Session: 07-19-01: Symposium on the Anniversary of the Timoshenko-Ehrenfest Beam Model and Other Refined Theories and Vibrations of Continuous Systems

Paper Number: 146901

146901 - Lower Bound Eigenvalues of a Timoshenko-Ehrenfest Vibrating Beam Using the Wittrick-Williams Algorithm 

The Wittrick-Williams algorithm excels in solving transcendental eigenvalue problems arising from the exact solutions of governing differential equations. Initially developed for structural mechanics, where eigenvalues are non-negative, the algorithm's versatility extends beyond. The author demonstrates that negative eigenvalues, present in other scientific disciplines, do not hinder the algorithm's effectiveness.

The exact solution to the axially loaded free vibration of the Timoshenko-Ehrenfest vibrating model was formulated in the early 1970s, marking a groundbreaking milestone in the study of vibration in continuous systems. This seminal paper has since inspired extensive research in this field. The paper introduces the dynamic stiffness matrix method, providing a precise means to converge with certainty upon any specific natural frequency of a plane frame. The constituent members are assumed to possess a uniform mass distribution, with the effects of axial load, rotatory inertia, and shear deflection meticulously described. Each effect can be isolated by setting the relevant parameter to zero.  This work was further developed in the 1980s, where researchers extended the dynamic stiffness matrix method to include the scenario of a beam embedded in an elastic medium. Additionally, subsequent research by the author explored the computation of negative eigenvalues for the Laplace operator, analogous to the axial vibration of a bar.

The analysis method presented is specifically tailored to the free vibration of structural systems that can be precisely analyzed through exact solutions of differential equations. This results in real eigenvalues that are bounded below. While this approach necessitates the first eigenvalue to be real, this paper illustrates that the first eigenvalue can be either positive (as commonly seen in structural mechanics) or negative. This property enables the utilization of the Wittrick-Williams algorithm to converge on any desired eigenvalue with machine accuracy.

An example is provided by computing the first eigenvalue of a Timoshenko-Ehrenfest beam with a negative stiffness spring. Since the first eigenvalue is negative, the problem is redefined as a vibrating beam in an elastic medium, ensuring the first eigenvalue becomes positive. Solving this modified problem reveals the necessary eigenvalues through a straightforward mathematical operation.

This methodology finds broad applicability in mathematical disciplines where negative eigenvalues arise, providing an efficient means to extract the necessary eigenvalues. The vibration behavior of frameworks is directly analogous to solving the problem of differential operators acting on graphs or quantum graphs, a topic of interest in mathematics. The spectral properties of quantum graphs can be obtained by the methods put forward by this paper.  Thus, this paper showcases the interdisciplinary nature of the work.

Presenting Author: Andrew Watson Loughborough University

Presenting Author Biography: Andrew obtained his undergraduate and higher degrees from Cardiff University, Wales. His PhD looked at the stability analysis and optimisation of light weight structures. After two post-doctoral appointments at Cardiff Andrew joined Loughborough University as a member of academic staff in 2004.
His research includes buckling and postbuckling of aerospace panels and vibration of Timoshenko beams. Buckling and vibration problems can be approached by using the Dynamic Stiffness Method along with the Wittrick-Williams algorithm. Vibrating structures can be modelled as quantum graphs and Andrew is currently researching higher order graphs to obtain the spectral results of tree shaped graphs all using the DSM.
Outside of this research Andrew has been looking at fossil fuels and other finite resources. To facilitate this he is developing analytical methods to optimise structures where the objective function can be mass, energy costs or environmental degradation. Jaguar Land Rover are funding a research studentship looking at thermal management of electric vehicles.
In his spare time he likes to keep up with current affairs and enjoys walking and swimming.

Authors:

Andrew Watson Loughborough University
Francesco Pellicano Università di Modena e Reggio Emilia
Antonio Zippo Università di Modena e Reggio Emilia
W Paul Howson Independent Consultant