Dynamics Of Flexible Multibody Systems Research Articles

A Stable Inversion Method for Feedforward Control of Constrained Flexible Multibody Systems

The inverse dynamics of flexible multibody systems is formulated as a two-point boundary value problem for an index-3 differential-algebraic equation (DAE). This DAE represents the equation of motion with kinematic and trajectory constraints. For so-called nonminimum phase systems, the remaining dynamics of the inverse model is unstable. Therefore, boundary conditions are imposed not only at the initial time but also at the final time in order to obtain a bounded solution of the inverse model. The numerical solution strategy is based on a reformulation of the DAE in index-2 form and a multiple shooting algorithm, which is known for its robustness and its ability to solve unstable problems. The paper also describes the time integration and sensitivity analysis methods that are used in each shooting phase. The proposed approach does not require a reformulation of the problem in input-output normal form, which is known from nonlinear control theory. It can deal with serial and parallel kinematic topology, minimum phase and nonminimum phase systems, and rigid and flexible mechanisms.

Methods & Prospects of Research on Flexible Multi-Body Dynamics

This paper summarized the research method of flexible multi-body system dynamics. Three stages of flexible multi-body system model are reviewed. The paper is especially summary on modeling method of flexible multi-body system, flexible multi-body dynamical equation numerical calculation and flexible multi-body system vibration control and so on. And the paper is looked forward to further study.

Interaction of flexible multibody systems with fluids analyzed by means of smoothed particle hydrodynamics

The present work deals with a computational approach to fluid-structure interaction (FSI) problems by coupling of flexible multibody system dynamics and fluid dynamics. Since the methods for the numerical modeling are well known, both for the structural and the fluid part, the focus of this work lies on the coupling formalism. Moreover, the applicability of the presented approach to arbitrary geometries and high structural stiffness is studied, as well as an easy model setup. No restriction should be made on the topology of the structure or the complexity of motion. For the fluid part a meshless method, known as smoothed particle hydrodynamics (SPH) is applied, which fulfills the above requirements. While an explicit time integration scheme in SPH provides a fast simulation of the fluid dynamics, advanced methods from flexible multibody dynamics provide a variety of benefits for the simulation of the solid part. Amongst these are specialized structural finite elements for both small and large deformation bodies, joints, stable implicit time-integration schemes, and model reduction techniques. A rule for the interaction between fluids and structures is derived from imposing a distributed potential over boundary segments of the structures, which the fluid particles respond to. The work is concluded by illustrative examples, demonstrating the successful coupling of flexible multibody systems with fluids.

Absolute nodal coordinate plane beam formulation for multibody systems dynamics

A new plane beam dynamic formulation for constrained multibody system dynamics is developed. Flexible multibody system dynamics includes rigid body dynamics and superimposed vibratory motions. The complexity of mechanical system dynamics originates from rotational kinematics, but the natural coordinate formulation does not use rotational coordinates, so that simple dynamic formulation is possible. These methods use only translational coordinates and simple algebraic constraints. A new formulation for plane flexible multibody systems are developed utilizing the curvature of a beam and point masses. Using absolute nodal coordinates, a constant mass matrix is obtained and the elastic force becomes a nonlinear function of the nodal coordinates. In this formulation, no infinitesimal or finite rotation assumptions are used and no assumption on the magnitude of the element rotations is made. The distributed body mass and applied forces are lumped to the point masses. Closed loop mechanical systems consisting of elastic beams can be modeled without constraints since the loop closure constraints can be substituted as beam longitudinal elasticity. A curved beam is modeled automatically. Several numerical examples are presented to show the effectiveness of this method.

With the Backlash Dynamics Simulation of a Crank-Slider Mechanism of the Internal Combustion Engine

On the basis of flexible multi-body system dynamics theory, we built flexible multi-body system dynamics models which include a backlash, and to a slider-crank mechanism as the research object, we made a preliminary study on the effect on the flexible components and the backlash of the kinematic pair on mechanical system dynamics characteristics. To consider the backlash of the kinematic pair and component of flexible space can show a preliminary research on the dynamic simulation, and focus on the backlash, friction and gravity field to influence in the dynamic characteristics of the system. The simulation results show that, due to the existence of backlash made the two components frequent collision in the process of the stretching, clearance, flexible and friction are closed, make the system nonlinear characteristics increased.

With the Backlash Dynamics Simulation of a Crank-Slider Mechanism

On the basis of flexible multi-body system dynamics theory, we built flexible multi-body system dynamics models which include a backlash, and to a slider-crank mechanism as the research object, we made a preliminary study on the effect on the flexible components and the backlash of the kinematic pair on mechanical system dynamics characteristics. To consider the backlash of the kinematic pair and component of flexible space can show a preliminary research on the dynamic simulation, and focus on the backlash, friction and gravity field to influence in the dynamic characteristics of the system. The simulation results show that, due to the existence of backlash made the two components frequent collision in the process of the stretching, clearance, flexible and friction are closed, make the system nonlinear characteristics increased.

