Method of experimentally identifying the complex mode shape of the self-excited oscillation of a cantilevered pipe conveying fluid

Abstract

The dynamics of a flexible cantilevered pipe conveying fluid have been researched for several decades. It is known that the flexible pipe undergoes self-excited vibration when the flow speed exceeds a critical speed. This instability phenomenon is caused by nonconservative forces. From a mathematical point of view, the system has a characteristic of non-selfadjointness and the linear eigenmodes can be complex and non-orthogonal to each other. As a result, such a mathematical feature of the system is directly related to the instability phenomenon. In this study, we propose a method of experimentally identifying the complex mode from experimentally obtained time histories and decomposing the linear mode into real and imaginary components. In nonlinear analysis, we show that the nonlinear effects of practical systems on the mode in the steady-state self-excited oscillation are small. The real and imaginary components identified using the proposed method for experimental steady-state self-excited oscillations are compared with those obtained in the theoretical analysis, thus validating the proposed identification method.

Appendices

Nonlinear term

The nonlinear term in Eq. (34) is expressed as

$$\begin{aligned} n(\varPhi _{n1})&=\Biggl \{-2i\omega _n^2{\overline{\varPhi }}_{n1}'\int _{0}^{s} \varPhi _{n1}'^2ds -iU_{ncr}\sqrt{\beta }\omega _n \varPhi _{n1}'^2{\overline{\varPhi }}_{n1}' \nonumber \\&-\frac{1}{2}U_{ncr}^2(\varPhi _1'^2{\overline{\varPhi }}_{n1}'' +2\varPhi _{n1}'{\overline{\varPhi }}_{n1}'\varPhi _{n1}'') \nonumber \\&-\frac{1}{2}\omega _n(\varPhi _{n1}'^2{\overline{\varPhi }}_{n1}+2\varPhi _{n1}'\varPhi _{n1} {\overline{\varPhi }}_{n1}')-\frac{9}{2}\varPhi _{n1}''^2{\overline{\varPhi }}_{n1}'' \nonumber \\&-3(\varPhi _{n1}'\varPhi _{n1}''{\overline{\varPhi }}_{n1}'''+\varPhi _{n1}'{\overline{\varPhi }}_{n1}'' \varPhi _{n1}'''+{\overline{\varPhi }}_{n1}'\varPhi _{n1}''\varPhi _{n1}''') \nonumber \\&-\frac{1}{2}(\varPhi _{n1}'^2{\overline{\varPhi }}_{n1}''''+2\varPhi _{n1}' {\overline{\varPhi }}_{n1}'\varPhi _{n1}'''') \nonumber \\&+\frac{1}{2}{\overline{\varPhi }}_{n1}''\int _{s}^{1}\Bigl (\int _{0}^{s} \varPhi _{n1}'^2 ds\Bigr )ds+\omega _n^2{\overline{\varPhi }}_{n1}''\int _{s}^{1}\varPhi _{n1}\varPhi _{n1}' ds \nonumber \\&+\omega _n^2\varPhi _{n1}''\int _{s}^{1}\Bigl (\varPhi _{n1}{\overline{\varPhi }}_{n1} +\varPhi _{n1}{\overline{\varPhi }}_{n1}'\Bigr ) ds \nonumber \\&-\frac{\gamma }{2}\Bigl ({\overline{\varPhi }}_{n1}''\int _{s}^{1}\varPhi _{n1}'^2ds +2\varPhi _{n1}''\int _{s}^{1} \varPhi _{n1}'{\overline{\varPhi }}_{n1}'ds\Bigr ) \nonumber \\&+\frac{\gamma }{2}(1-s)({\overline{\varPhi }}_{n1}''\varPhi _{n1}'^2+2\varPhi _{n1} {\overline{\varPhi }}_{n1}'\varPhi _{n1}'') \nonumber \\&+\frac{1}{2}\Bigl ({\overline{\varPhi }}_{n1}''\varPhi _{n1}''^2(1) +2\varPhi _{n1}''\varPhi _{n1}''(1){\overline{\varPhi }}_{n1}''(1)\Bigr )\Biggr \}|A_n|^2A_n. \nonumber \\ \end{aligned}$$

