Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Free vibration analysis of beams with various interfaces by using a modified matched interface and boundary method

  • Song, Zhiwei (School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology) ;
  • Li, Wei (School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology) ;
  • He, Xiaoqiao (Department of Architecture and Civil Engineering, City University of Hong Kong) ;
  • Xie, De (School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology)
  • Received : 2019.03.24
  • Accepted : 2019.04.30
  • Published : 2019.10.10
⟨ Previous Next ⟩

Abstract

This paper proposes a modified matched interface and boundary (MMIB) method to analyze the free vibration of beams with various interfaces caused by steps, intermediate rigid and elastic supports, intermediate concentrated masses and spring-mass systems, etc. A new strategy is developed to determine the parameters in the iterative computation of MMIB. The MMIB procedures are established to deal with boundary conditions and various interface conditions, which overcomes the shortcoming of the traditional MIB. A number of examples are utilized to illustrate the performance of MMIB method. Numerical results indicate that the MMIB method is a highly accurate and convergent approach for solving interface problems.

Keywords

Acknowledgement

Supported by : Hong Kong Special Administrative Region, Science and Technology Innovation Commission of Shenzhen Municipality

References

  1. Chai, Y., Gong, Z., Li, W., Li, T., Zhang, Q., Zou, Z. and Sun, Y. (2018), "Application of smoothed finite element method to twodimensional exterior problems of acoustic radiation", Int. J. Comput. Methods, 15, 1850029-1-35. https://doi.org/10.1142/S0219876218500299.
  2. Civalek O. (2013), "Nonlinear dynamic response of laminated plates resting on nonlinear elastic foundations by the discrete singular convolution-differential quadrature coupled approaches", Compos. Part B-Eng., 50,171-179. https://doi.org/10.1016/j.compositesb.2013.01.027.
  3. Civalek O. (2017), "Free vibration of carbon nanotubes reinforced (CNTR) and functionally graded shells and plates based on FSDT via discrete singular convolution method", Compos. Part B-Eng., 111, 45-59. https://doi.org/10.1016/j.compositesb.2016.11.030.
  4. De Rosa, M.A., Belles, P.M. and Maurizi, M.J. (1995), "Free vibration of stepped beams with intermediate elastic supports", J. Sound Vib., 181, 905-910. https://doi.org/10.1006/jsvi.1995.0177
  5. Duan, G.H. and Wang, X.W. (2013), "Free vibration analysis of multiple-stepped beams by the discrete singular convolution", Appl. Math. Comput., 219, 11096-11109. https://doi.org/10.1016/j.amc.2013.05.023.
  6. Duan, G.H. and Wang, X.W. (2014), "Vibration analysis of stepped rectangular plates by the discrete singular convolution algorithm", Int. J. Mech. Sci., 82, 100-109. https://doi.org/10.1016/j.ijmecsci.2014.03.004.
  7. Farghaly, S.H. and EI-Sayed, T.A. (2016), "Exact free vibration of multi-step Timoshenko beam system with several attachments", Mech. Syst. Signal Pr., 72-73, 525-546. https://doi.org/10.1016/j.ymssp.2015.11.025.
  8. Fornberg, B. (1998), "Calculation of weights in finite difference formulas", SIAM Rev., 40, 685-691. https://doi.org/10.1137/S0036144596322507.
  9. Jang, S. K. and Bert, C.W. (1989a), "Free vibrations of stepped beams: exact and numerical solutions", J. Sound Vib., 130(2), 342-346. https://doi.org/10.1016/0022-460X(89)90561-0
  10. Jang, S.K. and Bert, C. W. (1989b), "Free vibrations of stepped beams: higher mode frequencies and effects of steps on frequency", J Sound Vib., 132(1), 164-168. https://doi.org/10.1016/0022-460X(89)90882-1
  11. Lee, J.H. (2015), "Application of Chebyshev-tau method to the free vibration analysis of stepped beams", Int. J. Mech. Sci., 101-102, 411-410. https://doi.org/10.1016/j.ijmecsci.2015.08.012.
  12. Li, W., Song, Z. W. and Chai, Y. B. (2015), "Discrete singular convolution method for dynamic stability analysis of beams under periodic axial forces", J. Eng. Mech., 141, 04015033-1-13. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000931.
  13. Li, W., Song, Z. W., He, X, Q. and Xie, D. (2019), "A comparison study of HO-CFD and DSC-RSK for solving some classes of boundary-value and eigenvalue problems", Int. J. Comput. Methods. https://doi.org/10.1142/S0219876219500117.
  14. Lin, H. Y. (2008), "Dynamic analysis of a multi-span beam carrying a number of various concentrated elements", J. Sound Vib., 309, 262-275. https://doi.org/10.1016/j.jsv.2007.07.015.
  15. Lin, H. Y. (2009), "On the natural frequencies and mode shapes of a multi-span Timoshenko beam carrying a number of various concentrated elements", J. Sound Vib., 319, 593-605. https://doi.org/10.1016/j.jsv.2008.05.022.
  16. 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.
