A computationally efficient quasi-two-degree-of-freedom (Q2DOF) stochastic model and a stability analysis of barges in random seas are presented in this paper. Based on the deterministic 2DOF coupled roll-heave model with high-degree polynomial approximation of restoring forces and moments developed in Part I, an attempt is made to further reduce the DOF of the model for efficient stochastic stability analysis by decoupling the heave effects on roll motion, resulting in a one-degree-of-freedom (1DOF) roll-only model. Using the Markov assumption, stochastic differential equations governing the evolution of probability densities of roll-heave and roll responses for the two low-DOF models are derived via the Fokker-Planck formulation. Numerical results of roll responses for the 2DOF and 1DOF models, using direct simulation in the time domain and the path integral solution technique in the probability domain, are compared to determine the effects of neglecting the influence of heave on roll motion and assess the relative computational efforts required. It is observed that the 1DOF model is computationally very efficient and the 2DOF model response predictions are quite accurate. However, the nonlinear roll-heave coupling is found to be significant and needs to be directly taken into account, rendering the 1DOF roll-only model inadequate for practical use. The 2DOF model is impractical for long-duration real-time response computation due to the insurmountable computational effort required. By taking advantage of the observed strong correlation between measured heave and wave elevation in the experimental results, an accurate and efficient Q2DOF model is developed by expressing the heave response in the 2DOF model as a function of wave elevation, thus reducing the effective DOF to unity. This Q2DOF model is essential as it reduces the computational effort by a factor of $10−5$ compared to that of the 2DOF model, thus making practical stochastic analysis possible. A stochastic stability analysis of the barge under operational and survival sea states specified by the U.S. Navy is presented using the Q2DOF model based on first passage time formulation.

1.
Roberts
,
J. B.
,
1982a
, “
A Stochastic Theory for Nonlinear Ship Rolling in Irregular Seas
,”
J. Ship Res.
,
26
(
4
), pp.
229
245
.
2.
Roberts
,
J. B.
,
1982b
, “
Effect of Parametric Excitation on Ship Rolling Motion in Random Waves
,”
J. Ship Res.
,
26
(
4
), pp.
246
253
.
3.
Robert
,
J. B.
,
Dunne
,
J. F.
, and
Debonos
,
A.
,
1994
, “
Stochastic Estimation Methods for Non-Linear Ship Roll Motion
,”
Probab. Eng. Mech.
,
9
, pp.
83
93
.
4.
Dahle
,
E. A.
,
Myhaug
,
D.
, and
Dahl
,
S. J.
,
1988
, “
Probability of Capsizing in Steep and High Waves From the Side in Open Sea and Coastal Waters
,”
Ocean Eng.
,
15
(
2
), pp.
139
151
.
5.
Lin
,
H.
, and
Yim
,
S. C. S.
,
1995
, “
Chaotic Roll Motion and Capsizing of Ships Under Periodic Excitation With Random Noise
,”
Appl. Ocean. Res.
,
17
(
3
), pp.
185
204
.
6.
Falzarano
,
J. M.
,
Shaw
,
S. W.
, and
Troesch
,
A. W.
,
1992
, “
Application of Global Methods for Analyzing Dynamical Systems to Ship Rolling Motion and Capsizing
,”
Int. J. Bifurcation Chaos Appl. Sci. Eng.
,
2
(
1
), pp.
101
116
.
7.
Nayfeh
,
A. H.
, and
Sanchez
,
N. E.
,
1990
, “
Stability and Complicated Rolling Responses of Ships in Regular Beam Sea
,”
Int. Shipbuilding Congress
,
37
(
412
), pp.
331
352
.
8.
Kwon
,
S. H.
,
Kim
,
D. W.
, and
McGregor
,
R. C.
,
1993
, “
A Stochastic Roll Response Analysis of Ships in Irregular Waves
,”
Int. J. Offshore Polar Eng.
,
3
(
1
), pp.
32
34
.
9.
Cai, G. Q., Yu, S. J., and Lin, Y. K., 1994, “Ship Rolling in Random Sea,” Stochastic Dynamics and Reliability of Nonlinear Ocean Systems, ASME DE-Vol. 77, pp. 81–88.
10.
Chakrabarti, S. K., 1994, Hydrodynamics of Offshore Structures, Computational Mechanics Publications, Southampton.
11.
Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. T., 1986, Numerical Recipes, Cambridge University Press, Cambridge.
12.
Naess, A., and Johnsen, J. M., “Response Statistics of Nonlinear Dynamic Systems by Path Integrations,” IUTAM Symposium on Nonlinear Stochastic Mechanics, Torino, Italy, 1991, pp. 401–409.
13.
Naess
,
A.
, and
Johnsen
,
J. M.
,
1993
, “
Response Statistic of Nonlinear, Compliant Offshore Structures by the Path Integral Solution Method
,”
Probab. Eng. Mech.
,
8
,
91
106
.
14.
Wissel
,
C.
,
1979
, “
Manifolds of Equivalent Path Integral Solutions of the Fokker-Planck Equation
,”
Z. Phys. B
,
35
, pp.
185
191
.
15.
Wehner
,
M. F.
, and
Wolfer
,
W. G.
,
1983
, “
Numerical Evaluation of Path-Integral Solutions to Fokker-Planck Equations
,”
Phys. Rev. A
,
27
,
2663
2670
.