求解非定常非完整约束多体系统动力学的一种方法

通过对约束矩阵进行奇异值分解,将系统的动力学方程沿与约束相容和不相容的两个方向上投影,求解受非定常非完整约束的多体系统动力学问题,并给出了求约束反力的公式和避免违约的一种方法;这种解法不仅不依赖于描述系统的坐标选择,而且计算效率高。最后举了一个说明性例子。     

Nonholonomic LR Systems as Generalized Chaplygin Systems with an Invariant Measure and Flows on Homogeneous Spaces

We consider a class of dynamical systems on a compact Lie group G with a left-invariant metric and right-invariant nonholonomic constraints (so-called LR systems) and show that, under a generic condition on the constraints, such systems can be regarded as generalized Chaplygin systems on the principle bundle G \to Q = G/H, H being a Lie subgroup. In contrast to generic Chaplygin systems, the reductions of our LR systems onto the homogeneous space Q always possess an invariant measure. We study the case G = SO(n), when LR systems are ultidimensional generalizations of the Veselova problem of a nonholonomic rigid body motion which admit a reduction to systems with an invariant measure on the (co)tangent bundle of Stiefel varieties V(k, n) as the corresponding homogeneous spaces. For k = 1 and a special choice of the left-invariant metric on SO(n), we prove that after a time substitution the reduced system becomes an integrable Hamiltonian system describing a geodesic flow on the unit sphere S^n-1. This provides a first example of a nonholonomic system with more than two degrees of freedom for which the celebrated Chaplygin reducibility theorem is applicable for any dimension. In this case we also explicitly reconstruct the motion on the group SO(n).     

非完整动力学逆问题的一种提法和解法*

本文给出非完整动力学逆问题的一种提法和解法:已知某些积分,来求施加在系统上的非完整约束的形式;进而在已知系统动能表达式的情况下,来求加在系统上的广义约束反力;最后,给出例子说明解法的应用.     

Integrated motion measurement of rigid multibody systems

During the last years, integrated navigation systems based on gyros, accelerometers, and GPS receivers became powerful devices for the guidance of aircraft and ships. Comparable equipment using especially wheel sensors exists for cars. The kernel of such systems is a Kalman filter estimating the relevant vehicle motion. The filter design in turn requires a kinematical model to settle on the motion components considered and to describe the mechanical meaning of the measurements employed. Until now, usual models consider only one to six degrees of freedom of a single rigid body. The assumption of a solitary rigid body is not a consequence of the basic concept of integrated navigation; it reflects merely classical navigation requirements. In principle, determining the motion of multibody systems, representing certain vehicle types of varying shape, is possible if appropriate kinematical models and sensor arrangements are available. Based on the theory of integrated navigation systems, the paper outlines the fundamentals of designing integrated motion measurement systems for rigid multibody structures. The example of a double pendulum with a movable inertial support and with equipment of microelectromechanical inertial sensors and of small radar units illustrates this approach. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

多体系统动力学微分-代数方程L-稳定方法

针对多体系统动力学微分-代数方程形式，在时间区间上构造L-稳定方法，分别基于等距节点、Chebyshev节点和Legendre节点等非等距节点建立求解格式，依据Ehle定理及猜想，与Padé逼近式对比得到待定矩阵和向量，从而获得L-稳定求解公式，循环求解过程采用Newton迭代法计算.以平面双连杆机械臂系统为例，使用L-稳定方法进行数值仿真，通过改变时间区间节点数和步长对各个指标结果进行比较，并与经典Runge-Kutta法对比.结果表明，该方法具有稳定性好、精度高等优点，适用于长时间情况下的多体系统动力学仿真.     

Multidimensional viscous shocks II: The small viscosity limit

In this paper we prove the existence of curved, multidimensional viscous shocks and also justify the small‐viscosity limit. Starting with a curved, multidimensional (inviscid) shock solution to a system of hyperbolic conservation laws, we show that the shock can be obtained as a small‐viscosity limit of solutions to an associated parabolic problem (viscous shocks). The two main hypotheses are a natural Evans function assumption on the viscous profile, together with a restriction on how much the shock can deviate from flatness. The main tools are a conjugation lemma that removes x_N/? dependence from the linearization of the parabolic problem about the viscous profile, new degenerate Kreiss‐type symmetrizers used to prove an L² estimate for the linearized problem, and a finite‐regularity calculus of semiclassical and mixed type (classical‐semiclassical) pseudodifferential operators. © 2003 Wiley Periodicals, Inc.     