(A.1)

Non-selfadjointness of the system

We introduce $\varPsi _{n1}$ as the adjoint function of $\varPhi _{n1}$, which satisfies

$$\begin{aligned}&\int _{0}^{1} [\varPhi _{n1}''''+\{U_{ncr}^2-\gamma (1-s)\}\varPhi _{n1}''\nonumber \\&\quad +(2i\omega _{n} \sqrt{\beta }U_{ncr}+\gamma )\varPhi _{n1}'-\omega ^2\varPhi _{n1}]{\overline{\varPsi }}_{n1}ds=0. \end{aligned}$$

(B.1)

Considering the boundary conditions of Eq. (33), we note that $\varPsi _{n1}$ is the solution to the boundary value problem

$$\begin{aligned}&{\overline{\varPsi }}_{n1}''''+\{U_{ncr}^2 - \gamma (1-s)\}{\overline{\varPsi }}_{n1}''\nonumber \\&\quad +(-2i\omega _{n} \sqrt{\beta }U_{ncr}+\gamma ){\overline{\varPsi }}_{n1}'-\omega _{n} ^2{\overline{\varPsi }}_{n1}=0, \end{aligned}$$

(B.2)

$$\begin{aligned}&\left\{ \begin{array}{ll} {\overline{\varPsi }}_{n1}(0)=0 \\ {\overline{\varPsi }}_{n1}'(0)=0 \\ {\overline{\varPsi }}_{n1}^{''}(1)=-U_{ncr}^2{\overline{\varPsi }}_{n1}(1) \\ {\overline{\varPsi }}_{n1}^{'''}(1)=2i\omega _{n} \sqrt{\beta }U_{ncr}{\overline{\varPsi }}_{n1}(1)-U_{ncr}^2{\overline{\varPsi }}_{n1}'(1). \end{array} \right. \end{aligned}$$

(B.3)

According to Eqs. (B.2) and (B3), the differential equation satisfied by $\varPsi _{n1}$ does not correspond to Eqs. (32) and (33), which $\varPhi _{n1}$ satisfies. Therefore, $\varPsi _{n1}$ is not the eigenmode and this system has non-selfadjointness. When we assume the flow velocity to be zero in Eqs. (B.2) and (B.3), these equations correspond to Eqs. (32) and (33). Therefore, the fluid conveying in the pipe makes this system non-selfadjoint.

$\varPhi _{n1}$ and $\varPsi _{m1}$ satisfy $\int _0^1 \varPhi _{n1}{\overline{\varPsi }}_{m1}ds=\delta _{nm}$, where $\delta _{nm}$ is Kronecker’s delta. Hence, from this property, the adjoint function plays an important role in the analysis of non-selfadjoint system and is used to obtain the eigenvalue [26]. Also, the application of the orthogonality yields the solvability condition related to the amplitude equation in nonlinear non-selfadjoint systems [19]. In this manuscript, we use the adjoint function in order to derive the solvability condition of $\varPhi _{n3}$as in Sect. 3. But, the adjoint function is not necessary for obtaining the complex mode shape experimentally as mentioned in Sect. 4.

Identification of flexural rigidity

We need the flexural rigidity EI as a parameter when we calculate the eigenmode theoretically. In this study, we determined the flexural rigidity EI of the pipe so that the natural frequency of free vibration without internal flow for the first mode obtained experimentally is equal to that obtained theoretically. Using this flexural rigidity EI, we find that the frequency of the second mode obtained experimentally is similar to that calculated theoretically as shown in Table 4.

Table 4 Comparison of the frequencies of free vibration of the first and second modes in the experimental results with those in the theoretical results