Estimation of instrumental errors for an inertial navigation system using the Schuler period

The control mode based on using the Schuler period is a special operation mode of the INS-2000 platform inertial navigation system. This mode is used to check the consistency between the calibration parameters of INS sensors and their actual values. Several error equations are proposed for this mode. Some feasible estimation algorithms for INS instrumental errors are tested on model data.     

Finite element simulations of dynamics of multivariant martensitic phase transitions based on Ginzburg–Landau theory

A finite element approach is suggested for the modeling of multivariant stress-induced martensitic phase transitions (PTs) in elastic materials at the nanoscale for the 2-D and 3-D cases, for quasi-static and dynamic formulations. The approach is based on the phase-field theory, which includes the Ginzburg–Landau equations with an advanced thermodynamic potential that captures the main features of macroscopic stress–strain curves. The model consists of a coupled system of the Ginzburg–Landau equations and the static or dynamic elasticity equations, and it describes evolution of distributions of austenite and different martensitic variants in terms of corresponding order parameters. The suggested explicit finite element algorithm allows decoupling of the Ginzburg–Landau and elasticity equations for small time increments. Based on the developed phase-field approach, the simulation of the microstructure evolution for cubic-tetragonal martensitic PT in a NiAl alloy is presented for quasi-statics (i.e., without inertial forces) and dynamic formulations in the 2-D and 3-D cases. The numerical results show the significant influence of inertial effects on microstructure evolution in single- and polycrystalline samples, even for the traditional problem of relaxation of initial perturbations to stationary microstructure.     

A survey of equations of motion in terms of inertial quasi-velocities for serial manipulators

In this work we compare equations of motion using the so-called inertial quasi-velocities. As a result of these velocities we obtain two first-order decoupled equations of motion instead of one second-order differential equation of motion. The methods presented here, solve in a way, the problem of nonlinear dynamic decoupling. The first and the second method result from diagonalized Lagrangian robot dynamics (Jain and Rodriguez, IEEE Trans Robot Autom 11:571–584, 1995) and are known as normalized and unnormalized quasi-velocities. The third method described by Junkins and Schaub (J Astronaut Sci 45:279–295, 1997) offers eigenfactor quasi-coordinate velocities formulation for multibody dynamics. As a consequence of using transformation given by Loduha and Ravani (Trans ASME J Appl Mech 62:216–222, 1995) we obtain decoupled equations of motion in terms of modified generalized velocity components. Here we limit all these methods to serial manipulators. The novelty of this paper consists in physical interpretation of the quasi-velocities and discussion concerning equations of motion, the kinetic energy shaping, relationship between each of them and properties useful for simulation and control purposes. Also forward dynamics algorithms and their computational complexity in terms of new velocities are given. Simulation results illustrate the theoretical investigations. We conclude that all methods offer interesting possibilities for dynamic simulation and future control investigations.     

A gradient elasticity theory for second-grade materials and higher order inertia

Second-grade elastic materials featured by a free energy depending on the strain and the strain gradient, and a kinetic energy depending on the velocity and the velocity gradient, are addressed. An inertial energy balance principle and a virtual work principle for inertial actions are envisioned to enrich the set of traditional theoretical tools of thermodynamics and continuum mechanics. The state variables include the body momentum and the surface momentum, related to the velocity in a nonstandard way, as well as the concomitant mass-accelerations and inertial forces, which do intervene into the motion equations and into the force boundary conditions. The boundary traction is the sum of two parts, i.e. the Cauchy traction and the Gurtin–Murdoch traction, whereas the traction boundary condition exhibits the typical format of the equilibrium equation of a material surface (as known from the principles of surface mechanics) whereby the Gurtin–Murdoch traction (incorporating the inertial surface force) plays the role of applied surfacial force density. The body’s boundary surface constitutes a thin boundary layer which is in global equilibrium under all the external forces applied on it, a feature that makes it possible to exploit the traction Cauchy theorem within second-grade materials. This means that a second-grade material is formed up by two sub-systems, that is, the bulk material operating as a classical Cauchy continuum, and the thin boundary layer operating as a Gurtin–Murdoch material surface. The classical linear and angular momentum theorems are suitably extended for higher order inertia, from which the local motion equations and the moment equilibrium equations (stress symmetry) can be derived. For an isotropic material featured by four constants, i.e. the Lamé constants and two length scale parameters (Aifantis model), the dynamic evolution problem is characterized by a Hamilton-type variational principle and a solution uniqueness theorem. Closed-form solutions of the wave dispersion analysis problem for beam models are presented and compared with known results from the literature. The paper indicates a correct thermodynamically consistent way to take into account higher order inertia effects within continuum mechanics.     

