请润色英文：Vibration and noise reduction of motional structures is a conventional difficulty in a variety of industrial realms due to synchronous spatial motions presented In this case optimizing structure

Vibration and noise reduction are common challenges in various industrial fields due to synchronous spatial motions. To address this issue, optimizing the design of structures can offer a promising solution. Inspired by the concept of wave manipulation using phononic crystals (PCs), this study focuses on controlling the three-dimensional (3D) vibration transmission of a motional pipe that conveys fluid by introducing an axial periodic design.

The pipe is composed of alternating materials along the axial direction and undergoes a rigid rotating motion while fluid flows inside it. By utilizing the Rayleigh beam theory, a set of doubly-gyroscopic equations is derived to govern the in-plane, out-of-plane flexural, and axial motions of the pipe. These equations take into account the rotation gyroscopic force and fluid gyroscopic force. The spectral element technology is employed to solve this multi-dimensional system. The validity of the results is confirmed through finite element (FE) simulation.

The band structure, frequency response function (FRF), and elastic wave shapes are analyzed to illustrate the bandgap (BG) mechanism of the rotating PC pipe. The findings demonstrate the superior effectiveness of 3D vibration suppression. Extensive parametric discussions reveal that the rotating motion, flowing fluid, and pipe geometry all significantly impact the BG performance of the rotating PC pipe system