Analytical solution of basic equations set of atmospheric motion

On condition that the basic equations set of atmospheric motion possesses the best stability in the smooth function classes, the structure of solution space for local analytical solution is discussed, by which the third-class initial value problem with typicality and application is analyzed. The calculational method and concrete expressions of analytical solution about the well-posed initial value problem of the third kind are given in the analytic function classes. Near an appointed point, the relevant theoretical and computational problems about analytical solution of initial value problem are solved completely in the meaning of local solution. Moreover, for other type of problems for determining solution, the computational method and process of their stable analytical solution can be obtained in a similar way given in this paper.     

Impact of singularity of Navier-Stokes equation upon atmospheric motion equations

Some conclusions about the smooth function classes stability for the basic system of equations of atmospheric motion and instability for Navier-Stokes equation are summarized. On the basis of this, by taking the basic system of equations of atmospheric motion via Boussinesq approximation as example to explain in detail that the instability about some simplified models of the basic system of equations for atmospheric motion is caused by the instability of Navier-Stokes equation, thereby, a principle to guarantee the stability of simplified equation is drawn in simplifying the basic system of equations.     

NONLINEAR DYNAMICAL BIFURCATION AND CHAOTIC MOTION OF SHALLOW CONICAL LATTICE SHELL

The nonlinear dynamical equations of axle symmetry are established by the method of quasi-shells for three-dimensional shallow conical single-layer lattice shells. The compatible equations are given in geometrical nonlinear range. A nonlinear differential equation containing the second and the third order nonlinear items is derived under the boundary conditions of fixed and clamped edges by the method of Galerkin. The problem of bifurcation is discussed by solving the Floquet exponent. In order to study chaotic motion, the equations of free oscillation of a kind of nonlinear dynamics system are solved. Then an exact solution to nonlinear free oscillation of the shallow conical single-layer lattice shell is found as well. The critical conditions of chaotic motion are obtained by solving Melnikov functions, some phase planes are drawn by using digital simulation proving the existence of chaotic motion.     

RESTUDY OF THEORIES FOR ELASTIC SOLIDS WITH MICROSTRUCTURE

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Finite amplitude waves superimposed on pseudoplanar motions for Mooney-Rivlin viscoelastic solids

This paper deals exclusively with finite amplitude motions in viscoelastic materials for which the stress is the sum of a part corresponding to the classical Mooney-Rivlin incompressible isotropic elastic solid and of a dissipative part corresponding to the classical viscous incompressible fluid. Of particular interest is a finite pseudoplanar elliptical motion which is an exact solution of the equations of motion. Superposed on this motion is a finite shearing motion. An explicit exact solution is presented. It is seen that the basic pseudoplanar motion is stable with respect to the finite superposed shearing motion. Particular exact solutions are obtained for the classical neo-Hookean solid and also for the classical Navier-Stokes equations. Finally, it is noted that parallel results may be obtained for a basic pseudoplanar hyperbolic motion.     

Well posed formulations of holonomic mechanical system dynamics and design sensitivity analysis

Abstract

Manifold theoretic ordinary differential equations of motion for holonomic mechanical systems that depend on problem data, or design variables, are shown to be well posed; i.e., they have a unique solution that depends continuously on problem data. It is proved that these differential equations are equivalent to the d’Alembert variational formulation and the index 3 Lagrange multiplier formulation of differential-algebraic equations of motion, which are also shown to be well posed. These results provide a foundation for dynamic system design sensitivity analysis, which requires differentiability of solutions of the equations of motion with respect to design variables.     

