Analytical investigation of natural frequencies and mode shapes of a multi-span beam with a disc and spring supports

Abstract

Determining the frequency equation and mode shapes becomes challenging when long shafts are modeled as multi-span Euler–Bernoulli beams with an intermediate disc, particularly due to the coupling effects introduced at the span junction. During operation, such systems may encounter critical speeds where resonance occurs. In the present work, an analytical investigation of a multi-span beam–disc system with simply supported and elastically supported boundaries is carried out. Closed-form expressions for the frequency equation and corresponding mode shapes are derived. A key contribution of this study is the identification and analytical treatment of coupling terms at the span junction, enabling explicit characterization of inter-span influence on mode-shape behavior. The analysis incorporates mass and stiffness ratios to examine their effects on modal characteristics. Furthermore, the influence of spring stiffness on shifting natural frequencies is analytically demonstrated, providing insight into critical speed tuning. The developed formulation generalizes previously observed experimental behavior into a unified analytical framework, enabling predictive design and supporting further theoretical developments in vibration control applications.