A model kinetic equation for a gas with rotational degrees of freedom

A model kinetic equation for a gas with rotational degrees of freedom is obtained. By averaging of the distribution function over quantities corresponding to the rotational degrees of freedom this equation is reduced to a closed system of two kinetic equations, each of which is analogous to the kinetic equation of a monatomic gas.     

Rotational relaxation of nitrogen in a viscous shock layer at low reynolds numbers

The viscous shock-layer model is used to examine relaxation of rotational degrees of freedom of molecular nitrogen in flow of a rarefied gas near the stagnation flow line around a sphere. It is shown that in the strongly smeared shock-wave region the rotational degrees of freedom can exhibit substantial nonequilibrium, leading to the increase of temperature and an increase of shock-layer thickness as compared with the equilibrium values. The influence of rotational relaxation on the shock-wave structure is discussed, and boundaries are found for the flow regions when rotational relaxation plays on important role,A comparison is made between the results of numerical calculations and experimentally obtained density profiles available in the literature near the stagnation line in flow of a rarefied gas over a sphere [1, 2]. Quite good agreement is obtained between the results of the calculation and experimental data over a wide range of Reynolds numbers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 172–175, July–August, 1977.     

刚架结构频域子结构法的实验研究

以空间刚架为模型,对Jetmundsen频域子结构综合法进行了探索性实验,并研究了子结构的界面自由度完整性对最后综合结果的影响。实验中,频响函数矩阵中与线自由度响应和集中力输入相关的项通过加速度计与力传感器直接测量,而与转角自由度响应或纯力矩输入相关的项通过一种结合转角自由度间接测试和模态分析技术的方法获取。实验结果显示,实验子结构综合法能够有效地预报结构的动力学性能,而转角自由度的引入能在一定程度上改善预报质量。     

Direct Shocks in a Vibrationally Nonequilibrium Gas

Direct shocks in flows of a high-temperature diatomic gas with rotational and vibrational degrees of freedom are considered. Gas-dynamic parameters and populations of molecular vibrational levels behind a shock are studied for the case of disturbance of vibrational equilibrium in an incident flow.     

Theory of rotational relaxation in a diatomic gas

Journal of Applied Mechanics and Technical Physics - The three-dimensional problem of energy exchange between translational and rotational degrees of freedom in collisions of diatomic molecules...     

Investigation of the Nitrogen Shock Wave Structure on the Basis of Trajectory Calculations of the Molecular Interaction

The shock wave structure in a diatomic gas is investigated using the direct statistical simulation (Monte-Carlo) method. The energy exchange between translational and rotational degrees of freedom (TR-exchange) is calculated by solving the dynamic problem of the interaction between rigid-rotator molecules within the framework of classical mechanics. The density profiles calculated are compared with the experimental data and on this basis the nitrogen rotational relaxation time is estimated. The possibility of using simplified intermolecular interaction models, namely, the variable-diameter sphere model employed together with a phenomenological consideration of the TR-exchange, is studied. Gasdynamic parameter profiles in the shock wave are analyzed. Simple approximations of the velocity gradient and translational and rotational temperature profiles are obtained on the basis of a parametric calculation of the shock wave structure. This makes it possible approximately to describe the gasdynamic parameter profiles in terms of elementary functions.     

Slip and Macroparameter Jumps of a Polyatomic Gas with Rotational Degrees of Freedom on a Weakly Curved Spherical Surface

The problem of the slip of a temperature-inhomogeneous polyatomic gas along a spherical surface of small curvature is solved. The solution is obtained using the half-space moment method on the basis of a previously proposed model kinetic equation which takes into account the rotational degrees of freedom of the polyatomic gas. Both the first- and second-order (in the Knudsen number) slip coefficients and the polyatomic gas macroparameter jump coefficients on the phase interface are obtained. These coefficients are given as functions of the tangential momentum accommodation coefficients, the translational and rotational energy accommodation coefficients, and the Prandtl number. The kinetic coefficients are calculated for certain polyatomic gases.     

Sound absorption in a shock wave

It was shown in [1–4] that the reflection of a sound wave or its transmission through a shock front should be accompanied by attenuation or intensification of the wave is regarded as a discontinuity. In accordance with current representations [5, 6], a shock wave includes a viscous shock and a lengthy relaxation zone. Equilibrium is established with respect to translational and rotational degrees of freedom in the viscous shock and with respect to internal degrees of freedom in the relaxation zone. The result of the interaction of the shock and sound waves is determined by the relationship between the length of the sound wave and the width of the shock wave.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 90–94, May–June, 1986.     

Order and chaos in a three dimensional galaxy model

We explore the orbital dynamics of a realistic three dimensional model describing the properties of a disk galaxy with a spherically symmetric central dense nucleus and a triaxial dark matter halo component. Regions of phase space with regular and chaotic motion are identified depending on the parameter values for triaxiality of the dark matter halo and for breaking the rotational symmetry. The four dimensional Poincaré map of the three degrees of freedom system is analyzed by a study of its restriction to various two dimensional invariant subsets of its domain.     

