Skip to main content
SpringerLink
Log in

Probabilistic description of extreme oscillations and reliability analysis in rolling motion under stochastic excitation

  • Article
  • Published:
Science China Technological Sciences Aims and scope Submit manuscript

Abstract

Large-amplitude rolling motions, also regarded as extreme oscillations, are a great threat to marine navigation, which may lead to capsizing in ship motion. Therefore, it is important to quantify extreme oscillations, assess reliability of ship systems, and establish a suitable indicator to characterize extreme oscillations in ship systems. In this work, extreme events are investigated in a ship model considering a complex ocean environment, described by a single-degree-of-freedom nonlinear system with stochastic harmonic excitation and colored Gaussian noise. The stationary probability density function (PDF) of the system is derived through a probabilistic decomposition-synthesis method. Based on this, we infer the classical damage rate of the system. Furthermore, a new indicator, independent of the PDF, is proposed to quantify the damage related only to the fourth-order moment of the system and the threshold for extreme events. It is more universal and easier to determine as compared with the classical damage rate. A large damping ratio, a large noise intensity, or a short correlation time can reduce the damage rate and the value of the indicator. These findings provide new insights and theoretical guidance to avoid extreme oscillations and assess the reliability of practical ship movements.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Montewka J, Ehlers S, Goerlandt F, et al. A framework for risk assessment for maritime transportation systems—A case study for open sea collisions involving RoPax vessels. Reliab Eng Syst Safe, 2014, 124: 142–157

    Google Scholar 

  2. Lech K. Stability and safety of ships: holistic and risk approach. Reliab Risk Anal, 2008, 1: 95–105

    Google Scholar 

  3. Deng W, Zhang L, Zhou X, et al. Multi-strategy particle swarm and ant colony hybrid optimization for airport taxiway planning problem. Inf Sci, 2022, 612: 576–593

    Google Scholar 

  4. Xiang Y, Sheng J B, Wang L, et al. Research progresses on equipment technologies used in safety inspection, repair, and reinforcement for deepwater dams. Sci China Tech Sci, 2022, 65: 1059–1071

    Google Scholar 

  5. Lin H, Yim S C S. Chaotic roll motion and capsize of ships under periodic excitation with random noise. Appl Ocean Res, 1995, 17: 185–204

    Google Scholar 

  6. Xu X, Wiercigroch M. Approximate analytical solutions for oscillatory and rotational motion of a parametric pendulum. Nonlinear Dyn, 2007, 47: 311–320

    MathSciNet  MATH  Google Scholar 

  7. Xu Y, Liu Q, Guo G, et al. Dynamical responses of airfoil models with harmonic excitation under uncertain disturbance. Nonlinear Dyn, 2017, 89: 1579–1590

    MathSciNet  Google Scholar 

  8. Lin Y K, Cai G Q. Probabilistic Structural Dynamics: Advanced Theory and Applications. New York: McGraw-Hill, 2004

    Google Scholar 

  9. Boeck T, Sanjari S L, Becker T. Parametric instability of a magnetic pendulum in the presence of a vibrating conducting plate. Nonlinear Dynam, 2020, 102, 2039–2056

    Google Scholar 

  10. Ma J Z, Xu Y, Xu W, et al. Slowing down critical transitions via Gaussian white noise and periodic force. Sci China Tech Sci, 2019, 62: 2144–2152

    Google Scholar 

  11. Farokhi H, Xia Y, Erturk A. Experimentally validated geometrically exact model for extreme nonlinear motions of cantilevers. Nonlinear Dyn, 2022, 107: 457–475

    Google Scholar 

  12. Zhao L C, Zou H X, Gao Q H, et al. Design, modeling and experimental investigation of a magnetically modulated rotational energy harvester for low frequency and irregular vibration. Sci China Tech Sci, 2020, 63: 2051–2062

    Google Scholar 

  13. Kumarasamy S, Pisarchik A N. Extreme events in systems with discontinuous boundaries. Phys Rev E, 2018, 98: 032203

    Google Scholar 

  14. Liu Q, Xu Y, Li Y, et al. Fixed-interval smoothing of an aeroelastic airfoil model with cubic or free-play nonlinearity in incompressible flow. Acta Mech Sin, 2021, 37: 1168–1182

    MathSciNet  Google Scholar 

  15. Kingston S L, Thamilmaran K, Pal P, et al. Extreme events in the forced Liénard system. Phys Rev E, 2017, 96: 052204

    Google Scholar 

  16. Sudharsan S, Venkatesan A, Muruganandam P, et al. Emergence and mitigation of extreme events in a parametrically driven system with velocity-dependent potential. Eur Phys J Plus, 2021, 136: 129

    Google Scholar 

  17. de S. Cavalcante H L D, Oriá M, Sornette D, et al. Predictability and Suppression of Extreme Events in a Chaotic System. Phys Rev Lett, 2013, 111: 198701

    Google Scholar 

  18. Tian R L, Zhao Z J, Xu Y. Variable scale-convex-peak method for weak signal detection. Sci China Tech Sci, 2021, 64: 331–340

