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Two-to-one internal resonance of an inclined marine riser under harmonic excitations


In this paper, we study the two-to-one internal resonance of an inclined marine riser under harmonic excitations. The riser is modeled as an Euler–Bernoulli beam accounting for mid-plane stretching, self-weight, and an applied axial top tension. Due to the inclination, the self-weight load causes a static deflection of the riser, which can tune the frequency ratio between the third and first natural frequencies near two. The multiple-time-scale method is applied to study the nonlinear equation accounting for the system nonlinearity. The solution is then compared to a Galerkin solution showing good agreement. A further investigation is carried out by plotting the frequency response curves, the force response curves, and the steady-state response of the multiple-time-scale solution, in addition to the dynamical solution obtained by Galerkin, as they vary with the detuning parameters. The results reveal that the riser vibrations can undergo multiple Hopf bifurcations and experience quasi-periodic motion that can lead to chaotic behavior. These phenomena lead to complex vibrations of the riser, which can accelerate its fatigue failure.

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  1. Païdoussis, M.P., Price, S.J., De Langre, E.: Fluid-Structure Interactions: Cross-Flow-Induced Instabilities. Cambridge University Press, Cambridge (2010)

    Book  MATH  Google Scholar 

  2. Lacarbonara, W., Arafat, H.N., Nayfeh, A.H.: Non-linear interactions in imperfect beams at veering. Int. J. Non-Linear Mech. 40(7), 987–1003 (2005).

    Article  MATH  Google Scholar 

  3. Öz, H.R., Pakdemirli, M.: Two-to-one internal resonances in a shallow curved beam resting on an elastic foundation. Acta Mech. 185(3), 245–260 (2006).

    Article  MATH  Google Scholar 

  4. Nayfeh, A.H., Lacarbonara, W., Chin, C.-M.: Nonlinear normal modes of buckled beams: three-to-one and one-to-one internal resonances. Nonlinear Dyn. 18(3), 253–273 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  5. Chin, C., Nayfeh, A., Lacarbonara, W.: Two-to-one internal resonances in parametrically excited buckled beams. In: Proceedings of the 38th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials, pp. 7–10 (1997)

  6. Chin, C.-M., Nayfeh, A.H.: Three-to-one internal resonances in hinged-clamped beams. Nonlinear Dyn. 12(2), 129–154 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  7. Benedettini, F., Rega, G., Vestroni, F.: Modal coupling in the free nonplanar finite motion of an elastic cable. Meccanica 21(1), 38–46 (1986).

    Article  MATH  Google Scholar 

  8. Perkins, N.C.: Modal interactions in the non-linear response of elastic cables under parametric/external excitation. Int. J. Non-Linear Mech. 27(2), 233–250 (1992).

    Article  MATH  Google Scholar 

  9. Gattulli, V., Lepidi, M., Macdonald, J.H.G., Taylor, C.A.: One-to-two global-local interaction in a cable-stayed beam observed through analytical, finite element and experimental models. Int. J. Non-Linear Mech. 40(4), 571–588 (2005).

    Article  MATH  Google Scholar 

  10. Srinil, N., Rega, G.: Two-to-one resonant multi-modal dynamics of horizontal/inclined cables. Part II: internal resonance activation, reduced-order models and nonlinear normal modes. Nonlinear Dyn. 48(3), 253–274 (2007).

    Article  MATH  Google Scholar 

  11. Srinil, N., Rega, G., Chucheepsakul, S.: Two-to-one resonant multi-modal dynamics of horizontal/inclined cables. Part I: theoretical formulation and model validation. Nonlinear Dyn. 48(3), 231–252 (2007).

    Article  MATH  Google Scholar 

  12. Mansour, A., Mekki, O.B., Montassar, S., Rega, G.: Catenary-induced geometric nonlinearity effects on cable linear vibrations. J. Sound Vib. 413, 332–353 (2018).

