We develop a computational scheme to approximate the solutions of a mathematical model that describes the vibrations of a curved beam structure. The physical model also includes pairs of piezoceramic patches that can be viewed either as passive devices, changing only the structural properties of the beam, or as active devices used to control vibrations of the beam. As part of our numerical experiments, we compare schemes with a method that employs hybrid B-spline basis elements and a method that employs just one family of B-spline basis elements. We calculate the first four fundamental frequencies and mode shapes of the structure, and we simulate patch-activated vibrations. Finally, we devote a section to investigation of locking phenomena which have been reported for some shell problems.
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References
1.
1. Arnold, D. N.1981. "Discretization by Finite Elements of a Model Parameter Dependent Problem", Numerische Mathematik, 37:405-421.
2.
2. Archer, R. R.1996. "Small Vibrations of Thin Incomplete Circular Rings", Intern. J. Mech. Sci., 1:45-56.
3.
3. Akella, P., X. Chen, W. Cheng, D. Hughes and J. T. Wen. 1994. "Modeling and Control of Smart Structures with Bonded Piezoelectric Sensors and Actuators", Smart Materials and Structures, 3:344-353.
4.
4. Arnold, D. N. and R. S. Falk. 1989. "A Uniformly Accurate Finite Element Method for the Reissner-Mindlin Plate", SIAMJ. of Numerical Analysis, 26:1276-1290.
5.
5. Azvine, B., G. R. Tomlinson and R. J. Wynne. 1994. "Initial Studies of the Use of Active Constrained Layer Damping for Controlling Resonant Vibration", Presented at the SPIE Conference on Smart Structures and Materials, Orlando, Smart Materials and Structures, submitted.
6.
6. Brezzi, F., K.-J. Bathe and M. Fortin. 1989. "Mixed-Interpolated Elements for Reissner-Mindlin Plates", International J for Numerical Methods in Engineering, 28:1787-1801.
7.
7. Banks, H. T., D. J. Inman, D. J. Leo and Y. Wang. 1996. "An Experimentally Validated Damage Detection Theory in Smart Structures", CRSC-TR95-7, 1995. N. C. State University, J Sound and Vibration, 191:859-880.
8.
8. Banks, H. T., K. Ito and Y. Wang. 1995. "Well-Posedness for Damped Second Order Systems with Unbounded Input Operators", CRSCTR93-10, 1993. N. C. State University; Differential and Integral Equations, 8:587-606.
9.
9. Banks, H. T. and K. Kunisch. 1989. "Estimation Techniques for Distributed Parameter Systems", Birkhauiser, Boston.
10.
10. Banks, H. T., N. G. Medhin and Y. Zhang. 1996. "A Mathematical Framework for Curved Active Constrained Layer Structures: Well Posedness and Approximation", CRSC-TR95-32, 1995, N. C. State University; Numerical Functional Analysis & Optimizations, 17:1-22.
11.
11. Banks, H. T. and R. Smith. 1995. "Implementation Issues Regarding PDE-Based Controllers-Control of Transient and Periodic Plate Vibrations", CRSC-TR95-16,1995, N. C. State University, SIAMJ Control and Optimization, to appear.
12.
12. Babuska, I. and M. Suri. 1992. "On Locking and Robustness in the Finite Element Method", SIAM J. of Numerical Analysis, 29: 1261-1293.
13.
13. Banks, H. T., R. C. Smith and Y. Wang. 1996. "Smart Material Structures: Modeling, Estimation and Control", Masson/J. Wiley, Paris/ Chichester.
14.
14. Banks, H. T., R. C. Smith and Y. Wang. 1995. "The Modeling of Piezoceramic Patch Interactions with Shells, Plates, and Beams", CRSCTR92-12, 1992, N. C. State University; Quarterly of Applied Mathematics, 53:353-381.
15.
15. Crawley, E. F. and E. H. Anderson. 1989. "Detailed Models of Piezoceramic Actuation of Beams", AIAA Paper 89-1388-CP.
16.
16. Clark, R. W., Jr., C. R. Fuller and A. Wieks. 1991. "Characteristics of Multiple Piezoelectric Actuators for Structural Excitation", J. Acoust. Soc. Am., 90:346-357.
17.
17. Crawley, E. F. and J. de Luis. 1987. "Use of Piezoelectric Actuators as Elements of Intelligent Structures", AIAA Journal, 25:1373-1385.
18.
18. Dimitriadis, E. K., C. R. Fuller and C. A. Rogers. 1991. "Piezoelectric Actuators for Distributed Noise and Vibration Excitation of Thin Plates", ASME Journal of Vibration and Acoustics, 13:100-107.
19.
19. Dosch, J., D. J. Inman and E. Garcia. 1992. "A Self-Sensing Piezoelectric Actuator for Collocated Control", Journal of Intelligent Material Systems and Structures, 3:166-185.
20.
20. Fuller, C. R., G. P. Gibbs and R. J. Silcox. 1990. "Simultaneous Active Control of Flexural and Extensional Waves in Beams", Journal of Intelligent Material Systems and Structures, 1:235-247.
21.
21. Fripp, M. and N. Hagood. 1995. "Comparison of Electrostrictive and Piezoceramic Actuators for Vibration Suppression", Proceedings of the 1995 North American Conference on Smart Structures and Materials, February 26-March 3, San Diego, CA.
22.
22. Gandi, M. V. and B. S. Thompson. 1992. Smart Materials and Structures, Chapman and Hall, New York.
23.
23. Hagood, N. W. and A. von Flotow. 1992. "Damping of Structural Vibrations with Piezoelectric Materials and Passive Electrical Networks", Journal of Sound and Vibration, 146:243-268.
24.
24. Jia, J. and C. A. Rogers. 1989. "Formulation of a Laminated Shell Theory Incorporating Embedded Distributed Actuators", ASME, Reprinted from AD-Vol. 15, Adaptive Structures, Editor: B. K. Wada, Book No. H00533.
25.
25. Leino, Y. and J. Pitkaranta. 1994. "On the Membrane Locking of h-p Finite Elements in a Cylindrical Shell Problem", International J. for Numerical Methods in Engineering, 37:1053-1070.
26.
26. Rongong, J. A., J. R. Wright, G. R. Tomlinson and R. J. Wynne. 1995. Modeling of Hybrid Constrained Layer/Piezoceramic Approach to Active Damping, preprint.
27.
27. Suri, M., I. Babuska and C. Schwab. 1995. "Locking Effects in the Finite Element Approximation of Plate Models", Mathematics of Computation, 64:461-482.
28.
28. Tzou, H. S. and M. Gadre. 1989. "Theoretical Analysis of a Multi Layered Thin Shell Coupled with Piezoelectric Actuators for Distributed Vibration Controls", Journal of Sound and Vibration, 132:433-450.
29.
29. Zhang, Z.-Z.1995. "Locking and Robustness in the Finite Element Method for Circular Arch Problem", Numerische Mathematik, 69: 509-522.
