Singularity analysis of response bifurcation for a coupled pitch–roll ship model with quadratic and cubic nonlinearity

  • Xinye Li
  • Lijuan Zhang
  • Huabiao Zhang
  • Kai Li
Original Paper


Based on the coupling of roll and pitch motion of ships, a mathematical model with quadratic and cubic nonlinear terms is presented. Primary resonance is discussed by the averaging method when the frequency of excitation acting only on the pitch mode is near the natural frequencies of pitch mode and 1:2 internal resonance exists. By means of the slow-flow equations, the singularity analysis of the approximate solutions is investigated by means of bifurcation theory with constraint, from which one can find the effects of system parameters and excitation on the response. A total of 15 kinds of different persistent bifurcation modes were found. In terms of these results with stability analysis, the response can be classified into uncoupled periodic, coupled periodic and complex motions such as quasi-periodic or chaotic motions. By singularity analysis, saturation and jumping phenomena are also included in the present result which can be regarded as an extension of the available published works. The numerical simulations are in good agreement with the approximate analytical prediction.


Nonlinearly coupled pitch–roll ship model Primary resonance Internal resonance Singularity analysis 



This research is supported by the National Natural Science Foundation of China under Grant No. 10872063. The authors express sincere thanks to the reviewers for their valuable suggestions.

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this paper.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of Mechanical EngineeringHebei University of TechnologyTianjinChina
  2. 2.School of Mechanical EngineeringTianjin University of CommerceTianjinChina

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