Study On Some Problems Of Ship Structural Nonlinear Mechanics

Recent years, as the development trend of ship and ocean engineeringto deep ocean, supersize, extreme sea state, high power, new materials andacousto-optic electromagnetic coupling, the conventional linear elasticitytheory is more and more difficult to meet the design requirements. A plentyof non-linear mechanical problems has been proposed in ship and oceanengineering, for instance, the deep-sea space station, fluid structureinteraction analysis of the artificial island-very large floating structure,tank sloshing analysis of large LNG ship, the dynamics of3000metersdeep sea drilling platform, deep-water riser vortex-induced vibration,ultimate strength and fatigue and creep analysis of ten thousand metersROV, the design technology of coupling of radar absorbing and shapestealth mechanics and etc. This paper analyzes several non-linear dynamicsand non-linear structural mechanics problems involved in ship and oceanengineering analysis.Firstly, a highly accurate method based on the Hermite interpolation isproposed for the periodic motion of nonlinear conservation systems onaccount of the particularity of these systems. It is shown that a Hermiteinterpolation solution for a dynamical system can be obtained bytransforming the independent time variable to a vibration time. Thecorresponding transformed differential equation becomes well-conditionedfor a solution by the Hermite interpolation method (HIM). Convergence andaccuracy of the proposed HIM is superior to the traditional Qaisi’s powerseries method (PSM) for HIM using information of two points instead ofPSM one point. By a way of illustration, approximate analytical solutions ofa class of nonlinear oscillators are derived by the HIM. Results show thatthese solutions are simple in form with highly accuracy.Secondly, we studied the solutions of the damped Duffing, Helmholtzand Helmholtz-Duffing oscillators. It is interesting to see that both of the damped Duffing oscillator and Helmholtz oscillator possess solutions thatfollow closely to the undamped case, and even the solution procedures arealmost the same. However, the same phenomenon doesn’t appear to theHelmholtz-Duffing oscillator, for which we find that the solution cannot begiven exactly by applying the Painlevé test.Thirdly, we deepened the research of nonlinear pendulum and based onKirchhoff’s kinetic analogy we focused on the large deformation of slenderbeam which is also called the “elastica”:Explicit exact solution of the large deformations of a cantilever beamunder point load at the free tip is derived by using the Jacobi ellipticfunctions, and it is different from the conventional solution obtained by theelliptic integrals for it gives the rotation angle of any point in the beam bywhich the corresponding displacement can be obtained easily. Study showsthat the solution of the cantilever beam with a concentrated load at anyposition in the beam can be obtained by the exact solution; and symmetryanalysis shows that the exact solution can be applied to simple and fixedbeams with a concentrated load at middle point directly.Considering the integrity of our research, the exact solution of the largedeflections of a cantilever beam under rotational load at the end tip is alsogiven in this paper.At last, we studied another typical problem of the elastica:post-buckling of the slender flexible rod. Exact solutions for post-bucklingof elastica under four different typical boundary conditions, that is, two endshinged, one end free and one end fixed, two ends fixed and one end hingedand one end fixed, are given explicitly by the Jacobi elliptic functions. Bystudying of these solutions, some results are unified between the nonlinearand linear analysis of the buckling theory. However, a special phenomenonfor nonlinear analysis is the violent jumps for post-buckling of beams withone end hinged and one end fixed. Wang has revealed this phenomenon byperturbation method. And this paper will show that the violent jumps existnot only for the1st-order post-buckling, in fact, it also exists for higher-order situation. The values of “positive jumps” and “negative jump”point are given for the1st and2nd-order post-buckling.