Response Solutions for Degenerate Reversible Harmonic Oscillators with Zero-average Perturbation

In this paper,we consider a class of normally degenerate quasi-periodically forced reversible systems,obtained as perturbations of a set of harmonic oscillators,{x˙=y+∈f1(ωt,x,y),y˙=λx^(l)+∈f2(ωt,x,y),where 0≠λ∈R,l>1 is an integer and the corresponding involution G is(−θ,x,−y)→(θ,x,y).The existence of response solutions of the above reversible systems has already been proved in[22]if[f2(ωt,0,0)]satisfies some non-zero average conditions(See the condition(H)in[22]),here[·]denotes the average of a continuous function on T^(d).However,discussing the existence of response solutions for the above systems encounters difficulties when[f_(2)(ωt,0,0)]=0,due to a degenerate implicit function must be solved.This article will be doing work in this direction.The purpose of this paper is to consider the case where[f2(ωt,0,0)]=0.More precisely,with 2p<l,if f_(2)satisfies[f_(2)(ωt,0,0)]=[∂f_(2)(ωt,0,0)/∂x]=[∂^(2)f_(2)(ωt,0,0)/∂x2]=···=[∂p−1f2(ωt,0,0)∂xp−1]=0,eitherλ−1[∂pf2(ωt,0,0)∂xp]<0 as l−p is even orλ−1[∂pf2(ωt,0,0)∂xp]=0 as l−p is odd,we obtain the following results:(1)Forλ>˜0(seeλ˜in(2.2))and sufficiently small,response solutions exist for eachωsatisfying a weak non-resonant condition;(2)Forλ<˜0 and∗sufficiently small,there exists a Cantor set E∈(0,∗)with almost full Lebesgue measure such that response solutions exist for each∈E ifωsatisfies a Diophantine condition.In the remaining case whereλ−1[∂pf2(ωt,0,0)∂xp]>0 and l−p is even,we prove the system admits no response solutions in most regions.

EXACT SOLUTIONS FOR STATIONARY RESPONSES OF SEVERAL CLASSES OF NONLINEAR SYSTEMS TO PARAMETRIC AND/OR EXTERNAL WHITE NOISE EXCITATIONS

The exact solutions for stationary responses of one class of the second order and three classes of higher order nonlinear systems to parametric and/or external while noise excitations are constructed by using Fokkcr-Planck-Kolmogorov et/ualion approach. The conditions for the existence and uniqueness and the behavior of the solutions are discussed. All the systems under consideration are characterized by the dependence ofnonconservative fqrces on the first integrals of the corresponding conservative systems and arc catted generalized-energy-dependent f G.E.D.) systems. It is shown taht for each of the four classes of G.E.D. nonlinear stochastic systems there is a family of non-G.E.D. systems which are equivalent to the G.E.D. system in the sense of having identical stationary solution. The way to find the equivalent stochastic systems for a given G.E.D. system is indicated and. as an example, the equivalent stochastic systems for the second order G.E. D. nonlinear stochastic system are given. It is pointed out and illustrated with example that the exact stationary solutions for many non-G.E.D. nonlinear stochastic systems may he found by searching the equivalent G.E.D. systems.

Governing equations and numerical solutions of tension leg platform with finite amplitude motion

It is demonstrated that when tension leg platform （TLP） moves with finite amplitude in waves, the inertia force, the drag force and the buoyancy acting on the platform are nonlinear functions of the response of TLP, The tensions of the tethers are also nonlinear functions of the displacement of TLP. Then the displacement, the velocity and the acceleration of TLP should be taken into account when loads are calculated. In addition, equations of motions should be set up on the instantaneous position. A theo- retical model for analyzing the nonlinear behavior of a TLP with finite displacement is developed, in which multifold nonlinearities are taken into account, i.e., finite displace- ment, coupling of the six degrees of freedom, instantaneous position, instantaneous wet surface, free surface effects and viscous drag force, Based on the theoretical model, the comprehensive nonlinear differential equations are deduced. Then the nonlinear dynamic analysis of ISSC TLP in regular waves is performed in the time domain. The degenerative linear solution of the proposed nonlinear model is verified with existing published one. Furthermore, numerical results are presented, which illustrate that nonlinearities exert a significant influence on the dynamic responses of the TLP.

Arc-length technique for nonlinear finite element analysis

Nonlinear solution of reinforced concrete structures, particularly complete load-deflection response, requires tracing of the equilibrium path and proper treatment of the limit and bifurcation points. In this regard, ordinary solution techniques lead to instability near the limit points and also have problems in case of snap-through and snap-back. Thus they fail to predict the complete load-displacement response. The arc-length method serves the purpose well in principle, received wide acceptance in finite element analysis, and has been used extensively. However modifications to the basic idea are vital to meet the particular needs of the analysis. This paper reviews some of the recent developments of the method in the last two decades, with particular emphasis on nonlinear finite element analysis of reinforced concrete structures.

Analytic solution to the magnetotelluric response over anisotropic medium and its discussion

In the present paper,from the second order partial differential equations for solving the magnetotelluric(MT) fields of general anisotropic medium,we first obtained the second order partial differential equations for some anisotropic media with special conductivity(e.g.diagonal anisotropy,transverse anisotropy,azimuthal anisotropy,etc.) by simplifying the electrical conductivity tensor of anisotropic medium.And then we obtained the analytic solutions to MT fields for the case of transverse and azimuthal anisotropy through converting the conductivity parameter based on that of diagonal anisotropy.We further discussed the influence of the selection of integral limit and step length on precision in solving the analytic solutions for MT fields of isotropic medium.Finally,we presented the MT responses of two transverse and azimuthal anisotropic media as well as some applications of the analytic solutions to MT fields of anisotropic medium.