In high-speed motorsport power transmission systems, rotordynamic effects are often significant. Where several gear pairs and flexible transmission shafts are present, gyroscopic action can lead to considerable speed-driven changes in the natural frequencies of coupled lateral-torsional vibration modes. When system excitations such as gear meshing are considered, these changes must be taken into account in order to confidently model the vibrational responses that occur at the system critical speeds. This study applies an analytical finite element method to the targeted problem in order to study the system’s behaviour, providing a tool for design and diagnostics.