Computer system for kinematic and dynamic analysis synthesis of rigid and flexible multibody systems

In the paper an overview of a general numerical algorithm and program system library for deriving the kinematic constraint equations and dynamic equations of motion, as well as, computation of their first and second order partial derivatives with respect to kinematic parameters of motion, design parameters and mass and inertia characteristics for rigid and flexible multibody systems is presented. These are the main basic computational modules for implementation of kinematic and dynamic synthesis, optimization and design. The main theoretical basis consists in matrix methods for deriving the kinematic constraints and dynamic equations, as well as, the generalized Newton – Euler dynamic equations for rigid and flexible bodies, and finite element discretization in relative coordinates. Block–scheme of the computational procedures and problem oriented program compilation is presented. An example of kinematic synthesis of six–link path generating mechanism with singular points is presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the fast rotation of a heavy rigid body about a fixed point

Fast rotation of a symmetric heavy rigid body about a fixed point (the kinetic energy is large in comparison with the potential) is considered in cases when the resonance equations of Euler's motion /1, 2/ are approximately satisfied at the initial instant (the body is assumed to effect turns, ε is small, during time . It is shown that during that time ( ) a finite deviation from inertial motion takes place. Such mechanical effect is similar to the precession of a fast top, except that it is more “early” (in the considered time scale the top precession is slow). Approximate equations that define the motion in the principal order and are integrable in quadratures. The formal process of derivation of higher approximations is indicated, and a geometric interpretation of motions is given.     

Geometry of a five link mechanism with two degrees of freedom

The paper is devoted to the solution of straight and inverse geometrical tasks of five link mechanism with two degrees of freedom. The solution of the mentioned problem is very important in order to determine kinematic parameters of actuators. The problem can be divided into two parts. The first part is considered when we are given the coordinates of the output link of the mechanism and the necessity arises of determining the angles of rotation of actuators. On the other hand, it is very important to determine the position of the output link when the angles of rotation of the actuators are known. Here we consider that the mechanism is composed only of five classes of rotating kinematic pairs and the actuators are situated at the junctions of frames and links of the examined mechanism. The solution of the said problem is based on utilization of homogenous coordinates. On the basis of the obtained equations of motion, one can calculate the trajectories of motion of the output link as well angles of rotation of the actuators by taking into consideration preliminary given kinematic parameters of the mechanism. Here we also obtain equations for calculating values of speed and acceleration of the links of the mechanism. The calculations differ from known methods in simplicity and high performance, which would be useful for programming actuators mounted in the joints of the linkage.     

New kinematic parameters of the finite rotation of a rigid body

A new family of kinematic parameters for the orientation of a rigid body (global and local) is presented and described. All the kinematic parameters are obtained by mapping the variables onto a corresponding orientated subspace (hyperplane). In particular, a method of stereographically projecting a point belonging to a five-dimensional sphere S⁵ ⊂ R⁶ onto an orientated hyperplane R⁵ is demonstrated in the case of the classical direction cosines of the angles specifying the orientation of two systems of coordinates. A family of global kinematic parameters is described, obtained by mapping the Hopf five-dimensional kinematic parameters defined in the space R⁵ onto a four-dimensional orientated subspace R⁴. A correspondence between the five-dimensional and four-dimensional kinematic parameters defined in the corresponding spaces is established on the basis of a theorem on the homeomorphism of two topological spaces (a four-dimensional sphere S⁴ ⊂ R⁵ with one deleted point and an orientated hyperplane in R⁴). It is also shown to which global four-dimensional orientation parameters–quaternions defined in the space R⁴ the classical local parameters, that is, the three-dimensional Rodrigues and Gibbs finite rotation vectors, correspond. The kinematic differential rotational equations corresponding to the five-dimensional and four-dimensional orientation parameters are obtained by the projection method. All the rigid body kinematic orientation parameters enable one, using the kinematic equations corresponding to them, to solve the classical Darboux problem, that is, to determine the actual angular position of a body in a three-dimensional space using the known (measured) angular velocity of rotation of the object and its specified initial position.     

Finite-time Collapse of Three Point Vortices in the Plane

We investigate the finite-time collapse of three point vortices in the plane utilizing the geometric formulation of three-vortexmotion from Krishnamurthy, Aref and Stremler (2018) Phys. Rev. Fluids 3, 024702. In this approach, the vortex system is described in terms of the interior angles of the triangle joining the vortices, the circle that circumscribes that triangle, and the orientation of the triangle. Symmetries in the governing geometric equations of motion for the general three-vortex problem allow us to consider a reduced parameter space in the relative vortex strengths. The well-known conditions for three-vortex collapse are reproduced in this formulation, and we show that these conditions are necessary and sufficient for the vortex motion to consist of collapsing or expanding self-similar motion. The geometric formulation enables a new perspective on the details of this motion. Relationships are determined between the interior angles of the triangle, the vortex strength ratios, the (finite) system energy, the time of collapse, and the distance traveled by the configuration prior to collapse. Several illustrative examples of both collapsing and expanding motion are given.     

