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.

Motion equations of hemispherical resonator and analysis of frequency split caused by slight mass non-uniformity

The mass non-uniformity of hemispherical resonator is one of reasons for frequency split,and frequency split can cause gyroscope to drift.Therefore,it is of great significance to analyze the relationship between mass non-uniformity and frequency split,which can provide a theoretical basis for mass balance of imperfect resonator.The starting point of error mechanism analysis for gyroscope is the motion equations of resonator.Firstly,based on the Kirchhoff-Love hypothesis in the elastic thin shell theory,the geometric deformation equations of resonator are deduced.Secondly,the deformation energy equation of resonator is derived according to the vibration mode and relationship between the stress and strain of hemispherical thin shell.Thirdly,the kinetic energy equation of resonator is deduced by the Coriolis theorem.Finally,the motion equations of resonator are established by the Lagrange mechanics principle.The theoretical values of precession factor and natural frequency are calculated by the motion equations,which are substantially consistent with the ones by the finite element method and practical measurement,the errors are within a reasonable range.Simultaneously,the varying trend of natural frequency with respect to the geometrical and physical parameters of resonator by the motion equations is consistent with that by the finite element analysis.The above conclusions prove the correctness and rationality of motion equations.Similarly,the motion equations of resonator with mass non-uniformity are established by the same modeling method in case of ignoring the input angular rate and damping,and the state equations with respect to the velocity and displacement of vibration system are derived,then twonatural frequencies are solved by the characteristic equation.It is concluded that one of reasons for frequency split is the 4 th harmonic of mass non-uniformity,and thus much attention should be paid to minimizing the 4 th harmonic of mass non-uniformity in the course of mass balancing for imperfect resonator.

Advancing Hierarchical Equations of Motion for Efficient Evaluation of Coherent Two-dimensional Spectroscopy

To advance hierarchical equations of motion as a standard theory for quantum dissipative dynamics, we put forward a mixed Heisenberg-SchrSdinger scheme with block-matrix implementation on efficient evaluation of nonlinear optical response function. The new approach is also integrated with optimized hierarchical theory and numerical filtering algorithm. Different configurations of coherent two-dimensional spectroscopy of model excitonic dimer systems are investigated, with focusing on the effects of intermolecular transfer coupling and bi-exciton interaction.

Forward flight of a model butterfly: Simulation by equations of motion coupled with the Navier-Stokes equations

The forward flight of a model butterfly was stud- ied by simulation using the equations of motion coupled with the Navier-Stokes equations. The model butterfly moved under the action of aerodynamic and gravitational forces, where the aerodynamic forces were generated by flapping wings which moved with the body, allowing the body os- cillations of the model butterfly to be simulated. The main results are as follows： （1） The aerodynamic force produced by the wings is approximately perpendicular to the long-axis of body and is much larger in the downstroke than in the up- stroke. In the downstroke the body pitch angle is small and the large aerodynamic force points up and slightly backward, giving the weight-supporting vertical force and a small neg- ative horizontal force, whilst in the upstroke, the body an- gle is large and the relatively small aerodynamic force points forward and slightly downward, giving a positive horizon- tal force which overcomes the body drag and the negative horizontal force generated in the downstroke. （2） Pitching oscillation of the butterfly body plays an equivalent role of the wing-rotation of many other insects. （3） The body-mass- specific power of the model butterfly is 33.3 W/kg, not very different from that of many other insects, e.g., fruitflies and dragonflies.

EQUATIONS OF MOTION FOR NONHOLONOMIC MECHANICAL SYSTEMS WITH UNILATERAL CONSTRAINTS

In this paper, the equations of motion for nonholonomic mechanical system with unilateral holonomic constraints and unilateral nonholonomic constraints are presented, and an example to illustrate the application of the result is given.

EQUATIONS OF MOTION AND BOUNDARY CONDITIONS OF INCREMENTAL RATE TYPE FOR POLAR CONTINUA

The relations between various couple stress tensors and their change rates are derived. The equations of angular momentum and the corresponding boundary conditions of incremental rate type are presented. Thus the equations of motion and the boundary conditions of incremental rate type of Cauchy form, Piola form and Kirchhoff from for polar continua are obtained in combination of these results with those for classical continuum mechanics derived by kuang Zhenbang.

