Effect of the deformation in the inertia forces on the inverse dynamics of planar flexible mechanical systems

The inverse dynamics problem for articulated structural systems such as robotic manipulators is the problem of the determination of the joint actuator forces and motor torques such that the system components follow specified motion trajectories. In many of the previous investigations, the open loop control law was established using an inverse dynamics procedure in which the centrifugal and Coriolis inertia forces are linearized such that these forces in the flexible model are the same as those in the rigid body model. In some other investigations, the effect of the nonlinear centrifugal and Coriolis forces is neglected in the analysis and control system design of articulated structural systems. It is the objective of this investigation to study the effect of the linearization of the centrifugal and Coriolis forces on the nonlinear dynamics of constrained flexible mechanical systems. The virtual work of the inertia forces is used to define the complete nonlinear centrifugal and Coriolis force model. This nonlinear model that depends on the rate of the finite rotation and the elastic deformation of the deformable bodies is used to obtain the solution of the inverse dynamics problem, thus defining the joint torques that produce the desired motion trajectories. The effect of the linearization of the mass matrix as well as the centrifugal and Coriolis forces on the obtained feedforward control law is examined numerically. The results presented in this investigation are obtained using a slider crank mechanism with a flexible connecting rod.     

A Variational Approach to Dynamics of Flexible Multibody Systems

Abstract

This paper presents a variational formulation of constrained dynamics of flexible multibody systems, using a vector-variational calculus approach. Body reference frames are used to define global position and orientation of individual bodies in the system, located and oriented by position of its origin and Euler parameters, respectively. Small strain linear elastic deformation of individual components, relative to their body reference frames, is defined by linear combinations of deformation modes that are induced by constraint reaction forces and normal modes of vibration. A library of kinematic couplings between flexible and/or rigid bodies is defined and analyzed. Variational equations of motion for multibody systems are obtained and reduced to mixed differential-algebraic equations of motion. A space structure that must deform during deployment is analyzed, to illustrate use of the methods developed     

A nonlinear finite-element approach for kineto-static analysis of multibody systems

Methods that treat rigid/flexible multibody systems undergoing large motion as well as deformations are often accompanied with inefficiencies and instabilities in the numerical solution due to the large number of state variables, differences in the magnitudes of the rigid and flexible body coordinates, and the time dependencies of the mass and stiffness matrices. The kineto-static methodology of this paper treats a multibody mechanical system to consist of two collections of bulky (rigid) bodies and relatively flexible ones. A mixed boundary condition nonlinear finite element problem is then formulated at each time step whose known quantities are the displacements of the nodes at the boundary of rigid and flexible bodies and its unknowns are the deformed shape of the entire structure and the loads (forces and moments) at the boundary. Partitioning techniques are used to solve the systems of equations for the unknowns, and the numerical solution of the rigid multibody system governing equations of motion is carried out. The methodology is very much suitable in modelling and predicting the impact responses of multibody system since both nonlinear and large gross motion as well as deformations are encountered. Therefore, it has been adopted for the studies of the dynamic responses of ground vehicle or aircraft occupants in different crash scenarios. The kineto-static methodology is used to determine the large motion of the rigid segments of the occupant such as the limbs and the small deformations of the flexible bodies such as the spinal column. One of the most dangerous modes of injury is the amount of compressive load that the spine experiences. Based on the developed method, a mathematical model of the occupant with a nonlinear finite element model of the lumbar spine is developed for a Hybrid II (Part 572) anthropomorphic test dummy. The lumbar spine model is then incorporated into a gross motion occupant model. The analytical results are correlated with the experimental results from the impact sled test of the dummy/seat/restraint system. With this extended occupant model containing the lumbar spine, the gross motion of occupant segments, including displacements, velocities and accelerations as well as spinal axial loads, bending moments, shear forces, internal forces, nodal forces, and deformation time histories are evaluated. This detailed information helps in assessing the level of spinal injury, determining mechanisms of spinal injury, and designing better occupant safety devices.     

A Coordinate Reduction Technique for Dynamic Analysis of Spatial Substructures with Large Angular Rotations∗

A substructuring technique is presented for transient dynamic analysis of systems composed of interconnected rigid and elastic bodies that undergo large angular displacements. Displacement of elastic bodies is represented by superposition of local linear elastic deformation on large displacement of body reference coordinate systems. Elastic bodies are thus represented by combined sets of reference and local elastic generalized coordinates. Modal analysis and substructuring of individual elastic components allow for elimination of insignificant modes. Equations of motion and constraint are formulated in terms of mixed sets of modal and reference generalized coordinates. Planar and spatial linkages with flexible elements are presented to illustrate use of the method developed.     

