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Unusual Closed-Form Solutions for Beam Vibrations
Published in Isaac Elishakoff, Eigenvalues of Inhomogeneous Structures, 2004
It turns out that although each of the beams has different boundary conditions, and, moreover, each of them has different D(ξ) these beams have the same frequency. Two vibrating system which have the same natural frequencies are called isospectral. In our particular case, beams of different boundary conditions share the first natural frequency. Gottlieb (1991), Driscoll (1997) and others have constructed examples of isospectral structures. In particular, Gottlieb (1991) showed that clamped inhomogeneous circular plates have the same vibration spectrum as their homogeneous counterparts. In our cases, the second and other frequencies do not coincide. For example, the second natural frequency squared of the pinned beam is 10881.18bI/(AL4), while the clamped-clamped beam has a second natural frequency squared 5607.68I/(AL4). Clamped-free and clamped-pinned beams’ second natural frequencies squared are respectively 42727.97I/(AL4) and 8013.24I/(AL4). These values are obtained by the finite element method. The difference between the present work and those associated with the inverse vibration problem lies in our desire to obtain closed-form solutions to find any beam that has a polynomial mode shape.
On the equivariance properties of self-adjoint matrices
Published in Dynamical Systems, 2020
Michael Dellnitz, Bennet Gebken, Raphael Gerlach, Stefan Klus
If , then and it suffices to consider matrices of the form for (or, equivalently, ).If, furthermore, all eigenvalues are distinct, then the eigenvectors are determined up to the sign and we obtain the group Σ and thus the special case derived in [17].Since minimizing the Procrustes cost function corresponds to maximizing the cost function and vice versa, the results can be extended to the orthogonal relaxation of the quadratic assignment problem (QAP) [1,15].Given two undirected graphs and with adjacency matrices A and B, the graphs are isomorphic if they are isospectral and contains a permutation matrix, see also [16].
Robust bipartite output regulation of linear uncertain multi-agent systems
Published in International Journal of Control, 2022
Let . Then, as pointed out in Remark 2.1, under Assumptions 2.5 and 2.6, by Lemma 4 of Hu and Hong (2007), the eigenvalues of the matrix have positive real parts. Since and are isospectral, the eigenvalues of the matrix also have positive real parts. Also, by Aghbolagh et al. (2017), we have .
Estimation of nonclassical properties of multiphoton coherent states from optical tomograms
Published in Journal of Modern Optics, 2018
Pradip Laha, S. Lakshmibala, V. Balakrishnan
It is easily checked that (see Appendix 1), so that annihilates both and . In the restricted Hilbert space with basis (, we can therefore define the isospectral coherent state