    Article  Google Scholar 

  13. Nayfeh, A.H.: Nonlinear Interactions: Analytical, Computational and Experimental Methods. Wiley, Hoboken (2000)

    MATH  Google Scholar 

  14. Mazzilli, C.E., Sanches, C.T., Neto, O.G.B., Wiercigroch, M., Keber, M.: Non-linear modal analysis for beams subjected to axial loads: analytical and finite-element solutions. Int. J. Non-Linear Mech. 43(6), 551–561 (2008).

    Article  MATH  Google Scholar 

  15. Tien, W.-M., Namachchivaya, N.S., Bajaj, A.K.: Non-linear dynamics of a shallow arch under periodic excitation–I. 1:2 internal resonance. Int. J. Non-Linear Mech. 29(3), 349–366 (1994).

    Article  MATH  Google Scholar 

  16. Xiong, L.-Y., Zhang, G.-C., Ding, H., Chen, L.-Q.: Nonlinear forced vibration of a viscoelastic buckled beam with 2: 1 internal resonance. Math. Probl. Eng. (2014).

  17. Yi, Z., Wang, L., Kang, H., Tu, G.: Modal interaction activations and nonlinear dynamic response of shallow arch with both ends vertically elastically constrained for two-to-one internal resonance. J. Sound Vib. 333(21), 5511–5524 (2014).

    Article  Google Scholar 

  18. Ma, J., Gao, X., Liu, F.: Nonlinear lateral vibrations and two-to-one resonant responses of a single pile with soil-structure interaction. Meccanica 52(15), 3549–3562 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  19. Sahoo, B., Panda, L., Pohit, G.: Stability, bifurcation and chaos of a traveling viscoelastic beam tuned to 3: 1 internal resonance and subjected to parametric excitation. Int. J. Bifurc. Chaos 27(02), 1750017 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  20. Yu, T.-J., Zhang, W., Yang, X.-D.: Global dynamics of an autoparametric beam structure. Nonlinear Dyn. 88(2), 1329–1343 (2017).

    Article  Google Scholar 

  21. Zhu, B., Dong, Y., Li, Y.: Nonlinear dynamics of a viscoelastic sandwich beam with parametric excitations and internal resonance. Nonlinear Dyn. 1–38 (2018).

  22. Neto, O.B., Mazzilli, C.: Evaluation of multi-modes for finite-element models: systems tuned into 1: 2 internal resonance. Int. J. Solids Struct. 42(21), 5795–5820 (2005).

    Article  MATH  Google Scholar 

  23. Srinil, N., Wiercigroch, M., O’Brien, P.: Reduced-order modelling of vortex-induced vibration of catenary riser. Ocean Eng. 36(17–18), 1404–1414 (2009).

    Article  Google Scholar 

  24. Srinil, N.: Multi-mode interactions in vortex-induced vibrations of flexible curved/straight structures with geometric nonlinearities. J. Fluids Struct. 26(7–8), 1098–1122 (2010).

    Article  Google Scholar 

  25. Srinil, N.: Analysis and prediction of vortex-induced vibrations of variable-tension vertical risers in linearly sheared currents. Appl. Ocean Res. 33(1), 41–53 (2011).

    Article  Google Scholar 

  26. Chatjigeorgiou, I.K., Mavrakos, S.A.: Nonlinear resonances of parametrically excited risers–numerical and analytic investigation for \(\Omega \) = 2\(\omega \)1. Comput. Struct. 83(8–9), 560–573 (2005).

    Article  Google Scholar 

  27. Franzini, G., Mazzilli, C.: Non-linear reduced-order model for parametric excitation analysis of an immersed vertical slender rod. Int. J. Non-Linear Mech. 80, 29–39 (2016).

    Article  Google Scholar 

  28. Zhang, Y.-L., Chen, L.-Q.: Steady-state response of pipes conveying pulsating fluid near a 2: 1 internal resonance in the supercritical regime. Int. J. Appl. Mech. 6(05), 1450056 (2014).

    Article  Google Scholar 

  29. Wilson, J.F., Biggers, S.B.: Responses of submerged, inclined pipelines conveying mass. J. Eng. Ind. 96(4), 1141–1146 (1974).