A kinematic interpretation of the motion of a heavy rigid body with a fixed point

A kinematic interpretation of the motion of a rigid body with a fixed point is proposed using the rolling of a mobile hodograph, which describes, on the ellipsoid of inertia, a vector collinear with the vector of the angular velocity of the body, with respect to a fixed vector. On the basis of this, an interpretation of the motion of the body in the Steklov, Grioli, Dokshevich and Bobylev – Steklov solutions is obtained. A new formula is derived indicating the connection between the angle of precession and the polar angle of the equations of the fixed hodograph, indicated by Kharlamov.     

Model of motion of a controlled body in the base attached trihedron

The position of a controlled body in space is described in a moving Cartesian system of coordinates, which we call the base attached trihedron. A model of motion of the body in these axes is derived in the form of a system of differential equations allowing for the Earth's rotation and the variable curvature of the Earth's surface.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 61, pp. 112–118, 1987.     

On effects of discretization and the selection of constitutive equations on stability of continua

In classical continuum mechanics the set of basic equations consists of the equation of motion, the kinematic equation and the constitutive equations. The study concentrates on constitutive modeling and the effects of discretization on stability problems. The method of investigation is analytic, the spectra of linear mapping operators of continuous and discrete dynamical systems are studied. As results cases are found, when a hidden incursive nature of a material model leads to unstable behavior of the continuum. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Identification of material behavior via a Finite Element Model Updating strategy

In this paper an inverse and iterative method for the identification of material behavior is presented, based on the Finite Element Model Updating (FEMU) strategy. The FE simulations are performed with a commercial FE software code, using a self-implemented elastic material model at finite strain. The iterative identification procedure is based on an experimental test (numerical) whose measured kinematic values are compared to the corresponding simulated ones. Through an optimization algorithm the material parameters are varied in a way that the least-squares sum of the kinematic values is minimized and the optimal material parameters yielding the material behavior are identified. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite-element calculation of a plate compliant in transverse shear

A finite-element calculation of a plate with a low transverse shear stiffness is presented. As the basic kinematic parameters, the angles of transverse shear at each of four nodes of the finite element of the plate are selected. The results found for the stress-strain state of an isotropic homogeneous composite and a three-layer plates confirm that the finite-element model elaborated is also efficient in the cases of nonclassical boundary conditions for plates, including conditions for the angles of transverse shear.     

An algorithm for normalizing Hamiltonian systems in the problem of the orbital stability of periodic motions

A constructive procedure is proposed for constructing equations of perturbed motion convenient for investigating the orbital stability of periodic motion in an autonomous Hamiltonian system with two degrees of freedom. An algorithm for normalizing these equations is described, and formulae for evaluating the coefficients of the normal form are presented. The results are used to investigate the stability of motion in certain special cases of the regular Grioli precession of a heavy rigid body with one fixed point.     

自由漂浮空间非合作目标的运动预测

空间非合作目标的运动预测是航天器在轨服务中的一个重要问题.在获得非合作目标的运动预测结果后,追踪星即可规划运动轨迹以接近目标并对其进行捕获.该文提出了一种自由漂浮空间非合作目标的运动预测方法.该方法的核心思想是首先辨识出目标的姿态动力学参数和目标的质心运动学参数,然后利用参数辨识结果和目标的动力学方程实现对目标的运动预测.在姿态动力学参数的辨识过程中,首先对目标的惯性参数进行初步辨识,然后采用自适应无迹Kalman滤波器对姿态动力学参数进行粗略辨识,最后通过最优化方法进一步提高姿态动力学参数的辨识精度.该文通过数值仿真验证了所提运动预测方法的有效性.仿真结果表明,无论目标是做单轴旋转还是翻滚运动,所提运动预测方法都能够实现对目标的长时间高精度的运动预测.     

Stiffness matrix of the finite element of a plate compliant in transverse shear

A four-node rectangular finite element is elaborated for a plate calculated with account of transverse shear strains. On variational realization of the finite-element method, the stiffness matrix and the vector of equivalent nodal forces are obtained for the case of a surface load. The basic nodal kinematic parameters adopted also include the angles of transverse shear.     

