旋转充液腔体的全局稳定性及其受扰运动的定性分析

本文是[1]的继续.在本文中,我们研究了旋转充液腔体定常解的分布情况,确定了每一定常解的稳定性并给出了相应的稳定与不稳定区域,此外,对旋转充液腔体的受扰运动作出了全面的定性分析.     

轴对称充液腔体旋转的定态解及其稳定性

本文根据势能的极值条件,讨论了轴对称充液腔体绕定轴旋转的各种可能的平衡态,证明了充满粘性液体的腔体在倒立情形和下悬情形中都只存在绕铅垂的对称轴整体旋转形式的稳定终态解,并且应用关于连续系统的Лядунов直接方法研究了这种旋转状态在大扰动下的稳定性.本文还描述了充液腔体倒立旋转运动和旋转的球碗中小球的运动之间有趣的类比关系.有关结果为长期稳定性的真实性提供了一种理论依据.     

关于充液腔体旋转运动稳定问题的变分原理及其应用

对于任意形状、充满粘性液体的腔体绕惯性主轴整体旋转的一般情形,导得小扰动运动的一组线性微分一积分方程组.在稳定理论基础上发展的一系列方法都处理不了这个问题.本文应用文献[1]中的方法导得这个问题变分原理的一般形式,并得到一系列稳定判据,以前不少作者的结果均是本文的特例^[3-6]。本文的变分方法在充液腔体运动稳定性问题中具有较广泛的应用,对于线性非自伴本征值问题原则上都能应用。     

平面上旋转充液腔体稳定性问题的谱分析及状态空间算子的势理论

本文对充满均匀粘性液体的旋转对称腔体在光滑水平面上的平衡旋转态附近的振动模,导出本征方程?φ=0,对算子?进行谱分析,给出各模及总体的稳定性结论。本文还将Hilbert空间的算子势论推广到充液腔的复空间,给出与?相关的势泛函,从而获得二条泛函微分定理。本文方法比伴随变分更具普遍性。     

旋转充液腔体的非线性稳定理论及其对Columbus问题的应用

对于任意形状、充满粘性液体的腔体绕惯性主轴整体旋转的一般情形,本文首先导出大扰动运动方程组,然后按照直接方法建立了弱非线性稳定理论,并应用于Columbus问题,得到和实验完全一致的结论。     

低重环境下液体非线性晃动的稳态响应

在俯仰激励作用下,圆柱贮箱中液体晃动存在平面运动、旋转运动和平面运动中的旋转运动等,而这些运动的稳定、不稳定区间的分界线与贮箱的半径、充液深度、重力强度、表面张力系数和晃动阻尼等基本系统参数有关．据此,首先建立了液体非线性晃动的微分方程组,并借助变分原理建立了液体压力体积分形式的Lagrange函数;然后将速度势函数在自由液面处作波高函数的级数展开,通过变分从而导出自由液面运动学和动力学边界条件非线性方程组;最后用多尺度法求解非线性方程组,就重力强度对圆柱形贮箱中液体非线性晃动的全局稳态响应的影响进行了详细的理论分析,并发现系统软硬特性的变化、跳跃和滞后等非线性现象．     

Study on Discharge Coefficients of Drain Orifices in Closed Cavities北大核心CSCD

该文利用数值方法模拟了封闭腔体的排液流动,获取了排液孔的流量系数.通过量纲分析法研究了小孔流量系数的主要影响因素,拟合了计算小孔流量系数的经验公式.结果表明：当水头高度小于200 mm时,小孔流量系数随水头高度的增加而减小;当水头高度大于200 mm时,小孔流量系数稳定在0.61附近.不同厚径比的小孔流量系数表现为两种不同的形式：小厚径比的小孔呈现薄孔流动特性,流量系数为0.6左右;大厚径比的小孔呈现厚孔流动特性,流量系数为0.8左右.     

