Dynamics of Articulated Structures. Part I. Theory

ABSTRACT

A finite element based method is developed for geometrically nonlinear dynamic analysis of spatial articulated structures; i.e., structures in which kinematic connections permit large relative displacement between components that undergo small elastic deformation. Vibration and static correction modes are used to account for linear elastic deformation of components. Kinematic constraints between components are used to define boundary conditions for vibration analysis and loads for static correction mode analysis. Constraint equations between flexible bodies are derived in a systematic way and a Lagrange multiplier formulation is used to generate the coupled large displacement-small deformation equations of motion. A lumped mass finite element structural analysis formulation is used to generate deformation modes. An intermediate-processor is used to calculate time-independent terms in the equations of motion and to generate input data for a large-scale dynamic analysis code that includes coupled effects of geometric nonlinearity and elastic deformation. Examples are presented and the effects of deformation mode selection on dynamic prediction are analyzed in Part II of the paper.     

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     

Control structure interaction in the nonlinear analysis of flexible mechanical systems

The effect of the control structure interaction on the feedforward control law as well as the dynamics of flexible mechanical systems is examined in this investigation. An inverse dynamics procedure is developed for the analysis of the dynamic motion of interconnected rigid and flexible bodies. This method is used to examine the effect of the elastic deformation on the driving forces in flexible mechanical systems. The driving forces are expressed in terms of the specified motion trajectories and the deformations of the elastic members. The system equations of motion are formulated using Lagrange's equation. A finite element discretization of the flexible bodies is used to define the deformation degrees of freedom. The algebraic constraint equations that describe the motion trajectories and joint constraints between adjacent bodies are adjoined to the system differential equations of motion using the vector of Lagrange multipliers. A unique displacement field is then identified by imposing an appropriate set of reference conditions. The effect of the nonlinear centrifugal and Coriolis forces that depend on the body displacements and velocities are taken into consideration. A direct numerical integration method coupled with a Newton-Raphson algorithm is used to solve the resulting nonlinear differential and algebraic equations of motion. The formulation obtained for the flexible mechanical system is compared with the rigid body dynamic formulation. The effect of the sampling time, number of vibration modes, the viscous damping, and the selection of the constrained modes are examined. The results presented in this numerical study demonstrate that the use of the driving forees obtained using the rigid body analysis can lead to a significant error when these forces are used as the feedforward control law for the flexible mechanical system. The analysis presented in this investigation differs significantly from previously published work in many ways. It includes the effect of the structural flexibility on the centrifugal and Coriolis forces, it accounts for all inertia nonlinearities resulting from the coupling between the rigid body and elastic displacements, it uses a precise definition of the equipollent systems of forces in flexible body dynamics, it demonstrates the use of general purpose multibody computer codes in the feedforward control of flexible mechanical systems, and it demonstrates numerically the effect of the selected set of constrained modes on the feedforward control law.     

具有非线性homologous变形约束的桁架结构形态分析

研究了具有非线性homologous变形约束条件的桁架结构形态分析问题。在已有的线性homologous变形约束桁架形态分析的基础上,将结构的节点分成三类:homologous变形约束节点,形状可变节点和边界点。运用Moore-Penrose广义逆矩阵性质,将基础方程组解的存在条件表示为包含形状可变节点未知坐标的非线性方程组,为采用Newton-Raphson方法求解非线性方程组,对AA (A为任意矩阵,A 为A的Moore-Penrose广义逆矩阵)求偏导数,找到了满足保型要求的形态,给出的桁架算例说明了本文方法的有效性。     

A comparative study of joint formulations: application to multibody system tracked vehicles

This paper is focused on the dynamic formulation of mechanical joints using different approaches that lead to different models with different numbers of degrees of freedom. Some of these formulations allow for capturing the joint deformations using a discrete elastic model while the others are continuum-based and capture joint deformation modes that cannot be captured using the discrete elastic joint models. Specifically, three types of joint formulations are considered in this investigation; the ideal, compliant discrete element, and compliant continuum-based joint models. The ideal joint formulation, which does not allow for deformation degrees of freedom in the case of rigid body or small deformation analysis, requires introducing a set of algebraic constraint equations that can be handled in computational multibody system (MBS) algorithms using two fundamentally different approaches: constrained dynamics approach and penalty method. When the constrained dynamics approach is used, the constraint equations must be satisfied at the position, velocity, and acceleration levels. The penalty method, on the other hand, ensures that the algebraic equations are satisfied at the position level only. In the compliant discrete element joint formulation, no constraint conditions are used; instead the connectivity conditions between bodies are enforced using forces that can be defined in their most general form in MBS algorithms using bushing elements that allow for the definition of general nonlinear forces and moments. The new compliant continuum-based joint formulation, which is based on the finite element (FE) absolute nodal coordinate formulation (ANCF), has several advantages: (1) It captures modes of joint deformations that cannot be captured using the compliant discrete joint models; (2) It leads to linear connectivity conditions, thereby allowing for the elimination of the dependent variables at a preprocessing stage; (3) It leads to a constant inertia matrix in the case of chain like structure; and (4) It automatically captures the deformation of the bodies using distributed inertia and elasticity. The formulations of these three different joint models are compared in order to shed light on the fundamental differences between them. Numerical results of a detailed tracked vehicle model are presented in order to demonstrate the implementation of some of the formulations discussed in this investigation.     