    Article  Google Scholar 

  30. Alfosail, F.K., Nayfeh, A.H., Younis, M.I.: An analytic solution of the static problem of inclined risers conveying fluid. Meccanica 52(4), 1175–1187 (2016).

    MathSciNet  MATH  Google Scholar 

  31. Alfosail, F.K., Nayfeh, A.H., Younis, M.I.: Natural frequencies and mode shapes of statically deformed inclined risers. Int. J. Non-Linear Mech. 94, 12–19 (2017).

    Article  Google Scholar 

  32. Khalak, A., Williamson, C.H.K.: Motions, forces and mode transitions in vortex-induced vibrations at low mass-damping. J. Fluids Struct. 13(7), 813–851 (1999).

    Article  Google Scholar 

  33. Facchinetti, M.L., de Langre, E., Biolley, F.: Coupling of structure and wake oscillators in vortex-induced vibrations. J. Fluids Struct. 19(2), 123–140 (2004).

    Article  Google Scholar 

  34. Nayfeh, A.H.: Resolving controversies in the application of the method of multiple scales and the generalized method of averaging. Nonlinear Dyn. 40(1), 61–102 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  35. Nayfeh, A.H.: Introduction to Perturbation Techniques. Wiley, Hoboken (2011)

    MATH  Google Scholar 

  36. Cheney, E., Kincaid, D.: Numerical Mathematics and Computing. Cengage Learning, Boston (2012)

    MATH  Google Scholar 

  37. Balachandran, B., Nayfeh, A.: Cyclic motions near a Hopf bifurcation of a four-dimensional system. Nonlinear Dyn. 3(1), 19–39 (1992).

    Article  Google Scholar 

  38. . Wolfram Research, I.: Mathematica. In, vol. Version 10.1. Wolfram Research, Inc., Champaign, Illinois, (2015)

  39. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, Hoboken (2008)

    MATH  Google Scholar 

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We acknowledge the financial support of King Abdullah University of Science and Technology and Saudi Aramco.

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Correspondence to Mohammad I. Younis.

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Appendix A: Self-adjoint proof

In this appendix, we demonstrate by virtue of using integration by parts that the solution to the linear Eq. (13) is self-adjoint. Due to the internal resonance interaction, the solution to be utilized consists of two modes, namely \(\phi _m \left( x \right) \hbox {e}^{\pm i\omega _m T_0 }\) and \(\phi _n \left( x \right) \hbox {e}^{\pm i\omega _n T_0 }\) corresponding to modes m and n,  respectively. To verify that the problem is self-adjoint, we substitute the solutions from both modes in Eq. (13) and, multiply the equation with \(\phi _j \left( x \right) \hbox {e}^{\pm i\omega _j T_0 }\), then integrate by parts from \(x=0\) to \(x=1\) to obtain

