Abstract
Problems involving vibration occur in many areas of mechanical, civil and aerospace engineering: wave loading of offshore platforms, cabin noise in aircrafts, earthquake and wind loading of cable stayed bridges and high rise buildings, performance of machine tools — to pick only few examples. Human beings usually regard noise and vibration as uncomfortable. Beside this, an engineering structure can fail due to excessive vibration — the devastating effects of earthquakes on our society is a prime example of this fact. Due to this reasons over the years the aim of the vibration engineers has been to reduce vibration. In order to achieve this in an efficient and economic manner a good understanding of the physics of vibration phenomena in complex engineering structures is needed. In the last few decades, the sophistication of modern design methods together with the development of improved composite structural materials instilled a trend towards lighter structures. At the same time, there is also a constant demand for larger structures, capable of carrying more loads at higher speeds with minimum noise and vibration level as the safety/workability and environmental criteria become more stringent. Unfortunately, these two demands are conflicting and the problem cannot be solved without proper understanding of the vibration phenomena.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Bibliography
M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables. Dover Publications, Inc., New York, USA, 1965.
S. Adhikari. Dynamics of non-viscously damped linear systems. ASCE Journal of Engineering Mechanics, 128(3):328–339, March 2002.
S. Adhikari. Damping modelling using generalized proportional damping. Journal of Sound and Vibration, 293(1–2):156–170, May 2006.
S. Adhikari. Qualitative dynamic characteristics of a non-viscously damped oscillator. Proceedings of the Royal Society of London, Series-A, 461(2059):2269–2288, July 2005.
S. Adhikari. Damping Models for Structural Vibration. PhD thesis, Cambridge University Engineering Department, Cambridge, UK, September 2000.
S. Adhikari and J. Woodhouse. Identification of damping: part 2, non-viscous damping. Journal of Sound and Vibration, 243(1):63–88, May 2001.
R. J. Allemang and D. L. Brown. A correlation coefficient for modal vector analysis. In Proceedings of the 1st International Modal Analysis Conference (IMAC), pages 110–116, Orlando, FL, 1982.
K. F. Alvin, L. D. Peterson, and K. C. Park. Extraction of normal modes and full modal damping from complex modal parameters. AIAA Journal, 35(7): 1187–1194, July 1997.
R. L. Bagley and P. J. Torvik. Fractional calculus-a different approach to the analysis of viscoelastically damped structures. AIAA Journal, 21(5):741–748, May 1983.
M. Baruch. Identification of the damping matrix. Technical Report TAE No.803, Technion, Israel, Faculty of Aerospace Engineering, Israel Institute of Technology, Haifa, May 1997.
H. Benaroya. Random eigenvalues, algebraic methods and structural dynamic models. Applied Mathematics and Computation, 52:37–66, 1992.
M. A. Biot. Variational principles in irreversible thermodynamics with application to viscoelasticity. Physical Review, 97(6):1463–1469, 1955.
M. A. Biot. Linear thermodynamics and the mechanics of solids. In Proceedings of the Third U. S. National Congress on Applied Mechanics, pages 1–18, New York, 1958. ASME.
R. E. D. Bishop and W. G. Price. An investigation into the linear theory of ship response to waves. Journal of Sound and Vibration, 62(3):353–363, 1979.
D. R. Bland. Theory of Linear Viscoelasticity. Pergamon Press, London, 1960.
Norman Bleistein and Richard A. Handelsman. Asymptotic Expansions of Integrals. Holt, Rinehart and Winston, New York, USA, 1994.
W. E. Boyce. Random Eigenvalue Problems. Probabilistic methods in applied mathematics. Academic Press, New York, 1968.
T. K. Caughey. Classical normal modes in damped linear dynamic systems. Transactions of ASME, Journal of Applied Mechanics, 27:269–271, June 1960.
T. K. Caughey and M. E. J. O’Kelly. Classical normal modes in damped linear dynamic systems. Transactions of ASME, Journal of Applied Mechanics, 32:583–588, September 1965.
S. Y. Chen, M. S. Ju, and Y. G. Tsuei. Extraction of normal modes for highly coupled incomplete systems with general damping. Mechanical System and Signal Processing, 10(1):93–106, 1996.
J. D. Collins and W. T. Thomson. The eigenvalue problem for structural systems with statistical properties. AIAA Journal, 7(4):642–648, April 1969.