Helical flows of second grade fluid due to constantly accelerated shear stresses

The helical flows of second grade fluid between two infinite coaxial circular cylinders is considered. The motion is produced by the inner cylinder that at the initial moment applies torsional and longitudinal constantly accelerated shear stresses to the fluid. The exact analytic solutions, obtained by employing the Laplace and finite Hankel transforms and presented in series form in term of usual Bessel functions of first and second kind, satisfy both the governing equations and all imposed initial and boundary conditions. In the limiting case when α → 0, the solutions for Newtonian fluid are obtained for the same motion. The large-time solutions and transient solutions for second grade fluid are also obtained, and effect of material parameter α and kinematic viscosity ν is discussed. In the last, the effects of various parameters of interest on fluid motion as well as the comparison between second grade and Newtonian fluids are analyzed by graphical illustrations.     

声速及损耗系数的同时反演

引言 本文讨论有损层状介质的反演问题.有损介质的反演问题的研究,即使是一维情形,也未成熟,且无一般提法.G.Kristensson和R.J.Krueger的一系列文章,利用一般的二次双曲型方程,对有限厚度的介质进行反演,用到的响应信息是透射和反射数据,不     

Mathematical simulation of combined trajectory paths of a seven link biped robot

The following article focuses on biped robot simulation and control over combined trajectory paths with the aid of mathematical modeling methods focusing on the effects of hip height over torso’s modified motion. The mathematical simulation has been exploited to interpolate the combined trajectory of the robot path with the given breakpoints using inverse kinematic and dynamic methods to determine ZMP and stability treatments. After the robot’s combined path determination, a third-order spline is utilized because of its high precision and ability to calculate the kinematic, dynamic and control parameters. With the aid of this software, common parameters such as joint angles and inertial forces for the given specifications and nominal conditions are calculated and simulated.     

A Study of the Motions of an Autonomous Hamiltonian System at a 1:1 Resonance

We examine the motions of an autonomous Hamiltonian system with two degrees of freedom in a neighborhood of an equilibrium point at a 1:1 resonance. It is assumed that the matrix of linearized equations of perturbed motion is reduced to diagonal form and the equilibrium is linearly stable. As an illustration, we consider the problem of the motion of a dynamically symmetric rigid body (satellite) relative to its center of mass in a central Newtonian gravitational field on a circular orbit in a neighborhood of cylindrical precession. The abovementioned resonance case takes place for parameter values corresponding to the spherical symmetry of the body, for which the angular velocity of proper rotation has the same value and direction as the angular velocity of orbital motion of the radius vector of the center of mass. For parameter values close to the resonance point, the problem of the existence, bifurcations and orbital stability of periodic rigid body motions arising from a corresponding relative equilibrium of the reduced system is solved and issues concerning the existence of conditionally periodic motions are discussed.     

On the dynamics of a rigid body carrying a material point

In this paper we consider a system consisting of an outer rigid body (a shell) and an inner body (a material point) which moves according to a given law along a curve rigidly attached to the body. The motion occurs in a uniform field of gravity over a fixed absolutely smooth horizontal plane. During motion the shell may collide with the plane. The coefficient of restitution for an impact is supposed to be arbitrary. We present a derivation of equations describing both the free motion of the system over the plane and the instances where collisions with the plane occur. Several special solutions to the equations of motion are found, and their stability is investigated in some cases. In the case of a dynamically symmetric body and a point moving along the symmetry axis according to an arbitrary law, a general solution to the equations of free motion of the body is found by quadratures. It generalizes the solution corresponding to the classical regular precession in Euler??s case. It is shown that the translational motion of the shell in the free flight regime exists in a general case if the material point moves relative to the body according to the law of areas.     

Modeling the angular capability of the ball joints in a complex mechanism with two degrees of mobility

The paper puts forward a complex linkage mechanism with two degrees of mobility and three kinematic loops, which is used for the guiding (suspension and steering) system of the vehicles. The geometric parameters and the coordinates frames that define the mechanical system are presented, as well as the specific kinematic functions. For this complex mechanical system, the angular capability of the ball (spherical) joints is defined by two angles. The equations for these angles have been determined by matrix algebra tools, considering the transformation matrices between the bodies reference frames. The diagrams of the angular capability of the ball joints, which are represented in angular coordinates, describe the form, orientation and size of the sockets from the spherical casings. Wears, shocks, functional locks or the compromising of the joint strength can occur if scarce sockets are implemented. The risk points, in which the angular parameters have maximum values, have been determined, the simulation being performed for a real system (vehicle).     

An analysis of a mixed finite element method for the Navier-Stokes equations

Summary A simple mixed finite element method is developed to solve the steady state, incompressible Navier-Stokes equations in a neighborhood of an isolated—but not necessarily unique—solution. Convergence is established under very mild restrictions on the triangulation, and, when the solution is sufficiently smooth, optimal error bounds are obtained.