Comparative Analysis of the Generalized Omega Equation and Generalized Vertical Motion Equation

Research on vertical motion in mesoscale systems is an extraordinarily challenging effort.Allowing for fewer assumptions,a new form of generalized vertical motion equation and a generalized Omega equation are derived in the Cartesian coordinate system(nonhydrostatic equilibrium)and the isobaric coordinate system(hydrostatic equilibrium),respectively.The terms on the right-hand side of the equations,which comprise the Q vector,are composed of three factors:dynamic,thermodynamic,and mass.A heavy rain event that occurred from 18 to 19 July 2021 in southern Xinjiang was selected to analyze the characteristics of the diagnostic variable in the generalized vertical motion equation(Qz)and the diagnostic variable in the generalized Omega equation(Qp)using high-resolution model data.The results show that the horizontal distribution of the Qz-vector divergence at 5.5 km is roughly similar to the distribution of the Qp-vector divergence at 500 hPa,and that both relate well to the composite radar reflectivity,vertical motion,and hourly accumulated precipitation.The Qz-vector divergence is more effective in indicating weak precipitation.In vertical cross sections,regions with alternating positive and negative large values that match the precipitation are mainly concentrated in the middle levels for both forms of Q vectors.The temporal evolutions of vertically integrated Qz-vector divergence and Qp-vector divergence are generally similar.Both perform better than the classical quasigeostrophic Q vector and nongeostrophic Q vector in indicating the development of the precipitation system.

A FIELD METHOD FOR INTEGRATING THE EQUATIONS OF MOTION OF NONHOLONOMIC CONTROLLABLE SYSTEMS

This paper presents a field method for integrating the equations of motion of nonholonomic controllable systems. An example is given to illustrate the application of the method.

ONE TYPE OF INTEGRALS FOR THE EQUATIONS OF MOTION OF HIGHER-ORDER NONHOLONOMIC SYSTEMS

This paper presents one type of integrals and its condition of existence for the equations of motion of higher-order nonholonomic systems, including l-order integral (generalized energy integral), 2-order integral and p-order integral (p>2)All of these integrals can be constructed by the Lagrangian function of the system. Two examples are given to illustrate the application of the suggested method.

Mixed Quantum Classical Reaction Rates based on the Phase Space Formulation of the Hierarchical Equations of Motion

In a previous work[J.Chem.Phys.140,174105(2014)],we have shown that a mixed quantum classical(MQC)rate theory can be derived to investigate the quantum tunneling effects in the proton transfer reactions.However,the method is based on the high temperature approximation of the hierarchical equation of motion(HEOM)with the Debye-Drude spectral density,and results in a multistate Zusman type of equation.We now extend this theory to include quantum effects of the bath degrees of freedom.By writing the full HEOM into a multidimensional partial differential equation in phase space,we can define a new reaction coordinate,and the previous method can be generalized to the full quantum regime.The validity of the new method is demonstrated by using numerical examples,including the spin-Boson model,and the double well model for proton transfer reaction.The new method is found to resolve some key problems of the previous theory based on high temperature approximation,including possible numerical instability in long time simulation and wrong rate constant at low temperatures.

Adiabatic Terminator for Fermionic Hierarchical Equations of Motion

The hierarchical equation of motion method has become one of the most popular numerical methods for describing the dissipative dynamics of open quantum systems linearly coupled to environment.However,its applications to systems with strong electron correlation are largely restrained by the computational cost,which is mainly caused by the high truncation tier L required to accurately characterize the strong correlation effect.In this work,we develop an adiabatic terminator by decoupling the principal dissipation mode with the fastest dissipation rate from the slower ones.The adiabatic terminator leads to substantially enhanced convergence with respect to L as demonstrated by the numerical tests carried out on a single impurity Anderson model.Moreover,the adiabatic terminator alleviates the numerical instability problems in the long-time dissipative dynamics.

Higher-order differential variational principle and differential equations of motion for mechanical systems in event space

In this paper we study the higher-order differential variational principle and differential equations of motion for mechanical systems in event space. Based on the higher-order d＇Alembert principle of the system, the higher-order velocity energy and the higher-order acceleration energy of the system in event space are defined, the higher-order d＇Alembert- Lagrange principle of the system in event space is established, and the parametric forms of Euler-Lagrange, Nielsen and Appell for this principle are given. Finally, the higher-order differential equations of motion for holonomic systems in event space are obtained.

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 typ- icality 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 ofproblems for determining solution, the computational method and process of their stable analytical solution can be obtained in a similar way given in this paper.

THE EQUATIONS OF MOTION OF A MECHANICAL SYSTEM IN MATRIX FORM

This work recommends methods of construction of equations of motion of mechanical systems in matrix form. The use of a matrix form allows one to write an equation of dynamics in compact form, convenient for the in vestigation of multidimensional mechanical systems with the help of computers. Use is made of different methods of constructing equations of motion, based on the basic laws of dynamics as well as on the principles of D Alambert-Le range, Hamilton-Ostrogradski and Gauss.

