0
TECHNICAL PAPERS

Analytical Solution for Stochastic Response of a Fractionally Damped Beam

[+] Author and Article Information
Om P. Agrawal

Mechanical Engineering and Energy Processes, Southern Illinois University, Carbondale, IL 62901

J. Vib. Acoust 126(4), 561-566 (Dec 21, 2004) (6 pages) doi:10.1115/1.1805003 History: Received May 01, 2001; Revised September 01, 2002; Online December 21, 2004
Copyright © 2004 by ASME
Your Session has timed out. Please sign back in to continue.

References

Bagley,  R. L., and Torvik,  P. J., 1983, “Fractional Calculus—A Different Approach to the Analysis of Viscoelastically Damped Structures,” AIAA J., 21(5), pp. 741–748.
Bagley,  R. L., and Torvik,  P. J., 1983, “A Theoretical Basis for the Application of Fractional Calculus of Viscoelasticity,” J. Rheol., 27(3), pp. 201–210.
Bagley,  R. L., and Torvik,  P. J., 1985, “Fractional Calculus in the Transient Analysis of Viscoelastically Damped Structures,” AIAA J., 23(6), pp. 918–925.
Koeller,  R. C., 1984, “Application of Fractional Calculus to the Theory of Viscoelasticity,” ASME J. Appl. Mech., 51, pp. 299–307.
Mainardi,  F., 1994, “Fractional Relaxation in Anelastic Solids,” J. Alloys Compd., 211-1, pp. 534–538.
Shen,  K. L., and Soong,  T. T., 1995, “Modeling of Viscoelastic Dampers for Structural Applications,” J. Eng. Mech., 121, pp. 694–701.
Pritz,  T., 1996, “Analysis of Four-Parameter Fractional Derivative Model of Real Solid Materials,” J. Sound Vib., 195, pp. 103–115.
Papoulia,  K. D., and Kelly,  J. M., 1997, “Visco-Hyperelastic Model for Filled Rubbers Used in Vibration Isolation,” ASME J. Eng. Mater. Technol., 119, pp. 292–297.
Friedrich, C., Schiessel, H., and Blumen, A., 1999, “Constitutive Behavior Modeling and Fractional Derivatives,” Advances in the Flow and Rheology of Non-Newtonian Fluids—Part A, D. A. Siginer, R. P. Chabra, and De Kee, eds., Elsevier, Amsterdam, pp. 429–466.
Koh,  C. G., and Kelly,  J. M., 1990, “Application of Fractional Derivatives to Seismic Analysis of Base-Isolated Models,” Earthquake Eng. Struct. Dyn., 19, pp. 229–241.
Makris,  N., and Constantinou,  M. C., 1992, “Spring-Viscous Damper Systems for Combined Seismic and Vibration Isolation,” Earthquake Eng. Struct. Dyn., 21, pp. 649–664.
Lee,  H. H., and Tsai,  C. S., 1994, “Analytical Model for Viscoelastic Dampers for Seismic Mitigation of Structures,” Comput. Struct., 50(1), pp. 111–121.
Gorenflo, R., and Mainardi, F., 1997, “Fractional Calculus: Integral and Differential Equations of Fractional Order,” Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi, eds., Springer-Verlag-Wien New York, pp. 223–276.
Rossikhin,  Y. A., and Shitikova,  M. V., 1997, “Applications of Fractional Calculus to Dynamic Problems of Linear and Nonlinear Hereditary Mechanics of Solids,” Appl. Mech. Rev., 50(1), pp. 15–67.
Jones, D. I. G., 2001, Handbook of Viscoelastic Vibration Damping, Wiley, New York.
Mainardi, F., 1997, “Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics,” Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi, eds., Springer-Verlag New York, pp. 291–348.
Spanos,  P. D., and Zeldin,  B. A., 1997, “Random Vibration of Systems With Frequency—Dependent Parameters or Fractional Derivatives,” J. Eng. Mech., 123(3), pp. 290–292.
Agrawal,  O. P., 2002, “Stochastic Analysis of a 1-D System With Fractional Damping of Order 1/2,” ASME J. Vibr. Acoust., 124, pp. 454–460.
Suarez,  L. E., and Shokooh,  A., 1997, “An Eigenvector Expansion Method for the Solution of Motion Containing Fractional Derivatives,” ASME J. Appl. Mech., 64, pp. 629–635.
Metzler,  R., and Klafter,  J., 2000, “The Random Walk’s Guide to Anomalous Diffusion: A Fractional Dynamics Approach,” Phys. Rep., 339(1), pp. 1–77.
Agrawal,  O. P., 2003, “Response of a Diffusion-Wave System Subjected to Deterministic and Stochastic Fields,” Z. Angew. Math. Mech., 4, pp. 265–274.
Oldham, K. B., and Spanier, J., 1974, The Fractional Calculus, Academic Press, New York.
Miller, K. S., and Ross, B., 1993, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York.
Podlubny, I., 1999, Fractional Differential Equations, Academic Press, New York.
Lin, Y. K., 1965, Probabilistic Theory of Structural Dynamics, McGraw-Hill, New York.
Nigam, N. C., 1983, Introduction to Random Vibrations, MIT Press, Cambridge, MA.
Eldred,  L. B., Baker,  W. P., and Palazotto,  A. N., 1996, “Numerical Applications of Fractional Derivative Model Constitutive Relations for Viscoelatic Materials,” Comput. Struct., 60(6), pp. 875–882.
Lixia, Y., and Agrawal, O. P., 1998, “A Numerical Scheme for Dynamic Systems Containing Fractional Derivatives,” Proc. 1998 ASME Design Engineering Technical Conferences, September, Atlanta, ASME, New York.
Shabana, A. A., 1991, Theory of Vibration, Volume II: Discrete and Continuous Systems, Springer-Verlag, New York.

Figures

Grahic Jump Location
Variance function E[x2](ρA)2/q as a function of space and time
Grahic Jump Location
Variance function E[v2](ρA)2/q as a function of space and time
Grahic Jump Location
Covariance function E[xv](ρA)2/q as a function of space and time

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In