# 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  
In order to more explicitly reveal the influence of splitter blades on the resonance conditions, a lumped parameter model of the impeller is established as shown in Figure 1. Modal analysis is performed and the undamped equation of motion (EoM) of the \(i\)-th sector is given as Equation (1):  
\[\left\{ \begin{array}{l}m_{b}\ddot{q}_{1}^{i}+k_{b}\Big{(}q_{1}^{i}-q_{3}^{i} \Big{)}=0\\ am_{b}\ddot{q}_{2}^{i}+\beta k_{b}\Big{(}q_{2}^{i}-q_{4}^{i}\Big{)}=0\\ m_{d}\ddot{q}_{3}^{i}+k_{d}q_{3}^{i}+k_{b}\Big{(}q_{3}^{i}-q_{1}^{i}\Big{)}+k_ {c}\Big{(}q_{3}^{i}-q_{4}^{i-1}\Big{)}+k_{c}\Big{(}q_{3}^{i}-q_{4}^{i}\Big{)}= 0\\ m_{d}\ddot{q}_{4}^{i}+k_{d}q_{4}^{i}+k_{b}\Big{(}q_{4}^{i}-q_{2}^{i}\Big{)}+k_ {c}\Big{(}q_{4}^{i}-q_{3}^{i+1}\Big{)}+k_{c}\Big{(}q_{4}^{i}-q_{3}^{i}\Big{)}= 0\end{array} \right. \tag{1}\]  
in which \(m_{b}\), \(\alpha m_{b}\), and \(m_{d}\) represent masses of the main blade, the splitter blade, and the disk part, respectively, \(q_{1}\) and \(q_{2}\) represent the displacements of the main blade and the splitter blade, \(q_{3}\) and \(q_{4}\) correspond the two disk parts, \(k_{b}\) and \(\beta k_{b}\) are the stiffnesses of the main and the splitter blade, \(k_{c}\) is the coupling stiffness. Unlike the conventionally designed impellers which contain only one blade per sector, the impellers with splitters usually contain one main blade and one splitter blade per sector. Rewrite Equation (1) in matrix form  
\[m_{b}M_{1}\ddot{q}^{i}+k_{b}K_{1}q^{i}+k_{b}K_{2}q^{i-1}+k_{b}K_{3}q^{i+1}=0, \tag{2}\]  
and the nondimensional EoM of the \(N\)-sector impeller can be written as  
\[M\ddot{q}+Kq=0, \tag{3}\]  
Related properties of circulant matrices are utilized to solve the EoM [34]. Transformation of coordinates is introduced as  
\[q=(E_{N}\otimes I_{4})u,\]  
Figure 1: Lumped parameter model of the impeller with splitter blades. in which \(\otimes\) represents the direct product (or Kronecker product), \(E_{N}\) is \(N\times N\) Fourier matrix as shown in Equation (5) in which \(w_{N}=e^{j\frac{2n}{N}}\), and \(I_{4}\) is the \(4\times 4\) unit matrix. \[E_{N}=\left( \begin{array}{ccccc}e_{1}&e_{2}&\cdots&e_{N}\end{array} \right)= \frac{1}{\sqrt{N}}\left[ \begin{array}{ccccc}1&1&1&\ldots&1\\ 1&w_{N}&w_{N}^{2}&\ldots&w_{N}^{N-1}\\ 1&w_{N}^{2}&w_{N}^{4}&\ldots&w_{N}^{2(N-1)}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 1&w_{N}^{(N-1)}&w_{N}^{2(N-1)}&\ldots&w_{N}^{(N-1)(N-1)}\end{array} \right], \tag{5}\]  
The original EoM is block decoupled by substituting Equation (4) into Equation (3) and multiplying from the left by \(\left(E_{N}\otimes I_{4}\right)^{H}\). Hence, the eigen equation of the decoupled EoM is written as  
\[\left(\overline{K}_{i}-\overline{\omega}_{i}^{2}\overline{M}_{i}\right)\!u_ {i}=0, \tag{6}\]  
and the nondimensional natural frequency \(\overline{\omega}_{i}^{(p)}\) of the \(p\)-th mode of nodal diameter (ND) \(i-1\) and its corresponding eigenvector \(u_{i}^{(p)}\) can be obtained, \(i=1,2,\ldots N\), \(p=1,2,3,4\). Due to the symmetry characteristics of the eigenvalue and eigenvector structure of the cyclic symmetric structures [34], only the circumstance of \(i=1,2,\ldots\left\lfloor\frac{N}{2}\right\rfloor+1\) needs to be discussed, in which \(\left\lfloor\cdot\right\rfloor\) denotes floor function. Partition the eigenvector \(u_{i}^{(p)}\) into the blade part \(U_{Bi}^{(p)}\) and the disk part \(U_{Di}^{(p)}\) as shown in Equation (7) and the blade mode shape for the \(p\)-th mode of ND \(i-1\) can be written in the physical coordinate as Equation (8)  
\[u_{i}^{(p)}=\left[ \begin{array}{ccccc}&u_{Bi}^{(p)}&u_{SBi}^{(p )}&u_{BDi}^{(p)}&u_{SDi}^{(p)}\end{array} \right]^{\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\  
Denote \(W_{N}\) as the \(N\times N\) DFT sinusoid matrix then the DFT of \(q_{Bi}^{(p)}\) can be represented as Equation (9).