    Google Scholar 

  19. Sornette D. Dragon-kings, black swans, and the prediction of crises. Int J Terraspace Sci Eng, 2009, 2: 1–18

    Google Scholar 

  20. Hristopulos D T, Petrakis M P, Kaniadakis G. Finite-size effects on return interval distributions for weakest-link-scaling systems. Phys Rev E, 2014, 89: 052142

    Google Scholar 

  21. Zhao D, Li Y, Xu Y, et al. Extreme events in a class of nonlinear Duffing-type oscillators with a parametric periodic force. Eur Phys J Plus, 2022, 137: 314

    Google Scholar 

  22. Ghil M, Yiou P, Hallegatte S, et al. Extreme events: dynamics, statistics and prediction. Nonlin Processes Geophys, 2011, 18: 295–350

    Google Scholar 

  23. Ansmann G, Karnatak R, Lehnertz K, et al. Extreme events in excitable systems and mechanisms of their generation. Phys Rev E, 2013, 88: 052911

    Google Scholar 

  24. Mohamad M A, Sapsis T P. Probabilistic description of extreme events in intermittently unstable dynamical systems excited by correlated stochastic processes. SIAM-ASA J Uncertain, 2015, 3: 709–736

    MathSciNet  MATH  Google Scholar 

  25. Mohamad M A, Sapsis T P. Probabilistic response and rare events in Mathieu’s equation under correlated parametric excitation. Ocean Eng, 2016, 120: 289–297

    Google Scholar 

  26. Mohamad M A, Cousins W, Sapsis T P. A probabilistic decomposition-synthesis method for the quantification of rare events due to internal instabilities. J Comput Phys, 2016, 322: 288–308

    MathSciNet  MATH  Google Scholar 

  27. Lü C, Zhang Y, Wang X, et al. Frequency reliability-based robust design of the suspension device of a jarring machine with arbitrary distribution parameters. Mech Based Des Struct Machines, 2015, 43: 487–500

    Google Scholar 

  28. Liu Q, Xu Y, Kurths J, et al. Complex nonlinear dynamics and vibration suppression of conceptual airfoil models: A state-of-the-art overview. Chaos, 2022, 32: 062101

    MathSciNet  Google Scholar 

  29. Chen H, Miao F, Chen Y, et al. A hyperspectral image classification method using multifeature vectors and optimized KELM. IEEE J Sel Top Appl Earth Observat Remote Sens, 2021, 14: 2781–2795

    Google Scholar 

  30. Rice S O. Mathematical analysis of random noise. Bell Syst Technical J, 1944, 23: 282–332

    MathSciNet  MATH  Google Scholar 

  31. Cottone G, Di Paola M, Ibrahim R, et al. Stochastic ship roll motion via path integral method. Int J Naval Architecture Ocean Eng, 2010, 2: 119–126

    Google Scholar 

  32. Jin Y R, Qin C J, Zhang Z N, et al. A multi-scale convolutional neural network for bearing compound fault diagnosis under various noise conditions. Sci China Tech Sci, 2022, 65: 2551–2563

    Google Scholar 

  33. Joo H K, Mohamad M A, Sapsis T P. Extreme events and their optimal mitigation in nonlinear structural systems excited by stochastic loads: Application to ocean engineering systems. Ocean Eng, 2017, 142: 145–160

    Google Scholar 

  34. Li Y, Wei Z, Kapitaniak T, et al. Stochastic bifurcation and chaos analysis for a class of ships rolling motion under non-smooth perturbation and random excitation. Ocean Eng, 2022, 266: 112859

    Google Scholar 

  35. Liu Q, Xu Y, Xu C, et al. The sliding mode control for an airfoil system driven by harmonic and colored Gaussian noise excitations. Appl Math Model, 2018, 64: 249–264

    MathSciNet  MATH  Google Scholar 

  36. Xu Y, Gu R, Zhang H, et al. Stochastic bifurcations in a bistable Duffing–Van der Pol oscillator with colored noise. Phys Rev E, 2011, 83: 056215

    Google Scholar 

  37. Ge M Y, Wang G W, Jia Y. Influence of the gaussian colored noise and electromagnetic radiation on the propagation of subthreshold signals in feedforward neural networks. Sci China Tech Sci, 2021, 64: 847–857

    Google Scholar 

  38. France W N, Levadou M, Treakle T W, et al. An investigation of head-sea parametric rolling and its influence on container lashing systems. Mar Tech SNAME News, 2003, 40: 1–19

    Google Scholar 

  39. Kreuzer E, Sichermann W. The effect of sea irregularities on ship rolling. Comput Sci Eng, 2006, 8: 26–34

    Google Scholar 

  40. Kovacic I, Rand R, Mohamed Sah S. Mathieu’s equation and its generalizations: Overview of stability charts and their features. Appl Mech Rev, 2018, 70: 020802

    Google Scholar 

  41. Ren Z, Han X, Yu X, et al. Data-driven simultaneous identification of the 6DOF dynamic model and wave load for a ship in waves. Mech Syst Signal Process, 2023, 184: 109422

