Effect of Shape Memory Alloy Actuation on Parametric Instability in Pipes Conveying Pulsating Fluid

Abstract

Purpose

Vibration control of pipes conveying fluid at high pressure and pulsating velocity has been a key area of research for many engineering applications. The variation of velocity of flowing fluid with time leads to parametric instabilities in the transverse motion of the pipe. The objective of the present work is to effectively control the parametric instabilities of flexible pipes conveying pulsating fluid (sinusoidal velocity profile) by applying shape memory alloy (SMA) based actuation.

Methods

The Equivalent Coefficient of Thermal Expansion (ECTE) based SMA model has been employed to assimilate the actuator effect as part of the governing equation. Bolotin’s approach has been used to investigate the principal primary and secondary instability regions due to pulsating flow in a pipe with and without the actuator.

Results

It is observed that the extent of the instability region increases with a rise in flow velocity of the fluid, however, the inclusion of SMA wire actuator causes a significant reduction in the region of principal primaryinstability. The effect of other parameters like fluid-pipe mass ratio, and mean component of fluid velocity on both primary and secondary instabilities of the system is also addressed.

Conclusion

SMA actuators could be effectively used for the vibration control of pipes containing pulsating flows, as long as the harmonic component of the fluctuation of the flow is within a specific limit.

Appendix A Equation of Motion

The assumptions made while deriving the governing equation are as follows:

• 1. The flow is assumed to be of plug type.

• 2. The fluid is assumed to be incompressible.

• 3. The effect of pressurization of flowing fluid is neglected.

• 4. The damping and axial tension in the pipe are neglected.

• 5. The motion of the pipe is assumed to be in one plane (X-Z).

Consider a pinned-pinned flexible pipe conveying pulsating fluid, as shown in Fig. 1 including the SMA actuators. Let $l_p$ be the pipe length, $E_pI_p$ is flexural rigidity, $S_p$ is internal surface area per unit length, and $m_f$ is mass per unit length of the pipe conveying fluid of mass per unit length $m_p$. Let $U_f$ be the axial flow velocity, $A_f$ is the cross-sectional flow area, and P is the fluid gauge pressure.

Assume a small element of the pipe of length dx and the corresponding element of the enclosed fluid, as shown in Figs. 12 and  13, for deriving the governing equation of motion. Let $q_p$ is the shear stress on the internal surface of the pipe, F is the transverse force between pipe wall and fluid, $Q_p$ is the transverse shear force in the pipe, $f_a$ is the force induced in the SMA actuator, which in turn induces a compressive load in the pipe and $M_p$ is the bending moment.

For the fluid element shown in Fig. 12, force balance in X and Z directions, respectively, gives

$$\begin{aligned}&-A_f\frac{\partial P}{\partial x}-q_pS_p+F\frac{\partial z}{\partial x}-m_f\frac{\partial U_f}{\partial t}=0 \end{aligned}$$

(32)

$$\begin{aligned}&F+A_f\frac{\partial }{\partial x}\ \left( P\frac{\partial z}{\partial x}\right) +q_pS_p\frac{\partial z}{\partial x}+\ m_f\ {\left( \frac{\partial }{\partial t}+U_f\frac{\partial }{\partial x}\right) }^2z=0\ \end{aligned}$$

(33)

Fig. 12

Free body diagram of fluid element

For the pipe element shown in Fig. 13, force balance in X and Z directions gives (z-component of $f_a$ is neglected since the case is assumed to be of small vibrations)

Fig. 13

Free body diagram of pipe element

$$\begin{aligned}&\frac{\partial T}{\partial x}+q_p S_p-F\frac{\partial z}{\partial x}=0 \end{aligned}$$

(34)

$$\begin{aligned}&\frac{\partial Q_p}{\partial x}+F+\frac{\partial }{\partial x}\ \left( T\frac{\partial z}{\partial x}\right) +q_p S_p\frac{\partial z}{\partial x}-m_p\ \frac{{\partial }^2z}{\partial t^2}=0\ \end{aligned}$$

(35)

The relation between $Q_p$ and $M_p$ is given by

$$\begin{aligned} Q_p=-\frac{\partial M_p}{\partial x}=-E_pI_p\frac{{\partial }^3z}{\partial x^3} \end{aligned}$$

(36)

Combining the Eqs. 33, 35 and 36, we get

$$\begin{aligned} -E_pI_p\ \frac{{\partial }^4z}{\partial x^4}+\frac{\partial }{\partial x}\ \left[ \left( T-PA_f\right) \frac{\partial z}{\partial x}\right] -\ m_f\ {\left( \frac{\partial }{\partial t}+U_f\frac{\partial }{\partial x}\right) }^2z-m_p\ \frac{{\partial }^2z}{\partial t^2}=0 \end{aligned}$$

(37)

Adding the Eqs. 32 and 34 gives

$$\begin{aligned} \frac{\partial }{\partial x}\ \left( T-PA_f\right) =m_f\frac{\partial U_f}{\partial t} \end{aligned}$$

(38)

Integrating Eq. 38 from x to ${l_p}$, we get

$$\begin{aligned} {\left( T-PA_f\right) }_{x=l_p}-{\left( T-PA_f\right) }_x=m_f\frac{\partial U_f}{\partial t}\ \left( l_p-x\right) \end{aligned}$$

(39)

The pressure at x = $l_p$ is zero, since it is measured above atmosphere and the fluid discharges to atmosphere, i.e. ${(PA_f}_{x=l_p}=0$ in the above equation and ${\left( T\right) }_{x=l_p}=-f_a$ (compressive)

$$\begin{aligned} {\left( T\right) }_{x=l_p}\ -{\left( T-PA_f\right) }_x\ =\ m_f\frac{\partial U_f}{\partial t}\ \left( l_p-x\right) \end{aligned}$$

(40)

or

$$\begin{aligned} {-\left( T-PA_f\right) }_x=f_a+m_f\frac{\partial U_f}{\partial t}\ \left( l_p-x\right) \end{aligned}$$

(41)

Substituting Eq. 41 into Eq. 37, we get the equation of motion as

$$\begin{aligned}&E_pI_p\frac{{\partial }^4z}{\partial x^4}+\frac{\partial }{\partial x}\left[ \left\{ f_a+m_f\frac{\partial U_f}{\partial t}\left( l_p-x\right) \right\} \frac{\partial z}{\partial x}\right] +m_f\frac{{\partial }^2z}{\partial t^2}+2m_f U_f\ \frac{{\partial }^2z}{\partial x\partial t}\nonumber \\&\quad +m_f U_f^2\frac{{\partial }^2z}{\partial x^2}+m_p\ \frac{{\partial }^2z}{\partial t^2}=0 \end{aligned}$$

(42)

Simplifying the above equation further, the governing equation becomes

$$\begin{aligned}&E_pI_p\ \frac{{\partial }^4z}{\partial x^4}+\left[ m_f U_f^2+f_a+ m_f\frac{\partial U_f}{\partial t}\ \left( l_p-x\right) \right] \ \ \frac{{\partial }^2z}{\partial x^2}+2m_f U_f\ \frac{{\partial }^2z}{\partial x\partial t}\nonumber \\&\quad +(m_f+m_p)\ \frac{{\partial }^2z}{\partial t^2}=0 \end{aligned}$$

(43)

The equation of motion is converted to the non-dimensional form using the non-dimensional terms given in Sect. 2.1. Thus, we finally obtain

$$\begin{aligned}&\frac{{\partial }^4w}{\partial {\mu }^4}+u^2\frac{{\partial }^2w}{\partial {\mu }^2}+\left[ \beta \left( 1-\mu \right) \frac{\partial u}{\partial \tau }+{\mathcal {F}}\right] \frac{{\partial }^2w}{\partial {\mu }^2}+2\beta u\frac{{\partial }^2w}{\partial \mu \partial \tau }+\ \frac{{\partial }^2w}{\partial {\tau }^2}=0;\,\nonumber \\&\quad 0\le \ \mu \le 1 \end{aligned}$$

(44)