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 authors demonstrate 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 of continuous systems. The exact solution approach utilises the dynamic stiffness matrix method, providing a precise means to converge with certainty upon any specific natural frequency of a plane frame. This work was further developed in the following decade to include the scenario of a beam on an elastic foundation modelled as a Winkler foundation. More recently subsequent research by the first and fourth authors 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 supported by a negative stiffness spring. Since the first eigenvalue is negative, the problem is redefined as a vibrating beam on an elastic foundation, 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 current 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.