Energy-momentum conserving integration of parallel manipulators

In the framework of a rotationless formulation for multibody systems, we present an investigation of multibody mechanisms. The benefit of this kind of formulation is the design of energy-momentum conserving integration schemes, which facilitate a stable numerical integration of differential algebraic equations governing the motion of open-loop and closed-loop systems. We introduce a coordinate augmentation technique for the incorporation of rotational degrees of freedom and subsequently perform a size-reduction to lower the computational costs and to improve the numerical conditioning [1]. Furthermore the treatment of nonholonomic constraints is addressed [3]. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Contact of ropes and orthotropic rough surfaces

Contact between arbitrary curved ropes and arbitrary curved rough orthotropic surfaces has been revised from the geometrical point of view. Variational equations for the equilibrium of ropes on orthotropic rough surfaces are derived, first, using the consistent variational inclusion of frictional contact constraints via Karush-Kuhn-Tucker conditions expressed in Darboux basis. Then, the systems of differential equations are derived for both statics and dynamics of ropes on a rough surface depending on the sticking-sliding condition for orthotropic Coulomb's friction. Three criteria are found to be fulfilled during the static equilibrium of a rope on a rough surface: “no separation”, condition for dragging coefficient of friction and inequality for tangential forces at the end of the rope. The limit tangential loads still preserve the famous “Euler view” T = T₀e^ωs for the curves and surfaces of constant curvature. It is shown that the curve of the maximum tension of a rough orthotropic surface is geodesic. Equations of motion are derived in the case if the sliding criteria is fulfilled and there is “no separation”. Various cases possessing analytical solutions of the derived system, including Euler case and a spiral rope on a cylinder are shown as examples of application of the derived theory. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonexistence of nonpositively curved surfaces¶with one embedded end

We prove certain complete nonpositively curved surfaces arising from general relativity isometrically immersible in R ³ do not exist, assuming square integrable second fundamental form. We provide an example showing the sharpness of our conditions. Received: 19 February 1999 / Revised version: 2 December 1999     

On the Application of Accurate Distance Algorithms to Multibody Simulations

Distance algorithms are most frequently used in robotics to determine the distance between two obstacles in the environment of a robot or between a sensor point and an object. We extend the multibody simulation package MOBILE for an application of accurate algorithms for distance computation between objects represented by convex or non-convex polyhedra. These objects are represented by their vertices and oriented facets. As an application example, a multibody system is discussed where a sensor point moves close to a non-convex obstacle. The computed results show that the algorithms developed are suitable for accurate real-time multibody simulations. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonholonomic Mechanical Systems and Kaluza–Klein Theory

In the first part of the paper we present a new point of view on the geometry of nonholonomic mechanical systems with linear and affine constraints. The main geometric object of the paper is the nonholonomic connection on the distribution of constraints. By using this connection and adapted frame fields, we obtain the Newton forms of Lagrange–d’Alembert equations for nonholonomic mechanical systems with linear and affine constraints. In the second part of the paper, we show that the Kaluza–Klein theory is best presented and explained by using the framework of nonholonomic mechanical systems. We show that the geodesics of the Kaluza–Klein space, which are tangent to the electromagnetic distribution, coincide with the solutions of Lagrange–d’Alembert equations for a nonholonomic mechanical system with linear constraints, and their projections on the spacetime are the geodesics from general relativity. Any other geodesic of the Kaluza–Klein space that is not tangent to the electromagnetic distribution is also a solution of Lagrange–d’Alembert equations, but for affine constraints. In particular, some of these geodesics project exactly on the solutions of the Lorentz force equations of the spacetime.     

A Co-Simulation Approach for the 3D Dynamic Simulation of Vehicles Considering Sloshing in Cargo and Fuel Tanks

The sloshing of liquids in cargo and fuel tanks mounted on vehicles can have a significant influence on the vehicle's driving dynamics and stability. To evaluate and optimize the quality of tank designs, we propose a co-simulation approach that consists of a coupled multibody system simulation for the vehicle and a Discrete Element Method and Smoothed Particle Hydrodynamics simulation for the sloshing cargo. This approach is beneficial especially for the simulation of fluid cargos, as Smoothed Particle Hydrodynamics does not require additional models to track and reconstruct free fluid surfaces. By means of dynamic 3D simulations of a double lane change maneuvers we compare the two different cargo models and demonstrate the viability of the co-simulation approach. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Noisy Extensions of Nonholonomic Constraints

