Development of Mathematical Model of a Plate for Active Vibration Suppression

Abstract

A mathematical model of the transverse vibration of a plate to be used for free and forced vi bration control purposes is presented. For the analysis of plate flexural vibration, the eigenfunctions of a Poisson-Kirchoff plate have been used as basis functions in a Galerkin formulation. Separation of variables and a double Fourier expansion coupled with Navier's method are used to find the optimal location of the sensors/actuators. A quadratic control objective is defined as a measure of system performance. The control objective is composed of those error variables that are important to the design, and they are used to approx imate the high-order plant system by a lower order model. To guarantee that the error in the reduced model is smaller than the desired error, a minimum required number of modes in the plate model has been derived analytically.

Keywords

Plate vibration control Galerkin formulation Navier's method order reduction optimal sensors/actua tors piezoelectric active vibration actuator sensor

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References

Baz, A. and Poh, S. , 1988, "Performance of an active control system with piezoelectric actuators," Journal of Sound and Vibration 126, 327-343.

Chen, C.T. , 1970, Introduction to Linear System Theory, Holt, Rinehart and Winston, New York.

Fariborzi, E. , Golnaraghi, M.F. , and Heppler, G.R. , 1997, "Experimental control of free and forced structural vibration using linear coupling strategy," Journal of Smart Materials and Structures 6, 1-9.

Friswell, M.I. , 1990, "Candidate reduced order models for structural parameter estimation," Journal of Vibration and Acoustics 112, 93-97.

Hughes, P.C. and Skelton, R.E. , 1980, "Controllability and observability of linear matrix-second-order systems," Journal of Applied Mechanics 47(2), 415-420.

Hughes, P.C. and Skelton, R.E. , 1981, "Modal truncation for flexible spacecraft," Journal of Guidance and Control 4(3),4291-4297.

Hung, Y.S. and Muzlifah, 1990, "Hankel-norm model reduction with fixed modes" IEEE Transactions on Automatic Control 35(3), 373-377.

Kapania, R.K. and Stefano, R. , 1989, "Recent advances in analysis of laminated beam and plates ," AIAA Journal 27(7), 923-946.

Naylor, A.W. and Sell, G.R. , 1982, Linear Operator Theory in Engineering and Science, 2nd ed., Springer-Verlag, New York.

10.

Rajalingham, C. , Bhat, R.B. , and Xistris, G.D. , 1996, "Vibration of rectangular plates using plate characteristic functions," Journal of Sound and Vibration 193(2), 497-509.

11.

Salemi, P. , Golnaraghi, M.F. , and Heppler, G.R. , 1995, "Experimental control of forced and unforced structural vibration using a linear coupling strategy," in Proceedings of the SES'95 32nd Annual Technical Meeting.

12.

Skelton, R.E. and Hughes, P.C. , 1980, "Modal cost analysis for linear matrix-second-order systems ," Journal of Dynamic Systems, Measurement, and Control 102, 151-158.

13.

Szilard, R. , 1974, Theory and Analysis of Plates, Prentice Hall, Englewood Cliffs, New Jersey.

14.

Timoshenko, S. and Woinowsky-Krieger, S. , 1959, Theory of Plates and Shells , 2nd ed., McGraw-Hill, New York.

15.

Warburton, G.B. , 1953, "The vibration of rectangular plates," Journal of Applied Mechanics 11, 371-381.

16.

Young, D. , 1950, "Vibration of rectangular plates by the Ritz method ," Journal of Applied Mechanics 12(1), 448-453.

17.

Yousuff, A. , Wagie, D.A. , and Skelton, R.E. , 1985, "Linear system approximation via covariance equivalent realization," Journal of Mathematical Analysis and Applications 106, 91-115.