The Application of Thermomechanical Dynamics (TMD) to the Analysis of Nonequilibrium Irreversible Motion and a Low-Temperature Stirling Engine

We applied the method of Thermomechanical Dynamics (TMD) to a low-temperature Stirling engine, and the dissipative equation of motion and time-evolving physical quantities are self-consistently calculated for the first time in this field. The thermomechanical states of the heat engine are in Nonequilibrium Irreversible States (NISs), and time-dependent thermodynamic work W(t), internal energy E(t), energy dissipation or entropy Qd(t), and temperature T(t), are precisely studied and computed in TMD. We also introduced the new formalism, Q(t)-picture of thermodynamic heat-energy flows, for consistent analyses of NISs. Thermal flows in a long-time uniform heat flow and in a short-time heat flow are numerically studied as examples. In addition to the analysis of time-dependent physical quantities, the TMD analysis suggests that the concept of force and acceleration in Newtonian mechanics should be modified. The acceleration is defined as a continuously differentiable function of Class C2 in Newtonian mechanics, but the thermomechanical dynamics demands piecewise continuity for acceleration and thermal force, required from physical reasons caused by frictional variations and thermal fluctuations. The acceleration has no direct physical meaning associated with force in TMD. The physical implications are fundamental for the concept of the macroscopic phenomena in NISs composed of systems in thermal and mechanical motion.

NONLINEAR MOTION MECHANICS MODEL OF CURVED BLOOD VESSEL

A geometric model of curved blood vessels is established based on some reasonable hypotheses; the nonlinear motion mechanics model of the curved blood vessel is established according to basic mechanics laws. This model includes much more physiological factors. It couples the interaction of blood flow with mechanical factors such as the displacement, deformation, strain and stress etc. of the curved blood vessel. It is of great importance for investigating the circulation rules of the cardiovascular system and the nonlinear pulse wave propagation in curved blood vessels.

ON EQUATION OF DISCRETE SOLID PARTICLES' MOTION IN ARBITRARY FLOW FIELD AND ITS PROPERTIES

The forces on rigid particles moving in relation to fluid having been studied and the equation of modifications of their expressions under different flow conditions discussed, a general form of equation for discrete particles' motion in arbitrary flow field is obtained. The mathematical features of the linear form of the equation are clarified and analytical solution of the linearized equation is gotten by means of Laplace transform. According to above theoretical results, the effects of particles' properties on its motion in several typical flow field are studied, with some meaningful conclusions being reached.

Potential method of integration for solving the equations of mechanical systems

This paper is intended to apply a potential method of integration to solving the equations of holonomic and nonholonomic systems. For a holonomic system, the differential equations of motion can be written as a system of differential equations of first order and its fundamental partial differential equation is solved by using the potential method of integration. For a nonholonomic system, the equations of the corresponding holonomic system are solved by using the method and then the restriction of the nonholonomic constraints on the initial conditions of motion is added.

Characteristics of horizontal ground motion measures along principal directions

Ground motion records are often used to develop ground motion prediction equations （GMPEs） for a randomly oriented horizontal component, and to assess the principal directions of ground motions based on the Arias intensity tensor or the orientation of the major response axis. The former is needed for seismic hazard assessment, whereas the latter can be important for assessing structural responses under multi-directional excitations. However, a comprehensive investigation of the pseudo-spectral acceleration （PSA） and of GMPEs conditioned on different axes is currently lacking. This study investigates the principal directions of strong ground motions and their relation to the orientation of the major response axis, statistics of the PSA along the principal directions on the horizontal plane, and correlation of the PSA along the principal directions on the horizontal plane. For these, three sets of strong ground motion records, including intraplate California earthquakes, inslab Mexican earthquakes, and interface Mexican earthquakes, are used. The results indicate that one of the principal directions could be considered as quasi-vertical. By focusing on seismic excitations on the horizontal plane, the statistics of the angles between the major response axis and the major principal axis are obtained; GMPEs along the principal axes are provided and compared with those obtained for a randomly oriented horizontal component; and statistical analysis of residuals associated with GMPEs along the principal directions is carried out.

MOTION VELOCITY SMOOTH LINK IN HIGH SPEED MACHINING

To deal with over-shooting and gouging in high speed machining, a novel approach for velocity smooth link is proposed. Considering discrete tool path, cubic spline curve fitting is used to find dangerous points, and according to spatial geometric properties of tool path and the kinematics theory, maximum optimal velocities at dangerous points are obtained. Based on method of velocity control characteristics stored in control system, a fast algorithm for velocity smooth link is analyzed and formulated. On-line implementation results show that the proposed approach makes velocity changing more smoothly compared with traditional velocity control methods and improves productivity greatly.