Preferential Flow-Paths Detection for Heterogeneous Reservoirs Using a New Renormalization Technique

We have devised a renormalization scheme which allows very fast determination of preferential flow-paths and of up-scaled permeabilities of 2D heterogeneous porous media. In the case of 2D log-normal and isotropically distributed permeability-fields, the resulting equivalent permeabilities are very close to the geometric mean, which is in good agreement with a rigorous result of Matheron. It is also found to work well for geostatistically anisotropic media when comparing the resulting equivalent permeabilities with a direct solution of the finite-difference equations. The method works exactly as King's does, although the renormalization scheme was modified to obtain tensorial equivalent permeabilities using periodic boundary conditions for the pressure gradient. To obtain an estimation of the local fluxes, the basic idea is that if at each renormalization iteration all the intermediate renormalized permeabilities are stored in memory, we are able to compute -- ad reversum -- an approximation of the small-scale flux map under a given macroscopic pressure gradient. The method is very rapid as it involves a number of calculations that vary linearly with the number of elementary grid blocks. In this sense, the renormalization algorithm can be viewed as a rapid approximate pressure solver. The exact reference flow-rate map (for the finite-difference algorithm) was computed using a classical linear system inversion. It can be shown that the preferential flow paths are well detected by the approximate method, although errors may occur in the local flow direction.     

RESTUDY OF COUPLED FIELD THEORIES FOR MICROPOLAR CONTINUA (Ⅱ)-THERMOPIEZOELECTRICITY AND MAGNETOTHERMOELASTICITY

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｏｒｙｏｆｍｉｃｒｏｐｏｌｏａｒｔｈｅｒｍｏｅｌａｓｔｉｃｉｔｙｐｒｅｓｅｎｔｅｄｂｙＷ .Ｎｏｗａｃｋｉｉｓｒｅｓｔｕｄｉｅｄｉｎｏｕｒｐａｐｅｒ[1].Ｔｈｉｓｐａｐｅｒｉｓａｄｉｒｅｃｔｃｏｎｔｉｎｕａｔｉｏｎｏｆｒｅｆｅｒｅｎｃｅ [1 ] .Ｔｈｅｐｒｏｂｌｅｍｓｏｃｃｕｒｒｉｎｇｉｎｔｈｅｔｈｅｏｒｉｅｓｏｆｔｈｅｒｍｏｐｉｅｚｏｅｌｅｃｔｒｉｃｉｔｙａｎｄｍａｇｎｅｔｏｔｈｅｒｍｏｅｌａｓｔｉｃｉｔｙｆｏｒｍｉｃｒｏｐｏｌａｒｃｏｎｔｉｎｕａａｒｅｓｉｍｉｌａｒｔ…     

Dynamic buckling of a beam with transverse constraints

A nonlinear dynamic system with continuously distributed mass is studied using several approaches: experimentally, numerically as well as analytically. The nonlinearity of the system consists of geometrical constraints imposed on the motion. It is harmonically loaded and it is demonstrated that for certain choices of the loading parameters, periodic, quasi-periodic or chaotic behaviour may occur depending on the initial conditions. An important issue is to investigate the number of degrees of freedom needed in order to analytically model the system accurately enough that the important characteristics of the motion are retained in the solution. It is found that the impact conditions at the constraints are of crucial importance and a new approach is proposed for modelling of the impacts. The method is based on the fact that the free motion can be approximated with quite a few degrees of freedom, while at impact all the infinite number of degrees of freedom are considered.     

具有可积微分约束的力学系统的Lie对称性

研究具有可积微分约束的力学系统的Ｌｉｅ对称性与守恒量。采用两种方法：一是用不可积微分约束系统的方法;另一是用积分后降阶系统的方法,研究两种方法之间的关系。     

刚性椭球对固定面的三维摩擦碰撞

讨论刚性椭球对固定面的三维摩擦碰撞．以法向冲量为自变量，建立碰撞过程中接触点切向速度的微分方程．利用相平面的奇点理论对碰撞过程中切向滑动的全局性态作定性分析，并导出切向滑动的解析积分．     

