Skip to main content
SpringerLink
Log in

Variational Foundations of Modern Structural Dynamics

  • Conference paper
  • First Online:
Special Topics in Structural Dynamics, Volume 6

Abstract

The foundations of modern structural dynamic analysis are presented from a historical perspective. Underlying variational principles due to d’Alembert, Hamilton and Lagrange are reviewed, followed by subsequent key contributions to approximate analysis. Closely related procedures introduced by Ritz, Galerkin and Trefftz are described in terms of their original application to continuum boundary value problems. The role these procedures played in the development of the Finite Element Method and Matrix Structural Analysis is described. Finally, the continuing influence of variational principles in structural dynamics and mathematical physics in general is outlined.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Abbreviations

[Ap]:

Acoustic surface area matrix

[C]:

Damping matrix

[Cp]:

Acoustic compliance matrix

E:

Elastic stiffness property

F:

Force

[G]:

Transformation matrix

[K]:

Stiffness matrix

L:

Lagrangian

[M]:

Mass matrix

N.B.C:

Natural boundary condition

[P]:

Modal participation factor matrix

P.D.E:

Partial differential equation

{Q}:

Modal generalized forces

S:

Surface area

Sp :

Acoustic susceptance matrix

T:

Kinetic energy

U:

Potential or strain energy

V:

Volume

W:

Work

m:

Mass

{p}:

Acoustic pressure array

q:

Generalized coordinate (displacement)

t:

Time

u:

Displacement

x, y, z:

Position

[Φ]:

Modal matrix

[Γ]:

Force allocation matrix

[I]:

Identity matrix

Ψ:

Shape function

α, β:

Proportional damping constants

δ:

Variation

є:

Strain

λ:

Eigenvalue

ρ:

Mass density

ωn :

Natural frequency

ζn :

Critical damping ration

References

  1. Newton I (1689) Philosophiæ naturalis principia mathematica. London

    Google Scholar 

  2. Jean le Rond d’Alembert (1743) Trait e de Dynamique. Paris

    Google Scholar 

  3. Hamilton WR (1835) On a general method in dynamics. Phil Trans R Soc

    Google Scholar 

  4. Lagrange JL (1788) Mechanique Analitique. Desaint, Paris

    Google Scholar 

  5. Timoshenko SP (1953) History of strength of materials. McGraw-Hill, New York

    Google Scholar 

  6. Piersol A, Paez T (eds) (2010) Harris’ shock and vibration handbook, 6th edn. New York, McGraw-Hill

    Google Scholar 

  7. Ritz W (1909) Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik. Journal für die Reine und Angewandte Mathematik

    Google Scholar 

  8. Zienkiewicz OC, Taylor RL, Zhu JZ (2005) The finite element method, its basis and fundamentals, 6th edn. Elsevier, Boston

    Google Scholar 

  9. Pipes LA (1961) Matrix methods for engineering. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  10. Galerkin BG (1915) …Some questions of elastic equilibrium of rods and plates. Vestnik inzhenerov i tekhnikov 19

    Google Scholar 

  11. Nayfeh AH, Mook DT (1979) Nonlinear oscillations. Wiley, New York

    Google Scholar 

  12. Trefftz E (1926) Ein Gegenstuck zum Ritzschen Verfahren. In: Proceedings of the 2nd international congress of applied mechanics

    Google Scholar 

  13. Brebbia CA, Dominguez J (1992) Boundary elements an introductory course, 2nd edn. WIT Press, Boston

    Google Scholar 

  14. Castigliano A (1879) Theorie de l’equilibre des systemes elastiques. Turin

    Google Scholar 

  15. Rodden WP (2011) Theoretical and computational aeroelasticity. Crest Publishing, Burbank

    Google Scholar 

  16. Mach E (1883) The science of mechanics a historical account of its development. Open Court Publishing Co., Lasalle, IL USA

    Google Scholar 

  17. Vianello L (1898) Graphische Untersuchung der Knickfestigkeit gerarder Stabe. Z Ver Deut Ing 42

    Google Scholar 

  18. Stodola A (1905) Steam and gas turbines, 2nd edn. Van Nostrand Company, New York

    Google Scholar 

  19. Bathe K-J (1972) Large eigenvalue problems in dynamic analysis. ASCE J Eng Mech Div 98(6):1471–1485

    Google Scholar 

  20. Williams D (1945) Dynamic loads in aeroplanes under given impulsive load with particular reference to landing and gust loads on a large flying boat. Great Britain Royal Aircraft Establishment Reports SME 3309 and 3316

    Google Scholar 

  21. Hurty W (1965) Dynamic analysis of structural systems using component modes. AIAA J 3(4):678–685

    Google Scholar 

  22. Craig R, Bampton M (1968) Coupling of substructures for dynamic analysis. AIAA J 6(7):1313–1319

    Google Scholar 

  23. Toupin RA (1952) A variational principle for the mesh-type analysis of a mechanical system. J Appl Mech 74, 19:574

    Google Scholar 

  24. Coppolino RN (1975) A numerically efficient finite element hydroelastic analysis, volume 1: theory and results. NASA CR-2662, April 1975

    Google Scholar 

  25. Biot MA (1957) New methods in heat flow analysis with applications to flight structures. J Aeronaut Sci 24:857–873

    Google Scholar 

  26. MacNeal B, Bauer J, Coppolino R (1990) A general finite element vector potential formulation of electromagnetics using a time-integrated electric scalar potential. IEEE Trans Magn 26:1768–1770

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Measurement Analysis Corporation, 23850 Madison Street, Torrance, CA, 90505, USA

    Robert N. Coppolino

Authors
  1. Robert N. Coppolino
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Robert N. Coppolino .

Editor information

Editors and Affiliations

  1. Boeing Test & Evaluation, Seattle, Washington, USA

    Gary Foss

  2. Dept. of Mechanical Engineering, University of Massachusetts Lowell, Lowell, Massachusetts, USA

    Christopher Niezrecki

Rights and permissions

Reprints and permissions

Copyright information

© 2014 The Society for Experimental Mechanics, Inc.

About this paper

Cite this paper

Coppolino, R.N. (2014). Variational Foundations of Modern Structural Dynamics. In: Foss, G., Niezrecki, C. (eds) Special Topics in Structural Dynamics, Volume 6. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-04729-4_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-04729-4_3

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-04728-7

  • Online ISBN: 978-3-319-04729-4

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation