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Vibration control of a pipe conveying fluid under external periodic excitation using a nonlinear energy sink

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Abstract

This paper investigates the effects of a smooth nonlinear energy sink (NES) on the vibration suppression of a fixed-fixed pipe conveying fluid under excitation of an external harmonic load. Pipe is modeled using the Euler–Bernoulli beam theory, and the NES has an essentially nonlinear stiffness and a linear dam**. The required conditions that allow for saddle-node bifurcation, Hopf bifurcation and strongly modulated responses (SMRs) in the system are studied. The SMR phenomenon in the system response is considered as the most efficient regime of response for vibration mitigation. In addition, the effects of dam** value of the NES, location of the NES on the pipe, magnitude of the external force and the fluid velocity on the dynamical behavior of the system are investigated. The Runge–Kutta and complexification-averaging methods are employed for numerical and analytical solutions, respectively. Finally, the efficiency of an optimal NES in the energy reduction of the primary system is compared to that of an optimal linear absorber. It can be seen that reducing the distance between the NES and the pipe supports decreases the probability of occurrence of the SMR and weak modulated response; moreover, it provides suitable conditions for occurrence of the saddle-node bifurcation. Furthermore, increasing the fluid velocity decreases the amplitude of steady-state response of the system and extends the unstable region of the response. The results show that the middle of the pipe is the best position for connecting the NES to a fixed-fixed pipe conveying fluid under the external periodic excitation.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Tarbiat Modares University, P.O. Box 14115-177, Tehran, Iran

    Ali Ebrahimi Mamaghani, S. E. Khadem & Saeed Bab

Authors
  1. Ali Ebrahimi Mamaghani
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  2. S. E. Khadem
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  3. Saeed Bab
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Corresponding author

Correspondence to S. E. Khadem.

Appendices

Appendix 1

$$\begin{aligned} m_{11}= & {} \int _0^1 {\phi _1^2 (\bar{{x}})\hbox {d}\bar{{x}},\mu =\int _0^1 \bar{{U}}_f ^{2}\phi _1 (\bar{{x}})\left[ \frac{\hbox {d}^{2}\phi _1 (\bar{{x}})}{\hbox {d}\bar{{x}}^{2}}\right] \hbox {d}\bar{{x}}} ,k_{11}\\= & {} \int _0^1 {\phi _1 (\bar{{x}})\left[ \frac{\hbox {d}^{4}\phi _1 (\bar{{x}})}{\hbox {d}\bar{{x}}^{4}}\right] } \hbox {d}\bar{{x}}, \xi =\int _0^1 {\bar{{c}}_p \phi _1 ^{2}(\bar{{x}})} \hbox {d}\bar{{x}},\bar{{F}}\\= & {} \int _0^1 {\bar{{F}}\phi _1 (\bar{{x}})\hbox {d}\bar{{x}}} \end{aligned}$$

Appendix 2

$$\begin{aligned} \alpha _1= & {} 16\Omega ^{6}\left( {\begin{array}{l} \phi _{1d}^4 \alpha ^{2}\Omega ^{4}+2\phi _{1d}^3 \alpha \xi \Omega ^{4}-2\phi _{1d}^3 \alpha ^{2}\mu \Omega ^{2}\\ \quad +\phi _{1d}^2 \alpha ^{2}\xi ^{2}\Omega ^{2}+\phi _{1d}^2 \xi ^{2}\Omega ^{4}+\alpha ^{2}\sigma ^{2} \\ -2\phi _{1d}^2 \alpha ^{2}\sigma \Omega ^{2}+\phi _{1d}^2 \alpha ^{2}\mu ^{2} +\phi _{1d}^2 \mu ^{2}\Omega ^{2}\\ \quad +2\phi _{1d} \alpha ^{2}\mu \sigma +2\phi _{1d} \mu \sigma \Omega ^{2}+\sigma ^{2}\Omega ^{2} \\ \end{array}} \right) \\ \alpha _2= & {} 24\beta \Omega ^{4}\left( \phi _{1d}^3 \mu \Omega ^{2}-\phi _{1d}^2 \xi ^{2}\Omega ^{2}+\phi _{1d}^2 \sigma \Omega ^{2}\right. \\&\quad \left. -\,\phi _{1d}^2 \mu ^{2}-2\phi _{1d} \mu \sigma -\sigma ^{2} \right) \\ \alpha _3= & {} 9\beta ^{2}\left( \phi _{1d}^4 \Omega ^{4}-2\phi _{1d}^3 \mu \Omega ^{2}+\phi _{1d}^2 \xi ^{2}\Omega ^{2}\right. \\&\quad \left. -\,2\phi _{1d}^2 \sigma \Omega ^{2}+\phi _{1d}^2 \mu ^{2}+2\phi _{1d} \mu \sigma +\sigma ^{2} \right) \\ \alpha _4= & {} -16\phi _{1d}^2 F^{2}\Omega ^{10} \end{aligned}$$

Appendix 3

$$\begin{aligned} \eta _1= & {} \phi _{1d}^2 \alpha \varepsilon +\phi _{1d} \varepsilon \xi +\alpha \\ \eta _2= & {} \frac{1}{64\Omega ^{6}}(16\phi _{1d}^4 \alpha ^{2}\varepsilon ^{2}\Omega ^{6}+32\phi _{1d}^3 \alpha \varepsilon ^{2}\xi \Omega ^{6}\\&+\,27\phi _{1d}^4 \beta ^{2}\varepsilon ^{2}\phi _{2f}{}^{4}+16\phi _{1d}^2 \varepsilon ^{2}\xi ^{2}\Omega ^{6}\\&+\,48\phi _{1d}^3 \beta \varepsilon ^{2}\mu \Omega \phi _{2f}{}^{2} +32\phi _{1d}^2 \alpha ^{2}\varepsilon \Omega ^{6}\\&+\,48\phi _{1d}^2 \beta \varepsilon ^{2}\sigma \Omega ^{2}\phi _{2f}{} ^{2}+16\phi _{1d}^2 \varepsilon ^{2}\mu ^{2}\Omega ^{4}\\&+\,64\phi _{1d} \alpha \varepsilon \xi \Omega ^{6}+54\phi _{1d}^2 \beta ^{2}\varepsilon \phi _{2f}{} ^{4} \\&+\,32\phi _{1d} \varepsilon ^{2}\mu \sigma \Omega ^{4}+16\alpha ^{2}\Omega ^{6}+16\varepsilon ^{2}\sigma ^{2}\Omega ^{4}\\&+\,16\Omega ^{8}-48\beta \Omega ^{4}\phi _{2f}{} ^{2}+27\beta ^{2}\phi _{2f}{} ^{4}) \\ \eta _3= & {} \frac{\varepsilon }{64\Omega ^{6}}(16\phi _{1d}^3 \alpha ^{2}\varepsilon \xi \Omega ^{6}+16\phi _{1d}^2 \alpha \varepsilon \xi ^{2}\Omega ^{6}\\&+\,27\phi _{1d}^3 \beta ^{2}\varepsilon \xi \phi _{2f}{} ^{4}+16\phi _{1d}^2 \alpha \Omega ^{8}\\&+\,16\phi _{1d}^2 \alpha \varepsilon \mu ^{2}\Omega ^{4} +16\phi _{1d} \alpha ^{2}\xi \Omega ^{6}+16\phi _{1d} \xi \Omega ^{8}\\&+\,32\phi _{1d} \alpha \varepsilon \mu \sigma \Omega ^{4}-48\phi _{1d} \beta \xi \Omega ^{4}\phi _{2f}{} ^{2}\\&+27\phi _{1d} \beta ^{2}\xi \phi _{2f}{} ^{4}+16\alpha \varepsilon \sigma ^{2}\Omega ^{4}) \\ \eta _4= & {} \frac{\varepsilon ^{2}}{256\Omega ^{8}}(16\phi _{1d}^4 \alpha ^{2}\Omega ^{10}+32\phi _{1d}^3 \alpha \xi \Omega ^{10}\\&+\,27\phi _{1d}^4 \beta ^{2}\Omega ^{4}\phi _{2f}{} ^{4}-32\phi _{1d}^3 \alpha ^{2}\mu \Omega ^{8}\\&-\,48\beta \sigma ^{2}\Omega ^{4}\phi _{2f}{} ^{2} +16\phi _{1d}^2 \alpha ^{2}\xi ^{2}\Omega ^{8}\\&+\,16\phi _{1d}^2 \xi ^{2}\Omega ^{10}+48\phi _{1d}^3 \beta \mu \Omega ^{6}\phi _{2f}{} ^{2}\\&-\,32\phi _{1d}^2 \alpha ^{2}\sigma \Omega ^{8}-48\phi _{1d}^2 \beta \xi ^{2}\Omega ^{6}\phi _{2f}{} ^{2} \\&-\,54\phi _{1d}^3 \beta ^{2}\mu \Omega ^{2}\phi _{2f}{} ^{4}+16\phi _{1d}^2 \alpha ^{2}\mu ^{2}\Omega ^{6}\\&+\,27\phi _{1d}^2 \beta ^{2}\xi ^{2}\Omega ^{2}\phi _{2f}{} ^{4}+48\phi _{1d}^2 \beta \sigma \Omega ^{6}\phi _{2f}{} ^{2}\\&+\,27\beta ^{2}\sigma ^{2}\phi _{2f}{} ^{4} +16\phi _{1d}^2 \mu ^{2}\Omega ^{8}\\&-54\phi _{1d}^2 \beta ^{2}\sigma \Omega ^{2}\phi _{2f}{} ^{4}-48\phi _{1d}^2 \beta \mu ^{2}\Omega ^{4}\phi _{2f}{} ^{2}\\&+\,32\phi _{1d} \alpha ^{2}\mu \sigma \Omega ^{6}+32\phi _{1d}^ \mu \sigma \Omega ^{8} \\&+\,27\phi _{1d}^2 \beta ^{2}\mu ^{2}\phi _{2f}{} ^{4}-96\phi _{1d} \beta \mu \sigma \Omega ^{4}\phi _{2f}{} ^{2}\\&+\,16\alpha ^{2}\sigma ^{2}\Omega ^{6}+16\sigma ^{2}\Omega ^{8}+54\phi _{1d} \beta ^{2}\mu \sigma \phi _{2f}{} ^{4}) \end{aligned}$$

Appendix 4

$$\begin{aligned} v_1= & {} -\frac{729}{4096}\frac{\varepsilon \phi _{1d} \beta ^{4}\xi \left( {\phi _{1d}^2 \varepsilon +1} \right) ^{2}\left( {\phi _{1d}^4 \alpha \varepsilon ^{2}+\phi _{1d}^3 \varepsilon ^{2}\xi +2\phi _{1d}^2 \alpha \varepsilon +\alpha } \right) }{\Omega ^{12}}\\ v_2= & {} -\frac{81\varepsilon \phi _{1d} \beta ^{3}\xi \left( {\phi _{1d}^2 \varepsilon +1} \right) }{256\Omega ^{10}}\\&(\phi _{1d}^5 \alpha \varepsilon ^{3}\mu +\phi _{1d}^4 \alpha \varepsilon ^{3}\sigma -\phi _{1d}^4 \alpha \varepsilon ^{2}\Omega ^{2}\\&+\,\phi _{1d}^4 \varepsilon ^{3}\mu \xi +\phi _{1d}^3 \varepsilon ^{3}\sigma \xi \\&-\,\phi _{1d}^3 \varepsilon ^{2}\xi \Omega ^{2}+\phi _{1d}^3 \alpha \varepsilon ^{2}\mu +\phi _{1d}^2 \alpha \varepsilon ^{2}\sigma \\&-\,3\phi _{1d}^2 \alpha \varepsilon \Omega ^{2}-2\alpha \Omega ^{2}) \\ v_3= & {} -\frac{9\varepsilon \phi _{1d} \beta ^{2}}{256\Omega ^{8}}\{6\phi _{1d} ^{8}\alpha ^{3}\varepsilon ^{4}\xi \Omega ^{2}+15\phi _{1d} ^{7}\alpha ^{2}\varepsilon ^{4}\xi ^{2}\Omega ^{2}\\&+\,12\phi _{1d} ^{6}\alpha \varepsilon ^{4}\xi ^{3}\Omega ^{2}+22\alpha \xi \Omega ^{4}+3\phi _{1d} ^{7}\alpha ^{2}\varepsilon ^{4}\mu ^{2}\\&+\,24\phi _{1d} ^{6}\alpha ^{3}\varepsilon ^{3}\xi \Omega ^{2} +3\phi _{1d} ^{5}\varepsilon ^{4}\xi ^{4}\Omega ^{2}+6\phi _{1d} ^{6}\alpha ^{2}\varepsilon ^{4}\mu \sigma \\&+\,6\phi _{1d} ^{6}\alpha ^{2}\varepsilon ^{3}\mu \Omega ^{2}+6\phi _{1d} ^{6}\alpha \varepsilon ^{4}\mu ^{2}\xi \\&+\,6\alpha ^{3}\xi \Omega ^{2}-12\phi _{1d} \alpha \varepsilon ^{2}\mu \sigma \xi +3\phi _{1d} \alpha ^{2}\Omega ^{4} \\&+\,42\phi _{1d} ^{5}\alpha ^{2}\varepsilon ^{3}\xi ^{2}\Omega ^{2}+3\phi _{1d} ^{5}\alpha ^{2}\varepsilon ^{4}\sigma ^{2}\\&+\,6\phi _{1d} ^{5}\alpha ^{2}\varepsilon ^{3}\sigma \Omega ^{2}+3\phi _{1d} ^{5}\alpha ^{2}\varepsilon ^{2}\Omega ^{4}\\&+\,18\phi _{1d} ^{4}\alpha \varepsilon ^{3}\xi ^{3}\Omega ^{2}+6\phi _{1d} ^{5}\alpha ^{2}\varepsilon ^{3}\mu ^{2}\\&-\,6\alpha \varepsilon ^{2}\sigma ^{2}\xi +12\phi _{1d} ^{5}\alpha \varepsilon ^{4}\mu \sigma \xi -4\phi _{1d} ^{5}\alpha \varepsilon ^{3}\mu \xi \Omega ^{2}\\&+\,3\phi _{1d} ^{5}\varepsilon ^{4}\mu ^{2}\xi ^{2}-4\phi _{1d} ^{4}\alpha \varepsilon ^{3}\sigma \xi \Omega ^{2}\\&+\,6\phi _{1d} ^{4}\alpha \varepsilon ^{2}\xi \Omega ^{4}+6\phi _{1d} ^{4}\varepsilon ^{4}\mu \sigma \xi ^{2} \\&+\,36\phi _{1d} ^{4}\alpha ^{3}\varepsilon ^{2}\xi \Omega ^{2}+6\phi _{1d} ^{4}\alpha \varepsilon ^{4}\sigma ^{2}\xi \\&-\,10\phi _{1d}^4 \varepsilon ^{3}\mu \xi ^{2}\Omega ^{2}+12\phi _{1d} ^{4}\alpha ^{2}\varepsilon ^{3}\mu \sigma \\&+\,12\phi _{1d} ^{4}\alpha ^{2}\varepsilon ^{2}\mu \Omega ^{2}+39\phi _{1d} ^{3}\alpha ^{2}\varepsilon ^{2}\xi ^{2}\Omega ^{2} \\&+\,3\phi _{1d}^3 \varepsilon ^{4}\sigma ^{2}\xi ^{2}10\phi _{1d} ^{3}\varepsilon ^{3}\sigma \xi ^{2}\Omega ^{2}\\&+\,3\phi _{1d} ^{3}\varepsilon ^{2}\xi ^{2}\Omega ^{4}+6\phi _{1d} ^{3}\alpha ^{2}\varepsilon ^{3}\sigma ^{2}\\&+\,12\phi _{1d} ^{3}\alpha ^{2}\varepsilon ^{2}\sigma \Omega ^{2}+6\phi _{1d} ^{3}\alpha ^{2}\varepsilon \Omega ^{4}\\&+\,3\phi _{1d} \alpha ^{2}\varepsilon ^{2}\sigma ^{2} \\&-\,4\phi _{1d} ^{3}\alpha \varepsilon ^{2}\mu \xi \Omega ^{2}+6\phi _{1d} ^{2}\alpha \varepsilon ^{2}\xi ^{3}\Omega ^{2}\\&+\,3\phi _{1d} ^{3}\alpha ^{2}\varepsilon ^{2}\mu ^{2}+24\phi _{1d} ^{2}\alpha ^{3}\varepsilon \xi \Omega ^{2} -4\phi _{1d} ^{2}\alpha \varepsilon ^{2}\sigma \xi \Omega ^{2}\\&+\,28\phi _{1d} ^{2}\alpha \varepsilon \xi \Omega ^{4} +6\phi _{1d} ^{2}\alpha ^{2}\varepsilon \mu \Omega ^{2}-6\phi _{1d} ^{2}\alpha \varepsilon ^{2}\mu ^{2}\xi \\&+\,12\phi _{1d} \alpha ^{2}\varepsilon \xi ^{2}\Omega ^{2}+6\phi _{1d} \alpha ^{2}\varepsilon \sigma \Omega ^{2} +6\phi _{1d} ^{2}\alpha ^{2}\varepsilon ^{2}\mu \sigma \} \\ \end{aligned}$$
$$\begin{aligned} v_4= & {} -\frac{3\varepsilon \phi _{1d} \alpha \beta }{16\Omega ^{6}}\{\phi _{1d} ^{7}\alpha ^{2}\varepsilon ^{4}\mu \xi \Omega ^{2}\\&+\,\phi _{1d} ^{6}\alpha ^{2}\varepsilon ^{4}\sigma \xi \Omega ^{2}-\phi _{1d} ^{6}\alpha ^{2}\varepsilon ^{3}\xi \Omega ^{4}\\&+\,2\phi _{1d} ^{6}\alpha \varepsilon ^{4}\mu \xi ^{2}\Omega ^{2}+2\varepsilon ^{2}\sigma ^{2}\xi \Omega ^{2} \\&-\,2\phi _{1d} ^{5}\alpha \varepsilon ^{3}\xi ^{2}\Omega ^{4}+\phi _{1d} ^{5}\varepsilon ^{4}\mu \xi ^{3}\Omega ^{2}\\&+\,\phi _{1d} ^{6}\alpha \varepsilon ^{4}\mu ^{3}+2\phi _{1d} ^{5}\alpha ^{2}\varepsilon ^{3}\mu \xi \Omega ^{2}\\&+\,\phi _{1d} ^{4}\varepsilon ^{4}\sigma \xi ^{3}\Omega ^{2}-\phi _{1d} ^{4}\varepsilon ^{3}\xi ^{3}\Omega ^{4} \\&+\,3\phi _{1d} ^{5}\alpha \varepsilon ^{4}\mu ^{2}\sigma +\phi _{1d} ^{5}\alpha \varepsilon \mu \Omega ^{2}\\&+\,\phi _{1d} ^{5}\varepsilon ^{4}\mu ^{3}\xi +2\phi _{1d} ^{4}\alpha ^{2}\varepsilon ^{3}\sigma \xi \Omega ^{2}\\&-\,4\phi _{1d} ^{4}\alpha ^{2}\varepsilon ^{2}\xi \Omega ^{4}+2\phi _{1d} ^{4}\alpha \varepsilon ^{3}\mu \xi ^{2}\Omega ^{2} \\&+\,3\phi _{1d} ^{4}\alpha \varepsilon ^{4}\mu \sigma ^{2}+2\phi _{1d} ^{4}\alpha \varepsilon ^{3}\mu \sigma \Omega ^{2}\\&-\,\phi _{1d} ^{4}\alpha \varepsilon ^{2}\mu \Omega ^{4}+3\phi _{1d} ^{4}\varepsilon \epsilon ^{4}\mu ^{2}\sigma \xi \\&+\,\phi _{1d} ^{4}\varepsilon ^{3}\mu ^{2}\xi \Omega ^{2}+2\phi _{1d} ^{3}\alpha \varepsilon ^{3}\sigma \xi ^{2}\Omega ^{2} \\&-\,6\phi _{1d} ^{3}\alpha \varepsilon ^{2}\xi ^{2}\Omega ^{4}+\phi _{1d} ^{4}\alpha \varepsilon ^{3}\mu ^{3}\\&+\,\phi _{1d} ^{3}\alpha ^{2}\varepsilon ^{2}\mu \xi \Omega ^{2}+\phi _{1d} ^{3}\alpha \varepsilon ^{4}\sigma ^{3}\\&+\,\phi _{1d} ^{3}\alpha \varepsilon ^{3}\sigma ^{2}\Omega ^{2}-\phi _{1d} ^{3}\alpha \varepsilon ^{2}\sigma \Omega ^{4}-2\xi \Omega ^{6} \\&-\,\phi _{1d} ^{3}\alpha \varepsilon \Omega ^{6}+3\phi _{1d} ^{3}\varepsilon ^{4}\mu \sigma ^{2}\xi +2\phi _{1d} ^{3}\varepsilon ^{3}\mu \sigma \xi \Omega ^{2}\\&-\,\phi _{1d} ^{3}\varepsilon ^{2}\mu \xi \Omega ^{4}-2\phi _{1d} ^{2}\varepsilon ^{2}\xi ^{3}\Omega ^{4}+3\phi _{1d} ^{3}\alpha \varepsilon ^{3}\mu ^{2}\sigma \\&+\,\phi _{1d} ^{3}\alpha \varepsilon ^{2}\mu ^{2}\Omega ^{2}+\phi _{1d} ^{2}\alpha ^{2}\varepsilon ^{2}\sigma \xi \Omega ^{2}-5\phi _{1d} ^{2}\alpha ^{2}\varepsilon \xi \Omega ^{4}\\&+\,\phi _{1d} ^{2}\varepsilon ^{4}\sigma ^{3}\xi +\phi _{1d} ^{2}\varepsilon ^{3}\sigma ^{2}\xi \Omega ^{2}-\phi _{1d} ^{2}\varepsilon ^{2}\sigma \xi \Omega ^{4} \\&-\,\phi _{1d} ^{2}\varepsilon \xi \Omega ^{6}+3\phi _{1d} ^{2}\alpha \varepsilon ^{3}\mu \sigma ^{2}+2\phi _{1d} ^{2}\alpha \varepsilon ^{2}\mu \sigma \Omega ^{2}\\&-\,\phi _{1d} ^{2}\alpha \varepsilon \mu \Omega ^{4}+2\phi _{1d} ^{2}\varepsilon ^{2}\mu ^{2}\xi \Omega ^{2}-4\phi _{1d} \alpha \varepsilon \xi ^{2}\Omega ^{4} \\&+\,\phi _{1d} \alpha \varepsilon ^{3}\sigma ^{3}+\phi _{1d} \alpha \varepsilon ^{2}\sigma ^{2}\Omega ^{2}-\phi _{1d} \alpha \varepsilon \sigma \Omega ^{4}\\&-\,\phi _{1d} \alpha \Omega ^{6}+4\phi _{1d} \varepsilon ^{2}\mu \sigma \xi \Omega ^{2}-2\alpha ^{2}\xi \Omega ^{4}\\&+\,2\phi _{1d}^5 \alpha \varepsilon ^{4}\sigma \xi ^{2}\Omega ^{2}\} \\ v_5= & {} -\frac{\varepsilon \phi _{1d} \alpha }{16\Omega ^{4}}\phi _{1d} ^{4}\alpha ^{2}\varepsilon ^{2}\xi \Omega ^{2}+2\phi _{1d} ^{3}\alpha \varepsilon ^{2}\xi ^{2}\Omega ^{2}\\&+\,\phi _{1d}^2 \varepsilon ^{2}\xi ^{3}\Omega ^{2}+\phi _{1d} ^{3}\alpha \varepsilon ^{2}\mu ^{2}+2\phi _{1d}^2 \alpha ^{2}\varepsilon \xi \Omega ^{2} \\&+\,2\phi _{1d}^2 \alpha \varepsilon ^{2}\mu \sigma +2\phi _{1d} ^{2}\alpha \varepsilon \mu \Omega ^{2}+\phi _{1d} ^{2}\varepsilon ^{2}\mu ^{2}\xi \\&+\,2\phi _{1d} \alpha \varepsilon \xi ^{2}\Omega ^{2}+\phi _{1d} \alpha \varepsilon ^{2}\sigma ^{2}\\&+\,2\phi _{1d} \alpha \varepsilon \sigma \Omega ^{2}+\phi _{1d} \alpha \Omega ^{4}+2\phi _{1d} \varepsilon ^{2}\mu \sigma \xi \\&+\,\alpha ^{2}\xi \Omega ^{2}+\varepsilon ^{2}\sigma ^{2}\xi +2\varepsilon \sigma \xi \Omega ^{2}\\&+\,2\phi _{1d} \varepsilon \mu \xi \Omega ^{2}\}\times \{\phi _{1d} ^{4}\alpha ^{2}\varepsilon ^{2}\Omega ^{2}+2\phi _{1d} ^{3}\alpha \varepsilon ^{2}\xi \Omega ^{2} \\&+\,\phi _{1d} ^{2}\varepsilon ^{2}\xi ^{2}\Omega ^{2}+\varepsilon ^{2}\sigma ^{2}+\phi _{1d} ^{2}\varepsilon ^{2}\mu ^{2}+2\phi _{1d} \alpha \varepsilon \xi \Omega ^{2}a \\&+\,2\phi _{1d} \varepsilon ^{2}\mu \sigma +2\phi _{1d} ^{2}\alpha ^{2}\varepsilon \Omega ^{2}-2\phi _{1d} \varepsilon \mu \Omega ^{2}+\alpha ^{2}\Omega ^{2}\\&-\,2\varepsilon \sigma \Omega ^{2}+\xi \Omega ^{4}+\Omega ^{4}\} \end{aligned}$$

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Mamaghani, A.E., Khadem, S.E. & Bab, S. Vibration control of a pipe conveying fluid under external periodic excitation using a nonlinear energy sink. Nonlinear Dyn 86, 1761–1795 (2016). https://doi.org/10.1007/s11071-016-2992-x

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