Abstract

In the construction of modern large flexible space structures, active and passive damping devices are commonly installed at joints of coupled beams to achieve the suppression of vibration. In order to successfully control such dynamic structures, the function and behavior of dissipative joints must be carefully studied.
These dissipative joints are analyzed by first classifying them into types according to the discontinuities of physical variables across a joint. The four important physical variables for beams are displacement $( y )$, rotation $( \theta )$, bending moment $( M )$, and shear $( V )$. Dissipative joints can be classified into the following four types: (1) M and V are continuous, y and $\theta $ are discontinuous; (2) y and M are continuous, $\theta $ and V are discontinuous; (3) y and $\theta $ are continuous, M and V are discontinuous; (4) $\theta $ and V are continuous, y and M are discontinuous,according to the conjugacy of these variables. Mechanical designs have been achieved for all these dissipative joints of the linear passive type.
The spectrum of two identical coupled beams with a linear dissipative joint shows an interesting pattern. It is proven that there are two families of eigenvalues, asymptotically appearing alternately and parallel to the imaginary axis with eigenfrequencies spaced vertically with gap $O( {n^2 } )$.
This interesting spectral behavior has also been observed and studied in experiments conducted at the Modelling, Information Processing and Control Facility of the University of Wisconsin. Numerical simulations using the Legendre spectral method have also confirmed these spectral properties.
All of the aforementioned mechanical designs, experimental, and numerical results are presented in this paper.

MSC codes

  1. 73D30
  2. 73K12
  3. 35P20

Keywords

  1. beam joints
  2. Euler–Bernoulli beam equations
  3. asymptotic eigenfrequency analysis
  4. mechanical designs

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References

1.
G. Chen, S. G. Krantz, D. W. Ma, C. E. Wayne, H. H. West, S. J. Lee, The Euler-Bernoulli beam equation with boundary energy dissipationOperator methods for optimal control problems (New Orleans, La., 1986), Lecture Notes in Pure and Appl. Math., Vol. 108, Dekker, New York, 1987, 67–96
2.
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3.
Michel C. Delfour, John Lagnese, Michael P. Polis, Stabilization of hyperbolic systems using concentrated sensors and actuators, IEEE Trans. Automat. Control, 31 (1986), 1091–1096
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D. Gottlieb, S. G. Orszag, Numerical analysis of spectral methods: theory and applications, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977v+172
5.
W. D. Pilkey, Manual for the Response of Structural Members, Vol. I, Chicago, 1969
6.
P. Rideau, Masters Thesis, Controle d'un assemblage de poutres flexibles par des capteurs-actionneurs ponctuels, ètude du spectre du système, Thèse, L'Ecole Nat. Sup. des Mines de Paris, Sophia-Antipolis, France, 1985, Nov.
7.
D. L. Russell, On mathematical models for the elastic beam with frequency-proportional damping, to appear
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H. H. West, Analysis of Structures, John Wiley, New York, 1980

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Published In

cover image SIAM Journal on Applied Mathematics
SIAM Journal on Applied Mathematics
Pages: 1665 - 1693
ISSN (online): 1095-712X

History

Submitted: 11 April 1988
Accepted: 24 August 1988
Published online: 10 July 2006

MSC codes

  1. 73D30
  2. 73K12
  3. 35P20

Keywords

  1. beam joints
  2. Euler–Bernoulli beam equations
  3. asymptotic eigenfrequency analysis
  4. mechanical designs

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