Skip to main content
SpringerLink
Log in

Recursive formulations for the method of reverberation-ray matrix and the application

  • Published:
Science in China Series G: Physics, Mechanics and Astronomy Aims and scope Submit manuscript

Abstract

A recursive formulation is proposed for the method of reverberation-ray matrix (MRRM) to exactly analyze the free vibration of a multi-span continuous rectangular Kirchhoff plate, which has two opposite simply-supported edges. In contrast to the traditional MRRM, numerical stability is achieved by using the present new formulation for high-order frequencies or/and for plates with large span-to-width ratios. The heavy computational cost of storage and memory are also cut down. An improved recursive formulation is further proposed by modifying the iterative formula to reduce the matrix inversion operations. Numerical examples are finally given to demonstrate the effectiveness and efficiency of the proposed recursive formulae.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Veletsos A S, Newmark N M. Determination of natural frequencies of continuous plates hinged along two opposite edges. ASME J Appl Mech, 1956, 23: 97–102

    MATH  Google Scholar 

  2. Bishop R E D, Johnson D C. The Mechanics of Vibration. London: Cambridge University Press, 1960

    MATH  Google Scholar 

  3. Ungar E E. Free oscillations of edge-connected simply supported plate systems. ASME J Eng Industry, 1961, 83: 434–440

    Google Scholar 

  4. Dickinson S M, Warburton G B. Natural frequencies of plate systems using the edge-effect method. J Mech Eng Sci, 1967, 9: 318–329[DOI]

    Article  Google Scholar 

  5. Cheung Y K, Cheung M S. Flexural vibrations of rectangular and other polygonal plates. ASCE J Eng Mech Div, 1971, 97: 391–411

    Google Scholar 

  6. Wu C I, Cheung Y K. Frequency analysis of rectangular plates continuous in one or two directions. Earthq Eng Struct Dyn, 1974, 3: 3–14

    Article  Google Scholar 

  7. Sakata T. A reduction method for vibrating and buckling problems of orthotropic continuous plates. J Sound Vibr, 1976, 49: 45–52[DOI]

    Article  MATH  ADS  Google Scholar 

  8. Takahashi K, Chishaki T. Free vibrations of two-way continuous rectangular plates. J Sound Vibr, 1979, 62: 455–459[DOI]

    Article  ADS  Google Scholar 

  9. Azimi S, Hamilton J F, Soedel W. The receptance method applied to the free vibration of continuous rectangular plates. J Sound Vibr, 1984, 93: 9–29[DOI]

    Article  MATH  ADS  Google Scholar 

  10. Cortinez V H, Laura P A A. A note on vibrating membranes and plates with an internal support. J Sound Vibr, 1986, 104: 533–535[DOI]

    Article  ADS  Google Scholar 

  11. Kim S C, Dickinson S M. The flexural vibration of line supported rectangular plate systems. J Sound Vibr 1987, 114: 129–142[DOI]

    Article  ADS  Google Scholar 

  12. Liew K M, Lam K Y. Vibration analysis of multi-span plates having orthogonal straight edges. J Sound Vibr 1991, 147: 255–264[DOI]

    Article  ADS  Google Scholar 

  13. Golley B W, Petrolito J. Finite strip-elements for the dynamic analysis of orthotropic plate structures. J Sound Vibr, 1993, 163: 479–491 [DOI]

    Article  MATH  ADS  Google Scholar 

  14. Liew K M, Xiang Y, Kitipornchai S. Transverse vibration of thick rectangular plates-II. Inclusion of oblique internal line supports. Comput Struct, 1993, 49: 31–58[DOI]

    Article  Google Scholar 

  15. Zhou D. Eigenfreqeuncies of line supported rectangular plates. Int J Solids Struct, 1994, 31: 347–358[DOI]

    Article  MATH  Google Scholar 

  16. Kong J, Cheung Y K. Vibration of shear-deformable plates with intermediate line supports: A finite layer approach. J Sound Vibr, 1995, 184: 639–649 [DOI]

