Sage Journals: Discover world-class research

Abstract

The equations of motion of structures with elastic and viscoelastic materials in the time domain are derived. The development is consistent with the finite element formulation and leads to a system of equations where the matrices are symmetric, real, and composed of constant coefficients. A four-parameter fractional derivative model is used to model the frequency dependence of the linear viscoelastic material since experimental data can be fitted successfully over a wide frequency range. The resulting equations of motion are known as the elastic-viscoelastic equations of motion. Numerical procedures for solving the elastic-viscoelastic equations of motion in the time domain are developed, and a procedure based on the central-difference method that incorporates fractional derivatives is presented. The numerical stability of the procedure is presented, and criteria for selecting the size of the time step are given.

The closed-form, steady state solution of a single degree of freedom system is obtained in the frequency domain and is utilized to validate the results obtained by using the numerical procedures. The proper selection of the stiffness for viscoelastic dampers placed in elastic structural systems is discussed in order to ensure that the damper is effective in reducing dynamic amplification of the structure. The dynamic response of a multi-degree of freedom structure, obtained by using the numerical procedures, is used to demonstrate the effectiveness of the dampers in reducing the structural response to dynamic loading.

Get full access to this article

View all access options for this article.

References

Bagley, R. L. 1979 "Applications of Generalized Derivatives to Viscoelasticity," Air Force Materials Laboratory, TR-79-4103, and Dissertation, Air Force Institute of Technology.

Bagley, R. L. and Torvik, P. J. 1979

"A Generalized Derivative Model for an Elastomer Damper,"

Shock and Vibration Bulletin, Vol. 49, No. 2, pp. 135-143.

Bagley, R. L. and Torvik, P. J. 1983

"A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity,"

Journal of Rheology, Vol. 27, No. 3, pp. 201-210.

Bathe, K. J. 1982 Finite Element Procedures in Engineering Analysis, Prentice-Hall, Englewood Cliffs, New Jersey.

Christensen, R. M. 1971 Theory of Viscoelasticity: An Introduction, Academic Press, New York.

Escobedo-Torres, J. 1997 "Solution Methods for the Dynamic Response of Structures with Viscoelastic Materials," Ph.D. Dissertation, Lehigh University, Bethlehem, PA.

Escobedo-Torres, J. and Ricles, J. M. , 1998, "Four-Parameter Viscoelastic Models in Vibration Damping," Proceedings of the 12th ASCE Engineering Mechanics Conference, UCSD, San Diego, CA.

Foss, K. A. 1958

"Co-ordinates which Uncouple the Equations of Motion of Damped Linear Dynamic Systems,"

Journal of Applied Mechanics, Vol. 25, pp. 361-364.

Golla, D. F. , and Hughes, P. C. 1985

"Dynamics of Viscoelastic Structures-A Time Domain, Finite Element Formulation,"

Journal of Applied Mechanics, Vol. 52, pp. 897-906.

10.

Higgins, C., 1997, Ph.D. Dissertation, Lehigh University.

11.

Hughes, T. J. R. 1987 The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Englewood Cliffs, New Jersey.

12.

Johnson, C. D. , and Kienholz, D. A. 1982

"Finite Element Prediction of Damping in Structures with Constrained Viscoelastic Layers,"

AIAA Journal, Vol. 20, No. 9, pp. 1284-1292.

13.

Kasai, K. , Munshi, J. A. , Lai, M. , and Maison, B. F. 1993

"Viscoelastic Damper Hysteretic Model: Theory, Experiment, and Application,"

Seminar on Seismic Isolation, Passive Energy Dissipation, and Active Control, ATC-17-1, San Francisco.

14.

Koh, C. G. , and Kelly, J. M. 1990

"Application of Fractional Derivatives to Seismic Analysis of Base-Isolated Models,"

Earthquake Engineering and Structural Dynamics, Vol. 19, pp. 229-241.

15.

Lesieutre, G. A. , and Mignori, D. L. 1990. "Finite Element Modeling of Frequency Dependent Material Damping Using Augmenting Thermodynamic Fields," AIAA Journal of Guidance, Control, and Dynamics, Vol. 13, No. 6, pp. 1040-1050.

16.

McTavish, D. J. , and Hughes, P. C. 1993

"Modeling of Linear Viscoelastic Space Structures,"

Journal of Vibration, Acoustics, Stress, and Reliability in Design, Vol. 115, No. 1, pp. 103-110.

17.

Nashif, A. D. , Jones, D. I. G. , and Henderson, J. P. 1985 Vibration Damping, John Wiley & Sons, New York.

18.

Oldham, K. B. , and Spanier, J. 1974 The Fractional Calculus, Academic Press, New York.

19.

Padovan, J. 1987

"Computational Algorithms for FE Formulations Involving Fractional Operators,"

Computational Mechanics, Vol. 2, pp. 271-287.

20.

Sause, R. , Hemingway, G. , and Kasai, K. 1994 "Simplified Seismic Response Analysis of Viscoelastic-Damped Frame Structures," Proceedings of the 5th U.S. National Conference on Earthquake Engineering, EERI, Chicago, IL., Vol. 1, pp. 839-848.

21.

Slater, J. C. , Belvin, W. K. , and Inman, D. J. 1993 "A Survey of Modern Methods for Modeling Frequency Dependent Damping in Finite Element Models," The 11th International Modal Analysis Conference, Proc. SPIE Vol. 1923, Pt. 2, pp. 1508-1512.

22.

Sun, C. T. , and Lu, Y. P. 1995 Vibration Damping of Structural Elements, Prentice-Hall, Englewood Cliffs, New Jersey.

23.

Ungar, E. E. , and Kerwin, E. M. 1962

"Loss Factors of Viscoelastic Systems in Terms of Energy Concepts,"

The Journal of the Acoustical Society of America, Vol. 34, No. 7, pp. 954-947.

24.

Yiu, Y. C. 1993 "Finite Element Analysis of Structures with Classical Viscoelastic Materials," Collection of Technical Papers, 34th AIAA/ ASME/ASCE/AHS Structures, Structural Dynamics and Materials Conference, pp. 2110-2119.

25.

3M, 1996 "Product Information and Performance Data," The 3M Company.

The Fractional Order Elastic-Viscoelastic Equations of Motion: Formulation and Solution Methods

Abstract

Get full access to this article

References