Thomas rotation and Foucault pendulum under a simple unifying geometrical point of view

In this paper we show that there is a complete parallelism between the Foucault pendulum and the Thomas rotation phenomena by using the concept of parallel transport in a surface. In the case of the Foucault pendulum the surface is the ordinary sphere corresponding to the Earth sphere, whereas in the case of the Thomas rotation the surface is the pseudosphere corresponding to the space of relativistic velocities. Moreover, in both cases we use a simple method that reduces the problem to the parallel transport in a conical surface, and so, to the plane.     

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研究了受空气阻尼影响下傅科摆的运动规律,采用数值和解析的方法对其运动轨迹进行 了模拟和讨论,并与无空气阻力时傅科摆的运动进行了比较.     

MOTION OF THE DAMPED FOUCAULT PENDULUM

本文建立运动方程,得到了阻尼傅科摆的解析解,并对解的性质进行了一些探究.文中建立的模型是在理想单摆的基础上增加了科里奥利力和与速度成正比的阻尼力.     

Birkhoff力学的研究进展

Birkhoff力学是Hamilton力学的一个自然发展,是分析力学发展的一个新阶段,它广泛应用于力学、物理学和工程.本文总结Birkhoff力学的形成和发展,特别是近二十年所取得的成就.首先,从Birkhoff的《动力系统》中的有关段落开始,叙述Birkhoff力学的起源.其次,叙述这个力学的基本原理——Pfaff-Birkhoff原理以及这个力学的基本方程——Birkhoff方程的形成和发展.第三,简述Birkhoff力学的一些专门问题,包括约束Birkhoff系统,Birkhoff方程的积分方法,Birkhoff动力学逆问题,Birkhoff方程的运动稳定性,Birkhoff系统的几何方法,Birkhoff系统的全局分析等.最后,对Birkhoff力学的未来研究提出一些建议.      

String Pendulum Dynamics. Control in the Single-Axis Azimuth-Free Gyro Mode

An implementation of a generalized Foucault pendulum, where the basic mechanical elements are a massive solid ball and two weightless strings, is considered. A two-dimensional electromechanical model of the gyro is constructed. Channels for inertial data acquisition and for vibration control are formed. The stability of the reference manifold for the chosen operation mode is shown.     

Self-similar solutions in the problem of generalized vortex diffusion

The analysis of the group properties and the search for self-similar solutions in problems of mathematical physics and continuum mechanics have always been of interest, both theoretical and applied [1–3]. Self-similar solutions of parabolic problems that depend only on a variable of the type η = x/√t are classical fundamental solutions of the one-dimensional linear and nonlinear heat conduction equations describing numerous physical phenomena with initial discontinuities on the boundary [4]. In this study, the term “generalized vortex diffusion” is introduced in order to unify the different processes in mechanics modeled by these problems. Here, vortex layer diffusion and vortex filament diffusion in a Newtonian fluid [5] can serve as classical hydrodynamic examples. The cases of self-similarity with respect to the variable η are classified for fairly general kinematics of the processes, physical nonlinearities of the medium, and types of boundary conditions at the discontinuity points. The general initial and boundary value problem thus formulated is analyzed in detail for Newtonian and non-Newtonian power-law fluids and a medium similar in behavior to a rigid-ideally plastic body. New self-similar solutions for the shear stress are derived.     

Numerical study on the dynamics of droplet passing through a cylinder obstruction in confined microchannel flow

The droplet dynamics passing through a cylinder obstruction was investigated with direct numerical simulations with FE-FTM (Finite Element-Front Tracking Method). The effect of droplet size and capillary number (Ca) was studied for both Newtonian and viscoelastic fluids. In the case of Newtonian droplet immersed in Newtonian medium, the droplet breakup induced by the geometric hindrance depends on the droplet size. As Ca increases, the short droplets (1.3 times longer than the channel width) break up while passing through the obstruction. However, the breakup does not occur for longer droplets (1.8 times longer than the channel width). When the viscoelastic fluid characterized by the Oldroyd-B model is considered, the Newtonian droplet immersed in viscoelastic medium breaks up into two smaller droplets while passing through the cylinder obstruction with increasing De_m (Deborah number of the medium). We also show that the normal stress difference plays a key role on the droplet breakup and the droplet extension. The normal stress difference is enhanced in the negative wake region due to the droplet flow, which also promotes droplet extension in that region. This numerical study provides information not only on underlying physics of the droplet flows passing through a cylinder obstruction but also on the useful guidelines for microfluidic applications.     