基于共旋列式的空间梁的稳定性分析

基于独立于单元的共旋列式(EICR),将一种几何线性的无剪切锁死的Timoshenko梁单元扩展用于空间梁结构的几何非线性分析。考虑到三维分析中发生大转动时转动顺序的不可交换性,也即转动自由度不能作为向量采用加法规则更新,采用了四元变量来存储和更新转动自由度,使得共旋列式适用于位移任意大和转动任意大但应变很小的几何非线性分析。同时改进了Riks弧长法使之适用于带有大转动的三维几何非线性分析。给出了几个数值算例,结果表明本文方法具有较高的计算精度和效率。     

Dynamic response of the spherical pendulum subjected to horizontal Lissajous excitation

Nonlinear Dynamics - This paper examines the oscillations of a spherical pendulum with horizontal Lissajous excitation. The pendulum has two degrees of freedom: a rotational angle defined in the...     

基于参数变分原理的Cosserat连续体弹塑性分析

基于参数变分原理,提出了Cosserat模型弹塑性计算的算法,给出了基于Cosserat理论的参数最小势能原理,基于所提出的变分方程,建立了Cosserat理论弹塑性分析的参数二次规划模型,进一步将算法应用于平面应变软化问题计算中,获得的结果具有良好的非网格依赖性.     

PARAMETRIC VARIATIONAL PRINCIPLE BASED ELASTIC-PLASTIC ANALYSIS OF COSSERAT CONTINUUM

A new algorithm is developed based on the parametric variational principle for elastic-plastic analysis of Cosserat continuum. The governing equations of the classic elastic-plastic problem are regularized by adding rotational degrees of freedom to the conventional translational degrees of freedom in conventional continuum mechanics. The parametric potential energy principle of the Cosserat theory is developed, from which the finite element formulation of the Cosserat theory and the corresponding parametric quadratic programming model are constructed. Strain localization problems are computed and the mesh independent results are obtained.     

Digital volume correlation including rotational degrees of freedom during minimization

Digital volume correlation is a new experimental technique that allows the measurement of the full-field strain tensor in three dimensions. We describe the addition of rotational degrees of freedom into the minimization problem for digital volume correlation in order to improve the overall performance of the strain measurement. A parameterization of rotations that is particularly suited to the minimization problem is presented, based on the angle-axis representation of finite rotations. The partial derivative of both a normalized cross-correlation coefficient and the sum-of-squares correlation coefficient are derived for use with gradient-based minimization algorithms. The addition of rotation is shown to greatly reduce the measurement error when even small amounts of rigid body rotation are present in an artificially rotated test volume. In an aluminum foam sample loaded in compression, including rotational degrees of freedom produced smoother contours of minimum principal strain. Renderings of the aluminum foam architecture in areas of low, medium and high rotation showed material deformation pattern in detail.     

Analytical solution of the problem of sphere rotation in a rarefied molecular gas

A problem of sphere rotation in a rarefied molecular gas is solved in an isothermal approximation. The particle velocity profile in the rarefied molecular gas entrained by the rotating sphere is obtained with a second-order correction in terms of the Knudsen number. For a rarefied molecular gas, in contrast to a monatomic gas, the particle velocity is demonstrated to depend substantially on the Prandtl number if rotational degrees of freedom of molecules are taken into account.     

带转动自由度的内参型非协调元研究

非协调元虽然破坏了单元间位移的连续性，却能很好地反映弯曲类变形，然而在不增加单元结点自由度的情况下，非协调元的计算精度总是滞留在某一水平，无法得到较大改变。基于修正后的位移型Reissner泛函中引入独立转动场的变分原理，采用连续介质力学中的转动自由度的定义，转动场采用结点真实转角来插值，结合平面四结点单元讨论了有效附加非协调位移的合理形式，引入了适用于任何四边形单元的非协调位移函数，从而建立了一种带转动自由度的平面四结点内参型非协调元模型。本文单元能通过分片检验，并易于与带转动自由度的梁单元相容．教值算例表明具有较高的计算精度。     

Spherical flow of a diatomic gas from an evaporating drop

The spherical expansion of gas from an evaporating drop is investigated on the basis of the numerical solution to a model kinetic equation for a gas with rotational degrees of freedom. Examples considered are the stationary evaporation of a drop with given temperature into the vacuum and evaporation of a drop into a gas-filled space under the condition of an energy balance on the drop surface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 184–187, July–August, 1980.     

Influence of internal degrees of freedom on flow of a rarefied gas over a flat plate

Flow of a rarefied gas over a flat plate has been investigated numerically by a number of authors, using both the kinetic model equations (e.g., 1, 2]) and the Boltzmann equation [3, 6], In most cases a solution was found for a monatomic gas. The appreciable influence of the molecule structure on local and total aerodynamic characteristics and on the flow field over a flat plate at small angles of attack was noted in [1, 5, 7], where the authors examined various models for the rotational molecular degrees of freedom. In the present paper a two-point repulsion center model with constant collision cross section is used to investigate the influence of internal degrees of freedom of the molecule in flow over a plate, positioned parallel to (angle of attack = 0), and transverse to ( = 90 °) a rarefied gas stream. The data are compared with those calculated for a monatomic gas and from experimental results.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 151–156, November–December, 1978.     

Modelling and analysis of a single gimbal gyroscopic energy harvester

This work investigates the non-linear dynamics of a single gimbal gyroscopic energy harvester, excited by a harmonic moment about 1 and 2 axes simultaneously. The governing equations of motion are derived and a numerical model is developed to analyse the forced system response in all three rotational degrees of freedom. Simulations showing the effect of the harmonic forcing frequency and the gyroscopic damping are presented. The results identify the characteristic motion responses and available power of a single gimbal gyroscopic energy harvester including the development of the non-linear responses.