Bridge Between Inertial and Rotational Coordinate Systems

This chapter discusses a bridge for the knowledge with respect to the rotor-shaft vibration defined in an inertial coordinate system and the rotating structure vibration formulated in a rotating coordinate system. The equations of motion for rotor vibration discussed hitherto have been based on the description concerning the absolute complex displacement z = x + jy measured in an inertial (fixed, stationary) coordinate system. This description is requested from a practical viewpoint, because the vibrations measurement corresponds to displacement sensors (or gap sensors, displacement meters) placed on a stationary part of machine. Alternatively, this vibration can be measured by strain gauges fixed at a rotational coordinate system, as written by the displacement z r . These variables are mutually related by: \( z = z_{r} {\text{e}}^{{j\Omega t}} \) (Ω = rotational speed) Therefore, if an eigenvalue is λ in the inertial coordinate system and \( \lambda_{r} \) in the rotational coordinate system, these entities are mutually related by: $$ \lambda = \lambda_{r} + j\Omega $$ This chapter moves the viewpoint concerning vibration measurement from z to z r .