Modern topics and challenges in dynamic fracture

The field of dynamic fracture has been enlivened over the last 5 years or so by a series of remarkable accomplishments in different fields—earthquake science, atomistic (classical and quantum) simulations, novel laboratory experiments, materials modeling, and continuum mechanics. Important concepts either discovered for the first time or elaborated in new ways reveal wider significance. Here the separate streams of the literature of this progress are reviewed comparatively to highlight commonality and contrasts in the mechanics and physics.Much of the value of the new work resides in the new questions it has raised, which suggests profitable areas for research in the next few years and beyond. From the viewpoint of fundamental science, excitement is greatest in the struggle to probe the character of dynamic fracture at the atomic scale, using Newtonian or quantum mechanics as appropriate (a qualifier to be debated!). But lively interest is also directed towards modeling and experimentation at macroscales, including the geological, where the science of fracture is pulled at once by fundamental issues, such as the curious effects of friction, and the structural, where dynamic effects are essential to proper design or certification and even in manufacture.     

DYNAMICS IN NEWTONIAN-RIEMANNIAN SPACE-TIME (Ⅳ)

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曲面物理和力学

回顾了平坦空间中的经典物理和力学, 综述了卷曲空间中物理和力学的三个典型案例, 即生物膜力学、扁壳力学和曲面扩散动力学, 揭示了平坦空间与卷曲空间物理和力学的根本差异. 论文阐明了, 平坦空间的物理和力学只有一个基本微分算子------经典梯度, 而曲面物理和力学必然存在两个基本微分算子------经典梯度和形状梯度.     

Flow of viscoelastic fluids between rotating disks

Few boundary-value problems in fluid mechanics can match the attention that has been accorded to the flow of fluids, Newtonian and non-Newtonian, between parallel rotating disks rotating about a common axis or about distinct axes. An interesting feature which has been recently observed is the existence of solutions that are not axially symmetric even in the case of flow due to the rotation of disks about a common axis. In this article we review the recent efforts that have been expended in the study of both symmetric and asymmetric solutions in the case of both the classical linearly viscous fluid and viscoelastic fluids.The support of the Air Force Office of Scientific Research is gratefully acknowledged.     

DYNAMICS IN NEWTONIAN-RIEMANNIAN SPACE-TIME(Ⅳ)

Lagrangian mechanics in Newtonian-Riemannian space-time and relationship between Lagrangian mechanics and Newtonian mechanics, and between Lagrangian mechanics and Hamiltonian mechanics in N-R space-time are discussed.     

基于扭摆的微冲量测量方法及实验研究

超高微重力水平的卫星平台在空间引力波探测、地球重力场测量中发挥着重要的作用,脉冲微推力器可以帮助微重力卫星实现姿态控制.微冲量是评价脉冲微推力器性能的重要指标之一,常用的基于扭摆的微冲量测量方法有两种,方法一是根据单个冲量元瞬间作用于无阻尼扭摆后,扭摆转动最大角位移计算冲量,方法二是根据高固定频率的连续脉冲作用于有阻尼...     

Parallel domain decomposition method for finite element approximation of 3D steady state non‐Newtonian fluids

We introduce a stabilized finite element method for the 3D non‐Newtonian Navier–Stokes equations and a parallel domain decomposition method for solving the sparse system of nonlinear equations arising from the discretization. Non‐Newtonian flow problems are, generally speaking, more challenging than Newtonian flows because the nonlinearities are not only in the convection term but also in the viscosity term, which depends on the shear rate. Many good iterative methods and preconditioning techniques that work well for the Newtonian flows do not work well for the non‐Newtonian flows. We employ a Galerkin/least squares finite element method, with stabilization parameters adjusted to count the non‐Newtonian effect, to discretize the equations, and the resulting highly nonlinear system of equations is solved by a Newton–Krylov–Schwarz algorithm. In this study, we apply the proposed method to some inelastic power‐law fluid flows through the eccentric annuli with inner cylinder rotation and investigate the robustness of the method with respect to some physical parameters, including the power‐law index and the Reynolds number ratios. We then report the superlinear speedup achieved by the domain decomposition algorithm on a computer with up to 512 processors. Copyright © 2015 John Wiley & Sons, Ltd.     