倾斜层中的对流斑图及其临界条件

通过二维流体力学基本方程的数值模拟，探讨了Prandtl(普朗特)数Pr=6.99时，倾斜矩形腔体中的对流斑图和斑图转换的临界条件.根据倾角θ和相对Rayleigh(瑞利)数Ra_r的变化，倾斜矩形腔体中的对流斑图可以分为:单滚动圈对流斑图、充满腔体的多滚动圈对流斑图和过渡阶段的多滚动圈对流斑图.当θ一定时，随着Ra_r的减小，系统由充满腔体的多滚动圈对流斑图过渡到单滚动圈对流斑图.这时，对流振幅A和Nusselt(努塞尔)数Nu随着Ra_r的增加而增加.当Ra_r=9时，随着θ的增加，系统由充满腔体的多滚动圈对流斑图过渡到单滚动圈对流斑图，这时对流振幅A随着θ的增加而减小，Nusselt数Nu随着θ的增加而增加.在θ_c-Ra_r平面上对多滚动圈到单滚动圈对流斑图过渡的模拟结果表明， 在Ra_r=2时, 腔体中没有发现多滚动圈对流斑图.在Ra_r为2.5左右时，腔体中出现多滚动圈到单滚动圈对流斑图的过渡.当多滚动圈到单滚动圈对流斑图过渡的临界倾角θ_c<10°时，θ_{c随着Rar的减小而增加.当θc>10°时，θc随着Rar的增加而增加，在Rar≤5时，θc随着Rar的增加而迅速增加；当Rar>5时，θc随着Rar的增加而缓慢增加.θc与Rarθ的关系与Rar类似}     

充液弹性毛细管低温相变的力学分析

充液弹性毛细管广泛存在于生物体（如毛细血管、植物导管等）和工程领域（如微流控冰阀门、制冷系统热管、MEMS微通道谐振器等）.低温工作环境中,充液弹性毛细管内部的液柱会发生相变并引发冻胀效应,从而导致管壁的变形、损伤乃至断裂.该文建立并求解了考虑温度梯度、界面张力及液体冻胀作用的弹性毛细管平衡方程,分析了液柱低温相变过程中毛细管壁的径向和环向应力,发现管壁应力分布受热毛细弹性数和冻毛细弹性数的影响,且影响大小跟壁厚相关.该研究不仅有助于理解生物体内充液弹性毛细管冻胀失效机制,还可为MEMS微流控芯片的抗冻胀失效设计提供理论指导.     

汽车罐车横向运动液体晃动动力学特性模拟

针对部分充液罐车横向运动时罐体内液体的晃动问题,基于多相流模型,运用VOF法对罐车在高速转弯或紧急避让时罐内液体的晃动动力学特性进行了数值模拟.分析了罐车防波板数量、结构以及充液比、侧向加速度等因素对液体晃动动力学特性的影响.模拟结果表明:防波板横向布置可显著降低罐内液体对罐壁的侧向冲击力,且布置一块较大面积的防波板即可达到较好的防波效果;随着充液比的增大,液体横向晃动减小,并能快速趋于平稳;随着罐车侧向加速度的增大,液体横向晃动增大,进而影响车辆侧向稳定性.     

The stability boundaries of the steady motion of a satellite with a gyroscope

Parts of the asymptotic stability boundaries of the uniform motion of the centre of mass of a system of bodies consisting of an asymmetrical satellite with a three-axis gyroscope in a circular orbit are investigated by the second Lyapunov method. Terms of the Lyapunov function that are higher than the second order are enlisted for the investigation. The sign-definiteness criterion of inhomogeneous forms is employed for the corresponding function. Parts of the stability boundaries in which the steady motion investigated is asymptotically stable are established using the Lyapunov asymptotic stability theorem. Application of the Barbashin and Krasovskii theorems reveals parts of the stability boundaries in which the steady motion is unstable. It is established that the asymptotic stability of the steady motion investigated is solved by expanding the Lyapunov function to sixth-order terms.     

On the orbital stability of a periodic solution of the equations of motion of a Kovalevskaya gyroscope

The sufficient conditions for the orbital stability of a periodic solution of the equations of motion of a Kovalevskaya gyroscope in the case of Bobylev-Steklov integrability are obtained.

It is difficult to expect Lyapunov stability for the unsteady motions of a heavy solid having a fixed point since a dependence of the vibrations frequency on the initial conditions is characteristic for the simplest of them, i.e. periodic motions /1/. Moreover, a rougher property of periodic solutions of the Euler-Poisson equations, orbital stability /2/, is not the subject of special investigations in the dynamics of a solid. The algorithm of the present investigation utilizes the treatment ascribed Zhukovskii /3/ of orbital stability as the Lyapunov stability of motion for a special selection of the variable playing the part of time (see /4/ also) and the Chetayev method /5/ of constructing Lyapunov functions from the first integrals of the equations of perturbed motion. This latter circumstance enables the Chetayev method to be put in one series with the methods used in /1, 4, 6–9/, etc.     