  17. Lin, R. M. and Ng, T. Y. (2004), "Exact vibration modes of multiple-stepped beams with arbitrary steps and supports using elemental impedance method", Eng. Struct., 152, 24-34. https://doi.org/10.1016/j.engstruct.2017.07.095.
  18. Liu, G. R. and Wu, T. Y. (2001), "Vibration analysis of beams using the generalized differential quadrature rule and domain decomposition", J. Sound Vib., 246(3), 461-481. https://doi.org/10.1006/jsvi.2001.3667.
  19. Lu, Z. R., Huang, M., Liu, J. K., Chen, W. H. and Liao, W. Y. (2009), "Vibration analysis of multiple -stepped beams with the composite element model", J. Sound Vib., 322, 1070-1080. https://doi.org/10.1016/j.jsv.2008.11.041.
  20. Maiz, S., Bambill, D.V., Rossit, C. A. and Laura, P.A. A. (2007), "Transverse vibration of Bernoulli-Euler beams carrying point masses and taking into account their rotatory inertia: exact solution", J. Sound Vib., 303, 895-908. https://doi.org/10.1016/j.jsv.2006.12.028.
  21. Mao, Q. and Pietrzko, S. (2010), "Free vibration analysis of stepped beams by using a domian decomposition method", Appl. Math. Comput., 217, 3429-3441. https://doi.org/10.1016/j.amc.2010.09.010.
  22. Maurizi, M. J. and Bambill, D. V. (1987), "Free vibration of a clamped-clamped beam with an intermediate elastic support", J. Sound Vib., 119, 173-176. https://doi.org/10.1016/0022-460X(87)90196-9
  23. Naguleswaran, S. (2002), "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.
  24. Song, Z. W., Li, W. and Liu G. R. (2012), "Stability and nonstationary vibration analysis of beams subjected to periodic axial forces using discrete singular convolution", Stru. Eng. Mech., 44, 487-499. https://doi.org/10.12989/sem.2012.44.4.487
  25. Song, Z. W., Chen, Z. G., Li, W. and Chai, Y. B. (2016), "Dynamic stability analysis of beams with shear deformation and rotary inertia subjected to periodic axial forces by using discrete singular convolution method", J. Eng. Mech., 142, 04015099-1-18. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001023.
  26. Wang, B., Xia, K. L. and Wei, G. W. (2015), "Second order solving 3D elasticity equations with complex interfaces", J. Comput. Phys., 294, 405-438. https://doi.org/10.1016/j.jcp.2015.03.053.
  27. Wang, X. W. and Wang, Y. L. (2013), "Free vibration analysis of multiple-stepped beams by the differential quadrature element method", Appl. Math. Comput., 219, 5802-5810. https://doi.org/10.1016/j.amc.2012.12.037.
  28. Wang, X.W. and Yuan, Z.X. (2017), "Discrete singular convolution and Taylor series expansion method for free vibration analysis of beams and rectangular plates with free boundaries", Int. J. Mech. Sci., 122,184-191. https://doi.org/10.1016/j.ijmecsci.2017.01.023.
  29. Wei, G.W. (1999), "Discrete singular convolution for the solution of the Fokker-Planck equations", J. Chem. Phys., 110, 8930-8942. https://doi.org/10.1063/1.478812.
  30. Wu, J. S. and Chou, H. M. (1999), "A new approach for determining the natural frequencies and mode shapes of a uniform beam carrying any number of sprung masses", J. Sound Vib., 220, 451-468. https://doi.org/10.1006/jsvi.1998.1958.
  31. Yu, S.N. and Wei, G.W. (2007), "Three-dimensional matched interface and boundary (MIB) method for treating geometric singularities", J. Comput. Phys., 227, 602-632. https://doi.org/10.1016/j.jcp.2007.08.003.
  32. Yu, S. N., Zhou, Y. C. and Wei, G. W. (2007), "Matched interface and boundary (MIB) method for elliptic problems with sharpedged interfaces", J. Comput. Phys., 224, 729-756. https://doi.org/10.1016/j.jcp.2006.10.030.
  33. Yu, S. N., Xiang, Y. and Wei, G. W. (2009), "Matched interface and boundary (MIB) method for the vibration analysis of plates", Commun. Numer. Methods Eng., 25, 923-950. https://doi.org/10.1002/cnm.1130.
  34. Zhao, S. and Wei, G. W. (2004), "High order FDTD methods via derivative matching for Maxwell's equations with material interfaces", J. Comput. Phys., 200, 60-103. https://doi.org/10.1016/j.jcp.2004.03.008.
  35. Zhao, S. and Wei, G. W. (2009), "Matched interface and boundary (MIB) for the implementation of boundary conditions in highorder central finite differences", Int. J. Numer. Meth. Eng., 77, 1690-1730. https://doi.org/10.1002/nme.2473.
  36. Zhou, Y. C. and Wei, G. W. (2006), "On the fictitious-domain and interpolation formulations of the matched interface and boundary (MIB) method", J. Comput. Phys., 219, 228-246. https://doi.org/10.1016/j.jcp.2006.03.027.
  37. Zhou, Y. C., Zhao, S., Feig, M. and Wei, G. W. (2006), "High order matched interface and boundary (MIB) schemes for elliptic equations with discontinuous coefficients and singular sources", J. Comput. Phys., 213, 1-30. https://doi.org/10.1016/j.jcp.2005.07.022.

Cited by

  1. Development of Matched Interface and Boundary Method for Buckling Analysis of Plates with Various Interfaces vol.12, pp.8, 2019, https://doi.org/10.1142/s1758825120500866
  2. Damage detection using both energy and displacement damage index on the ASCE benchmark problem vol.77, pp.2, 2019, https://doi.org/10.12989/sem.2021.77.2.151