Investigation on the choice of boundary conditions and shape functions for flexible multi-body system

The objective of this investigation is to examine the correctness and efficiency of the choice of boundary conditions when using assumed mode approach to simulate flexible multi-body systems. The displacement field due to deformation is approximated by the Rayleigh-Ritz assumed modes in floating frame of reference (FFR) formulation. The deformations obtained by the absolute nodal coordinate (ANC) formulation which are transformed by two sets of reference coordinates are introduced as a criterion to verify the accuracy of the simulation results by using the FFR formulation. The relationship between the deformations obtained from different boundary conditions is revealed. Numerical simulation examples demonstrate that the assumed modes with cantilevered-free, simply-supported and freefree boundary conditions without inclusion of rigid body modes are suitable for simulation of flexible multi-body system with large over all motion, and the same physical deformation can be obtained using those mode functions, differ only by a coordinate transformation. It is also shown that when using mode shapes with statically indeterminate boundary conditions, significant error may occur. Furthermore, the slider crank mechanism with rigid crank is accurate enough for investigating boundary condition problem of flexible multi-body system, which cost significant less simulating time.     

Boundary element method for buckling eigenvalue problem and its convergence analysis

The conditions for determining solution of buckling eigenvalue problem are discussed. The corresponding system of integral equations with constraint conditions and boundary variational equations with Lagrange multiplier are established. The theorems on the existence and uniqueness of the solution for these problems are given. The corresponding boundary element method is constructed and the error estimation for the approximation solution is obtained. Finally the numerical example is given. Foundation item: the National Natural Science Foundation Pre-research Project (T4107015) Biography: Ding Rui (1969-)     

Flexible Multibody Simulation and Choice of Shape Functions

The approach most widely used for the modelling of flexible bodies in multibody systems has been called the floating frame of reference formulation. In this methodology the flexible body motion is subdivided into a reference motion and deformation. The displacement field due to deformation is approximated by the Ritz method as a product of known shape functions and unknown coordinates depending on time only. The shape functions may be obtained using finite-element-models of flexible bodies in multibody systems, resulting in a detailed system representation and a high number of system equations. The number of system equations of such a nodal approach can be reduced considerably using a modal representation of deformation. This modal approach, however, leads to the fundamental problem of selecting the shape functions.The floating frame of reference formulation is reviewed here using a generic flexible body model, from which the various body models used in multibody simulations may be derived by formulation of specific constraint equations. Special attention is given in this investigation to the following subjects: The separation of flexible body motion into a reference motion and deformation requires the definition of a body reference frame, which in turn affects the choice of shape functions. Some alternatives will be outlined together with their advantages and disadvantages. Assuming the body deformation to be small, the system equations can be linearized. This may require considering geometric stiffening terms. The problem of how to compute these terms has been solved in literature on the instability of structures under critical loads. For finite element models the geometric stiffening terms are obtained from the tangential stiffness matrix. The generality of the flexible body model allows the definition of an object oriented data base to describe the system bodies. Such a data base includes a general interface between multibody- and finite-element-codes. By combining eigenfunctions and static deformation modes to represent body deformation one obtains a set of so-called quasi-comparison functions. When selected properly these functions can be shown to improve the representation of stresses significantly.     

A FINITE VOLUME SOLVER FOR THE SHALLOW WATER EQUATIONS IN VARYING BED ENVIRONMENTS

Abstract

A numerical scheme for solving the shallow-water equations is presented. An analogy is made between flows governed by shallow-water equations and the Euler system of equations used in gas dynamics. An emphasis is placed on the difference presented by the bathymetry in hydraulic systems. The discretization of the governing equations is based on Roe's flux difference-splitting solver, initially developed for solving inviscid compressible flows. The spatial discretization is handled within a finite-volume context by using triangles or quadrilaterals as the basic control-volume cells. This approach enables an easy and flexible treatment of general geometries. A development of the boundary conditions tailored for the current scheme is given. Fundamental validation tests are presented.     

