Session: 07-23-01: 100th Anniversary of the Timoshenko-Ehrenfest Beam Model

Paper Number: 95959

95959 - A Reformulated Transfer Matrix Method for the Dynamic Response of Multistep Euler-Bernoulli and Timoshenko-Ehrenfest Beams 

Stepped beams constitute an important class of engineering structures whose vibration response has been widely studied. Many of the existing methods for studying stepped beams manifest serious difficulties as the number of segments or the frequency of excitation increase.  In this paper, we focus on the Transfer Matrix Method (TMM), which provides a simple and elegant formulation for multi-step beams. The main idea in the TMM is to model each step in the beam as a uniform element whose vibration configurations are spanned by the segment’s local eigenfunctions. Utilizing these local expressions, the boundary conditions at the ends of the multistep beam as well as the continuity and compatibility conditions across each step are used to obtain the nonlinear eigenvalue problem. Also, and perhaps more importantly, we provide a reformulation for multi-step beams that avoids the numerical singularity problems that have plagued most of the earlier efforts. Further, this reformulation is extended to accommodate short beam segments better represented by a Timoshenko beam model.

While many studies on TMM present formulations that theoretically accommodate beams with an arbitrary number of steps, in practice it is only possible to analyze beams with a limited number of steps. There are two reasons for this limitation, one is related to the numerical instability of the prevalent eigenmode expressions, and the second is a consequence of the repeated matrix multiplications in the TMM. Specifically, many of the tools that leverage Euler-Bernoulli beam (EB) or Timoshenko beam (TB) theory with multi-step beams utilize the conventional expressions for the modes which suffer from an inherent numerical instability due to the presence of hyperbolic trigonometric terms.  These terms are unbounded and as the frequency gets higher or the mode shape approaches the far end of the beam, the resulting round-off errors lead to numerical instabilities that preclude the analysis for high frequencies. This problem is exacerbated when beam theory is utilized to analyze multi-step beams with TMM since the numerical stabilities occur sooner because the determinant in the resulting matrix includes products of the hyperbolic terms. While some prior work has attempted to mitigate this issue, this problem remains unsolved even for beams with a small number of segments. In fact, the issue of numerical instabilities in the eigenvalue problem even for a single step beam was only recently resolved for the Euler-Bernoulli beam and for the Timoshenko beam.

 

In this paper we reformulate the solutions to the eigenvalue problem starting with basis functions introduced for Euler-Bernoulli beams and generalized here for Timoshenko beam models as well that do not involve hyperbolic trig functions. The resulting novel expressions are then combined with the transfer matrix method to obtain the eigenvalue problem for multistep beams. This approach enables the computation of the eigenvalues and the eigenmodes for beams with long spans, and large number of segments. The resulting eigenvalues are verified against finite element computations. Our results show that it is possible to reliably obtain the eigenmodes of multistep Euler-Bernoulli beams. For the Timoshenko beam, while our approach enables the computation of the eigenmodes well beyond what current analytical methods allow, it does have some limitations.  Specifically, because the eigenmode expressions for the Timoshenko beam explicitly contains the square of the frequency, the resulting expressions for multistep beams are ill-conditioned, and numerical errors grow with the number of beam segments.

Presenting Author: Firas Khasawneh Michigan State University

Presenting Author Biography: Firas Khasawneh earned his bachelor&#39;s degree from the Jordan University for Science and Technology, his masters degree at University of Missouri - Columbia, and his PhD at Duke University.<br/><br/>Prof. Khasawneh&#39;s broad research interest include studying dynamical systems with nonlinearities, stochasticity, and time delays. These studies include examining sensory signals (for example from additive and subtractive manufacturing processes), using innovative tools from topological data analysis, and machine learning. The objective of these investigations is to classify and predict the system behavior in order to enhance the efficiency and accuracy of these processes.

Authors:

Daniel Segalman Dept Mechanical Engineering, Michigan State University
Firas Khasawneh Michigan State University