Experimental Validation of Frame-Dependent Gyroscopic Frequency Splitting in a Freely Rotating Tire Using Wireless MEMS Accelerometers and Scanning Laser Doppler Vibrometer

Gyroscopic effects split circumferential traveling-wave resonances of rotating structures into forward and backward branches. This work first analyzes the splitting in the co-rotating (Lagrangian) frame to provide physical intuition for the evolution of the two branches with spin speed. A transformation to the inertial (Eulerian) frame is then derived, showing that the observed frequencies are shifted by a kinematic Doppler-like term that acts with opposite sign on the forward and backward waves, leading to different Campbell-diagram slopes depending on the observation frame. The resulting framework is validated experimentally on a freely rotating, unloaded tire using two complementary sensing modalities: wireless on-tire accelerometers (co-rotating view) and a scanning laser Doppler vibrometer (inertial view). A frequency-domain SVD-based identification (FDD/ODS-SVD) is used to extract poles and deformation patterns over a range of spin speeds, enabling Campbell diagrams in both frames. The application of the proposed transformation maps the co-rotating branches onto the inertial observations, yielding consistent forward/backward splitting between the two measurement systems.