A comparative study of pressure correction and block-implicit finite volume algorithms on parallel computers

A comparison of a new parallel block-implicit method and the parallel pressure correction procedure for the solution of the incompressible Navier–Stokes equations is presented. The block-implicit algorithm is based on a pressure equation. The system of non-linear equation s is solved by Newton's method. For the solution of the linear algebraic systems the Bi-CGSTAB algorithm with incomplete lower–upper (ILU) decomposition of the matrix is applied. Domain decomposition serves as a strategy for the parallelization of the algorithms. Different algorithms for the parallel solution of the linear system of algebraic equations in conjunction with the pressure correction procedure are proposed. Three different flows are predicted with the parallel algorithms. Results and efficiency data of the block-implicit method are compared with the parallel version of the pressure correction algorithm. The block-implicit method is characterized by stable convergence behaviour, high numerical efficiency, insensitivity to relaxation parameters and high spatial accuracy. © 1997 John Wiley & Sons, Ltd.     

Inertial Manifolds and Forms for Stochastically Perturbed Retarded Semilinear Parabolic Equations

We construct inertial manifolds for a class of random dynamical systems generated by retarded semilinear parabolic equations subjected to additive white noise. These inertial manifolds are finite-dimensional invariant surfaces, which attract exponentially all trajectories. We study the corresponding inertial forms, i.e., the restriction of the stochastic equation to the inertial manifold. These inertial forms are finite-dimensional Ito equations and they completely describe the long-time dynamics of the system under consideration. The existence of inertial manifolds and the properties of inertial forms allow us to show that under mild additional conditions the system has a global (random) attractor in the sense of the theory of random dynamical systems.     

Random drift estimation for laser gyroscopes using filters based on the error equations of inertial navigation systems

An algorithm based on the error equations of inertial navigation systems is proposed to estimate the random components of errors for laser gyroscopes. The estimation is performed using the navigation parameters of an inertial system or using the output signals of laser gyroscopes. In the second case, the proposed algorithm is a low-pass filter. Its frequency analysis is compared with other filters.     

捷联惯导量化误差建模研究

从分析惯性器件量化误差产生的机理入手,建立了含量化噪声的捷联惯导系统误差微分方程,并用该方程的离散形式研究了系统误差方程中量化噪声的统计特性(功率谱密度的计算问题)。最后通过数值实例,定量分析了量化噪声对系统精度的影响和量化噪声建模不当引起的模型误差。     

Boundary domain integral method for transport phenomena in porous media

A boundary domain integral method (BDIM) for the solution of transport phenomena in porous media is presented. The complete, so‐called modified Navier–Stokes equations (Brinkman‐extended Darcy formulation with inertial term included) have been used to describe the fluid motion in porous media. Velocity–vorticity formulation (VVF) of the conservative equations is employed. In this paper, the proposed numerical scheme is tested on a particular case of natural convection and the results of flow and heat transfer characteristics of a fluid in a vertical porous cavity heated from the side and saturated with Newtonian fluid are presented in detail. Copyright © 2001 John Wiley & Sons, Ltd.     