A chassis-based kinematic formulation for flexible multi-body vehicle dynamics

This research proposed an efficient implicit integration method for the real-time simulation of flexible multi-body vehicle dynamics models. The equations of motion for the flexible bodies were formulated with respect to the moving chassis-body reference frame instead of the fixed inertial reference frame. The proposed approach does not require evaluation of system Jacobian and its LU-decomposition in time loof of simulation. This is one of the key aspects that enable high computational efficiency of the proposed method. The numerical simulation results of the proposed approach were matched up with those of the conventional approach but the computation time can be reduced by applying the proposed method. The joint constraint and generalized force equations are the same as the equations for rigid vehicle dynamics models because the joints and forces between flexible bodies are connected by the RBE (rigid body element). On the various driving conditions, the numerical simulation results show that the proposed approach yields almost exact solutions compared to the conventional approach. And the proposed approach spends only 22.9% of conventional approach on computation time under CPU 3.2 GHz personal computer.     

Effect of the order of the finite element and selection of the constrained modes in deformable body dynamics

Finite elements with different orders can be used in the analysis of constrained deformable bodies that undergo large rigid body displacements. The constrained mode shapes resulting from the use of finite elements with different orders differ in the way the stiffness of the body bending and extension are defined. The constrained modes also depend on the selection of the boundary conditions. Using the same type of finite element, different sets of boundary conditions lead to different sets of constrained modes. In this investigation, the effect of the order of the element as well as the selection of the constrained mode shapes is examined numerically. To this end, the constant strain three node triangular element and the quadratic six node triangular element are used. The results obtained using the three node triangular element are compared with the higher order six node triangular element. The equations of motion for the three and six node triangular elements are formulated from assumed linear and quadratic displacement fields, respectively. Both assumed displacement fields can describe large rigid body translational and rotational displacements. Consequently, the dynamic formulation presented in this investigation can also be used in the large deformation analysis. Using the finite element displacement field, the mass, stiffness, and inertia invariants of the three and six-node triangular elements are formulated. Standard finite element assembly techniques are used to formulate the differential equations of motion for mechanical systems consisting of interconnected deformable bodies. Using a multibody four bar mechanism, numerical results of the different elements and their respective performance are presented. These results indicate that the three node triangular element does not perform well in bending modes of deformation.     

基于李群局部标架的多柔体系统动力学建模与计算

多柔体系统动力学主要研究由多个具有运动学约束、存在大范围相对运动的柔性部件构成的动力学系统的建模、计算和控制.多柔体系统不仅具有柔体大变形导致的几何非线性,更具有大范围刚体运动引起的几何非线性,其非线性程度远高于计算结构力学所研究的几何非线性问题.本文基于李群局部标架(local frame of Lie group,...     

Modeling the motion of connected bodies

Spatial motion of mechanical systems consisting of jointed rigid bodies is considered. The methods previously developed to solve the dynamic equations of “carrier + loads” systems analytically for the accelerations of individual bodies are used to obtain the equations of motion of systems with complicated branched structure. The following practically important cases are analyzed: (i) a central carrier with peripheral loaded carriers and (ii) a chain and a ring consisting of loaded carriers. Numerical results are given for the stress-strain state of an elastic spacecraft represented as a chain of rigid bodies connected by elastic joints.     

Dynamics of flexible multibody systems with tree topologies

The dynamic equations of flexible multibody systems with tree topological configuration are derived by using the Jourdain's principle. The independent joint coordinates are introduced to describe the large displacements of the bodies, and the modal coordinates are used to describe small deformations of flexible bodies based on the consistent mass finite element method and normal vibration mode analysis. The minimum differential equations are developed, which are compatible with the equations of multi-rigid body systems or structural dynamics. The stiff problem in the numerical integration is thus alleviated effectively. The method used in this paper can be extended to deal with systems with other topological configurations. Finally, the validity and feasibility of the presented mathematical model are demonstrated by a numerical example of a manipulator with two elastic links. The project supported by National Natural Science Foundation of China     

Unilateral Multibody Dynamics

Contact processes may be represented by local discretization, by a rigid body approach or by a mixed method using both ideas. For the dynamics of mechanical systems a rigid body approach is described achieving good results also for multiple contact problems. This paper considers mainly contacts in multi-body systems where the corresponding contact constraints vary with time thus generating structure-variant systems. The equations of motion for dynamical systems with such an unilateral behavior are discussed, solution methods and applications are presented.     

A geometric solution to the general single contact frictionless problem by combining concepts of analytical dynamics and b-calculus

This study presents a systematic approach, leading to a new set of equations of motion for a class of mechanical systems subject to a single frictionless contact constraint. To achieve this goal, some fundamental concepts of b-geometry are utilized and adapted to the general framework of Analytical Dynamics. These concepts refer to the theory of manifolds with boundary and provide a suitable and strong theoretical foundation. First, the boundary is defined within the original configuration manifold of the system by the equality in the unilateral constraint. Then, an appropriate vector bundle is considered, involving only smooth vector fields, even at the boundary. After determining the essential geometric properties (i.e., the metric and the connection) near the boundary, Newton’s law of motion is applied. In this way, the equations of motion during the contact phase are derived as a system of ordinary differential equations. These equations possess a special form inside a thin boundary layer. In particular, the essential dynamics of the systems examined is found to be governed by a single second order ordinary differential equation, which is investigated fully. Moreover, a critical comparison of the present formulation with the classical formulations applied to systems with a frictionless contact is performed. Finally, the effect of the dominant parameters on the dynamics during the contact phase and the steps for the application process to mechanical systems are illustrated by two selected examples, referring to contact of a particle and a rigid body with a plane.     

