This study presents a comprehensive investigation into the dynamic behaviour of a simply supported static shaft subjected to a rotating force at constant speed, formulated within the framework of Timoshenko beam theory. By employing energy-based methods, the governing system equations are derived and subsequently solved through a combination of Navier’s analytical approach in the spatial domain and Newmark’s numerical integration scheme in the time domain. The primary objective of this research is to identify and analyze the critical speed of peripheral waves generated in shafts under rotary moving forces, a phenomenon that poses significant challenges in the design of high-speed racing car tyres and other advanced mechanical systems. At the critical rotational speed, circumferential flexural waves resonate with the excitation, resulting in a continuous escalation of dynamic response and potential instability. The study further explores the influence of geometric and material parameters, including shaft slenderness ratio, force positioning, and material properties, on the onset of resonance. Comparative validation against existing analytical and numerical studies confirms the accuracy of the proposed methodology and highlights its applicability to practical engineering problems. The results reveal that steel demonstrates the highest critical rotational speed among the three materials examined, thereby reinforcing its suitability for high-speed applications where dynamic stability is essential. Overall, the findings contribute to a deeper understanding of shaft dynamics under rotary excitation and provide valuable insights for the optimization of mechanical components in automotive and aerospace engineering.
Dehghan et al. (Sat,) studied this question.