First order sensitivity analysis of flexible multibody systems using absolute nodal coordinate formulation

Design sensitivity analysis of flexible multibody systems is important in optimizing the performance of mechanical systems. The choice of coordinates to describe the motion of multibody systems has a great influence on the efficiency and accuracy of both the dynamic and sensitivity analysis. In the flexible multibody system dynamics, both the floating frame of reference formulation (FFRF) and absolute nodal coordinate formulation (ANCF) are frequently utilized to describe flexibility, however, only the former has been used in design sensitivity analysis. In this article, ANCF, which has been recently developed and focuses on modeling of beams and plates in large deformation problems, is extended into design sensitivity analysis of flexible multibody systems. The Motion equations of a constrained flexible multibody system are expressed as a set of index-3 differential algebraic equations (DAEs), in which the element elastic forces are defined using nonlinear strain-displacement relations. Both the direct differentiation method and adjoint variable method are performed to do sensitivity analysis and the related dynamic and sensitivity equations are integrated with HHT-I3 algorithm. In this paper, a new method to deduce system sensitivity equations is proposed. With this approach, the system sensitivity equations are constructed by assembling the element sensitivity equations with the help of invariant matrices, which results in the advantage that the complex symbolic differentiation of the dynamic equations is avoided when the flexible multibody system model is changed. Besides that, the dynamic and sensitivity equations formed with the proposed method can be efficiently integrated using HHT-I3 method, which makes the efficiency of the direct differentiation method comparable to that of the adjoint variable method when the number of design variables is not extremely large. All these improvements greatly enhance the application value of the direct differentiation method in the engineering optimization of the ANCF-based flexible multibody systems.

Dynamic factor analysis considering elastic boom effects in heavy lifting operations

The dynamic factor is the ratio of the maximum dynamic load to the static load acting on the wire ropes between the boom of a floating crane and a cargo. In this paper, the dynamic factor is analyzed based on dynamic simulations of a floating crane and a cargo, considering an elastic boom. For the simulation, we designed a multibody system that consists of a floating crane barge, an elastic boom, and a cargo connected to the boom through wire ropes. The dynamic equations of motion of the system are based on flexible multibody system dynamics. Six-degree-of-freedom motions are considered for the floating crane and for the cargo, and three-dimensional deformations for the elastic boom. The hydrostatic force, the hydrodynamic force, the gravitational force, and the wire rope forces are considered as external forces. The dynamic factor is obtained by numerically solving the equation. The effects of the elastic boom on heavy cargo lifting are discussed by comparing the simulation results of an elastic boom and a rigid boom.

A topological optimization method for flexible multi-body dynamic system using epsilon algorithm

In a flexible multi-body dynamic system the typical topological optimization method for structures cannot be directly applied, as the stiffness varies with position. In this paper, the topological optimization of the flexible multi-body dynamic system is converted into structural optimization using the equivalent static load method. First, the actual boundary conditions of the control system and the approximate stiffness curve of the mechanism are obtained from a flexible multi-body dynamical simulation. Second, the finite element models are built using the absolute nodal coordination for different positions according to the stiffness curve. For efficiency, the static reanalysis method is utilized to solve these finite element equilibrium equations. Specifically, the finite element equilibrium equations of key points in the stiffness curve are fully solved as the initial solution, and the following equilibrium equations are solved using a reanalysis method with an error controlled epsilon algorithm. In order to identify the efficiency of the elements, a non-dimensional measurement is introduced. Finally, an improved evolutional structural optimization (ESO) method is used to solve the optimization problem. The presented method is applied to the optimal design of a die bonder. The numerical results show that the presented method is practical and efficient when optimizing the design of the mechanism.

Galerkin meshfree methods applied to the nonlinear dynamics of flexible multibody systems

Meshfree Galerkin methods have been developed recently for the simulation of complex mechanical problems involving large strains of structures, crack propagation, or high velocity impact dynamics. At the present time, the application of these methods to multibody dynamics has not been made despite their great advantage in some situations over standard finite element techniques.