Application of Radau IIA algorithms to flexible multibody system with holonomic constraints

The paper designs new 2- and 3-stage Radau IIA algorithms to integrate the dynamic responses of flexible multibody system with holonomic constraints. The total translation, the incremental rotation and associated velocities are selected as unknowns to avoid the linearization of angular acceleration which makes it possible to parameterize the finite rotation by using the Wiener–Milenkovi? parameters. The new algorithms release the heavy computational burden through the simplified Newton iterations and are stabilized by the preferable h-scaling technique. Contributions of the paper include: (1) For 2-stage algorithm, the resulting block triangular equations are solved efficiently by an inner iteration scheme. (2) For 3-stage algorithm, the full-size linear system is decoupled into a real and a complex subsystems which reduces the size of the system dramatically. (3) A new scheme is designed to predict the truncation error from the associated deferred correction equations without overestimation. Finally, numerical simulations show the 2- and 3-stage Radau IIA algorithms have excellent stability and convergence properties and behavior in a computationally efficient manner.     

Mixed Convection in a Ventilated Cavity Filled with a Triangular Porous Layer

Accurate prediction of the macroscopic flow parameters needed to describe flow in porous media relies on a good knowledge of flow field distribution at a much smaller scale—in the pore spaces. The extent of the inertial effect in the pore spaces cannot be underestimated yet is often ignored in large-scale simulations of fluid flow. We present a multiscale method for solving Oseen’s approximation of incompressible flow in the pore spaces amid non-periodic grain patterns. The method is based on the multiscale finite element method [MsFEM Hou and Wu in J Comput Phys 134:169–189, 1997)] and is built in the vein of Crouzeix and Raviart elements (Crouzeix and Raviart in Math Model Numer Anal 7:33–75, 1973). Simulations of inertial flow in highly non-periodic settings are conducted and presented. Convergence studies in terms of numerical errors relative to the reference solution are given to demonstrate the accuracy of our method. The weakly enforced continuity across coarse element edges is shown to maintain accurate solutions in the vicinity of the grains without the need for any oversampling methods. The penalisation method is employed to allow a complicated grain pattern to be modelled using a simple Cartesian mesh. This work is a stepping stone towards solving the more complicated Navier–Stokes equations with a nonlinear inertial term.     

捷联惯性系粗对准误差及数值问题分析

针对捷联惯导系统惯性系粗对准算法的对准误差进行了详细地分析。首先,依据惯性器件输出模型和粗对准算法模型,经过严格数学推导,得到了对准误差的解析表达式。其次,从求解矩阵方程的角度考察粗对准过程,得出对准过程等效于病态方程组的求解问题。理论分析表明,一方面惯性系粗对准算法的理论对准精度取决于惯性器件的精度,另一方面,对准误差上限取决于具体的算法和量测相对误差。在此基础上,进行了简单而有效的计算机仿真,结果与理论分析及工程经验吻合。     

Fundamental solutions of the streamfunction–vorticity formulation of the Navier–Stokes equations

A new boundary element procedure is developed for the solution of the streamfunction–vorticity formulation of the Navier–Stokes equations in two dimensions. The differential equations are stated in their transient version and then discretized via finite differences with respect to time. In this discretization, the non-linear inertial terms are evaluated in a previous time step, thus making the scheme explicit with respect to them. In the resulting discretized equations, fundamental solutions that take into account the coupling between the equations are developed by treating the non-linear terms as in homogeneities. The resulting boundary integral equations are solved by the regular boundary element method, in which the singular points are placed outside the solution domain.     

Design sensitivity analysis for parameters affecting geometry,elastic–viscoplastic material constant and boundary condition by consistent tangent operator-based boundary element method

This paper presents a design sensitivity analysis method by the consistent tangent operator concept-based boundary element implicit algorithm. The design variables for sensitivity analysis include geometry parameters, elastic–viscoplastic material parameters and boundary condition parameters. Based on small strain theory, Perzyna’s elastic–viscoplastic material constitutive relation with a mixed hardening model and two flow functions is considered in the sensitivity analysis. The related elastic–viscoplastic radial return algorithm and the formula of elastic–viscoplastic consistent tangent operator are derived and discussed. Based on the direct differentiation approach, the incremental boundary integral equations and related algorithms for both geometric and elastic–viscoplastic sensitivity analysis are developed. A 2D boundary element program for geometry sensitivity, elastic–viscoplastic material constant sensitivity and boundary condition sensitivity has been developed. Comparison and discussion with the results of this paper, analytical solution and finite element code ANSYS for four plane strain numerical examples are presented finally.     