    Article  MATH  ADS  Google Scholar 

  17. Zhou D, Cheung Y K. Free vibration of line supported rectangular plates using a set of static beam functions. J Sound Vibr, 1999, 223: 231–245[DOI]

    Article  Google Scholar 

  18. Wei G W, Zhao Y B, Xiang Y. Discrete singular convolution and its application to the analysis of plates with internal supports, part 1: Theory and algorithm. Int J Numer Methods Eng, 2002, 55: 913–946 [DOI]

    Article  MATH  MathSciNet  Google Scholar 

  19. Xiang Y, Zhao Y B, Wei G W. Levy solutions for vibration of multi-span rectangular plates. Int J Mech Sci, 2002, 44: 1195–1218[DOI]

    Article  MATH  Google Scholar 

  20. Lv C F, Lee Y Y, Lim C W, et al. Free vibration of long-span continuous rectangular Kirchhoff plates with internal rigid line supports. J Sound Vibr, 2006, 297: 351–364[DOI]

    Article  ADS  Google Scholar 

  21. Thomson W T. Matrix solution for the vibration of non-uniform beams. ASME J Appl Mech, 1950, 17: 337–339

    MATH  MathSciNet  Google Scholar 

  22. Pestel E C, Leckie F A. Matrix Methods in Elasto Mechanics. New York: McGraw-Hill, 1963

    Google Scholar 

  23. Knopoff L. A matrix method for elastic wave problems. Bull Seismol Soc Am, 1964, 54: 431–438

    Google Scholar 

  24. Kausel E, Roësset J. Stiffness matrices for layered soils. Bull Seismol Soc Am, 1981, 71: 1743–1761

    Google Scholar 

  25. Howard S M, Pao Y-H. Analysis and experiments on stress waves in planar trusses. J Eng Mech, 1998, 124: 884–891[DOI]

    Article  Google Scholar 

  26. Pao Y-H, Keh D C, Howard S M. Dynamic response and wave propagation in plane trusses and frames. AIAA J, 1999, 37: 594–603 [DOI]

    Article  Google Scholar 

  27. Pao Y-H, Su X Y, Tian J Y. Reverberation matrix method for propagation of sound in a multilayered liquid. J Sound Vibr, 2000, 230: 743–760[DOI]

    Article  ADS  Google Scholar 

  28. Su X Y, Tian J Y, Pao Y-H. Application of the reverberation-ray matrix to the propagation of elastic waves in a layered solid. Int J Solids Struct, 2002, 39: 5447–5463[DOI]

    Article  MATH  Google Scholar 

  29. Tan E L. Stiffness matrix method with improved efficiency for elastic wave propagation in layered anisotropic media. J Acoust Soc Am, 2000, 118: 3400–3403[DOI]

    Article  ADS  Google Scholar 

  30. Leissa A W. Vibration of plates, NASA SP-160. Washington: Office of Technology Utilization, NASA, 1969

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Civil Engineering, Zhejiang University, Hangzhou, 310027, China

    Jun Zhu, WeiQiu Chen, GuiRu Ye & ChaoFeng Lü

Authors
  1. Jun Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. WeiQiu Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. GuiRu Ye
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. ChaoFeng Lü
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to WeiQiu Chen.

Additional information

Supported by the National Natural Science Foundation of China (Grant Nos. 10725210, 10832009, and 10432030), the National Basic Research Program of China (Grant No. 2009CB623204), the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20060335107), and the Program for New Century Excellent Talents in University (Grant No. NCET-05-0510)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhu, J., Chen, W., Ye, G. et al. Recursive formulations for the method of reverberation-ray matrix and the application. Sci. China Ser. G-Phys. Mech. Astron. 52, 293–302 (2009). https://doi.org/10.1007/s11433-009-0037-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11433-009-0037-x

Keywords

Search

Navigation