# Investigation of the Influence of Splitter Blades on the Resonance Conditions of Impellers
## 1 Introduction  
Centrifugal compressors are widely used in gas turbine engines, turbochargers, and oil and gas industries [1; 2; 3]. The impeller is the vital component of the compressor. It can be classified as two types according to whether it has splitter blades: the conventionally designed impeller and the impeller with splitter blades. Research shows that adding splitter blades not only helps reduce blade loadings but also can improve the compressor performance [4; 5; 6]. Thus, the impellers with splitter blades are more and more used in engineering [7; 8; 9]. For both conventionally designed impellers and impellers with splitter blades, high cycle fatigue (HCF) failure due to the forced vibration under the aerodynamic excitation is the main failure form [10; 11; 12]. The excitation from the impeller-diffuser interaction is the main excitation acting on the impeller. Because the impeller-diffuser radial gap is becoming smaller by design [13; 14], the interaction is getting more intense and the resulting unsteady aerodynamic loads are bigger. Moreover, the five-axis machine is now a common tool for impeller machining and the impeller is machined as one piecewith very low structural damping [15; 16]. As a result, these make the forced vibration of the impeller more prominent [17; 18]. Especially when resonance occurs, the resulting excessive vibration stress contributes more to the HCF of the impeller, which threatens the engine's reliability and integrity [19; 20]. Therefore, it is essential to investigate the resonance conditions of the impeller for identifying the potential resonances within the frequency range of interest. Besides, during the design phase of the compressor, the resonance conditions can also be utilized for choosing the appropriate number of diffuser vanes to avoid the resonances [21]. The two types of impellers both belong to cyclic symmetric structures. Previous research has studied the resonance conditions of this kind of cyclic symmetric structure under traveling wave excitation, but only the conventionally designed impellers have been studied [22; 23; 24]. The research indicates that only when both the following conditions are simultaneously satisfied, can the resonance occur: (1) the frequency matching condition: the excitation frequency \(\omega_{e}\) equals the natural frequency of the impeller \(\omega_{n}\); (2) the shape matching condition: the harmonic excitation periodic shape matches the natural mode shape. From that, it can be seen that the resonance conditions of impellers are closely associated with two aspects of both the structural natural characteristics and excitation characteristics. However, due to the existence of the splitter blades, there are differences between these two types of impellers in the above two aspects. For instance, in most designs of impellers with splitter blades, splitter blades have the same blade profiles as the main blades but with shorter chord length. They are usually placed at the midpitch of the two main blades. In terms of aerodynamic forcing, the spatial harmonic contents of the unsteady pressure distribution at the impeller exit for impellers with splitter blades are different from those for conventionally designed ones [25]. Besides, the load amplitudes of the main and the splitter blades are usually different too [25; 26; 27]. Despite the above differences, very little attention has been paid to the influence of splitter blades on the resonance conditions of impellers. In most research, the vibration characteristics of the impellers are only analyzed based on the duplicated sector by utilizing the structural cyclic symmetry, and the influence of splitter blades is ignored. For example, [PERSON] et al. [28] proposed a cyclic symmetry method for analyzing the impeller's dynamic characteristics, but the cyclosymmetry of which was demonstrated only from the perspective of the main blade mode shapes. [PERSON] et al.