Skip to main content

Approximate Mode Shape for Damped Structures

Part of the Structural Integrity book series (STIN,volume 8)


Mode shape is very important in dynamic analysis of the structures. It can be employed to assess dynamic interaction between a structure and its supports to avoid sudden failure. However, unlike undamped structures, exact mode shapes for damped structures are difficult to obtain due to the eigenvectors complexity. In practice, damped structures cannot be shunned and they are available in many engineering applications. Some undamped structures may become damped structures during the operations. Such structures include pipes conveying fluid and because of their roles globally, their dynamic analysis becomes vital to check their integrity to prevent abrupt failures. In this paper, different methods of obtaining approximate mode shapes of composite pipe conveying fluid were investigated. The pipe is modeled using the extended Hamilton’s theory and discretized using wavelet-based finite element method. The pipe complex modal characteristics were obtained by solving the generalized eigenvalue problem and its mode shapes were computed.


  • Mode shapes
  • Composite fluid pipe
  • Damped structures
  • Complex eigenvectors
  • Wavelets

This is a preview of subscription content, access via your institution.

Buying options

USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-030-21894-2_2
  • Chapter length: 6 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
USD   149.00
Price excludes VAT (USA)
  • ISBN: 978-3-030-21894-2
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   199.99
Price excludes VAT (USA)
Hardcover Book
USD   279.99
Price excludes VAT (USA)
Fig. 1.


  1. Hu, Y.-J., Zhu, W.: Vibration analysis of a fluid-conveying curved pipe with an arbitrary undeformed configuration. Appl. Math Model 64, 624–642 (2018)

    MathSciNet  CrossRef  Google Scholar 

  2. Liu, M., Wang, Z., Zhou, Z., Qu, Y., Yu, Z., Wei, Q., Lu, L.: Vibration response of multi-span fluid-conveying pipe with multiple accessories under complex boundary conditions. Eur J Mech A Solids 72, 41–56 (2018)

    MathSciNet  CrossRef  Google Scholar 

  3. Yun-dong, L., Yi-ren, Y.: Vibration analysis of conveying fluid pipe via He’s variational iteration method. Appl Math Model 43, 409–420 (2017)

    MathSciNet  CrossRef  Google Scholar 

  4. Sarkar, A., Paidoussis, M.P.: A cantilever conveying fluid: coherent modes versus beam modes. Int J Non-Linear Mech 39(3), 467–481 (2004)

    CrossRef  Google Scholar 

  5. Wang, L.: Vibration and instability analysis of tubular nano- and micro-beams conveying fluid using nonlocal elastic theory. Physica E 41(10), 1835–1840 (2009)

    CrossRef  Google Scholar 

  6. Oke, W.A., Khulief, Y.A.: Effect of internal surface damage on vibration behavior of a composite pipe conveying fluid. Compos Struct 194, 104–118 (2018)

    CrossRef  Google Scholar 

  7. Young, D.F., Munson, B.R., Okiishi, T.H., Huebsch, W.W.: A Brief Introduction to Fluid Mechanics, 5th edn. Wiley, USA (2011)

    Google Scholar 

  8. Bansal, R.K.: A textbook of fluid mechanics and hydraulic machines, 9th edn. Laxmi Publications (P) Ltd, New Delhi, India (2010)

    Google Scholar 

  9. Oke, W.A., Khulief, Y.A.: Vibration analysis of composite pipes using the finite element method with B-spline wavelets. J Mech Sci Technol 30(2), 623–635 (2016)

    CrossRef  Google Scholar 

  10. Hajianmaleki M, Qatu MS Advances in composite materials—analysis of natural and man-made materials. InTech (2011)

    Google Scholar 

  11. Qatu, M.S., Iqbal, J.: Transverse vibration of a two-segment cross-ply composite shafts with a lumped mass. Compos Struct 92(5), 1126–1131 (2010)

    CrossRef  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Wasiu A. Oke .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Verify currency and authenticity via CrossMark

Cite this paper

Oke, W.A., Adeyemi, O.A., Bello, K.A., Adegbenjo, A. (2019). Approximate Mode Shape for Damped Structures. In: Gdoutos, E. (eds) Proceedings of the Second International Conference on Theoretical, Applied and Experimental Mechanics. ICTAEM 2019. Structural Integrity, vol 8. Springer, Cham.

Download citation

  • DOI:

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-21893-5

  • Online ISBN: 978-3-030-21894-2

  • eBook Packages: EngineeringEngineering (R0)