Spectral analysis of Euler–Bernoulli beam model with distributed damping and fully non-conservative boundary feedback matrix

Article type: Research Article

Authors: Shubov, Marianna A.^{; *}

Affiliations: Department of Mathematics and Statistics, University of New Hampshire, 33 Academic Way, Durham, NH 03824, USA. E-mail: marianna.shubov@gmail.com

Correspondence: [*] Corresponding author. E-mail: marianna.shubov@gmail.com.

Abstract: The distribution of natural frequencies of the Euler–Bernoulli beam resting on elastic foundation and subject to an axial force in the presence of several damping mechanisms is investigated. The damping mechanisms are: (i) an external or viscous damping with damping coefficient (−a0(x)), (ii) a damping proportional to the bending rate with the damping coefficient a1(x). The beam is clamped at the left end and equipped with a four-parameter (α, β, κ1, κ2) linear boundary feedback law at the right end. The 2×2 boundary feedback matrix relates the control input (a vector of velocity and its spacial derivative at the right end) to the output (a vector of shear and moment at the right end). The initial boundary value problem describing the dynamics of the beam has been reduced to the first order in time evolution equation in the state Hilbert space of the system. The dynamics generator has a purely discrete spectrum (the vibrational modes). Explicit asymptotic formula for the eigenvalues as the number of an eigenvalue tends to infinity have been obtained. It is shown that the boundary control parameters and the distributed damping play different roles in the asymptotical formulas for the eigenvalues of the dynamics generator. Namely, the damping coefficient a1 and the boundary controls κ1 and κ2 enter the leading asymptotical term explicitly, while damping coefficient a0 appears in the lower order terms.

Keywords: Non-selfadjoint operator, dynamics generator, vibrational modes, distributed damping, boundary control parameters, spectral asymptotics

DOI: 10.3233/ASY-211722

Journal: Asymptotic Analysis, vol. 129, no. 1, pp. 75-112, 2022