# Assessment of Novel Modal Testing Methods for Structures Rotating in Water
## 1 Introduction  
Hydraulic turbines are complex structures presenting a runner submerged in water and a shaft in air that are designed to operate at their best efficiency point (BEP). However, the introduction of intermittent renewable energy sources in the electrical grid, such as solar and wind power, is forcing them to operate for longer periods of time in off-design and transient conditions [1; 2; 3]. Under these circumstances, hydraulic turbines are subjected to more deleterious phenomena and transient events [1; 2; 3], which can cause fatigue cracks and high-amplitude rotor vibrations if a resonance occurs [4; 5]. In order to avoid resonances, it is of paramount importance to measure and precisely calculate the natural frequencies of hydraulic turbines in air and water under non-rotating and rotating conditions. Previous studies showed that there are many similarities between the mode shapes of some specific runners, such as Francis runners, and those of a disk [6; 7; 8; 9]. More specifically, some modes of both a Francis runner and a disk are characterized by the number of nodal diameters (ND) and circles (NC). The present investigation was thus performed on a structure composed of a disk clamped to a shaft supported by two bearings, resembling a hydraulic turbine rotor system. The modal responses of disk-like structures with or without blades in air have been investigated extensively. For instance, the presence of mistuned blades or shaft misalignments has been found to alter their natural frequencies [10; 11]. It has also been observed that high rotating speeds increase the natural frequencies [12; 13; 14; 15; 16]. However, other authors have concluded that this effect might be neglected at low rotating speeds [17]. Similarly, the modal response of disk-like structures in water has also attracted the attention of researchers. It has been proven experimentally and numerically that their natural frequencies decrease compared to those in air. The added mass effect has been considered to be primarily responsible for this frequency drop [17; 18; 19]. It has also been observed that the added mass is altered when a solid body is close to a rigid surface, resulting in an additional decrease in its natural frequencies [18; 19]. Additionally, the rotation has been found to influence the natural frequencies and mode shapes of submerged disks, even at low rotating speeds. For instance, the ND vibration modes evolve from standing mode shapes under non-rotating conditions to traveling mode shapes under rotating conditions. Under non-rotating conditions, the displacements of each point of the disk are either in phase or out of phase, while under rotating conditions the displacements are phase-shifted, resulting in a wave traveling over the disk [17; 20; 21; 22]. Understanding how the modal response and, more specifically, the natural frequencies are transmitted from a rotating frame to a stationary frame is indispensable in order to design and optimize structural health monitoring systems in hydraulic turbines, as they have a runner submerged in water which is not easily accessible. [PERSON] et al. [23] established an analytical correlation between the natural frequencies of a rotating disk in air measured from the rotating and from the stationary frames and then proved it experimentally. It was concluded that the relation between the measurements from the two different reference frames depended on the rotating speed and the mode shape. Later, [PERSON] et al. [24; 25] investigated the correlation between the natural frequencies of a submerged disk, measured from the rotating frame using accelerometers and from the stationary frame using a laser displacement sensor. In air, the laser displacement sensor measured two shifted natural frequencies for each single natural frequency measured from an on-board position. Nonetheless, in water, the laser displacement sensor only measured one shifted natural frequency for each single natural frequency measured on board.