The present study uses a typical spinning flow pipeline model to demonstrate these findings. A dynamic representation of a pipeline with spinning flow with elastic limitations on both sides is started to be constructed. This phenomenon can be achieved by implementing an oil drilling pipeline in alignment with the Euler-Bernoulli beam theory. The subsequent analysis of the pipeline's stability is facilitated by discretising the coupled control equations and solving the eigenfrequency problem through the Galerkin method. The enhanced Hamilton principle is applied to generate partial differential equations governing the pipe's two transverse vibrations. The fourth-order Galerkin technique is then utilized to truncate these equations. Finally, the discrete complex modal functions are introduced to obtain the intrinsic frequency and period solutions of the system. It reveals the method through which some critical factors, including mass ratio, flow rate, and rotational velocity, affect the system's stability and vibration characteristics. The paper's findings establish a theoretical framework for vibration control and stability design of spinning pipes in moving fluid applications with complex restrictions in practical engineering contexts.