Exact computation of lower bound eigenvalues of vibrating beams

<p dir="ltr">The body of this paper considers a clamped free Bernoulli-Euler beam from which the natural frequencies corresponding to in-plane flexure can be determined easily. A discrete lateral negative stiffness support is applied to the cantilever at its free end. The effect of the spring support is to reduce the first eigenvalue to below zero. This paper shows that by redefining the problem as a vibrating beam on a distributed elastic foundation, providing rotational restraint, ensures the first eigenvalue becomes positive. For the problem considered the elastic foundation is equivalent to tensile loading. The modified problem leads to the first eigenvalue being positive. However, the first eigenvalue of the original problem can also be computed with certainty by solving the original governing differential equation with no elastic foundation which will have an initial negative eigenvalue. Negative eigenvalues typically signal instability in structural systems — such as buckling or divergence — and arise in practical scenarios ranging from slender aerospace components under axial load to MEMS devices incorporating negative-stiffness elements for enhanced sensitivity. These modes are also intentionally exploited in compliant mechanisms and metamaterials designed for vibration isolation or energy absorption. Mathematically, such eigenvalues correspond to bound states in quantum graphs, where they reflect localized or unstable modes in a network of differential operators. This spectral analogy allows tools from quantum graph theory to inform the analysis of engineered structures with non-standard supports, reinforcing the interdisciplinary relevance of the methods developed in this paper.</p>