Skip to main content
SpringerLink
Log in

Recursive solution for dynamic response of one-dimensional structures with time-dependent boundary conditions

  • Published:
Journal of Mechanical Science and Technology Aims and scope Submit manuscript

Abstract

A recursive solution method is derived for the transient response of one-dimensional structures subjected to the general form of time-dependent boundary conditions. Unlike previous solution methods that assumed the separation of variables, the present method involves formulating and solving the dynamic problems using the summation of two single-argument functions satisfying the motion equation. Based on boundary and initial conditions, a recursive procedure is derived to determine the single-argument functions. Such a procedure is applied to the general form of boundary conditions, and an analytical solution is derived by solving the recursive equation. The present solution method is implemented for base excitation problems, and the results are compared with those of the previous analytical solution and the Finite element (FE) analysis. The FE results converge to the present analytical solution, although considerable error is found in predicting a solution method on the basis of the separation of variables. The present analytical solution predicts the transient response for wave propagation problems in broadband excitation frequencies.

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. D. Beskos and G. Narayanan, Dynamic response of frameworks by numerical Laplace transform, Computer Methods in Applied Mechanics and Engineering, 37 (1983) 289–307.

    Article  MATH  Google Scholar 

  2. Y. Ni, W. Lou and J. Ko, A hybrid pseudo-force/Laplace transform method for non-linear transient response of a suspended cable, Journal of Sound and Vibration, 238 (2000) 189–214.

    Article  Google Scholar 

  3. A. Papoulis, A new method of inversion of the Laplace transform, Quart. Appl. Math, 14 (1957) 124.

    MathSciNet  Google Scholar 

  4. F. Durbin, Numerical inversion of Laplace transforms: an efficient improvement to Dubner and Abate’s method, The Computer Journal, 17 (1974) 371–376.

    Article  MATH  MathSciNet  Google Scholar 

  5. S. S. Rao, The finite element method in engineering, Butterworth-heinemann (2005).

    Google Scholar 

  6. M. Eisenberger, Dynamic stiffness matrix for variable cross-section Timoshenko beams, Communications in Numerical Methods in Engineering, 11 (1995) 507–513.

    Article  MATH  Google Scholar 

  7. J. Banerjee, Dynamic stiffness formulation for structural elements: a general approach, Computers & Structures, 63 (1997) 101–103.

    Article  MATH  Google Scholar 

  8. R. Langley, Application of the dynamic stiffness method to the free and forced vibrations of aircraft panels, Journal of Sound and Vibration, 135 (1989) 319–331.

    Article  MathSciNet  Google Scholar 

  9. J. Banerjee and A. Sobey, Dynamic stiffness formulation and free vibration analysis of a three-layered sandwich beam, International Journal of Solids and Structures, 42 (2005) 2181–2197.

    Article  Google Scholar 

  10. U. Lee, Vibration analysis of one-dimensional structures using the spectral transfer matrix method, Engineering Structures, 22 (2000) 681–690.

    Article  Google Scholar 

  11. J. F. Doyle, Wave propagation in structures, Springer (1989).

    Google Scholar 

  12. S. S. Rao, Vibration of continuous systems, John Wiley & Sons (2007).

    Google Scholar 

  13. S. Pal, Explicit finite element modeling of joint mechanics, ProQuest (2008).

    Google Scholar 

  14. F. Moser, L. J. Jacobs and J. Qu, Modeling elastic wave propagation in waveguides with the finite element method, Ndt & E International, 32 (1999) 225–234.

    Article  Google Scholar 

  15. D. Roy Mahapatra and S. Gopalakrishnan, A spectral finite element model for analysis of axial–flexural–shear coupled wave propagation in laminated composite beams, Composite Structures, 59 (2003) 67–88.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Aerospace Research Institute, Ministry of Science, Research and Technology, Tehran, 14665-834, Iran

    Mohammad Tahaye Abadi

Authors
  1. Mohammad Tahaye Abadi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohammad Tahaye Abadi.

Additional information

Recommended by Associate Editor Eung-Soo Shin

Mohammad Tahaye Abadi is an Associate Professor of Mechanical Engineering. He received his Ph.D. from Amirkabir University of Technology (Tehran Polytechnic), Iran in 2003. His research interests include composite materials, material characterization, viscoelastic materials, shock and vibration, and structural health monitoring.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Abadi, M.T. Recursive solution for dynamic response of one-dimensional structures with time-dependent boundary conditions. J Mech Sci Technol 29, 4105–4111 (2015). https://doi.org/10.1007/s12206-015-0904-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12206-015-0904-5

Keywords

Search

Navigation