Error estimate of the modified point-mass trajectory model of an artillery shell

The article deals with the motion of an axially symmetric spinning artillery shell in the gravity field under the action of the system of aerodynamic forces and moments adopted in ballistics. As the starting point, the system of differential equations of motion of the shell is taken, which is obtained from the original “accurate” system by its linearization in the variables describing the angular motion of the symmetry axis and by additional linearization in the angle between the velocity vector of the center of mass and the vertical plane (l-system). This article examines the system of differential equations of the translational motion and axial rotation of the shell which describes its modified point-mass trajectory model as applied to l-system (m-system). By small parameter methods, an estimate is obtained for the difference of the solution of l-system with given initial data and the solution of m-system with the same initial data for the variables of translational motion and axial rotation. This analytical evaluation is built in such a way that it corresponds with certain numerical estimates for components of the translational motion and axial rotation. It is observed that, under accepted assumptions, m-system and l-system determine the translational motion of the shell with the same order of the error as compared to the original “accurate” nonlinear system of equations of motion of the shell. But m-system does not contain rapidly oscillating variables describing the angular motion of the symmetry axis, and so its numerical integration requires tens of times less computational resources than the numerical integration of l-system. Numerical simulation data are represented.     

Nonlinear resonance of a subsatellite on a short constant tether

The nonlinear resonant behavior of a subsatellite on a short constant tether during station-keeping phase is investigated in this paper. The nonlinear dynamic equations of in-plane motion of the system are derived based on Kane’s method first. Then an approach of multiple scales expressed in matrix form is employed in solving the simplified nonlinear system of cubic nonlinearity near its local equilibrium position. Analysis shows that there exists a three-to-one resonance in such a nonlinear system with two degrees of freedom. Afterward, the approximate solution up to third order determined analytically by the Weierstrass elliptic function is obtained and the comparison between the approximate and numerical solutions presented as well. The results show that the approximate solution is coincide well with the numerical solution of original system. The nonlinear resonance of the subsatellite on short tether exhibits coexistent quasiperiodic motions or a quasiperiodic oscillation near local equilibrium position.     

Universal equations of three-dimensional boundary layer theory

We consider a parametric method for investigating three-dimensional laminar motion of an incompressible fluid in a boundary layer on a curved surface. It is found that the problem solution in the general case depends on four series of parameters, constructed from two components of the outer flow velocity and the two Lamé coefficients characterizing the shape of the immersed surface. From the general equations of the three-dimensional boundary layer we obtain a system of two universal equations which do not contain the characteristics of the outer flow. This system may be solved once and for all. As an example we consider the problem of the laminar boundary layer on the walls of an axisymmetric channel in the case of swirling outer flow. For this case we obtain numerical solutions of the system of universal equations in the local two-parameter approximation.     

Nonlinear Dynamics of a Rigid Unbalanced Rotor in Journal Bearings. Part I: Theoretical Analysis

The dynamic behaviour of a rigid rotor supported on plain journal bearings was studied, focusing particular attention on its nonlinear aspects. Under the hypothesis that the motion of the rotor mass center is plane, the rotor has five Lagrangian co-ordinates which are represented by the co-ordinates of the mass center and the three angular co-ordinates needed to express the rotor's rotation with respect to its center of mass. In such conditions, the system is characterised not only by the nonlinearity of the bearings but also by the nonlinearity due to the trigonometric functions of the three assigned angular co-ordinates. However, if two angular co-ordinates have values that are generally quite small because of the small radial clearances in the bearings, the system is de facto linear in these angular co-ordinates. Moreover, if the third angular co-ordinate is assumed to be cyclic [18], the number of degrees of freedom in the system is reduced to four and nonlinearity depends solely on the presence of the journal bearings, whose reactions were predicted with the -film, short bearing model. After writing the equations of motion in this way and determining a numerical routine for a Runge–Kutta integration the most significant aspects of the dynamics of a symmetrical rotor were studied, in the presence of either pure static or pure couple unbalance and also when both types of unbalance were present. Two categories of rotors, whose motion is prevailingly a cylindrical whirl or a conical whirl, were put under investigation.     