$$\begin{aligned}&\left[ {\phi _j}{\phi _m}^{\prime \prime \prime } - {\phi _j}^\prime {\phi _m}^{\prime \prime } + {\phi _j}^{\prime \prime }{\phi _m}^\prime - {\phi _j}^{\prime \prime \prime }{\phi _m}+\, {{\bar{T}}_\mathrm{s}}{\phi _j}{\phi _m}^\prime \right. \nonumber \\&\quad \left. + {{\bar{T}}_\mathrm{s}}{\phi _j}^\prime {\phi _m} - \bar{\sigma }x{\phi _j}{\phi _m}^\prime \right. \left. +\, \bar{\sigma }{{\left( {x{\phi _j}} \right) }^\prime }{\phi _m} - \bar{\sigma }{\phi _j}{\phi _m} \right] _0^1 \nonumber \\&\quad \left[ {\phi _j}{\phi _n}^{\prime \prime \prime } - {\phi _j}^\prime {\phi _n}^{\prime \prime } + {\phi _j}^{\prime \prime }{\phi _n}^\prime - {\phi _j}^{\prime \prime \prime }{\phi _n} + {{\bar{T}}_\mathrm{s}}{\phi _j}{\phi _n}^\prime \right. \nonumber \\&\quad \left. +\, {{\bar{T}}_\mathrm{s}}{\phi _j}^\prime {\phi _n} - \bar{\sigma }x{\phi _j}{\phi _n}^\prime + \bar{\sigma }{{\left( {x{\phi _j}} \right) }^\prime }{\phi _n} - \bar{\sigma }{\phi _j}{\phi _n} \right] _0^1 \nonumber \\&\quad + 2\eta \left[ {\phi _j}{y_\mathrm{s}}{{^\prime }^\prime }\left( {\mathop \int \limits _0^1 {y_\mathrm{s}}^\prime {\phi _m}^\prime \mathrm{{d}}x} \right) \right. \nonumber \\&\quad \left. + {\phi _m}{y_\mathrm{s}}{{^\prime }^\prime }\left( {\mathop \int \limits _0^1 {y_\mathrm{s}}^\prime {\phi _j}^\prime \mathrm{{d}}x} \right) \right] _0^1 \nonumber \\&\quad +\, 2\eta \left[ {{\phi _j}{y_\mathrm{s}}{{^\prime }^\prime }\left( {\mathop \int \limits _0^1 {y_\mathrm{s}}^\prime {\phi _n}^\prime \mathrm{{d}}x} \right) + {\phi _n}{y_\mathrm{s}}^\prime \left( {\mathop \int \limits _0^1 {y_\mathrm{s}}^\prime {\phi _j}^\prime \mathrm{{d}}x} \right) } \right] _0^1 \nonumber \\&\quad \int \limits _0^1 {\phi _m}\left[ - {\omega _m}^2{\phi _j} + {\phi _j}^{iv} + {{\bar{T}}_\mathrm{s}}{\phi _j}^{\prime \prime } - \bar{\sigma }{\left( {x{\phi _j}} \right) }^{\prime \prime } \right. \nonumber \\&\quad \left. +\, \bar{\sigma }{\phi _j}^\prime - 2\eta {y_\mathrm{s}}^{\prime \prime }\left( {\mathop \int \limits _0^1 {y_\mathrm{s}}^\prime \left( x \right) {\phi _j}^\prime \mathrm{{d}}x} \right) \right] \;\mathrm{{d}}x \nonumber \\&\quad \int \limits _0^1 {\phi _n}\left[ - {\omega _n}^2{\phi _j} + {\phi _j}^{iv} + {{\bar{T}}_\mathrm{s}}{\phi _j}^{\prime \prime } - \bar{\sigma }{{\left( {x{\phi _j}} \right) }^{\prime \prime }}\right. \nonumber \\&\quad \left. +\, \bar{\sigma }{\phi _j}^\prime - 2\eta {y_\mathrm{s}}^{\prime \prime }\left( {\mathop \int \limits _0^1 {y_\mathrm{s}}^\prime \left( x \right) {\phi _j}^\prime \mathrm{{d}}x} \right) \right] \;\mathrm{{d}}x \end{aligned}$$

We observe from Eq. (1) that the solution \(\phi _j \left( x \right) \hbox {e}^{\pm i\omega _j T_0 }\) satisfies the eigenvalue problem Eq. (10) and it is self-adjoint.

Fig. 16
figure 16

Time histories of \(a_2 \), \(a_4 \), and \(a_5 \) modal coefficients from the Galerkin solution

Appendix B: definition of third-order solvability condition coupling coefficients

In this appendix, we provide the definition of the coupling coefficients \(K_1 \), \(K_2 \), \(K_3 \), and \(K_4 \) pertaining to Eqs. (30) and (31) given by

