Eigenfrequencies of curved Euler-Bernoulli beam structures with dissipative joints
Author:
William H. Paulsen
Journal:
Quart. Appl. Math. 53 (1995), 259-271
MSC:
Primary 73D30; Secondary 73K12
DOI:
https://doi.org/10.1090/qam/1330652
MathSciNet review:
MR1330652
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Abstract: In this paper, we will compute asymptotically the eigenfrequencies for the in-plane vibrations of an Euler-Bernoulli beam system with dissipative joints, which allow the beams to be curved into an arc of a circle. This enhances the author’s previous result for structures involving straight beams, given in his preprint “Eigenfrequencies of the non-collinearly coupled Euler-Bernoulli beam system with dissipative joints". Matrix techniques are used to combine asymptotic analysis with a form of the wave propagation method.
C. Aganovic and Z. Tutez, A justification of the one-dimensional model of an elastic beam, Math. Methods Appl. Sci. 8, 1–14 (1986)
W. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, Orlando, FL, 1986
G. Chen, M. C. Delfour, A. M. Krall, and G. Payne, Modeling, stabilization, and control of serially connected beams, SIAM J. Control Optim. 25, 526–546 (1987)
G. Chen, S. G. Krantz, D. W. Ma, C. E. Wayne, and H. H. West, The Euler-Bernoulli beam equation with boundary energy dissipation, Operator Methods for Optimal Control Problems, Marcel Dekker, New York, 1987, pp. 67–96
G. Chen, S. G. Krantz, D. L. Russell, C. E. Wayne, H. H. West, and M. P. Colman, Analysis, designs and behavior of dissipative joints for coupled beams, SIAM J. Appl. Math. 49, 1665–1693 (1989)
G. Chen, S. G. Krantz, D. L. Russell, C. E. Wayne, H. H. West, and J. Zhou, Modeling, analysis and testing of dissipative beam joints—experiments and data smoothing, Math. Comput. Modelling 11, 1011–1016 (1988)
G. Chen and H. Wang, Asymptotic locations of eigenfrequencies of Euler-Bernoulli beam with nonhomogeneous structural and viscous damping coefficients, SIAM J. Control Optim. 29, 347–367 (1991)
G. Chen and J. Zhou, The wave propagation method for the analysis of boundary stabilization in vibration structures, SIAM J. Appl. Math. 50, 1254–1283 (1990)
P. G. Ciarlet, Plates and Junctions in Elastic Multi-structures, Springer-Verlag, New York, 1990
J. B. Keller and S. I. Rubinow, Asymptotic solution of eigenvalue problems, Ann. of Physics 9, 24–75 (1960)
A. M. Krall, Asymptotic stability of the Euler-Bernoulli beam with boundary control, J. Math. Anal. Appl. 137, 288–295 (1989)
S. G. Krantz and W. Paulsen, Asymptotic eigenfrequency distributions for the N-beam Euler-Bernoulli coupled beam equation with dissipative joints, J. Symbolic Comput. 11, 369–418 (1991)
W. H. Paulsen, Eigenfrequencies of non-collinearly coupled beams with dissipative joints, Proc. 31st IEEE Conf. Decision and Control (Tucson, AZ, 1992), Vol. 3, IEEE Control Systems Soc., New York, 1992, pp. 2986–2991
W. H. Paulsen, Eigenfrequencies of the non-collinearly coupled Euler-Bernoulli beam system with dissipative joints, preprint
W. D. Pilkey, Manual for the response of structural members. Vol. I, Illinois Inst. Tech. Res. Inst. Project J6094, Chicago, IL, 1969
C. Aganovic and Z. Tutez, A justification of the one-dimensional model of an elastic beam, Math. Methods Appl. Sci. 8, 1–14 (1986)
W. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, Orlando, FL, 1986
G. Chen, M. C. Delfour, A. M. Krall, and G. Payne, Modeling, stabilization, and control of serially connected beams, SIAM J. Control Optim. 25, 526–546 (1987)
G. Chen, S. G. Krantz, D. W. Ma, C. E. Wayne, and H. H. West, The Euler-Bernoulli beam equation with boundary energy dissipation, Operator Methods for Optimal Control Problems, Marcel Dekker, New York, 1987, pp. 67–96
G. Chen, S. G. Krantz, D. L. Russell, C. E. Wayne, H. H. West, and M. P. Colman, Analysis, designs and behavior of dissipative joints for coupled beams, SIAM J. Appl. Math. 49, 1665–1693 (1989)
G. Chen, S. G. Krantz, D. L. Russell, C. E. Wayne, H. H. West, and J. Zhou, Modeling, analysis and testing of dissipative beam joints—experiments and data smoothing, Math. Comput. Modelling 11, 1011–1016 (1988)
G. Chen and H. Wang, Asymptotic locations of eigenfrequencies of Euler-Bernoulli beam with nonhomogeneous structural and viscous damping coefficients, SIAM J. Control Optim. 29, 347–367 (1991)
G. Chen and J. Zhou, The wave propagation method for the analysis of boundary stabilization in vibration structures, SIAM J. Appl. Math. 50, 1254–1283 (1990)
P. G. Ciarlet, Plates and Junctions in Elastic Multi-structures, Springer-Verlag, New York, 1990
J. B. Keller and S. I. Rubinow, Asymptotic solution of eigenvalue problems, Ann. of Physics 9, 24–75 (1960)
A. M. Krall, Asymptotic stability of the Euler-Bernoulli beam with boundary control, J. Math. Anal. Appl. 137, 288–295 (1989)
S. G. Krantz and W. Paulsen, Asymptotic eigenfrequency distributions for the N-beam Euler-Bernoulli coupled beam equation with dissipative joints, J. Symbolic Comput. 11, 369–418 (1991)
W. H. Paulsen, Eigenfrequencies of non-collinearly coupled beams with dissipative joints, Proc. 31st IEEE Conf. Decision and Control (Tucson, AZ, 1992), Vol. 3, IEEE Control Systems Soc., New York, 1992, pp. 2986–2991
W. H. Paulsen, Eigenfrequencies of the non-collinearly coupled Euler-Bernoulli beam system with dissipative joints, preprint
W. D. Pilkey, Manual for the response of structural members. Vol. I, Illinois Inst. Tech. Res. Inst. Project J6094, Chicago, IL, 1969
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© Copyright 1995
American Mathematical Society