    Google Scholar 

  42. Xu Y, Ma J, Wang H, et al. Effects of combined harmonic and random excitations on a Brusselator model. Eur Phys J B, 2017, 90: 194

    MathSciNet  Google Scholar 

  43. Hänggi P, Jung P. Colored noise in dynamical systems. Adv Chem Phys, 2007, 89: 239–326

    Google Scholar 

  44. Zhu W Q. Random Vibration (in Chinese). Beijing: Science Press, 1992

    Google Scholar 

  45. Sun J Q. Stochastic Dynamics and Control. Amsterdam: Elsevier, 2006

    Google Scholar 

  46. Zhang X Y, Xu Y, Liu Q, et al. Rate-dependent tipping-delay phenomenon in a thermoacoustic system with colored noise. Sci China Tech Sci, 2020, 63: 2315–2327

    Google Scholar 

  47. Krawiecki A, Hołyst J A, Helbing D. Volatility clustering and scaling for financial time series due to attractor bubbling. Phys Rev Lett, 2002, 89: 158701

    Google Scholar 

  48. Kallner A. Laboratory Statistics: Methods in Chemistry and Health Sciences. Waltham: Elsevier, 2017

    Google Scholar 

  49. Roberts J B. An approach to the first-passage problem in random vibration. J Sound Vib, 1968, 8: 301–328

    MATH  Google Scholar 

  50. Zhang X, Xu Y, Liu Q, et al. Rate-dependent bifurcation dodging in a thermoacoustic system driven by colored noise. Nonlinear Dyn, 2021, 104: 2733–2743

    Google Scholar 

  51. Coe T E, Xing J T, Shenoi R A, et al. A simplified 3-D human body-seat interaction model and its applications to the vibration isolation design of high-speed marine craft. Ocean Eng, 2009, 36: 732–746

    Google Scholar 

  52. Olausson K, Garme K. Prediction and evaluation of working conditions on high-speed craft using suspension seat modelling. P I Mech Eng M-J Eng, 2015, 229: 281–290

    Google Scholar 

  53. Biswas S, Bhattacharjee J K. On the properties of a class of higher-order Mathieu equations originating from a parametric quantum oscillator. Nonlinear Dyn, 2019, 96: 737–750

    MATH  Google Scholar 

  54. Shi Y, Li L, Yang J, et al. Center-based Transfer Feature Learning With Classifier Adaptation for surface defect recognition. Mech Syst Signal Process, 2023, 188: 110001

    Google Scholar 

  55. Ray A, Rakshit S, Basak G K, et al. Understanding the origin of extreme events in El Niño southern oscillation. Phys Rev E, 2020, 101: 062210

    MathSciNet  Google Scholar 

  56. Li H, Xu Y, Yue X, et al. Transition-event duration in one-dimensional systems under correlated noise. Physica A-Statistical Mech its Appl, 2019, 532: 121764

    MathSciNet  MATH  Google Scholar 

  57. Yurchenko D, Naess A, Alevras P. Pendulum’s rotational motion governed by a stochastic Mathieu equation. Probab Eng Mech, 2013, 31: 12–18

    Google Scholar 

  58. Liu Q, Xu Y, Kurths J. Bistability and stochastic jumps in an airfoil system with viscoelastic material property and random fluctuations. Commun Nonlinear Sci Numer Simul, 2020, 84: 105184

    MathSciNet  MATH  Google Scholar 

  59. Ma J, Liu Q, Xu Y, et al. Early warning of noise-induced catastrophic high-amplitude oscillations in an airfoil model. Chaos, 2022, 32: 033119

    MathSciNet  Google Scholar 

  60. Xie W C, So R M C. Parametric resonance of a two-dimensional system under bounded noise excitation. Nonlinear Dyn, 2004, 36: 437–453

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710072, China

    Dan Zhao, YongGe Li, Yong Xu & Qi Liu

  2. MOE Key Laboratory for Complexity Science in Aerospace, Northwestern Polytechnical University, Xi’an, 710072, China

    Yong Xu

  3. Potsdam Institute for Climate Impact Research, Potsdam, 14412, Germany

    Jürgen Kurths

  4. Department of Physics, Humboldt University Berlin, Berlin, 12489, Germany

    Jürgen Kurths

Authors
  1. Dan Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. YongGe Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yong Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Qi Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Jürgen Kurths
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to YongGe Li.

Additional information

This work was supported by the National Natural Science Foundation of China (Grant No. 12072264), and the Key International (Regional) Joint Research Program of the National Natural Science Foundation of China (Grant No. 12120101002). Li Y G thanks the support of the National Natural Science Foundation of China (Grant No. 12272296), the Natural Science Foundation of Chongqing, China (Grant No. cstc2021jcyj-msxmX0738), and the Natural Science Foundation of Guangdong Province, China (Grant No. 2023A1515012329).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhao, D., Li, Y., Xu, Y. et al. Probabilistic description of extreme oscillations and reliability analysis in rolling motion under stochastic excitation. Sci. China Technol. Sci. 66, 2586–2596 (2023). https://doi.org/10.1007/s11431-022-2388-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11431-022-2388-4

Search

Navigation