Internal Resonances for the Heave Roll and Pitch Modes of a Spar Platform Considering Wave and Vortex Exciting Loads in Heave Main Resonance

Abstract

The objective of this paper is to study the nonlinear coupling internal resonance of the heave, roll, and pitch response performance of a spar platform when their frequencies are in the ratio of 2:1:1 under wave and vortex exciting loads. The three degree-of-freedom (DOF) nonlinear coupled equations are established by considering a time-varying wet surface with a first-order wave force in heave and pitch and a vortex-induced force in roll. The first-order steady-state response is solved using the multi-scale method in heave main resonance. The multiple solutions of the motion equations are discussed using the analytic method and a numerical simulation. A sensitivity analysis is conducted to test the influence of the damping and internal detuning parameter. The regions of multiple solutions are found, and the jump phenomenon exists with the changes of the wave excitation. The regions of multiple solutions depend on the values of damping and detuning parameter.

Keywords

Spar platform / 3-DOF nonlinear equations / Multiple scales / Sub-harmonic response / Jump phenomenon / Sensitivity analysis

References

[1] Froude W. Remarks on Mr. Scott Russell’s paper on rolling. Trans Inst Naval Res, 1863, 4: 232-275

[2] Haslum HA, Faltinsen OM. Alternative shape of spar platforms for use in hostile areas, 1999, Houston: Offshore Technology Conference

[3] Hong YP, Lee DY, Choi YH, Hong SK, Kim SE, (2005) An experimental study on the extreme motion responses of a spar platform in the heave resonant waves. In Proceedings of the Fifteenth International Offshore and Polar Engineering Conference. Seoul, Korea 225–232. https://doi.org/10.3969/j.issn.1007-7294.2016.h1.008

[4] Koo BJ, Kim MH, Randall RE. Mathieu instability of a spar platform with mooring and risers. Ocean Eng, 2004, 31(17): 2175-2208

[5] Li W, Tang Y, Liu L, Liu S, Cai R. Heave-roll-pitch coupled nonlinear internal resonance response of a spar platform considering wave and vortex exciting loads. J Ocean Univ China, 2017, 16(2): 209-222

[6] Li W, Tang Y, Liu L, Li Y, Wang B. Internal resonances for heave, roll and pitch modes of a spar platform considering wave and vortex-induced loads in the main roll resonance. China Ocean Eng, 2017, 31(4): 408-417

[7] Liu L, Incecik A, Zhang Y, Tang Y. Analysis of heave motions of a truss spar platform with semi-closed moon pool. Ocean Eng, 2014, 92: 162-174

[8] Liu L, Zhou B, Tang Y. Study on the nonlinear dynamical behavior of deepsea Spar platform by numerical simulation and model experiment. J Vib Control, 2014, 20(10): 1528-1537

[9] Mook DT, Marshall LR, Nayfeh AH. Subharmonic and superharmonic resonances in the pitch and roll modes of ship motions. J Hydronaut, 1974, 8(1): 32-40

[10] Nayfeh AH. On the undesirable roll characteristics of ships in regular seas. J Ship Res, 2011, 32(2): 92-100

[11] Nayfeh AH, Mook DT, Marshall LR. Nonlinear couple of pitch and roll modes in ship motions. J Hydronaut, 1973, 7(4): 145-152

[12] Neves MA, Sphaier SH, Mattoso BM, Rodríguez CA, Santos AL,Vileti VL, Torres FG, 2008. On the occurrence of Mathieu instabilities of vertical cylinders. The 27th International Conference on Offshore Mechanics and Arctic Engineering, Estoril, 619–627. https://doi.org/10.1115/OMAE2008-57567

[13] Rho JB, Choi HS, Shin HS, Park IK (2003) An experimental study for mooring effects on the stability of spar platform. In Proceedings of the 13th International Offshore and Polar Engineering Conference, Honolulu, 285–288

[14] Rho JB, Choi HS, Shin HS, Park IK. A study on Mathieu-type instability of conventional spar platform in regular waves. Int J Offshore Polar Eng, 2005, 15(2): 104-108

[15] Sayed M, Hamed YS. Stability and response of a nonlinear coupled pitch-roll ship model under parametric and harmonic excitations. Nonlinear Dyn, 2011, 64(3): 207-220

[16] Shen W, Tang Y. Stochastic analysis of nonlinear coupled heave-pitch motion for the truss spar platform. J Mar Sci Appl, 2011, 10(4): 471-477

[17] Yang HZ, Xu PJ. Effect of hull geometry on parametric resonances of spar in irregular waves. Ocean Eng, 2015, 99: 14-22

[18] Zhao JR (2010) Study on nonlinear coupling dynamic response of a classic spar platform. Ph.D. thesis, School of Civil Engineering, Tianjin University, Tianjin (in Chinese)

[19] Zhao JR, Tang LZ, Tang YG, Wang WJ. Internal resonant responses of heave-pitch coupled motions of classic spar platform. J Tianjin Univ, 2009, 42(3): 201-207 (in Chinese)

[20] Zhao JR, Tang YG, Shen WJ. A study on the combination resonance response of a classic spar platform. J Vib Control, 2010, 16(14): 2083-2107