# Investigation of the Influence of Splitter Blades on the Resonance Conditions of Impellers
## 2 Theoretical Analysis
### Resonance Conditions of Impellers with Splitter Blades  
When the work done by the unsteady aerodynamic forces acting on the impeller is greater than zero after one cycle of the main harmonic, the resonance is excited, which leads to an increasing vibration amplitude. The resonance conditions will be derived based on this. From Section 2.2, we have already known that the RMV obtained based on all blades contains two harmonic contents. Thus, the displacement of the \(j\)-th blade can be written as  
\[X_{j}=A_{1}\cos(\omega_{n}t+\varphi_{t1})\cos(h_{1}\Delta\theta j+\varphi_{t1 })+A_{2}\cos(\omega_{n}t+\varphi_{t2})\cos(h_{2}\Delta\theta j+\varphi_{t2}), \tag{20}\]  
in which \(A_{1}\) and \(A_{2}\) are the amplitudes of the corresponding \(h_{1}\)-th and \(h_{2}\)-th harmonic components respectively, \(\varphi_{t}\) and \(\varphi_{h}\) are generic phases for the time and the angle variables. The speed of the \(j\)-th blade can be obtained as the partial derivative of the displacement with respect to time as shown in Equation (21), in which \(\varphi_{j}^{\prime}\) is the generic phase angle. \[ \begin{array}{l}V_{j}=\frac{A_{1}\omega_{n}}{2}\bigg{[}\cos\!\left(\omega_{n}t -h_{1}\Delta\theta j+\varphi_{j11}^{\prime}\right)+\cos\!\left(\omega_{n}t+h_{ 1}\Delta\theta j+\varphi_{j12}^{\prime}\right)\!\bigg{]}\\ +\frac{A_{2}\omega_{n}}{2}\bigg{[}\cos\!\left(\omega_{n}t-h_{2}\Delta\theta j+ \varphi_{j21}^{\prime}\right)+\cos\!\left(\omega_{n}t+h_{2}\Delta\theta j+ \varphi_{j22}^{\prime}\right)\!\bigg{]}\end{array}  \tag{21}\]  
The work done to the whole impeller is  
\[W=\sum_{i=1}^{N}W_{i}, \tag{22}\]  
which is equal to the sum of the work done to each sector. The work done to the \(i\)-th sector \(W_{i}\) is written as  
\[W_{i}=\int_{0}^{T}\!\!\left(f_{j}V_{j}+f_{j+1}V_{j+1}\right)\!dt, \tag{23}\]  
in which  
\[ \begin{array}{l}f_{i}V_{i}+f_{i+1}V_{i+1}\\ =\frac{F_{\text{inf}}A_{1}\omega_{n}}{2}\cos(n\Omega t+n\Delta\theta j)\times \left[\cos\!\left(\omega_{n}t-h_{1}\Delta\theta j+\varphi_{j11}^{\prime}\right) +\cos\!\left(\omega_{n}t+h_{1}\Delta\theta j+\varphi_{j12}^{\prime}\right) \right]\\ +\frac{F_{\text{inf}}A_{2}\omega_{n}}{2}\cos(n\Omega t+n\Delta\theta j)\times \left[\cos\!\left(\omega_{n}t-h_{2}\Delta\theta j+\varphi_{j21}^{\prime}\right) +\cos\!\left(\omega_{n}t+h_{2}\Delta\theta j+\varphi_{j22}^{\prime}\right) \right]\\ +\frac{F_{\text{inf}}A_{1}\omega_{n}}{2}\cos(n\Omega t+n\Delta\theta(j+1)) \times\left[\cos\!\left(\omega_{n}t-h_{1}\Delta\theta(j+1)+\varphi_{j21}^{ \prime}\right)+\cos\!\left(\omega_{n}t+h_{1}\Delta\theta(j+1)+\varphi_{j22}^{ \prime}\right)\right]\\ +\frac{F_{\text{inf}}A_{2}\omega_{n}}{2}\cos(n\Omega t+n\Delta\theta(j+1)) \times\left[\cos\!\left(\omega_{n}t-h_{2}\Delta\theta(j+1)+\varphi_{j21}^{ \prime}\right)+\cos\!\left(\omega_{n}t+h_{2}\Delta\theta(j+1)+\varphi_{j22}^{ \prime}\right)\right]\!.\end{array}  \tag{24}\]  
Similar to [PERSON] et al. 's research [42], the above equation yields the following results  
\[n\Omega=\omega_{n}, \tag{25}\]  
\[n=\pm h_{1}+2 kN\text{ or }n=\pm h_{2}+2 kN \tag{26}\]  
Considering the physical meaning of the excitation order, \(n\) has to be positive. Thus, Equation (26) is rewritten as  
\[n=|\pm h+2 kN|, \tag{27}\]  
in which \(h\) is taken as \(h_{1}\) or \(h_{2}\). Equations (25) and (27) constitute the resonance conditions of impellers with splitter blades. They indicate that only when the excitation frequency equals the natural frequency and the excitation order matches either harmonic index of the two harmonics, can the resonance occur. Due to the existence of the splitter blades, the blade-based RMV of the impeller contains two harmonic components and the amplitudes of the two harmonics are usually different. This indicates that the excitations corresponding to the different harmonic components will lead to the different resonance states even for the same mode, e.g., the excitation corresponding to the dominant harmonic may arouse a larger resonant response.