J. F. Doyle. Wave Propagation in Structures. Springer Verlag, New York, 1989.
I. Elishakoff and Y. J. Ren. Large Variation Finite Element Method for Stochastic Problems. Oxford University Press, Oxford, U.K., 2003.
Isaac Elishakoff. Eigenvalues of Inhomogeneous Structures: Unusual Closed-Form. CRC Press, Boca Raton, FL, USA, 2005.
D. J. Ewins. Modal Testing: Theory and Practice. Research Studies Press, Baldock, England, second edition, 2000.
N. J. Ferguson and W. D. Pilkey. Literature review of variants of dynamic stiffness method, Part 1: The dynamic element method. The Shock and Vibration Digest, 25(2):3–12, 1993a.
N. J. Ferguson and W. D. Pilkey. Literature review of variants of dynamic stiffness method, Part 1: Frequency-dependent matrix and other. The Shock and Vibration Digest, 25(4):3–10, 1993b.
M. I. Friswell and J. E. Mottershead. Finite Element Model Updating in Structural Dynamics. Kluwer Academic Publishers, The Netherlands, 1995.
M. Géradin and D. Rixen. Mechanical Vibrations. John Wiely & Sons, New York, NY, second edition, 1997. Translation of: Théorie des Vibrations.
R. Ghanem and P.D. Spanos. Stochastic Finite Elements: A Spectral Approach. Springer-Verlag, New York, USA, 1991.
D. F. Golla and P. C. Hughes. Dynamics of viscoelastic structures-a time domain finite element formulation. Transactions of ASME, Journal of Applied Mechanics, 52: 897–906, December 1985.
T. K. Hasselsman. A method of constructing a full modal damping matrix from experimental measurements. AIAA Journal, 10(4):526–527, 1972.
Francois M. Hemez and Yakov Ben-Haim. The Good, The Bad, and The Ugly of Predictive Science. In Kenneth M. Hanson and François M. Hemez, editors, Proceedings of the 4th International Conference on Sensitivity Analysis of Model Output, pages 181–193, Santa Fe, New Mexico, USA, March 2004. see http://library.lanl.gov/.
R. A. Ibrahim. Structural dynamics with parameter uncertainties. Applied Mechanics Reviews, ASME, 40(3):309–328, 1987.
S. R. Ibrahim. Dynamic modeling of structures from measured complex modes. AIAA Journal, 21(6):898–901, June 1983.
Norman L. Johnson and Samuel Kotz. Distributions in Statistics: Continuous Univariate Distributions-2. The Houghton Mifflin Series in Statistics. Houghton Mifflin Company, Boston, USA, 1970.
J. N. Kapur and H. K. Kesavan. Entropy Optimization Principles With Applications. Academic Press, San Diego, CA, 1992.
M. Kleiber and T. D. Hien. The Stochastic Finite Element Method. John Wiley, Chichester, 1992.
G. A. Lesieutre and D. L. Mingori. Finite element modeling of frequency-dependent material properties using augmented thermodynamic fields. Journal of Guidance, Control and Dynamics, 13:1040–1050, 1990.
R. H. Lyon and R. G. Dejong. Theory and Application of Statistical Energy Analysis. Butterworth-Heinmann, Boston, second edition, 1995.
N. M. M. Maia and J. M. M. Silva, editors. Theoretical and Experimental Modal Analysis. Engineering Dynamics Series. Research Studies Press, Taunton, England, 1997. Series Editor, J. B. Robetrs.
C. S. Manohar and S. Adhikari. Dynamic stiffness of randomly parametered beams. Probabilistic Engineering Mechanics, 13(1):39–51, January 1998.
C. S. Manohar and S. Gupta. Modeling and evaluation of structural reliability: Current status and future directions. In K. S. Jagadish and R. N. Iyengar, editors, Research reviews in structural engineering, Golden Jubilee Publications of Department of Civil Engineering, Indian Institute of Science, Bangalore. University Press, 2003.
C. S. Manohar and R. A. Ibrahim. Progress in structural dynamics with stochastic parameter variations: 1987 to 1998. Applied Mechanics Reviews, ASME, 52(5):177–197, May 1999.
A. M. Mathai and Serge B. Provost. Quadratic Forms in Random Variables: Theory and Applications. Marcel Dekker, Inc., 270 Madison Avenue, New York, NY 10016, USA, 1992.
D. J. McTavish and P. C. Hughes. Modeling of linear viscoelastic space structures. Transactions of ASME, Journal of Vibration and Acoustics, 115:103–110, January 1993.