We propose several stochastic extensions of nonholonomic constraints for mechanical systems and study the effects on the dynamics and on the conservation laws. Our approach relies on a stochastic extension of the Lagrange–d’Alembert framework. The mechanical system we focus on is the example of a Routh sphere, i.e., a rolling unbalanced ball on the plane. We interpret the noise in the constraint as either a stochastic motion of the plane, random slip or roughness of the surface. Without the noise, this system possesses three integrals of motion: energy, Jellet and Routh. Depending on the nature of noise in the constraint, we show that either energy, or Jellet, or both integrals can be conserved, with probability 1. We also present some exact solutions for particular types of motion in terms of stochastic integrals. Next, for an arbitrary nonholonomic system, we consider two different ways of including stochasticity in the constraints. We show that when the noise preserves the linearity of the constraints, then energy is preserved. For other types of noise in the constraint, e.g., in the case of an affine noise, the energy is not conserved. We study in detail a class of Lagrangian mechanical systems on semidirect products of Lie groups, with “rolling ball type” constraints. We conclude with numerical simulations illustrating our theories, and some pedagogical examples of noise in constraints for other nonholonomic systems popular in the literature, such as the nonholonomic particle, the rolling disk and the Chaplygin sleigh.     

受单面约束的非完整力学系统的运动方程

给出同时受有单面完整约束和单面非完整约束的非完整力学系统的运动方程,并举例说明其应用     

Realizing nonholonomic dynamics as limit of friction forces

The classical question whether nonholonomic dynamics is realized as limit of friction forces was first posed by Carathéodory. It is known that, indeed, when friction forces are scaled to infinity, then nonholonomic dynamics is obtained as a singular limit.Our results are twofold. First, we formulate the problem in a differential geometric context. Using modern geometric singular perturbation theory in our proof, we then obtain a sharp statement on the convergence of solutions on infinite time intervals. Secondly, we set up an explicit scheme to approximate systems with large friction by a perturbation of the nonholonomic dynamics. The theory is illustrated in detail by studying analytically and numerically the Chaplygin sleigh as an example. This approximation scheme offers a reduction in dimension and has potential use in applications.     

滚装船在波浪中横摇时船上滑动重载荷的同步效应

由于波浪和内部滑动车辆共同作用,使滚装船的横摇加剧,这是许多滚装船发生倾覆的重要原因之一．对由滚装船和多辆滑动车辆组成的浮基多体系统,取滚装船的横摇角和多辆自由滑动车辆在甲板上的横向位移为此系统的自由度,通过对船舶在波浪中所受表观重力和表观浮力的分析,导出波浪作用力矩,运用多体系统动力学方法,建立了系统的动力学方程．以某型海峡滚装渡轮为例,对在多辆车自由滑动和波浪共同作用下的滚装船浮基多体系统的横摇响应和车辆位移响应进行了数值计算,得出了多个自由滑动的重载荷因相互碰撞在舷侧舱壁的约束下随着时间的延长其运动将趋于同步的结论．     

变质量非完整系统的形式不变性与Lie对称性

研究变质量非完整系统的形式不变性和Lie对称性．给出变质量非完整系统在无限小变换下形式不变性和Lie对称性的定义、判据及存在守恒量的定理,得到形式不变性和Lie对称性的关系,并举例说明结果的应用．     

Controllability of a Nonholonomic Macroeconomic System

This paper studies optimal control problems and sub-Riemannian geometry on a nonholonomic macroeconomic system. The main results show that a nonholonomic macroeconomic system is controllable either by trajectories of a single-time driftless control system (single-time bang–bang controls), or by nonholonomic geodesics or by sheets of a two-time driftless control system (two-time bang–bang controls). They are strongly connected to the possibility of describing a nonholonomic macroeconomic system via a Gibbs–Pfaff equation or by four associated vector fields, based on a contact structure of the state space and our isomorphism between thermodynamics and macroeconomics that praises three laws of a nonholonomic macroeconomic system.     

Skew loops and quadric surfaces

A skew loop is a closed curve without parallel tangent lines. We prove: The only complete surfaces in R ³ with a point of positive curvature and no skew loops are the quadrics. In particular: Ellipsoids are the only closed surfaces without skew loops. Our efforts also yield results about skew loops on cylinders and positively curved surfaces. Received: January 7, 2002 RID="*" ID="*"The first author was partially supported by the NSF grant DMS-0204190.     

Equations of Motion for Constrained Multibody Systems and their Control

This paper presents some recent advances in the dynamics and control of constrained multibody systems. The constraints considered need not satisfy the D’Alembert principle and therefore the results are of general applicability. They show that, in the presence of constraints, the constraint force acting on the multibody system can always be viewed as made up of the sum of two components whose explicit form is provided. The first of these components consists of the constraint force that would have existed were all the constraints ideal; the second is caused by the nonideal nature of the constraints, and though it needs specification by the mechanician who is modeling the specific system at hand, it has a specific form. The general equations of motion obtained herein provide new insights into the simplicity with which Nature seems to operate. They point toward the development of new and novel approaches for the exact control of complex multibody nonlinear systems. In honor of Bob kalaba, friend, colleague, and mentor.