Internal resonant interactions of three free surface-waves in a circular cylindrical basin

IntroductionTheproblemofwavesofliquidinabasinhasbeenusedasaphysicalmodelinstudiesoftidalwaves ,seicheinlakes,oceanwaves,sloshingoffluidinacontainer (inindustry) .Itisstillanareaofcurrentlyvigorousresearchduetoitstheoreticalsignificance.McGoldrick[1]provedthattherewasthesecond_orderresonantinteractionsamongcapillarygravitysurface_waves.Hasselmann[2 ]studiedthenonlinearinteractionsofeveryfrequencyspectrumcomponentincontinuousspectrum ,andhederivedaformulathatdescribedtheenergytransformationsamo…     

Explicit criteria of absolute stability for the real second canonical form of control system

In this paper,some explicit criteria of absolute stability for the trivial solution of the real second canonical form of nonlinear control system are given,which include and improve the criteria in paper[1].By applying these criteria to the well-known equation of the longitudinal motion of aircraft,some results are obtained,which include and improve the corresponding results in papers[1,2,3,4].     

The criterions for absolute stability of direct control systems of lurie type

In this paper, we give necessary and sufficient conditions for absolute stability of several classes of direct control systems, and discuss the absolute stability of the first canonical form of control system. The corresponding results in references [3,5,6] and [7] are improved.This paper is completed under the direct guidance and encouragement of professor Liao Xiaoxin and professor Deng Zhong-qi. The writer especially acknowledges their help.     

On the criterion for the absolute stability of the control system

In this paper, firstly we give the criterion for the absolute stability of the second canonical form for the control system, including the equation of the longitudinal motions of a plane as a particular example. The corresponding result in [8], [9] is a particular example given in this paper. Secondly, we give the criteria for the absolute stability of the first canonical form in the usual case and in the critical case. Finally, we give some criteria for the absolute stability of the general form for the direct control system.All the results in this paper merely depend upon the relations between the parameters of the system itself to give an explicit algebraic discriminant.     

Necessary and sufficient criteria for absolute stability of the direct control system

ＮＥＣＥＳＳＡＲＹＡＮＤＳＵＦＦＩＣＩＥＮＴＣＲＩＴＥＲＩＡＦＯＲＡＢＳＯＬＵＴＥＳＴＡＢＩＬＩＴＹＯＦＴＨＥＤＩＲＥＣＴＣＯＮＴＲＯＬＳＹＳＴＥＭＺｈａｎｇＪｉ－ｙｅ（张继业）;ＳｈｕＺｈｏｎｇ－ｚｈｏｕ（舒仲周）（ＤｅｐａｒｔｍｅｎｔｏｆＥｎｇｉ...     

Necessary and sufficient conditions for the absolute stability of discrete type lurie control system

ＮＥＣＥＳＳＡＲＹＡＮＤＳＵＦＦＩＣＩＥＮＴＣＯＮＤＩＴＩＯＮＳＦＯＲＴＨＥＡＢＳＯＬＵＴＥＳＴＡＢＩＬＩＴＹＯＦＤＩＳＣＲＥＴＥＴＹＰＥＬＵＲＩＥＣＯＮＴＲＯＬＳＹＳＴＥＭＺｈａｎｇＪｉｙｅ（张继业）（ＲｅｃｅｉｘｅｄＯｃｔ．５．１９９４Ｃｏｍｍｕ...     

The stability of zero solution of second order system of variable coefficients with four non-linear terms

This paper discusses the following non-linear systems of second order coefficients: This paper is the generalization of works [1] and [2].     

On the absolute stability of a class of indirect control systems

ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒｔｈｅｉｎｄｉｒｅｃｔｃｏｎｔｒｏｌｓｙｓｔｅｍ ｘ =Ａｘ Ｂφ(σ) , σ =ＣＴｘ ρφ(σ) ,(1 )ｗｈｅｒｅｘ∈Ｒｎ,σ ∈Ｒ ,Ａｉｓｎ×ｎｓｔａｂｌｅｍａｔｒｉｘ ,ＢａｎｄＣａｒｅｎｄｉｍｅｎｓｉｏｎａｌｖｅｃｔｏｒｓ,ｔｈｅｓｙｍｂｏｌＴｓｔａｎｄｓｆｏｒｔｒａｎｓｐｏｓｉｔｉｏｎ ,ρｉｓａｃｏｎｓｔａｎｔ,ｆｕｎｃｔｉｏｎ φ(σ) :Ｒ →Ｒｉｓｃｏｎｔｉｎｕｏｕｓａｎｄｓａｔｉｓｆｉｅｓｃｏｎｄｉｔｉｏｎσφ(σ) >0　　 ( σ∈Ｒ ,σ≠ 0 ) . (2 )Ｍａｎｙｓｐｅ…     

Havelock型格林函数振荡项数值积分的稳定性研究

Havelock型格林函数的传播项被积函数是高频振荡且奇异的复变函数,文献[4]引入变量代换获得了一种兼具积分效率和精度的积分方法, 本文研究了该方法的积分稳定性,发现该方法仍存在如下的积分困难: (1) θ=γ时复函数中分母为零引起的计算溢出;(2) θ=π/2是复函数在y及z方向偏导数的无穷间断点;(3) 场点与源点横坐标相同时伪奇异点变为真奇点。针对这些积分困难,采用极限公式计算θ=γ处复函数的值避免计算溢出;在保证积分精度的前提下采用截断方法略去θ=π/2邻近区域的积分消除无穷间断处的奇异;针对(3)采用分区法处理以避开原被积函数的高频振荡,并消除奇异性。伪奇异性存在的条件是场点必须在点源传播波的传播范围内,伪奇异点最多为2个。     

硅微谐振加速度计高精度相位闭环控制系统设计与实现

硅微谐振加速度计因具有小体积优势和高精度潜力,成为硅微惯性传感器研制的热点之一。高精度相位闭环控制系统是决定硅微谐振加速度计精度水平的重要因素。在分析硅微谐振加速度计工作机理的基础上,从闭环控制系统设计的角度,分析了相位闭环控制回路的原理,提出了一种可以消除匀加速误差的高精度三阶无静差相位闭环控制方案。给出了设计思路,研究了环路性能测试方法,讨论了闭环系统相位误差的来源与抑制方法。所设计的闭环回路在0.1 Hz处静态增益为170 dB,启动时间小于20 ms,实测带宽为432 Hz,全温范围内相位闭环回路相差变化0.84°,系统参数满足设计指标。     

A stability in the large investigation for a non-linear second order two-degree-of-freedom system with periodic excitation

An extension to an algorithm due to Simpson has been developed for the analysis of a non-linear second order two-degree-of-freedom system with external periodic excitation. The form of equations considered arises from the study of mechanical systems with a single concentrated weak non-linearity and the method assumes a solution made up of harmonic terms whose amplitudes vary slowly in time. The system considered is such that in the absence of external excitation, it possesses a stable equilibrium point and an unstable limit cycle arising from a sub-critical Hopf bifurcation. When forcing is applied, the stable equilibrium point may then be replaced by a stable periodic attractor, and the limit cycle by an unstable multi-periodic attractor. The method has been applied to the problem of locating these attractors, and if they exist, of finding the stable attractor's basin of attraction in terms of initial conditions. The method reduces the problem from a search in four-dimensional phase space to a search for a boundary in a plane defined by amplitudes a₁ and a₂ in the assumed form of the solution.The method was applied to three non-linear systems in which the non-linearity was due to either a linear spring with a small amount of cubic hardening or a linear spring with freeplay. Agreement was shown to be good in those cases where the non-linearity was weak. However, the method would not be expected to give such accurate results if the non-linear effect was more significant. This was illustrated for a case involving the freeplay non-linearity.     

On the global asymptotic stability of periodic solutions of a class of ecological system

In this paper, we have made reaearches on the mathematical models which have three populations of mutual action: We have obtained the sufficient conditions respectively for the systems(*)and(**)for existence and uniqueness of single positive periodic solutions which are globally asymptotically stable.     

On the asymptotic solutions for a class of second order differential equations with slowly varying coefficients

In this paper we study the asymptotic expansions of the solutions for a class of secondorder ordinary differential equations with slowly varying coefficients.The defect of theknown works on these problems is noted,and the results in[1—4]are improved andextended by means of the modified method of multiple scales.     

New numerical method for volterra integral equation of the second kind in piezoelastic dynamic problems

The elastodynamic problems of piezoelectric hollow cylinders and spheres under radial deformation can be transformed into a second kind Volterra integral equation about a function with respect to time, which greatly simplifies the solving procedure for such elastodynamic problems. Meanwhile, it becomes very important to find a way to solve the second kind Volterra integral equation effectively and quickly. By using an interpolation function to approximate the unknown function, two new recursive formulae were derived, based on which numerical solution can be obtained step by step. The present method can provide accurate numerical results efficiently. It is also very stable for long time calculating.     

Constructions of the general solution for a system of partial differential equations with variable coefficients

Solving partial differential equations has not only theoretical significance,but alsopractical value.In this paper.by the property of conjugate operator.we give a method toconstruct the general solutions of a system of partial differential equations.     

Uniform convergence for the difference scheme in the conservation form of ordinary differential equation with a small parameter

In this paper, we consider a singular perturbation boundary problem for a self-adjoint ordinary differential equaiton. We construct a class of difference schemes with fitted factors, and give the sufficient conditions under which the solution of difference scheme converges uniformly to the solution of differential equation. From this we propose several specific schemes under weaker conditions, and give much higher order of uniform convergence.     

On the existence of periodic solutions for nonlinear system with multiple delays

IntroductionInthispaper,westudyT_periodicsolutionsofthefollowingnonlinearsystemwithmultipledelays x(t) =f(t,x(t) ,x(t-τ1(t) ) ,… ,x(t -τm(t) ) ) ,(1 )wherex(t) ∈C(R ,R) ,fiscontinuous,f(t+T ,·) =f(t,·) ,τi(t) (i=1 ,2 ,… ,m)arecontinuousperiodicfunctionsofperiodT .AlemmaisintroducedfordiscussingtheexistenceofT_periodicsolutionofsystem (1 ) .LetXbeaBanachSpace ,considerthefollowingoperatorequation :Lx =λNx　　 (λ∈ [0 ,1 ] ) ,whereL :DomL∩X→Xisalinearoperator,λ∈ [0 ,1 ]isapa…     

Exact solutions of the equations of motion for an incompressible viscoelastic Maxwell medium

The unsteady plane-parallel motion of a incompressible viscoelastic Maxwell medium with constant relaxation time is considered. The equations of motion of the medium and the rheological relation admit an extended Galilean group. The class of solutions of this system which are partially invariant with respect to the subgroup of the indicated group generated by translation and Galilean translation along one of the coordinate axes is studied. The system does not have invariant solutions, and the set of partially invariant solutions is very narrow. A method for extending the set of exact solutions is proposed which allows finding solutions with a nontrivial dependence of the stress tensor elements on spatial coordinates. Among the solutions obtained by this method, the solutions describing the deformation of a viscoelastic strip with free boundaries is of special interest from a point of view of physics. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 16–23, March–April, 2009.