https://doi.org/10.1016/S0022-460X(03)00300-6
  24. Levinson, M. (1981), "A new rectangular beam theory", J. Sound Vib., 74, 81-87. https://doi.org/10.1016/0022-460X(81)90493-4
  25. Nefovska-Danilovic, M. and Petronijevic, M. (2015), "In-plane free vibration and response analysis of isotropic rectengular plates using the dynamic stiffness method", Comput. Struct., 152, 82-95. https://doi.org/10.1016/j.compstruc.2015.02.001
  26. Ozkaya, E. and Oz, H.R. (2002), "Determination of natural frequencies and stability regions of axially moving beams using artificial neural networks method", J. Sound Vib., 252, 782-789. https://doi.org/10.1006/jsvi.2001.3991
  27. Reddy, J.N. (1984), "A simple higher-order theory for laminated composite plates", J. Appl. Mech., 51, 745-752. https://doi.org/10.1115/1.3167719
  28. Reddy, J.N., Wang, C.M. and Lee, K.H. (1997), "Relationships between bending solutions of classical and shear deformation beam theories", Int. J. Solid. Struct., 34, 3373-3384. https://doi.org/10.1016/S0020-7683(96)00211-9
  29. Semnani, S.J., Attarnejad, R. and Firouzjaei, R.K. (2013), "Free vibration analysis of variable thickness thin plates by two-dimensional differential transform method", Acta Mechanica, 224, 1643-1658. https://doi.org/10.1007/s00707-013-0833-2
  30. Soldatos, K.P. and Sophocleous, C. (2001), "On shear deformable beam theories: The frequency and normal mode equations of the homogenous orthotropic Bickford beam", J. Sound Vib., 242, 215-245. https://doi.org/10.1006/jsvi.2000.3367
  31. Su, H. and Banerjee, J.R. (2015), "Development of dynamic stiffness method for free vibration of functionally graded Timoshenko beams", Computers and Structures, 147, 107-116. https://doi.org/10.1016/j.compstruc.2014.10.001
  32. Wattanasakulpong, N. and Charoensuk, J. (2015), "Vibration characteristics of stepped beams made of FGM using differential transformation method", Meccanica, 50, 1089-1101. https://doi.org/10.1007/s11012-014-0054-3
  33. Wickert, J.A. and Mote, C.D. (1989), "On the energetics of axially moving continua", J. Acoust. Soc. Am., 85, 1365-1368. https://doi.org/10.1121/1.397418
  34. Yan, Q.Y., Ding, H. and Chen, L.Q. (2014), "Periodic responses and chaotic behaviors of an axially accelerating viscoelastic Timoshenko beam", Nonlin. Dyn., 78, 1577-1591. https://doi.org/10.1007/s11071-014-1535-6
  35. 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, 453-476. https://doi.org/10.12989/sem.2009.31.4.453
  36. Yesilce, Y. (2010), "Differential transform method for free vibration analysis of a moving beam", Struct. Eng. Mech., 35, 645-658. https://doi.org/10.12989/sem.2010.35.5.645
  37. Yesilce, Y. (2011), "Free vibrations of a Reddy-Bickford multi-span beam carrying multiple spring-mass systems", Shock Vib., 18, 709-726. https://doi.org/10.1155/2011/892736
  38. Yesilce, Y. (2013), "Determination of natural frequencies and mode shapes of axially moving Timoshenko beams with different boundary conditions using differential transform method", Adv. Vib. Eng., 12, 89-108.
  39. Yesilce, Y. (2015), "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", Struct. Eng. Mech., 53, 537-573. https://doi.org/10.12989/sem.2015.53.3.537
  40. Zhou, J.K. (1968), Differential transformation and its applications for electrical circuits, Huazhong University Press, Wuhan, China.

Cited by

  1. Modeling and analysis of an axially acceleration beam based on a higher order beam theory vol.53, pp.10, 2018, https://doi.org/10.1007/s11012-018-0840-4
  2. Optimal extended homotopy analysis method for Multi-Degree-of-Freedom nonlinear dynamical systems and its application vol.61, pp.1, 2016, https://doi.org/10.12989/sem.2017.61.1.105
  3. Differential transform method and Adomian decomposition method for free vibration analysis of fluid conveying Timoshenko pipeline vol.62, pp.1, 2016, https://doi.org/10.12989/sem.2017.62.1.065
  4. Dynamic stiffness formulations for harmonic response of infilled frames vol.68, pp.2, 2016, https://doi.org/10.12989/sem.2018.68.2.183
  5. Free vibration analysis of transmission lines based on the dynamic stiffness method vol.6, pp.3, 2016, https://doi.org/10.1098/rsos.181354
  6. Investigation on hygro-thermal vibration of P-FG and symmetric S-FG nanobeam using integral Timoshenko beam theory vol.8, pp.4, 2020, https://doi.org/10.12989/anr.2020.8.4.293