Stable computation of mode shapes of uniform Euler-Bernoulli beams subject to classical and non-classical boundary conditions via Lie symmetries

Abstract

Studying structural elements undergoing transverse vibration is crucial for scientific and industrial applications' design, safety, and efficiency. The Euler-Bernoulli beam model addresses a broad class of slender beam-like structures. The underlying assumption of neglecting shearing and rotary inertia effects at the beam's cross-section is reasonable. Displacement solutions of free vibrating uniform Euler-Bernoulli beams are usually defined using trigonometric and hyperbolic terms, prone to numerical instabilities at high frequencies as the wavelength becomes small. This conference paper proposes a Lie symmetry approach for determining numerically stable expressions systematically for the mode shape functions of Euler-Bernoulli beams subject to classical and non-classical boundary conditions. These mode shape solutions are computed by considering the invariance properties under a Lie group of transformations of the differential equation of a vibrating beam and its boundary conditions. The mode shapes are written as functions of constant parameters that do not exhibit instabilities when numerically assessed. Such stability provides appropriate mode shape solutions for high-frequency analysis, relevant for enhancing the accuracy of vibrating-beam solutions through the superimposition of different mode shapes within the range of validity of the Euler-Bernoulli beam model or building symbolic approximate mode shape solutions for rectangular plates.