Shallow waves in a two-layer vortex fluid under a lid

A mathematical model of the vortex motion of an ideal two-layer fluid in a narrow straight channel is considered. The fluid motion in the Eulerian-Lagrangian coordinate system is described by quasilinear integrodifferential equations. Transformations of a set of the equations of motion which make it possible to apply the general method of studying integrodifferential equations of shallow-water theory, which is based on the generalization of the concepts of characteristics and the hyperbolicity for systems with operator functionals, are found. A characteristic equation is derived and analyzed. The necessary hyperbolicity conditions for a set of equations of motion of flows with a monotone-in-depth velocity profile are formulated. It is shown that the problem of sufficient hyperbolicity conditions is equivalent to the solution of a certain singular integral equation. In addition, the case of a strong jump in density (a heavy fluid in the lower layer and a quite lightweight fluid in the upper layer) is considered. A modeling that results in simplification of the system of equations of motion with its physical meaning preserved is carried out. For this system, the necessary and sufficient hyperbolicity conditions are given. Novosibirsk State University, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 68–80, May–June, 1999.     

On the dynamic behaviour of a flexible beam carrying a moving mass

The behaviour of a system containing a mass traveling on a cantilever beam is considered. The mass is induced to move by an applied force as opposed to the case which has been considered in most literature where the position of the moving mass is assumed to be known and independent of the motion of the beam. Furthermore, the system to be discussed has the unique characteristic that the motions of the mass and the beam are coupled. The mathematical model of the system includes two coupled nonlinear integral/partial differential equations which are impossible to solve analytically and are difficult to solve numerically in their original form. As a remedy, the solution is discretized into space and time functions and the equations of motion are reduced to a set of ordinary differential equations. The shape function is chosen so that it satisfies the boundary conditions of the beam as well as the transient conditions imposed by the traveling mass. This choice of the shape function, which considers the mass-beam interaction, provides an improvement over the conventional method of using a simple cantilever beam mode shapes.The ordinary differential equations of motion using the improved shaped functions, are solved numerically to obtain the dynamic behaviour of the system. The results illustrate the validity of the model, and demonstrate the advantages of the improved model to the un-improved equations.     

Motion equations in redundant coordinates with application to inverse dynamics of constrained mechanical systems

The basis for any model-based control of dynamical systems is a numerically efficient formulation of the motion equations, preferably expressed in terms of a minimal set of independent coordinates. To this end the coordinates of a constrained system are commonly split into a set of dependent and independent ones. The drawback of such coordinate partitioning is that the splitting is not globally valid since an atlas of local charts is required to globally parameterize the configuration space. Therefore different formulations in redundant coordinates have been proposed. They usually involve the inverse of the mass matrix and are computationally rather complex. In this paper an efficient formulation of the motion equations in redundant coordinates is presented for general non-holonomic systems that is valid in any regular configuration. This gives rise to a globally valid system of redundant differential equations. It is tailored for solving the inverse dynamics problem, and an explicit inverse dynamics solution is presented for general full-actuated systems. Moreover, the proposed formulation gives rise to a non-redundant system of motion equations for non-redundantly full-actuated systems that do not exhibit input singularities.     

On the existence of indeterminate solutions to the equations of motion under non-associated flow

Under certain conditions, an indeterminate solution exists to the equations of motion for dynamic elastic–plastic deformation of materials using constitutive laws based on non-associated flow that suggests that an initially unbounded dynamic perturbation in the stress can develop from a quiescent state on the yield surface. The existence of this indeterminate solution has been alleged to discourage use of non-associated flow rules for both dynamic and quasi-static analysis theoretically. It is shown in this paper that the indeterminate solution that may solve the equations of motion is intrinsically dynamic, and it determinately goes to zero in the quasi-static limit regardless of other indeterminate parameters. Consequently, the existence of this unstable dynamic solution has no impact on stability and use of non-associated flow rules for analysis of the quasi-static problem. More importantly, for dynamic applications, it is also shown that the indeterminate solution solves the equations of motion only if critical restrictions are applied to the constitutive equations such that the effective modulus during loading is constant and the direction of the perturbation is unidirectional over a finite time interval. It is shown that common components of the constitutive laws used in metal forming and deformation analysis are inconsistent with these restrictions. So, these common models can be generalized to include non-associated flow for analysis of the dynamic problem without concern that the solution will become indeterminate.