L. Meirovitch. Principles and Techniques of Vibrations. Prentice-Hall International, Inc., New Jersey, 1997.
C. Minas and D. J. Inman. Identification of a nonproportional damping matrix from incomplete modal information. Transactions of ASME, Journal of Vibration and Acoustics, 113:219–224, April 1991.
J. E. Mottershead. Theory for the estimation of structural vibration parameters from incomplete data. AIAA Journal, 28(3):559–561, 1990.
A. Muravyov. Analytical solutions in the time domain for vibration problems of discrete viscoelastic systems. Journal of Sound and Vibration, 199(2):337–348, 1997.
N. C. Nigam. Introduction to Random Vibration. The MIT Press, Cambridge, Massachusetts, 1983.
Athanasios Papoulis and S. Unnikrishna Pillai. Probability, Random Variables and Stochastic Processes. McGraw-Hill, Boston, USA, fourth edition, 2002.
M. Paz. Structural Dynamics. CBS Publishers, New Delhi, 1985.
E. S. Pearson. Note on an approximation to the distribution of non-central x 2. Biometrica, 46:364, 1959.
D. P. Pilkey and D. J. Inman. Survey of damping matrix identification. In Proceedings of the 16th International Modal Analysis Conference (IMAC), volume 1, pages 104–110, 1998.
R. H. Plaut and K. Huseyin. Derivative of eigenvalues and eigenvectors in non-self adjoint systems. AIAA Journal, 11(2):250–251, February 1973.
W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipes in C. Cambridge University Press, Cambridge, 1992.
S. Anantha Ramu and R. Ganesan. A galerkin finite element technique for stochastic field problems. Computer Methods in Applied Mechanics and Engineering, 105:315–331, 1993.
Lord Rayleigh. Theory of Sound (two volumes). Dover Publications, New York, 1945 re-issue, second edition, 1877.
T. S. Sankar, S. A. Ramu, and R. Ganesan. Stochastic finite element analysis for high speed rotors. Transactions of ASME, Journal of Vibration and Acoustics, 115:59–64, 1993.
A. Sarkar and C. S. Manohar. Dynamic stiffness of a general cable element. Archive of Applied Mechanics, 66:315–325, 1996.
J. Vom Scheidt and Walter Purkert. Random Eigenvalue Problems. North Holland, New York, 1983.
Schlesinger. Terminology for model credibility. Simulation, 32,(3):103104, 1979.
A. Sestieri and R. A. Ibrahim. Analysis of errors and approximations in the use of modal coordinates. Journal of Sound and Vibration, 177(2):145–157, 1994.
M. Shinozuka and F. Yamazaki. Stochastic finite element analysis: an introduction. In S. T. Ariaratnam, G. I. Schueller, and I. Elishakoff, editors, Stochastic structural dynamics: Progress in theory and applications, London, 1998. Elsevier Applied Science.
J. M. Montalvao E Silva and Nuno M. M. Maia, editors. Modal Analysis and Testing: Proceedings of the NATO Advanced Study Institute, NATO Science Series: E: Applied Science, Sesimbra, Portugal, 3–15 May 1998.
N. Wagner and S. Adhikari. Symmetric state-space formulation for a class of non-viscously damped systems. AIAA Journal, 41(5):951–956, 2003.
R. Wong. Asymptotic Approximations of Integrals. Society of Industrial and Applied Mathematics, Philadelphia, PA, USA, 2001. First published by Academic Press Inc. in 1989.
J. Woodhouse. Linear damping models for structural vibration. Journal of Sound and Vibration, 215(3):547–569, 1998.
Kemin Zhou, John C. Doyle, and Keith Glover. Robust and Optimal Control. Prentice-Hall Inc, Upper Saddle River, New Jersey 07458, 1995.
O. C. Zienkiewicz and R. L. Taylor. The Finite Element Method. McGraw-Hill, London, fourth edition, 1991.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 CISM, Udine
About this chapter
Cite this chapter
Adhikari, S. (2007). Models, Verification, Validation, Identification and Stochastic Eigenvalue Problems. In: Elishakoff, I. (eds) Mechanical Vibration: Where do we Stand?. International Centre for Mechanical Sciences, vol 488. Springer, Vienna. https://doi.org/10.1007/978-3-211-70963-4_14
Download citation
DOI: https://doi.org/10.1007/978-3-211-70963-4_14
Publisher Name: Springer, Vienna
Print ISBN: 978-3-211-68586-0
Online ISBN: 978-3-211-70963-4
eBook Packages: EngineeringEngineering (R0)