Finite element modeling of contact conditions in multibody system dynamics

In this paper a new method is developed for the dynamic analysis of contact conditions in flexible multibody systems undergoing a rolling type of motion. The relative motion between the two contacting bodies is treated as a constraint condition describing their kinematic and geometric relations. Equations of motion of the system are presented in a matrix form making use of Kane's equations and finite element method. The method developed has been implemented in a general purpose program called DARS and applied to the simulation and analysis of a rotating wheel on a track. Both the bodies are assumed flexible and discretized using a three dimensional 8-noded isoparametric elements. The time variant constraint conditions are imposed on the nodal points located at the peripheral surfaces of the bodies under consideration. The simulation is carried out under two different boundary conditions describing the support of the track. The subsequent constraint forces associated with the generalized coordinates of the system are computed and plotted. The effects of friction are also discussed.     

非局部杆纵向振动的特性研究

基于非局部理论，研究弹性杆在任意边界约束条件下的纵向振动特性．根据Chebyshev 谱级数建立非局部弹性杆的纵向位移形式．在杆的两端引入纵向约束弹簧，通过设置弹簧刚度系数，模拟经典边界及弹性边界．建立非局部杆的能量表达式，由瑞利-里兹法得到齐次线性方程组，求解对应的矩阵特征值与特征向量问题获得非局部杆的固有频率和振型．通过数值仿真计算，研究非局部特征系数与边界约束条件对非局部杆振动频率的影响．结果表明本文方法合理简便，具有良好的精度，且适用于任意弹性边界条件．     

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.     

Nonlinear dynamic analysis of Timoshenko beams by BEM. Part II: applications and validation

In this two-part contribution, a boundary element method is developed for the nonlinear dynamic analysis of beams of arbitrary doubly symmetric simply or multiply connected constant cross section, undergoing moderate large displacements and small deformations under general boundary conditions, taking into account the effects of shear deformation and rotary inertia. In Part I the governing equations of the aforementioned problem have been derived, leading to the formulation of five boundary value problems with respect to the transverse displacements, to the axial displacement and to two stress functions. These problems are numerically solved using the Analog Equation Method, a BEM based method. In this Part II, numerical examples are worked out to illustrate the efficiency, the accuracy and the range of applications of the developed method. Thus, the results obtained from the proposed method are presented as compared with those from both analytical and numerical research efforts from the literature. More specifically, the shear deformation effect in nonlinear free vibration analysis, the influence of geometric nonlinearities in forced vibration analysis, the shear deformation effect in nonlinear forced vibration analysis, the nonlinear dynamic analysis of Timoshenko beams subjected to arbitrary axial and transverse in both directions loading, the free vibration analysis of Timoshenko beams with very flexible boundary conditions and the stability under axial loading (Mathieu problem) are presented and discussed through examples of practical interest.

     

On geometrically exact post-buckling of composite columns with interlayer slip—The partially composite elastica

This paper is focused on the geometrically exact elastic stability analysis of two interacting kinematically constrained, flexible columns. Possible applications are to partially composite or sandwich columns. A partially composite column composed of two inextensible elastically connected sub-columns is considered. Each sub-column is modeled by the Euler–Bernoulli beam theory and connected to each other via a linear constitutive law for the interlayer slip. The paper discusses the validity of parallel and translational kinematics beam assumptions with respect to the interlayer constraint. Buckling and post-buckling behavior of this structural system are studied for cantilever columns (clamped-free boundary conditions). A variational formulation is presented in order to derive relevant boundary conditions in a geometrically exact framework. The exact post-buckling behavior of this partially composite beam-column is investigated analytically and numerically. The Euler elastica problem is obtained in the case of non-composite action. The “partially composite elastica” is then treated analytically and numerically, for various values of the interaction connection parameter. An asymptotic expansion is performed to evaluate the symmetrical pitchfork bifurcation, and comparisons are made with some exact numerical results based on the numerical treatment of the non-linear boundary value problem. A boundary layer phenomenon, similar to that also observed for the linearized bending analysis of partially composite beams, is observed for large values of the connection parameter. This boundary layer phenomenon is investigated with a straightforward asymptotic expansion, that also is valid for large rotations. Finally, the paper analyses the effect of some additional imperfection eccentricities in the loading mode, that lead to some pre-bending phenomena.     

Free vibration of composite laminated plate with complicated cutout

Abstract

This paper presents the free vibration analysis of a composite laminated square plate with complicated cutout. The problem formulation is based on the higher order shear deformation plate theory HDST C⁰ coupled with a curved quadrilateral p-element. The elements of the stiffness and mass matrices are calculated analytically. The curved edges are accurately represented using the blending function method. A calculation program is developed to determine the fundamental frequencies for different physical and mechanical parameters such as the cutout shape, plate thickness, fiber orientation angle, and boundary conditions. The results obtained show a good agreement with the available solutions in the literature. New results for the fundamentals frequencies of a composite laminated plate with complicated cutout are presented.     

