# Analysis of the Mode Shapes of Kaplan Runners
## 1 Introduction  
In recent years, new renewable energies (NRE), such as wind and solar, have experienced a strong entry into power generation [1, 2]. Because the energy generated is random in nature, NRE can produce grid instabilities. Hydropower plants are the only ones that generate different levels of power depending on the demand for electricity (the generation has to match the demand for the stability of the electricity grid). As a result of this flexibility, hydroelectric power is an ideal complement to NREs. Because of this, hydraulic turbine units often undergo frequent start and stop cycles [3] and operate under out-of-design operating conditions. When the machine works in these operating conditions, fluid instabilities are induced. Due to this, turbulence and high-level pressure pulsations due cavitation, load variation and rotor-stator interaction can lead to undesirable excitations in the runner. These excitations can generate high vibration amplitudes in the runner, which can induce a useful life shortening or a premature failure in the worst cases [4, 5, 6, 7, 8, 9, 10]. In order to avoid the critical operating conditions, it is of paramount importance to analyze the dynamic response of the runners. Many investigations were carried out to understand the dynamic response of the turbine runners and the parameters that affect it. For instance, [PERSON] et al. [11] made a numerical modal analysis of a Francis runner considering the added mass effect. The mode shapes and natural frequencies were extracted in this study. Even though the natural frequencies calculated decrease significantly when the runner is submerged in water, the mode shapes undergo only small changes. [PERSON] et al. [12] made an experimental study about the added mass effect on a Francis runner in still water. It was found that the frequency decrease of a submerged Francis runner is caused by the added mass. Inthese works, it can be noticed that the mode shapes of the runner, extracted by numerical methods and experimentally, are mainly dominated by the deformation of the band and the blades. The global shape of these vibration modes can be found in simpler geometries, such as a thick-walled circular cylinder, like the one studied by [PERSON] et al. [13]. Similarly, for experimental procedures and analytical studies, it can be convenient to analyze simplified models of complex structures with similar dynamic response. [PERSON] et al. [14] studied the dynamic behaviour of a pump-turbine runner by numerical simulation. The runner was modeled as a disk-blade-disk structure that had the same mode shapes. It was concluded that the blade number and the shape of the blades can modify the natural frequencies depending on the mode shape. Even simpler geometries were used to model these kinds of runners. For instance, [PERSON] et al. [15] studied experimentally and analytically a rotating and submerged disk. In that study the rotation effects on the submerged disk and the transmission of the vibration to the casing were studied. The natural frequencies and mode shapes were measured from the casing. [PERSON] et al. [16] made an experimental study about the effect of the casing in a submerged disk, in order to understand the effect of a rigid and a flexible enclosure. It was found that while frequencies of the disk decrease when the disk gets close to the rigid wall, the opposite occurs when the disk gets close to the non-rigid wall. [PERSON] et al. [17] made a modal analysis of a spinning disk in a dense fluid as a model for a high-head hydraulic runner. The mode shapes were extracted and, an expression for co- and counter-rotating wave angular frequency was analytically determined. For the study of the dynamic response of Kaplan runners, the numerical simulation of the Kaplan turbine is often built and just a few mode global mode shapes are described [18; 19; 20; 21; 22]. According to the author's knowledge, no general description of the mode shapes of a Kaplan runner exist. In this paper the global mode shapes of Kaplan runners are investigated.