Eigenfrequencies of the non-collinearly coupled Euler-Bernoulli beam system with dissipative joints

Paulsen, William H.

doi:10.1090/qam/1466142

Author: William H. Paulsen
Journal: Quart. Appl. Math. 55 (1997), 437-457
MSC: Primary 73K12; Secondary 35P20, 73D30, 73K05
DOI: https://doi.org/10.1090/qam/1466142
MathSciNet review: MR1466142
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Abstract: In this paper, we will compute asymptotically the eigenfrequencies for the in-plane vibrations of the general non-collinear Euler-Bernoulli beam equation with dissipative joints. Many different kinds of dampers are allowed, even within the same structure. This generalizes a previous result for collinear structures. Matrix techniques are used to combine asymptotic analysis with the wave propagation method. We will find that if the lengths of the beams are rational, there will be a finite number of “streams” of eigenfrequencies, and, like the collinear case, each lies asymptotically to a vertical line.

Z. Tutek and I. Aganović, A justification of the one-dimensional linear model of elastic beam, Math. Methods Appl. Sci. 8 (1986), no. 4, 502–515. MR 870989, DOI https://doi.org/10.1002/mma.1670080133
William M. Boothby, An introduction to differentiable manifolds and Riemannian geometry, 2nd ed., Pure and Applied Mathematics, vol. 120, Academic Press, Inc., Orlando, FL, 1986. MR 861409
G. Chen, M. C. Delfour, A. M. Krall, and G. Payre, Modeling, stabilization and control of serially connected beams, SIAM J. Control Optim. 25 (1987), no. 3, 526–546. MR 885183, DOI https://doi.org/10.1137/0325029
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 (New Orleans, La., 1986) Lecture Notes in Pure and Appl. Math., vol. 108, Dekker, New York, 1987, pp. 67–96. MR 920570
G. Chen, S. G. Krantz, D. L. Russell, C. E. Wayne, H. H. West, and M. P. Coleman, Analysis, designs, and behavior of dissipative joints for coupled beams, SIAM J. Appl. Math. 49 (1989), no. 6, 1665–1693. MR 1025953, DOI https://doi.org/10.1137/0149101
Goong Chen, Steven G. Krantz, David L. Russell, C. Eugene Wayne, Harry H. West, and Jianxin Zhou, Modelling, analysis and testing of dissipative beam joints—experiments and data smoothing, Math. Comput. Modelling 11 (1988), 1011–1016. Mathematical modelling in science and technology (St. Louis, MO, 1987). MR 959971, DOI https://doi.org/10.1016/0895-7177%2888%2990645-0
Han Kun Wang and Goong Chen, Asymptotic locations of eigenfrequencies of Euler-Bernoulli beam with nonhomogeneous structural and viscous damping coefficients, SIAM J. Control Optim. 29 (1991), no. 2, 347–367. MR 1092732, DOI https://doi.org/10.1137/0329019

G. Chen and H. H. West, private communication

Goong Chen and Jianxin Zhou, The wave propagation method for the analysis of boundary stabilization in vibrating structures, SIAM J. Appl. Math. 50 (1990), no. 5, 1254–1283. MR 1061349, DOI https://doi.org/10.1137/0150076

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. Physics 9, 24–75 (1960)

Allan M. Krall, Asymptotic stability of the Euler-Bernoulli beam with boundary control, J. Math. Anal. Appl. 137 (1989), no. 1, 288–295. MR 981938, DOI https://doi.org/10.1016/0022-247X%2889%2990289-8
Steven G. Krantz and William H. Paulsen, Asymptotic eigenfrequency distributions for the $N$-beam Euler-Bernoulli coupled beam equation with dissipative joints, J. Symbolic Comput. 11 (1991), no. 4, 369–418. MR 1107880, DOI https://doi.org/10.1016/S0747-7171%2808%2980111-3

W. D. Pilkey, Manual for the Response of Structural Members, Vol. I, Illinois Inst, of Tech. Res. Inst. Project J6094, Chicago, IL, 1969