## 3 Solution Procedure  
### Spectral Element Method  
The SEM is a reliable semi-analytical approach, which has been proven quite efficient for the dynamical analysis of discontinuous systems. In this subsection, the SEM will be developed to treat the fluid-conveying pipes with periodically varying cross-section. Three steps will be taken in turn: (i) the dynamic stiffness matrix of a uniform pipe is derived; (ii) the local stiffness matrix for a single cell is determined; (iii) the global (spectral) stiffness matrix for the whole periodic pipe is obtained. The specific procedures are as follows:  
**First**, for a fluid-conveying uniform pipe with length \(l\), the deflection and slope of the two ends can be expressed as  
\[w_{1}=w(0,\ t),\ \theta_{1}=\theta(0,\ t),\ w_{2}=w(l,\ t),\ \theta_{2}=\theta(l,\ t). \tag{7}\]  
Substitution of Equation (5) into Equation (7) leads to the following matrix form of Equation (7)  
\[\mathbf{U}=\mathrm{e}^{\mathrm{i}\omega t}\mathbf{A}\mathbf{C}, \tag{8}\]  
Figure 1: Mechanical model of a fluid-conveying PC pipe with periodically varying cross-section: (**a**) model with \(n\) cells, (**b**) a single cell, (**c**) cross-section of the pipe. where  
\[ \begin{array}{c}\mathbf{U}=\left[\begin{array}{cccc}w_{1}&\theta_{1}&w_{2}& \theta_{2}\end{array}\right]^{\mathsf{T}},\ \mathbf{C}=\left[\begin{array}{cccc}C_{1}&C_{2}&C_{3}&C_{4}\end{array} \right]^{\mathsf{T}},\\ \mathbf{A}=\left[\begin{array}{cccc}1&1&1&1\\ \mathrm{i}k_{1}&\mathrm{i}k_{2}&\mathrm{i}k_{3}&\mathrm{i}k_{4}\\ \mathrm{e}^{\mathrm{i}k_{1}l}&\mathrm{i}^{\mathrm{i}k_{2}l}&\mathrm{e}^{ \mathrm{i}k_{3}l}&\mathrm{i}^{\mathrm{i}k_{4}l}\\ \mathrm{i}k_{1}\mathrm{e}^{\mathrm{i}k_{1}l}&\mathrm{i}_{2}\mathrm{e}^{ \mathrm{i}k_{2}l}&\mathrm{i}_{3}\mathrm{e}^{\mathrm{i}k_{3}l}&\mathrm{i}_{4} \mathrm{e}^{\mathrm{i}k_{4}l}\end{array}\right].\end{array}  \tag{9}\]  
Similarly, the general force of the two ends can be expressed as  
\[Q_{1}=-Q(0,\ t),\ M_{1}=-M(0,\ t),\ Q_{2}=Q(l,\ t),\ M_{2}=M(l,\ t). \tag{10}\]  
Substituting Equation (6) into Equation (10), the matrix form of Equation (10) can be yielded as  
\[\mathbf{F}=\mathrm{e}^{\mathrm{i}\omega t}\mathbf{B}\mathbf{C}, \tag{11}\]  
where  
\[ \begin{array}{c}\mathbf{B}=EI\left[\begin{array}{cccc}Q_{1}&M_{1}&Q_{2}&M_ {2}\end{array}\right]^{\mathsf{T}},\\ \mathbf{B}=EI\left[\begin{array}{cccc}-\mathrm{i}k^{3}&-\mathrm{i}k^{3}&- \mathrm{i}k^{3}_{3}&-\mathrm{i}k^{3}_{4}\\ k^{2}_{1}&k^{2}_{2}&k^{2}_{3}&k^{2}_{2}\\ ik^{3}_{3}\mathrm{e}^{\mathrm{i}k_{1}l}&\mathrm{i}k^{3}_{2}\mathrm{e}^{ \mathrm{i}k_{2}l}&\mathrm{i}k^{3}_{3}\mathrm{e}^{\mathrm{i}k_{3}l}&\mathrm{i} ^{3}_{4}\mathrm{e}^{\mathrm{i}k_{4}l}\\ -k^{2}_{1}\mathrm{e}^{\mathrm{i}k_{1}l}&-k^{2}_{2}\mathrm{e}^{\mathrm{i}k_{2}l }&-k^{2}_{3}\mathrm{e}^{\mathrm{i}k_{3}l}&-k^{2}_{4}\mathrm{e}^{\mathrm{i}k_ {4}l}\end{array}\right].\end{array}  \tag{12}\]  
Eliminating the coefficient matrix \(\mathbf{C}\) from Equations (8) and (11), the relationship between the general force and displacement can be determined as  
\[\mathbf{F}=\mathbf{SU}, \tag{13}\]  
where  
\[\mathbf{S}=\mathbf{B}\mathbf{A}^{-1} \tag{14}\]  
is defined as the dynamic stiffness matrix of a uniform pipe conveying fluid, which depends on the frequency, wavenumber, flow velocity, and physical and geometrical properties of the pipe and fluid. **Second**, on the basis of Equation (14), the dynamic stiffness matrices corresponding to segments A and B with different geometrical properties can be respectively obtained as  
\[\left\{ \begin{array}{c}\mathbf{S}_{\mathrm{A}}=\mathbf{B}_{\mathrm{A}} \mathbf{A}_{\mathrm{A}}^{-1}\\ \mathbf{S}_{\mathrm{B}}=\mathbf{B}_{\mathrm{B}}\mathbf{A}_{\mathrm{B}}^{-1} \end{array} \right. \tag{15}\]  
where  
\[\begin{array}{c}\mathbf{A}_{\mathrm{A}}=\left[