# Investigation of the Influence of Splitter Blades on the Resonance Conditions of Impellers
## 2 Theoretical Analysis
### Lumped Parameter Model of the Impeller with Splitter Blades
\[W_{N}=\sqrt{N}E_{N}, \tag{9}\]  
\[Q_{Bi}^{(p)}=W_{2N}\cdot q_{Bi}^{(p)}=W_{2N}\cdot\textit{real}\Big{(}e_{i} \otimes U_{Bi}^{(p)}\Big{)}, \tag{10}\]  
Partition \(U_{Bi}^{(p)}\) into real and imaginary parts:  
\[U_{Bi}^{(p)}=\left[ \begin{array}{c}u_{\textit{MB}i}^{(p)}\\ u_{\textit{SB}i}^{(p)}\end{array} \right]=\left[ \begin{array}{c}a_{\textit{MB} }+jb_{\textit{MB}}\\ a_{\textit{SB}}+jb_{\textit{SB}}\end{array} \right], \tag{11}\]  
Then the DFT result \(Q_{Bi}^{(p)}\) can be rewritten as Equation (12)  
\[Q_{Bi}^{(p)}=F_{1}-F_{2}, \tag{12}\]  
in which  
\[F_{1}=W_{2N}\cdot\textit{real}\left[ \begin{array}{c}a_{\textit{MB}}\\ a_{\textit{SB}}\\ \vdots\\ w_{N}^{(k-1)(i-1)}a_{\textit{MB}}\\ w_{N}^{(k-1)(i-1)}a_{\textit{SB}}\\ \vdots\\ w_{N}^{(N-1)(i-1)}a_{\textit{MB}}\\ w_{N}^{(k-1)(i-1)}a_{\textit{SB}}\end{array} \right],\,F_{2}=W_{2N}\cdot img \left[ \begin{array}{c}0\\ 0\\ \vdots\\ w_{N}^{(k-1)(i-1)}b_{\textit{MB}}\\ w_{N}^{(k-1)(i-1)}b_{\textit{SB}}\\ \vdots\\ w_{N}^{(N-1)(i-1)}b_{\textit{MB}}\\ w_{N}^{(N-1)(i-1)}b_{\textit{SB}}\end{array} \right] \tag{13}\]  
The \((2k-1)\)th and the 2 kth element in \(F_{1}\) and \(F_{2}\) are written as Equations (14) and (15), \(k=1,2,\ldots N\). \[\left\{ \begin{array}{ll}F_{2k-1}^{1}&=\sqrt{2N}a_{\textit{MB}}\sum_{j=1}^{N} w_{2N}^{2(k-1)\cdot 2(j-1)}\cdot\textit{real}\Big{(}w_{N}^{(j-1)(i-1)}\Big{)}\\ &+\sqrt{2N}a_{\textit{SB}}\sum_{j=1}^{N}w_{2N}^{2(k-1)\cdot(2j-1)}\cdot \textit{real}\Big{(}w_{N}^{(j-1)(i-1)}\Big{)}\\ F_{2k}^{1}&=\sqrt{2N}a_{\textit{MB}}\sum_{j=1}^{N}w_{2N}^{(2k-1)\cdot 2(j-1)} \cdot\textit{real}\Big{(}w_{N}^{(j-1)(i-1)}\Big{)}\\ &+\sqrt{2N}a_{\textit{SB}}\sum_{j=1}^{N}w_{2N}^{(2k-1)\cdot(2j-1)}\cdot \textit{real}\Big{(}w_{N}^{(j-1)(i-1)}\Big{)}\end{array} ,\right. \tag{14}\]  
\[\left\{ \begin{array}{ll}F_{2k-1}^{2}&=\sqrt{2N}b_{\textit{MB}}\sum_{j=1}^{N} w_{2N}^{2(k-1)\cdot 2(j-1)}\cdot img\Big{(}w_{N}^{(j-1)(i-1)}\Big{)}\\ &+\sqrt{2N}b_{\textit{SB}}\sum_{j=1}^{N}w_{2N}^{2(k-1)\cdot(2j-1)}\cdot img \Big{(}w_{N}^{(j-1)(i-1)}\Big{)}\\ F_{2k}^{2}&=\sqrt{2N}b_{\textit{MB}}\sum_{j=1}^{N}w_{2N}^{2(k-1)\cdot 2(j-1)} \cdot img\Big{(}w_{N}^{(j-1)(i-1)}\Big{)}\\ &+\sqrt{2N}b_{\textit{SB}}\sum_{j=1}^{N}w_{2N}^{(2k-1)\cdot(2j-1)}\cdot img \Big{(}w_{N}^{(j-1)(i-1)}\Big{)}\end{array} ,\right. \tag{15}\]  
The \(Q_{Bi}^{(p)}\) obtained by DFT is a \(2N\times 1\) real vector as shown in Equation (16), in which the elements distribute symmetrically around the center of the \((N+1)\)th element except for the first element. Thus, only the first \(N+1\) elements are discussed here due to their independence. \[Q_{Bi}^{(p)}=\Big{(} \begin{array}{ccccccccc}Q_{1}&Q_{2}&\ldots&Q_{N}&Q_{N+1}& Q_{N+2}&\ldots&Q_{2N}\end{array} \Big{)}^{T}, \tag{16}\]  
When the number of sectors \(N=2p\) is even, only the circumstances of \(i=1,\;2,\ldots p,\;p+1\) need to be discussed, which correspond to the \(0,\;1,\ldots,\;p\) ND modes. Further, when \(i\in[2m-1]m=1,2,\ldots,\lfloor p/2\rfloor+1\), only the \(i\)th and the \((N-i+2)\)th ones of the first \(N+1\) elementsare nonzero and the rest are all zeros because they are independent of \(a\) or \(b\), which can be calculated by Equations (14) and (15). The only two nonzero elements \(Q_{i}\) and \(Q_{N-i+2}\) correspond to the \(h_{1}\)th harmonic and \(h_{2}\)th harmonic, in which \(h_{1}=i-1\) and \(h_{2}=N-i+1\). Obviously,  
\[h_{1}+h_{2}=N. \tag{17}\]  
When \(i\in\{2m|m=1,2,\ldots,\lfloor(p+1)/2\rfloor\}\), here is the same situation that only the \(i\)th and the \((N-i+2)\)th element of \(Q_{Bi}^{(p)}\) are nonzero. When the number of sectors \(N=2p-1\) is odd, only the circumstances of \(i=1,2,\ldots p\) need to be discussed which correspond to the \(0,\ 1,\ \ldots,p-1\) ND modes. The same results can be reached. The details are omitted for the sake of brevity. The above analysis shows that the RMV of the tuned impeller with splitter blades contains two harmonic components rather than one. This is due to the existence of the splitter blades.