## 2 Prediction of Flow Velocity in Pipe from Vibration Responses  
Figure 2 illustrates the flexural vibration of the fluid-conveying pipe under the clamped-clamped boundary condition. Harmonic excitation is applied to the center of the pipe. The equation of motion for this fluid-conveying pipe is [22]  
\[D\frac{\partial^{4}w_{n}}{\partial x^{4}}+m_{s}\frac{\partial^{2}w_{n}}{ \partial t^{2}}+m_{f}\frac{\partial^{2}w_{n}}{\partial t^{2}}+2m_{f}U\frac{ \partial^{2}w_{n}}{\partial x\partial t}+m_{f}U^{2}\frac{\partial^{2}w_{n}}{ \partial x^{2}}=0, \tag{1}\]  
Figure 1: Alternative method for measurement of flow velocity of pipe by using force excitation and accelerometer signals. Figure 2: Analytical model of the flexural vibration of clamped-clamped fluid-conveying pipe with accelerometers attached to the outer surface at different positions. where \(n=1,2\) indicates the index of the pipe; \(w_{n}\) are the flexural displacements; \(D\) is the bending stiffness of the pipe; \(m_{s}\) and \(m_{f}\) are the masses per unit length of the pipe and fluid; \(U\) is the flow velocity, respectively. For simple harmonic excitation, the general solution of Equation (1) is assumed to be \(w_{n}(x,t)=\hat{A}e^{-\mathrm{i}\hat{k}_{b}x}e^{\mathrm{i}\omega t}\), where \(\hat{k}_{b}\) is the wavenumber related to the frequency. By substituting the general solution into Equation (1), the characteristic equation was derived as  
\[D\hat{k}_{b}^{4}-m_{f}U^{2}\hat{k}_{b}^{2}+2m_{f}U\omega\hat{k}_{b}-(m_{f}+m_{ s})\omega^{2}=0. \tag{2}\]  
Equation (2) is in the form of a depressed quartic equation. The four wavenumbers \(\hat{k}_{b,1}\), \(\hat{k}_{b,2}\), \(\hat{k}_{b,3}\), and \(\hat{k}_{b,4}\) were calculated using [PERSON]'s method. Figure 3 illustrates the real and imaginary parts of these wavenumbers with increasing frequency. The fluid-conveying pipe without flow velocity comprised the wavenumbers of flexural vibration of a uniform beam. With the flow velocity, \(\hat{k}_{b,1}\) and \(\hat{k}_{b,2}\) changed two distinct real roots, and \(\hat{k}_{b,3}\) and \(\hat{k}_{b,4}\) gained two complex roots. At higher excitation frequencies, the imaginary parts of \(\hat{k}_{b,1}\) and \(\hat{k}_{b,2}\) converged on zero, and the real parts of \(\hat{k}_{b,3}\) and \(\hat{k}_{b,4}\) converged on any value. Otherwise, the four wavenumbers are assumed to be  
\[\hat{k}_{b,1}=k_{1},\ \hat{k}_{b,2}=k_{2},\ \hat{k}_{b,3}=k_{r3}+\mathrm{i}k_{ i3},\ \hat{k}_{b,4}=k_{r4}+\mathrm{i}k_{i4}. \tag{3}\]  
Using the calculated wavenumbers, the steady state solutions were obtained as  
\[\hat{w}_{n}(x)=\hat{A}_{n,1}e^{-\mathrm{i}\hat{k}_{b,1}x}+\hat{A}_{n,2}e^{- \mathrm{i}\hat{k}_{b,2}x}+\hat{A}_{n,3}e^{-\mathrm{i}\hat{k}_{b,3}x}+\hat{A}_{ n,4}e^{-\mathrm{i}\hat{k}_{b,4}x}. \tag{4}\]  
To facilitate the determination of coefficients of Equation (4) by wave propagation analysis, eight boundary conditions were applied as  
\[ \begin{array}{c}\hat{w}_{1}(-L)=0,\ \frac{\partial\hat{w}_{1}(-L)}{ \partial x}=0,\ \hat{w}_{2}(L)=0,\ \frac{\partial\hat{w}_{2}(L)}{ \partial x}=0,\ \hat{w}_{1}(0^{-})=\hat{w}_{2}(0^{-}),\\ \frac{\partial\hat{w}_{1}(0^{-})}{\partial x}=\frac{\partial\hat{w}_{2}(0^{ \top})}{\partial x},\ \frac{\partial^{2}\hat{w}_{1}(0^{-})}{\partial x^{2}}=\frac{ \partial^{2}\hat{w}_{2}(0^{\top})}{\partial x^{2}},\ D^{-\frac{3}{2}\partial x ^{3}}=D^{\frac{3}{2}\partial_{2}(0^{+})}+F_{o}\end{array}  \tag{5}\]  
where \(F_{o}\) is the force amplitude applied to the pipe. The resulting matrix system of the equation was obtained as  
\[