On the orbital stability of pendulum-like oscillations and rotations of a symmetric rigid body with a fixed point

We deal with the problem of orbital stability of planar periodic motions of a dynamically symmetric heavy rigid body with a fixed point. We suppose that the center of mass of the body lies in the equatorial plane of the ellipsoid of inertia. Unperturbed periodic motions are planar pendulum-like oscillations or rotations of the body around a principal axis keeping a fixed horizontal position. Local coordinates are introduced in a neighborhood of the unperturbed periodic motion and equations of the perturbed motion are obtained in Hamiltonian form. Regions of orbital instability are established by means of linear analysis. Outside the above-mentioned regions, nonlinear analysis is performed taking into account terms up to degree 4 in the expansion of the Hamiltonian in a neighborhood of unperturbed motion. The nonlinear problem of orbital stability is reduced to analysis of stability of a fixed point of the symplectic map generated by the equations of the perturbed motion. The coefficients of the symplectic map are determined numerically. Rigorous results on the orbital stability or instability of unperturbed motion are obtained by analyzing these coefficients. The orbital stability is investigated analytically in two limiting cases: small amplitude oscillations and rotations with large angular velocities when a small parameter can be introduced.     

判别函数定号性和变号性的几个定理及其对睡陀螺轴运动条件稳定性的应用

本文给出判别函数定号性和变号性的几个定理并给出其对睡陀螺轴运动条件稳定性的应用.从而得出睡陀螺轴运动条件稳定的必充条件.它与章动角稳定和运动方程全部变量稳定的必充条件吻合.     

On the non-linear stability of a thermo-diffusive fluid mixture in a mixed problem

A linearization principle for a steady motion of a thermo-diffusive fluid mixture in a bounded domain with mixed boundary conditions is proved. Sufficient conditions are given for the coincidence of linear and non-linear stability parameters of the motionless state in a layer. An existence theorem for a steady motion in a layer is also given.     

On the orbital stability of planar periodic motions of a rigid body in the Bobylev-Steklov case

We deal with the problem of orbital stability of pendulum-like periodic motions of a heavy rigid body with a fixed point. We suppose that a mass geometry corresponds to the Bobylev-Steklov case. The stability problem is solved in nonlinear setting. In the case of small amplitude oscillations and rotations with large angular velocities the small parameter can be introduced and the problem can be investigated analytically. In the case of unspecified oscillation amplitude or rotational angular velocity the problem is reduced to analysis of stability of a fixed point of the symplectic map generated by the equations of the perturbed motion. The coefficients of the symplectic map are determined numerically. Rigorous results on the orbital stability or instability of unperturbed motion are obtained by analyzing these coefficients.     

On the Stability of Periodic Motions of an Autonomous Hamiltonian System in a Critical Case of the Fourth-order Resonance

The problem of orbital stability of a periodic motion of an autonomous two-degreeof- freedom Hamiltonian system is studied. The linearized equations of perturbed motion always have two real multipliers equal to one, because of the autonomy and the Hamiltonian structure of the system. The other two multipliers are assumed to be complex conjugate numbers with absolute values equal to one, and the system has no resonances up to third order inclusive, but has a fourth-order resonance. It is believed that this case is the critical one for the resonance, when the solution of the stability problem requires considering terms higher than the fourth degree in the series expansion of the Hamiltonian of the perturbed motion.Using Lyapunov’s methods and KAM theory, sufficient conditions for stability and instability are obtained, which are represented in the form of inequalities depending on the coefficients of series expansion of the Hamiltonian up to the sixth degree inclusive.     

On the stability of periodic trajectories of a planar Birkhoff billiard

The inertial motion of a material point is analyzed in a plane domain bounded by two curves that are coaxial segments of an ellipse. The collisions of the point with the boundary curves are assumed to be absolutely elastic. There exists a periodic motion of the point that is described by a two-link trajectory lying on a straight line segment passed twice within the period. This segment is orthogonal to both boundary curves at its endpoints. The nonlinear problem of stability of this trajectory is analyzed. The stability and instability conditions are obtained for almost all values of two dimensionless parameters of the problem.     

On the motion of a solid in a magnetic field

The problem of the motion of a magnetic solid in a constant uniform magnetic field, taking gyromagnetic effects into account, is considered. The equations of motion are derived, the Hamiltonian structure is studied, and the cases of integrability indicated. Certain classes of stationary motions are studied and their stability examined.

The gyromagnetic effects arise because the electrons have magnetic and mechanical spin moments /1/. The rotation of the body causes it to become magnetized (the Barnett effect) and when a freely suspended body is magnetized, it begins to rotate (the Einsteinde Haas effect). It is found that gyromagnetic phenomena must be taken into account when analysing the motion of gyroscopic precision systems.