$$\begin{aligned} K_1= & {} 2\eta \mathop \int \limits _0^1 \left( {\mathop \int \limits _0^1 \phi _n ^{\prime }\psi _1 ^{\prime }\hbox {d}x} \right) \phi _m y_\mathrm{s} ^{\prime \prime }\hbox {d}x\nonumber \\&+ \, 2\eta \mathop \int \limits _0^1 \left( {\mathop \int \limits _0^1 \phi _m ^{\prime }\psi _4 ^{\prime }\hbox {d}x} \right) \phi _m y_\mathrm{s} ^{\prime \prime }\hbox {d}x\nonumber \\&+\, 2\eta \mathop \int \limits _0^1 \left( {\mathop \int \limits _0^1 \left( {\phi _n ^{\prime }} \right) ^{2}\hbox {d}x} \right) \phi _m \phi _m ^{\prime \prime }\hbox {d}x \nonumber \\&+\, 2\eta \mathop \int \limits _0^1 \left( {\mathop \int \limits _0^1 y_\mathrm{s} ^{\prime }\psi _4 ^{\prime }\hbox {d}x} \right) \phi _m \phi _m ^{\prime \prime }\hbox {d}x\nonumber \\&+\, 4\eta \mathop \int \limits _0^1 \left( {\mathop \int \limits _0^1 \phi _m ^{\prime }\phi _n ^{\prime }\hbox {d}x} \right) \phi _m \phi _n ^{\prime \prime }\hbox {d}x\nonumber \\&+\, 2\eta \mathop \int \limits _0^1 \left( {\mathop \int \limits _0^1 y_\mathrm{s} ^{\prime }\psi _1 ^{\prime }\hbox {d}x} \right) \phi _m \phi _n ^{\prime \prime }\hbox {d}x \nonumber \\&+\, 2\eta \mathop \int \limits _0^1 \left( {\mathop \int \limits _0^1 y_\mathrm{s} ^{\prime }\phi _n ^{\prime }\hbox {d}x} \right) \phi _m \psi _1 ^{\prime \prime }\hbox {d}x\nonumber \\&+\, 2\eta \mathop \int \limits _0^1 \left( {\mathop \int \limits _0^1 y_\mathrm{s} ^{\prime }\phi _m ^{\prime }\hbox {d}x} \right) \phi _m \psi _4 ^{\prime \prime }\hbox {d}x \nonumber \\ K_2= & {} 2\eta \mathop \int \limits _0^1 \left( {\mathop \int \limits _0^1 \phi _m ^{\prime }\psi _3 ^{\prime }\hbox {d}x} \right) \phi _m y_\mathrm{s} ^{\prime \prime }\hbox {d}x\nonumber \\&+\, 3\eta \mathop \int \limits _0^1 \left( {\mathop \int \limits _0^1 \left( {\phi _m ^{\prime }} \right) ^{2}\hbox {d}x} \right) \phi _m \phi _m ^{\prime \prime }\hbox {d}x\nonumber \\&+\, 2\eta \mathop \int \limits _0^1 \left( {\mathop \int \limits _0^1 y_\mathrm{s} ^{\prime }\psi _3 ^{\prime }\hbox {d}x} \right) \phi _m \phi _m ^{\prime \prime }\hbox {d}x \nonumber \\&+\, 2\eta \mathop \int \limits _0^1 \left( {\mathop \int \limits _0^1 y_\mathrm{s} ^{\prime }\phi _m ^{\prime }\hbox {d}x} \right) \phi _m \psi _3 ^{\prime \prime }\hbox {d}x \end{aligned}$$
$$\begin{aligned} K_3= & {} 2\eta \mathop \int \limits _0^1 \left( {\mathop \int \limits _0^1 \phi _n ^{\prime }\psi _1 ^{\prime }\hbox {d}x} \right) \phi _m y_\mathrm{s} ^{\prime \prime }\hbox {d}x\nonumber \\&+\, 2\eta \mathop \int \limits _0^1 \left( {\mathop \int \limits _0^1 \phi _m ^{\prime }\psi _4 ^{\prime }\hbox {d}x} \right) \phi _m y_\mathrm{s} ^{\prime \prime }\hbox {d}x\nonumber \\&+\, 2\eta \mathop \int \limits _0^1 \left( {\mathop \int \limits _0^1 \left( {\phi _n ^{\prime }} \right) ^{2}\hbox {d}x} \right) \phi _m \phi _m ^{\prime \prime }\hbox {d}x \nonumber \\&+\, 2\eta \mathop \int \limits _0^1 \left( {\mathop \int \limits _0^1 y_\mathrm{s} ^{\prime }\psi _4 ^{\prime }\hbox {d}x} \right) \phi _m \phi _m ^{\prime \prime }\hbox {d}x\nonumber \\&+\, 4\eta \mathop \int \limits _0^1 \left( {\mathop \int \limits _0^1 \phi _m ^{\prime }\phi _n ^{\prime }\hbox {d}x} \right) \phi _m \phi _n ^{\prime \prime }\hbox {d}x\nonumber \\&+\, 2\eta \mathop \int \limits _0^1 \left( {\mathop \int \limits _0^1 y_\mathrm{s} ^{\prime }\psi _1 ^{\prime }\hbox {d}x} \right) \phi _m \phi _n ^{\prime \prime }\hbox {d}x \nonumber \\&+\, 2\eta \mathop \int \limits _0^1 \left( {\mathop \int \limits _0^1 y_\mathrm{s} ^{\prime }\phi _n ^{\prime }\hbox {d}x} \right) \phi _m \psi _1 ^{\prime \prime }\hbox {d}x\nonumber \\&+\, 2\eta \mathop \int \limits _0^1 \left( {\mathop \int \limits _0^1 y_\mathrm{s} ^{\prime }\phi _m ^{\prime }\hbox {d}x} \right) \phi _m \psi _4 ^{\prime \prime }\hbox {d}x \end{aligned}$$
$$\begin{aligned} K_4= & {} 2\eta \mathop \int \limits _0^1 \left( {\mathop \int \limits _0^1 \phi _n ^{\prime }\psi _2 ^{\prime }\hbox {d}x} \right) \phi _m y_\mathrm{s} ^{\prime \prime }\hbox {d}x\nonumber \\&+\, 2\eta \mathop \int \limits _0^1 \left( {\mathop \int \limits _0^1 \phi _m ^{\prime }\psi _4^{\prime }\hbox {d}x} \right) \phi _m y_\mathrm{s} ^{\prime \prime }\hbox {d}x\nonumber \\&+\, 2\eta \mathop \int \limits _0^1 \left( {\mathop \int \limits _0^1 \left( {\phi _n ^{\prime }} \right) ^{2}\hbox {d}x} \right) \phi _m \phi _m ^{\prime \prime }\hbox {d}x \nonumber \\&+\, 2\eta \mathop \int \limits _0^1 \left( {\mathop \int \limits _0^1 y_\mathrm{s} ^{\prime }\psi _4 ^{\prime }\hbox {d}x} \right) \phi _m \phi _m ^{\prime \prime }\hbox {d}x\nonumber \\&+\, 4\eta \mathop \int \limits _0^1 \left( {\mathop \int \limits _0^1 \phi _m ^{\prime }\phi _n ^{\prime }\hbox {d}x} \right) \phi _m \phi _n ^{\prime \prime }\hbox {d}x\nonumber \\&+\, 2\eta \mathop \int \limits _0^1 \left( {\mathop \int \limits _0^1 y_\mathrm{s} ^{\prime }\psi _2 ^{\prime }\hbox {d}x} \right) \phi _m \phi _n ^{\prime \prime }\hbox {d}x \nonumber \\&+\, 2\eta \mathop \int \limits _0^1 \left( {\mathop \int \limits _0^1 y_\mathrm{s} ^{\prime }\phi _n ^{\prime }\hbox {d}x} \right) \phi _m \psi _1 ^{\prime \prime }\hbox {d}x\nonumber \\&+\, 2\eta \mathop \int \limits _0^1 \left( {\mathop \int \limits _0^1 y_\mathrm{s} ^{\prime }\phi _m ^{\prime }\hbox {d}x} \right) \phi _m \psi _4 ^{\prime \prime }\hbox {d}x \end{aligned}$$

Appendix C: Time history of modal coefficients pertaining to the Galerkin solution

In this appendix, we plot the modal coefficients corresponding to the riser solution in Fig. 13, which has weak or negligible contribution (Fig. 16).

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Alfosail, F.K., Younis, M.I. Two-to-one internal resonance of an inclined marine riser under harmonic excitations. Nonlinear Dyn 95, 1301–1321 (2019).

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