GENERALIZATION OF KIRCHHOFF KINETIC ANALOGY TO THIN ELASTIC SHELLS 1)

将弹性细杆的"Kirchhoff动力学比拟"方法推广到弹性薄壳,使弹性薄壳的变形在物理概念上和刚体的运动对应, 在数学表述上等同,从而可以用刚体动力学的理论和方法研究弹性薄壳的变形,为连续的弹性薄壳提供新的离散化方法. 在直法线假设下,在弹性中面上构筑空间正交轴系, 此轴系沿坐标线"运动"的角速度构成两自变量的弯扭度. 沿两个坐标线的弯扭度表达了弹性薄壳的变形和位形,证明了弯扭度之间以及弯扭度与中面切矢间的相容关系. 用Euler角和Lam$\acute{e}$系数表达了非完整约束和中面位形的微分方程,用弯扭度和Lam$\acute{e}$系数表达了应变和应力以及内力及其本构方程.导出了用分布内力集度表达的弹性薄壳在变形后位形上的平衡偏微分方程组,方程的形式与刚体动力学的Euler方程和弹性细杆的Kirchhoff方程具有相似性,实现了Kirchhoff动力学比拟对弹性薄壳的推广.总结了弹性薄壳静力学和刚体动力学以及弹性细杆静力学在概念上的比拟关系.最后给出了一个算例. 为研究弹性薄壳的变形和运动提供新的建模方法和研究思路.也可进一步推广到弹性薄壳动力学.     

Kirchhoff动力学比拟对弹性薄壳的推广

将弹性细杆的"Kirchhoff动力学比拟"方法推广到弹性薄壳,使弹性薄壳的变形在物理概念上和刚体的运动对应,在数学表述上等同,从而可以用刚体动力学的理论和方法研究弹性薄壳的变形,为连续的弹性薄壳提供新的离散化方法.在直法线假设下,在弹性中面上构筑空间正交轴系,此轴系沿坐标线"运动"的角速度构成两自变量的弯扭度.沿两个...     

An anti-disturbance PD control scheme for attitude control and stabilization of flexible spacecrafts

This paper studies the attitude control problem of spacecrafts with flexible appendages. It is well known that the unwanted vibration modes, model uncertainty and space environmental disturbances may cause degradation of the performance of attitude control systems for a flexible spacecraft. In this paper, the vibration from flexible appendages is modeled as a derivative-bounded disturbance to the attitude control system of the rigid hub. A disturbance-observer-based control (DOBC) is formulated for feedforward compensation of the elastic vibration. The model uncertainty and space environmental disturbances as well as other noises are merged into an “equivalent” disturbance. We design a composite controller with a hierarchical architecture by combining DOBC and PD control, where DOBC is used to reject the vibration effect from the flexible appendages. Numerical simulations are performed to demonstrate that by using the composite hierarchical control law, disturbances can be effectively attenuated and the robust dynamic performances be enhanced.     

Dynamics of a spacecraft with large flexible appendage constrained by multi-strut passive damper

This paper is concerned with the dynamics of a spacecraft with multi-strut passive damper for large flexible appendage.The damper platform is connected to the spacecraft by a spheric hinge,multiple damping struts and a rigid strut.The damping struts provide damping forces while the rigid strut produces a motion constraint of the multibody system.The exact nonlinear dynamical equations in reducedorder form are firstly derived by using Kane’s equation in matrix form.Based on the assumptions of small velocity and small displacement,the nonlinear equations are reduced to a set of linear second-order differential equations in terms of independent generalized displacements with constant stiffness matrix and damping matrix related to the damping strut parameters.Numerical simulation results demonstrate the damping effectiveness of the damper for both the motion of the spacecraft and the vibration of the flexible appendage,and verify the accuracy of the linear equations against the exact nonlinear ones.     

Unilateral problems of dynamics

Summary Contact processes may be described by local discretizations, by rigid representation or by mixed methods incorporating both ideas. A rigid body approach is proposed for the dynamics of mechanical systems, achieving good results also for multiple-contact problems. Contacts in multibody systems are mainly considered, with the corresponding contact constraints varying with time, thus generating structure-variant systems. The equations of motion for dynamical systems with such unilateral behavior are discussed, solution methods and applications are presented. Received 3 March 1999; accepted for publication 5 May 1999     

考虑刚弹耦合作用的柔性多体连续系统动力学建模

基于Ｈａｍｉｌｔｏｎ原理建立起一般柔性体连续系统的动力学建模方法,进而以水平面内作大范围回转运动的柔性梁为例,在Ｅｕｌｅｒ－Ｂｅｒｎｏｕｌｌｉ梁模型的假设前提下,根据轴向不可伸长的柔性梁的几何约束条件;推导出作大范围刚体运动的柔性梁连续系统的一致线性化振动微分方程．采用假设模态法对其离散化,导出考虑刚弹耦合作用的柔性梁有限维离散化动力学模型．最后给出仿真算例,验证了该方法的有效性．