Dynamics of a pipe conveying fluid flexibly restrained at the ends

The subject of this paper is the study of dynamics and stability of a pipe flexibly supported at its ends and conveying fluid. First, the equation of motion of the system is derived via the extended form of Hamilton׳s principle for open systems. In the derivation, the effect of flexible supports, modelled as linear translational and rotational springs, is appropriately considered in the equation of motion rather than in the boundary conditions. The resulting equation of motion is then discretized via the Galerkin method in which the eigenfunctions of a free-free Euler–Bernoulli beam are utilized. Thus, a general set of second-order ordinary differential equations emerges, in which, by setting the stiffness of the end-springs suitably, one can readily investigate the dynamics of various systems with either classical or non-classical boundary conditions. Several numerical analyses are initially performed, in which the eigenvalues of a simplified system (a beam) with flexible end-supports are obtained and then compared with numerical results, so as to verify the equation of motion, in its simplified form. Then, the dynamics of a pinned-free pipe conveying fluid is systematically investigated, in which it is found that a pinned-free pipe conveying fluid is generally neutrally stable until it becomes unstable via a Hopf bifurcation leading to flutter. The next part of the paper is devoted to studying the dynamics of a pinned-free pipe additionally constrained at the pinned end by a rotational spring. A wide range of dynamical behaviour is seen as the mass ratio of the system (mass of the fluid to the fluid+pipe mass) varies. It is surprising to see that for a range of rotational spring stiffness, an increase in the stiffness makes the pipe less stable. Finally, a pipe conveying fluid supported only by a translational and a rotational spring at the upstream end is considered. For this system, the critical flow velocity is determined for various values of spring constants, and several Argand diagrams along with modal shapes of the unstable modes are presented. The dynamics of this system is found to be very complex and often unpredictable (unexpected).     

充液管道系统的模态分析

考虑管道与液体之间的泊松耦合与连接耦合,推导了低频情况下的充液直管轴向、横向振动传递矩阵与弯管单元的传递矩阵,引入了弯管弯曲因子,对－“L”形充液管系进行了研究,使用ANSYS有限元分析软件与传递矩阵法计算了两种边界条件下充液、充所管系的模态频率与模态振型,结果表明,气体与管道之间的耦合作用对管系模态的影响可忽略;而对充液管系而言,因液耦合作用不能忽略。     

Conforming radial point interpolation method for spatial shell structures on the stress-resultant shell theory

The implementation of the conforming radial point interpolation method (CRPIM) for spatial thick shell structures is presented in this paper. The formulation of the discrete system equations is derived from a stress-resultant geometrically exact theory of shear flexible shells based on the Cosserat surface. A discrete singularity-free mapping between the five degrees of freedom of the Cosserat surface and the normal formulation with six degrees of freedom is constructed by exploiting the geometry connection between the orthogonal group and the unit sphere. A radial basis function is used in both the construction of shape functions based on arbitrarily distributed nodes as well as in the surface approximation of general spatial shell geometries. The major advantage of the CRPIM is that the shape functions possess a delta function property and the interpolation function obtained passes through all the scattered points in the influence domain. Thus, essential boundary conditions can be easily imposed, as in finite element method. A range of shape parameters is studied to examine the performance of CRPIM for shells, and optimal values are proposed. The phenomena of shear locking and membrane locking are illustrated by presenting the membrane and shear energies as fractions of the total energy. Several benchmark problems for shells are analyzed to demonstrate the validity and efficiency of the present CRPIM. The convergence rate of the results using a Gaussian (EXP) radial basis is relatively high compared to those using a multi-quadric (MQ) radial basis for the shell problems.     

An automatic constraint violation stabilization method for differential/ algebraic equations of motion in multibody system dynamics

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Variable penalty method for finite element analysis of incompressible flow

A new scheme is applied for increasing the accuracy of the penalty finite element method for incompressible flow by systematically varying from element to element the sign and magnitude of the penalty parameter λ, which enters through ?.v + p/λ = 0, an approximation to the incompressibility constraint. Not only is the error in this approximation reduced beyond that achievable with a constant λ, but also digital truncation error is lowered when it is aggravated by large variations in element size, a critical problem when the discretization must resolve thin boundary layers. The magnitude of the penalty parameter can be chosen smaller than when λ is constant, which also reduces digital truncation error; hence a shorter word-length computer is more likely to succeed. Error estimates of the method are reviewed. Boundary conditions which circumvent the hazards of aphysical pressure modes are catalogued for the finite element basis set chosen here. In order to compare performance, the variable penalty method is pitted against the conventional penalty method with constant λ in several Stokes flow case studies.