An Analysis of Journal Orbits for Nonlinear Dynamic Bearing Systems

An analysis of journal centre orbits is presented in this paper based on a non-isothermal non-Newtonian fluid model for dynamically loaded bearing systems. A spectral element approach is used to solve a full set of coupled equations (kinematics and constitutive) governing the flow of the lubricant, and an operator-splitting spectral element technique is used to evaluate the dynamic energy equation. The motion of the journal is calculated on the basis of Newtonian mechanics incorporated with a simple cavitation model. The stability of the journal orbits is investigated under a wide range of the rotation speeds of journal. The unstable orbits arise as a sub-harmonic motion when the journal rotation speed is increased beyond a critical value. The influences of the oscillation speeds of the applied loads on the journal orbits are examined. The numerical simulations demonstrate that both the rotation speed of the journal and the oscillation speed of the applied load play an important role in determining the pattern of the journal orbits. The effects of square-wave and rotating applied loads on the journal orbits are also investigated. Received 22 April 1998 and accepted 26 May 1999     

Rotational modes of motion for an aerodynamic pendulum with a vertical rotation axis

An aerodynamic pendulum placed in a steady horizontal air flow is studied in connection with modeling a wind-receiving element of a vertical-axis wind turbine (VAWT). When modeling the medium effects on the pendulum, the following two approaches are combined: the quasi-static one based on stationary wind tunnel experiments and the unsteady one based on the added mass effects. The existence of stable and unstable rotational modes is analyzed analytically (via the Poincaré-Pontryagin method) and numerically. The dependence on the parameters responsible for the viscous friction at the rotation axis and for the added mass effects is taken into account.     

On the objective stress rate in co-moving coordinate system

The objective stress rate is a rather important problem in mechanics of finite deformation. In this paper, the objective stress rate in co-moving coordinate is derived by applying nonlinear geometric field theory of deformation. Problems, such ax targe extension coupled with rotation, and large shear deformation, are exemplified by using the new formula. Comparing with Jaumann’s stress rate and other formulae presented in current literature, the new result appears to be the reasonable one in co-moving coordinate system.     

Dispersion of linear gravity waves on a viscoelastic fluid in an horizontal canal

A characteristic equation is derived that describes the spatial decay of linear surface gravity waves on Maxwell fluids. Except at small frequencies, the derived dispersion relation is different from the temporal decay dispersion relation which is normally studied within fluid mechanics. The implications for waves on viscous Newtonian fluids using the two different dispersion relations is briefly discussed. The wave number is measured experimentally as function of the frequency in a horizontal canal. Seven Newtonian fluids and four viscoelastic liquids with constant viscosity have been used in the experiments. The spatial decay theory for Newtonian fluids fits well to the experimental data. The model and experiments are used to determine limits for the Maxwell fluid time numbers for the four viscoelastic liquids. As a result of low viscosity it was not possible within this study to obtain these time numbers from oscillatory experiments. Therefore, a comparison of surface gravity wave experiments with theory is applicable as a method to evaluate memory times of low viscosity viscoelastic fluids.     

GENERAL METHOD FOR ANALYZING GEOMETRIC COMPONENTS OF A PLANE SYSTEM

平面体系机动分析是正确求解结构力学后续问题的基础. 综合各种类型题目,明确3 个基本组成规则的具体应用,总结平面体系几何组成分析的一般方法,使分析过程可按照一定的步骤进行,对理解和掌握该部分内容具有指导意义.     

两自由度参数激励摆旋转运动分析和实验

研究具有支撑参数激励摆系统的支撑结构振动对摆旋转的影响,其中支撑结构是受到扭簧约束的刚性悬臂梁,参数激励摆与刚性悬臂梁的悬臂段铰接．首先,通过拉格朗日方程建立了系统两自由度的动力学方程．其次,利用多尺度法对建立的模型进行理论分析,得到悬臂梁的振动与上摆不同运动形式的关系,从而得到上摆不同运动形式下的参数平面分类和悬臂梁在上摆转动时的振动频响．最后,通过建立实验装置,观察理论预测,实验结果验证了理论分析的正确性．实验与理论对照得到,当参数激励频率接近悬臂梁的一阶固有频率时,悬臂梁的振幅变大,会破坏摆的转动稳定性．