Technical overview of the equivalent static loads method for non-linear static response structural optimization

Linear static response structural optimization has been developed fairly well by using the finite element method for linear static analysis. However, development is extremely slow for structural optimization where a non linear static analysis technique is required. Optimization methods using equivalent static loads (ESLs) have been proposed to solve various structural optimization disciplines. The disciplines include linear dynamic response optimization, structural optimization for multi-body dynamic systems, structural optimization for flexible multi-body dynamic systems, nonlinear static response optimization and nonlinear dynamic response optimization. The ESL is defined as the static load that generates the same displacement field by an analysis which is not linear static. An analysis that is not linear static is carried out to evaluate the displacement field. ESLs are evaluated from the displacement field, linear static response optimization is performed by using the ESLs, and the design is updated. This process proceeds in a cyclic manner. A variety of problems have been solved by the ESLs methods. In this paper, the methods are completely overviewed. Various case studies are demonstrated and future research of the methods is discussed.

Numerical and experimental studies on impact dynamics of a planar flexible multibody system

In this paper a computational methodology on impact dynamics of the flexible multibody system is presented. First, the floating frame of reference approach and nodal coordinates on the basis of finite element formulation are used to describe the kinematics of planar deformable bodies. According to the kinematic description of contact conditions, the contact constraint equations of planar flexible bodies are derived. Based on the varying topology technique the impact dynamic equations for a planar multibody system are established. Then the initial conditions of the equations in each contact stage are determined according to the discontinuity theory in continuum mechanics. The experiments between the aluminum rods are performed to check the correctness of the proposed method. Through the comparison between the numerical and experimental results the proposed method is validated. Experimental results also show that the impulse momentum method cannot accurately predict the complex impact dynamic phenomena and the continuous model may lead to a serious error when used to simulate the impact problems with significant wave propagation effects.

중량물의 동적 거동에 미치는 크레인 붐(boom)의 탄성 영향 분석

In this paper, in order to analyze the dynamic response of a floating crane when it lifts a heavy cargo, the boom of the floating crane is considered as an elastic beam. The boom is divided into elements based on finite element formulation and the floating frame of reference formulation and nodal coordinates are employed to model the boom as a flexible body. As an extension of the previous study, in order to consider spatial motion in waves, the coupled equations of motions of the 6 degree of freedom (DOF) floating crane and 6 DOF cargo are developed based on the flexible multibody system dynamics. The 3 dimensional deformation of the elastic boom is considered with 18 DOF. The dynamic simulation of the floating crane and the cargo is performed under regular wave conditions with various cargo weights. Finally, the effects of the elastic boom on lifting cargo are discussed by comparing the simulation results between the elastic boom and a rigid boom.

붐(Boom)의 탄성을 고려한 해상크레인의 비선형 정적/동적 거동을 위한 수치 해석

해상크레인은 크레인을 탑재한 선박으로서, 조선소에서 대형 블록이나 구조물의 탑재 및 해상 운송 작업에 사용된다. 본 논문에서는 해상크레인과 중량물의 전후 동요(Surge), 상하 동요(Heave), 종 동요(Pitch)에 대한 정적/동적 거동을 분석하였다. 이 때, 유연 다물체계 동역학을 적용하여 해상크레인의 붐(boom)을 탄성으로 고려하였으며, 플로팅 프레임(floating frame)과 노드 좌표(nodal coordinates)를 사용하였다. 질량 행렬, 탄성 강성 행렬, 2 차 속도 벡터, 일반화 좌표 방향으로 작용하는 외력 등을 고려하여 모든 운동이 연성된 비선형 운동 방정식을 구성하였다. 외력으로는 비선형 유체정역학 힘, 선형화된 유체동역학 힘, wire rope 힘, 계류력이 고려되었다. 수치 해석을 위해 Hilber-Hughes-Taylor 방법을 비선형 운동방정식에 적용하였다. 정적 거동 분석을 통한 정적 평형 자세를 고려한 경우와 고려하지 않은 경우에 대해 결과를 비교하였으며, 수치 해석 방법에 대한 정적/동적 거동 분석 결과를 비교하였다.

붐의 탄성효과를 고려한 해상크레인의 유연 다물체 동역학 해석

This study analyzes the dynamic response of a floating crane with a cargo considering an elastic boom to evaluate(or for evaluation of) its flexibility effect on their dynamic response. Flexible multibody system dynamics is applied in order to establish a dynamic equation of motion of the multibody system, which consists of flexible and rigid bodies. In addition, a floating reference frame and nodal coordinates are used to model the boom as a flexible body. The study also simulates the coupled surge, pitch, and heave motions of the floating crane carrying the cargo with three degrees of freedom by numerically solving the equation. Finally, the simulation results of the elastic and rigid booms are comparatively analyzed and the effects of the flexible boom are discussed.

Analytical mechanics solution for mechanism motion and elastic deformation hybrid problem of beam system

Based on the dynamics of flexible multi-body systems and finite element method, a beam system dynamics model is built for solving motion–deformation mixed problem and tracing the whole process of mechanism motion. Kinetic control equation and constraint equation in which, mechanism motion and elastic deformation is described using hybrid coordinates, and the spatial position matrix of the element is described using Euler quaternion are derived. Numerical examples show that the method can trace and solve the track and internal force of the system.

