Galerkin’s method revisited and corrected in the problem of Jaworski and Dowell
Introduction
Numerous structures, like aircraft wings, helicopter rotor blades, spacecraft antennae, robot arms can be modeled as beams. Free vibration of beams is a classic subject in structural mechanics originating at about 1735 when Daniel Bernoulli and Leonhard Euler investigated the vibration of uniform and homogeneous beams [5]. Since then, numerous papers, review papers and books, have been written on this subject. Among the several, it is important to note the work of Young and Felgar [21], which provides tables with the numerical solution for beams in different boundary conditions, the one by Duncan [4] in which normalized orthogonal deflection functions for beams can be found; and the books devoted to structural dynamics by Gorman [6] or by Karnovsky and Lebed [10].In another work, Karnovsky and Lebed [9] introduced also the Krylov-Duncan method as an approach to obtain the natural frequencies by solving an eigenvalue problem;
Beams with discontinuous cross-sectional areas, i.e. stepped beams, were also investigated by means of various approaches. Specifically, the Cauchy iteration method was applied by Taleb and Suppiger [18] yielding upper bounds for natural frequencies; lower bounds were found by Buckens [1] using a decomposition method; the variational component method with Lagrange multipliers was used by Klein [11] to satisfy geometric continuity conditions between different steps. Rayleigh-Ritz energy approach was used by Yuan and Dickinson [20] in conjunction with artificial spring constraints between beam components. This approach was utilized also by Maurizi and Belles [16].
The above papers resorted to approximate methods. Exact solution was provided by Levinson [13] for a single-stepped beam simply supported at its both ends. Jang and Bert [7] summarized results for a single, centrally located stepped beam. Naguleswaran [17] dealt with beams with up to three steps, whereas Lu et al [14], Mao [15], Duan and Wang [2] and Wang X-W. and Wang Y-L [19] dealt with the case of multiple stepped beams.
Jaworski and Dowell [8] conducted a thorough investigation of a beam with 13 steps using approximate methods. Namely, they applied the Rayleigh-Ritz method as well as the finite element method, and conducted extensive experiments to validate their results. For the implementation of the Rayleigh-Ritz method, the authors used the exact modes of vibration of cantilever beam, whereas they built FEM models using ANSYS and various element available in its library (BEAM4, BEAM188, SHELL93 and SOLID45). The above results were contrasted with the experimental results.
In this paper, we conducted two analyses that complement the work by Jaworski and Dowell [8]. Namely, we first used the so-called Krylov-Duncan functions to compute the exact natural frequencies of a stepped beam and next two versions of the Galerkin method. In particular, for the Galerkin method we implemented both the straightforward method, in which the basis functions exist and are evaluated for each step of the beam and next combined, and the rigorous version of the method which is based on generalized functions of the mass and stiffness of the beam which exist over the entire beam domain. In this latter case we used the Heaviside unit step function, the Dirac’s delta function and its derivative, as well as the doublet function, to formulate the characteristic equation of the free vibration problem of the stepped beam.
Section snippets
Basic equations
We are interested to evaluate the natural frequencies of a multi-step beam as shown in Fig. 1.
The beam is a cantilever made of a single material so the elastic modulus and the mass density are constants. The beam is composed by two alternating sections, namely section A and section B.
We study the free vibrations of this beam in both vertical x-y and horizontal x-z planes as shown in Fig. 2.
The Euler-Bernoulli differential equation governing the flexural vibrations in one principal plane of
Exact solution
The evaluation of the exact solution consists in the demand that not all four coefficients for each component vanish simultaneously. In our study we have 13 different segments for the multi-step beam resulting in 52 unknowns. The solution should satisfy continuity conditions between the segments and the boundary conditions at the outer sections of the beams (first and the 13th components).
For each discontinuity, we have four compatibility conditions namely continuity of vertical
Straightforward Galerkin method
The Galerkin method is a numerical strategy to solve differential equations in a discrete manner:
By introducing the axial coordinate in non-dimensional form Eq. (12) can be represented as:
In order to apply the Galerkin method in its straightforward version, we have to express the vertical displacement in terms of the so-called comparison functions as:where are unknown constants. Now we substitute the expression
Rigorous Galerkin method
The rigorous version of the Galerkin method does require generalized functions over the entire domain of the beam length (0 < x < L). Starting from Eqs. (1), (3) we obtain:
Introducing a non-dimensional axial coordinate and looking for a solution true for any time value, we obtain:
In order to implement the rigorous Galerkin method we represent the flexural rigidity and the mass of the system as generalized functions:
Comparison functions for Galerkin method
To compare the straightforward and rigorous version of the Galerkin method with the exact solution of some beam problems, first comparison functions must be assumed. Comparison functions are supposed to represent well the solution of the differential equation while satisfying all the boundary conditions of the problem.
According to Jaworski and Dowell [8], good comparison functions for the problem at hand, consist in the mode shapes for the homogeneous cantilevered beam:
Conclusions
In this study we have analyzed the free vibrations of a cantilever homogeneous non-uniform beam with different methods. In particular, we have compared two different versions of Galerkin method, namely the straightforward version which is generally used in literature and the rigorous one proposed in this work, with the exact solution based on the Krylov-Duncan functions. While the straightforward approach considers basis functions with domain existing only over each segment, the rigorous
Dedication
This paper is dedicated to the blessed memory of Professor Simon G. Braun, the former Editor-in-Chief of this journal, and the former colleague of one of the authors (I.E.), at the Technion--Israel Institute of Technology, Haifa, Israel.
CRediT authorship contribution statement
Isaac Elishakoff: Conceptualization, Methodology, Validation, Formal analysis, Investigation. Marco Amato: Methodology, Validation, Formal analysis, Investigation, Software, Data curation. Alessandro Marzani: Methodology, Validation, Formal analysis, Investigation.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
Gratefully acknowledge financial support from the European Union’s Horizon2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement ‘INSPIRE - Innovative ground interface concepts for structure protection’ PITN-GA-2019-813424-INSPIRE.
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