This paper studies the stability and controllability of Euler-Bernoulli beams whose bending vibration is controlled through intelligent constrained layer (ICL) damping treatments proposed by Baz (1993) and Shen (1993, 1994). First of all, the homogeneous equation of motion is transformed into a first order matrix equation in the Laplace transform domain. According to the transfer function approach by Yang and Tan (1992), existence of nontrivial solutions of the matrix equation leads to a closed-form characteristic equation relating the control gain and closed-loop poles of the system. Evaluating the closed-form characteristic equation along the imaginary axis in the Laplace transform domain predicts a threshold control gain above which the system becomes unstable. In addition, the characteristic equation leads to a controllability criterion for ICL beams. Moreover, the mathematical structure of the characteristic equation facilitates a numerical algorithm to determine root loci of the system. Finally, the stability and controllability of Euler-Bernoulli beams with ICL are illustrated on three cantilever beams with displacement or slope feedback at the free end.

Issue Section:
Research Papers
Topics:
Stability, Laplace transforms, Algorithms, Cantilever beams, Damping, Displacement, Equations of motion, Feedback, Poles (Building), Transfer functions, Vibration
1.
Agnes, G. S., and Napolitano, 1993, “Active Constrained Layer Viscoelastic Damping,” Proceedings 34th SDM Conference, pp. 3499–3506.
2.
Bailey, T., Gruzen, A., and Maden, P., 1988, “RCS/Piezoelectric Distributed Actuator Study,” AFAL-TR-88-038.
3.
Baz, A., 1993, “Active Constrained Layer Damping,” Proceedings of Damping ’93 Conference, San Francisco, CA, Vol. 3, pp. IBB 1–23.
4.
Christensen, R. M., 1982, Theory of Viscoelasticity, Academic Press, Inc., New York, pp. 21–25.
5.
Douglas
B. E.
, and
Yang
J.
,
1978
, “
Transverse Compressional Damping in the Vibratory Response of Elastic-Viscoelastic-Elastic Beams
,”
AIAA Journal
, Vol.
16
pp.
925
–
930
.
6.
Duclos, T. G., Coulter, J. P., and Miller, L. R., 1988, “Applications for Smart Materials in the Field of Vibration Control,” ARO Smart Materials, Structures, and Mathematical Issues Workshop Proceedings, Virginia Polytechnic Institute and State University, Blacksburg, VA, Sept. 15–16, pp. 132–146.
7.
Flu¨gge, W., 1975, Viscoelasticity, Spring-Verlag: Berlin, pp. 22–23.
8.
Shen
I. Y.
,
1993
, “
Intelligent Constrained Layer; An Innovative Approach
,”
ASME Intelligent Structures, Materials, and Vibrations
, DE-Vol.
58
pp.
75
–
82
.
9.
Shen
I. Y.
,
1994
, “
Hybrid Damping Through Intelligent Constrained Layer Treatments
,”
ASME JOURNAL OF VIBRATION AND ACOUSTICS
, Vol.
116
pp.
341
–
349
.
10.
Van Nostrand, W. C., Knowles, G. J., and Inman, D. J., 1993, “Active Constrained Layer Damping for Micro-satellites,” Dynamics and Control of Structures in Space, Vol. II, Kirk, C. L. and Hughes, P. C., eds., pp. 667–681.
11.
Van Nostrand, W. C., Knowles, G. J., and Inman, D. J., 1994, “Finite Element Methods for Active Constrained Layer Damping,” Passive Damping, North American Conference on Smart Structures and Materials, Orlando, FL., pp. 126–137.
12.
Yang
B.
, and
Tan
C. A.
,
1992
, “
Transfer Functions of One-Dimensional Distributed Parameter Systems
,”
ASME Journal of Applied Mechanics
, Vol.
59
pp.
1009
–
1014
.
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