Exact and numerically stable expressions for Euler-Bernoulli and Timoshenko beam modes

Abstract

In this work we present a general procedure for deriving exact, analytical, and numerically stable expressions for the characteristic equations and the eigenmodes of the Timoshenko and the Euler-Bernoulli beam models. This work generalizes the approach recently described in Gonçalves et al. (P.J.P. Goncalves, A. Peplow, M.J. Brennan, Exact expressions for numerical evaluation of high order modes of vibration in uniform Euler-Bernoulli beams, Applied Acoustics 141 (2018) 371-373), which allows the numerical stabilization for the case of the Timoshenko beam model. Our results enable the reliable computation of the eigenvalues and the eigenmodes of both beam models for any number of modes. In addition to presenting the necessary details for stabilizing the solutions to the eigenvalue problem for the two beam models, we also tabulate the results for a large number of the common boundary conditions so that one can compare the predictions of all those models. Therefore, another contribution of our work is the presentation of both the conventional as well as the novel, numerically stabilized results for both the Euler-Bernoulli and the Timoshenko beam models in one manuscript with consistent notation, and for the most common boundary conditions. The code for the stabilized Timoshenko expressions as well as for the finite element verification are made available through the Mendeley Data repository http://dx.

Rapidly decaying Fourier-like bases

“…As indicated in Eqs. (12) and more generally from (27b), these RDF bases allow the determination of the mean square N-th derivative of a function in terms of a spectral moment of order 2N. For example, mean square slopes can be determined from the second moment of the spectra associated with the RDF bases presented in Secs.…”

“…Such properties are indicators of strong similarities to Fourier bases and give RDF bases potential value for a variety of applications that involve functions defined over finite domains. Indeed, certain subsets of RDF bases are already used in the study of elastic deformations of finite domains in different geometries [11][12][13][14][15].…”

“…It should be noted that the RDF elements for N = 2 correspond directly to the vibrational modes of an unclamped one-dimensional bar [11][12][13]. As alluded to in Sec.…”

“…In analogy with the one-dimensional solutions in Eqs. (15), the solutions given by 2 V m n (r, φ) happen to describe the elastic modes of unclamped thin disks [11][12][13], and have been used to study deformable mirrors [14]. Furthermore, in two-dimensional paraxial imaging systems with MSF structures, these solutions are a natural basis for the assessment of the perturbation model [15]; the basis required for analogous studies of non-paraxial imaging systems is obtained by solving Eq.…”

“…In this way, we finally resolve the nature of the spectra referred to in [9] that gives access to an intuitive characteristic scale of a function's variations. Interestingly, some of these RDF bases are useful in a range of other applications [11][12][13][14][15]. Fig.…”