Dynamics of spatial flexible multibody systems with clearance and lubricated spherical joints

A computational methodology for analysis of spatial flexible multibody systems, considering the effects of the clearances and lubrication in the system spherical joints, is presented. The dry contact forces are evaluated through a Hertzian-based contact law, which includes a damping term representing the energy dissipation. The frictional forces are evaluated using a modified Coulomb’s friction law. In the case of lubricated joints, the resulting lubricant forces are derived from the corresponding Reynolds’ equation. An absolute nodal formulation is utilized in flexible body formulation. The generalized -α method is used to solve the resulting equations of motion. The effectiveness of the methodology is demonstrated by two numerical examples.

Geometric nonlinear effects on the planar dynamics of a pivoted flexible beam encountering a point-surface impact

Flexible-body modeling with geometric nonlinearities remains a hot topic of research by applications in multibody system dynamics undergoing large overall motions. However, the geometric nonlinear effects on the impact dynamics of flexible multibody systems have attracted significantly less attention. In this paper, a point-surface impact problem between a rigid ball and a pivoted flexible beam is investigated. The Hertzian contact law is used to describe the impact process, and the dynamic equations are formulated in the floating frame of reference using the assumed mode method. The two important geometric nonlinear effects of the flexible beam are taken into account, i.e., the longitudinal foreshortening effect due to the transverse deformation, and the stress stiffness effect due to the axial force. The simulation results show that good consistency can be obtained with the nonlinear finite element program ABAQUS/Explicit if proper geometric nonlinearities are included in the floating frame formulation. Specifically, only the foreshortening effect should be considered in a pure transverse impact for efficiency, while the stress stiffness effect should be further considered in an oblique case with much more computational effort. It also implies that the geometric nonlinear effects should be considered properly in the impact dynamic analysis of more general flexible multibody systems.

A study of the inverse dynamics of a hybrid polishing kinematical machine tool based on the flexible multi-body systems

This paper presents a novel 5-DOF Hybrid Polishing Kinematical Machine Tool (HPKMT), which is made up of '3 axes parallel + 2 axes series' to obtain more stable machining result in the elastic polishing on the free-form surfaces. The dynamic characteristics of HPKMT are studied in detail by means of the theory of dynamics of flexible multi-body systems. Its characteristics of the kinematics and dynamics need be known deeply to control HPKMT effectively. The analysis of the inverse of mechanism is the important part in the research of dynamics. Adopting the same experiment condition with simulation analysis, the inverse kinematical trace of the three sliders can be obtained. It is shown that the simulation result is almost the same as the measured result from the experiment.

Multibody Formulation with Large Bending Finite Elements (LBFE)

The goal of this paper is to present a flexible multibody formulation for Euler-Bernoulli beams involving large displacements. This method is based on a discretisation of internal and kinetic energies. The beam is represented by its line of centroids and each section is oriented by a frame defined by three Euler angles. We apply a finite element formulation to describe the evolution of these angles along the neutral fibre and describe the internal energy. The kinetic energy is approximated as the one of two rigid bars tangent to the neutral fibre at the ends of the beam. We derive the equations of motion from a Lagrange formulation. These equations are solved using the Newmark method or/and the Newton-Raphson technique. We solve some very classic problems taken from the literature as the curved beam presented by Simo [Simo, J. C., ‘A three-dimensional finite-strain rod model. the three-dimensional dynamic problem. Part I’, Comput. Meths. Appl. Mech. Engrg.49, 1985, 55–70; Simo, J. C. and Vu-Quoc, L., ‘A three-dimensional finite-strain rod model, Part II: Computationals aspects’, Comput. Meths. Appl. Mech. Engrg.58, 1988, 79–116] and Lee [Lee, Kisu, ‘Analysis of large displacements and large rotations of three-dimensional beams by using small strains and unit vectors’, Commun. Numer. Meth. Engrg.13, 1997, 987–997] or the rotational rod presented by Avello [Avello, A., Garcia de Jalon, J., and Bayo, E., ‘Dynamics of flexible multibody systems using cartesian co-ordinates and large displacement theory’, Int. J. Num. Methods in Engineering32, 1991, 1543–1563] and Simo [Simo, J. C. and Vu-Quoc, L., ‘On the dynamics of flexible beams under large overall motions – the planar case. Part I’ Jour. of Appl. Mech.53, 1986, 849–854; Simo, J. C. and Vu-Quoc, L., ‘On the dynamics of flexible beams under large overall motions – the planar case. Part II’, Jour. of Appl. Mech.53, 1986, 855–863].