基于惯性参考系基准的快速传递对准方法

研究了以惯性参考系为基准的新型传递对准方法。推导了计算惯性坐标系和计算体坐标系传递对准动态误差模型,并给出了相应的"速度+姿态"观测方程。基于惯性参考系的对准方法通过链式法则将子惯导输出的姿态矩阵描述为三个变换矩阵之积,其中两个变换矩阵通过对准时间和主惯导提供的位置信息可得到精确求解,剩余的变换矩阵(子惯导体坐标系至惯性坐标系间的变换矩阵)通过子惯导陀螺仪的输出进行解算,其误差通过传递对准估计得到的失准角进行补偿。对提出的两种对准方法进行摇摆实验验证,方位对准误差优于4’(1)。与传统基于导航坐标系的方法相比,基于惯性坐标系的方法直接将误差定位到惯性坐标系上,具有算法简便的特点。     

Approximate inertial manifolds for reaction-diffusion equations in high space dimension

The concept of approximate inertial manifolds was introduced by Foiaset al. (1987) in the case of the two-dimensional Navier-Stokes equations. These manifolds are finite dimensional smooth manifolds such that the orbits enter a very thin neighborhood of the manifold after a transient time; this concept replaces the one of inertial manifold when either an inertial manifold does not exist or its existence is not known. Our aim in this paper is to prove that approximate inertial manifolds exist for reaction-diffusion equations in high space dimension by opposition with exact inertial manifolds whose existence has only been proved in low dimension and for which nonexistence results have been obtained in space dimensionn=4.     

Inertial stage of bar buckling under longitudinal compression

The problem of dynamic buckling of a bar under the influence of a compressive force is solved taking into account inertial and elastic forces in different stages of the process. The duration of the inertial stage is determined. It is shown that in solids and gas–liquid media, the duration of the inertial stage for real parameters of structural members can be longer than the duration of impact loading.     

A combined Lagrangian method for simulation of axisymmetric gas-particle vortex flows

A combined fully Lagrangian approach for meshless modeling of unsteady axisymmetric vortex flows of a gas-particle mixture with an incompressible carrier phase is developed. The approach proposed is based on the combination of a meshless vortex method for calculating axisymmetric flows of the carrier phase described by the Navier–Stokes (or Euler) equations and the full Lagrangian method for calculating the parameters of the dispersed phase. The combination of these methods reduces the problem of modeling the two phase flows to the solution of a high-order system of ordinary differential equations for the coordinates of toroidal vortex elements in the carrier phase and the particle trajectories, the velocity components, and the components of the Jacobian of transformation from the Eulerian to the Lagrangian variables in the dispersed phase. The application of the method is illustrated by modeling the behavior of an admixture of inertial Stokes particles with a small mass concentration in unsteady flows like solitary vortex rings in a viscous carrier phase and groups of vortex rings in an effectively inviscid carrier phase.     

轴向瞬间阶梯载荷下圆柱壳动力屈曲的双特征参数分析

对于轴向瞬间阶梯载荷下圆柱壳的弹性非轴对称动力屈曲问题，将临界应力和屈曲惯性项指数参数作为双特征参数求解。由能量转换和守恒准则，导出压缩波阵面上的屈曲变形附加约束条件。失稳控制方程、边界条件和波阵面上的连续条件，连同此附加约束条件构成求解两个特征参数和动力失稳模态的完备定解条件。由伽辽金法得出求解双特征参数问题的数值方法。     

Settling velocity of quasi-neutrally-buoyant inertial particlesVitesse de sédimentation de particules inertielles dotées de flottabilité quasi neutre

We investigate the sedimentation properties of quasi-neutrally buoyant inertial particles carried by incompressible zero-mean fluid flows. We obtain generic formulae for the terminal velocity in generic space-and-time periodic (or steady) flows, along with further information for flows endowed with some degree of spatial symmetry such as odd parity in the vertical direction. These expressions consist in space-time integrals of auxiliary quantities that satisfy partial differential equations of the advection–diffusion–reaction type, which can be solved at least numerically, since our scheme implies a huge reduction of the problem dimensionality from the full phase space to the classical physical space.