Dynamics Of Flexible Multibody Systems Research Articles

Friction Models and Stress Recovery Methods in Vehicle Dynamics Modelling

This paper presents two examples of calculations for vehicles with flexible bodies by using mixed multibody and finite element methods. The first example deals with dynamics computations for a bimodal train with a flexible cistern, whereas the second example concerns the dynamics calculations for the PW-6 glider. In the first example, the influence of the chosen friction model on the train dynamics calculations results was discussed. The second example presents several methods of stress calculations and a comparison of results. The achieved conclusions may be used as suggestions towards a modelling method choice for a given problem.Both issues being discussed are of great importance in dynamics of flexible multibody systems modelling practice and durability assessment.

Positional FEM formulation for flexible multi-body dynamic analysis

This paper presents a simple formulation to deal with flexible multi-body dynamic systems by the finite element method. The proposed methodology is based on the minimum potential energy theorem written regarding nodal positions. Velocity, acceleration and strain are achieved directly from positions, not displacements. A non-dimensional space is created and the relative curvature and fibers length are calculated for both reference and deformed configurations and used to calculate the strain energy at general points. The classical Newmark equations are used to integrate time. Damping is introduced into the mechanical system by a rheonomic energy functional. The final formulation has the advantage of being simple and easy to teach, when compared to classical counterparts. The behavior of a bench-mark problem (spin-up maneuver) is studied regarding the influence of mass representation on its overall transient and steady-state behavior. Three other examples are presented to show the applicability of the technique, namely, a flexible slider–crank mechanism, a flexible beam flight and a Peaucellier-type mechanism. The results are compared with other authors’ numerical solutions.

Optimization of Flexible Multibody Dynamic Systems Using the Equivalent Static Load Method

Recently, an algorithm for dynamic response optimization transforming dynamic loads into equivalent static loads has been proposed. In later research, it was proved that the solution obtained by the algorithm satisfies the Karush-Kuhn-Tucker necessary conditions. In the present research, the proposed algorithm is applied to the optimization of flexible multibody dynamic systems. The equivalent static load is derived from the equations of motion for a flexible multibody dynamic system. The equivalent load is utilized in sequential static response optimization of the flexible mutibody dynamic system. In the end, the converged solution of the sequential static response optimization is the solution of the original dynamic response optimization. Some standard examples are solved to show the feasibility and efficiency of the proposed method. The control arm of an automobile suspension system is optimized as a practical problem. The results are discussed regarding the application of the proposed algorithm to flexible multibody dynamic systems.

동하중을 받는 구조물의 최적화에 관한 연구동향

Various aspects of structural optimization techniques under transient loads are extensively reviewed. The mam themes of the paper are treatment of time dependent constraints, calculation of design sensitivity, and approximation Each subject is reviewed with the corresponding papers that have been published since 1970s. The treatment of time dependent constraints in both the direct method and the transformation method is discussed. Two ways of calculating design sensitivity of a structure under transient loads are discussed-direct differentiation method and adjoint variable method. The approximation concept mainly focuses on response surface method in crashworthiness and local approximation with the intermediate variable. Especially, as an approximated optimization technique, Equivalent Static Load method Which takes advantage of the well-established static response optimization technique is introduced And as an application area of dynamic response optimization technique, the structural optimization In flexible multibody dynamic systems is re-viewed in the viewpoint of the above three themes.

The Poincaré-Chetayev equations and flexible multibody systems

Problems of the dynamics of flexible multibody systems (FMBSs) and its relation to the fundamental system of equations obtained by Poincare about 100 years ago [1] are considered. These equations, called the Poincare-Chetayev equations, are now well known as the basis of the Lagrange reduction theory. By extending these equations to the case of the motion of a Cosserat medium it is shown that in the dynamics of FMBS it is possible to use two principal systems of equations. It is proved that a generalized Newton-Euler model of FMBSs in projections onto floating axes and the partial differential equations of the non-linear, geometrically exact theory within the Galilean approach comprise the Poincare-Chetayev equations.

Study on the dynamic characteristics of a virtual-axis hybrid polishing machine tool by flexible multibody dynamics

This paper establishes the dynamic equations of the spatial rigid-flexible coupling multibody systems for a five-coordinate virtual-axis hybrid polishing machine tool, namely model JDYP51 developed by the authors. The dynamic characteristics of the hybrid machine tool are studied in detail by means of the theory of dynamics of a flexible multibody system and the finite element models of the whole machine tool and each leg respectively. The dynamic simulation of the machine shows that the elastic deformation of each leg is very small during the movement of the machine and the influence of the flexibility of each leg on the trace of moving platform is not obvious due to the characteristics of joints. At the beginning of the movement of the machine, the fundamental frequency of parallel mechanism easily leads to vibration of the machine. Therefore, the stable status of polishing can be obtained at all periods of the movement of the machine except the beginning of the movement.

Experimental Validation of a Flexible MBS Dynamic Formulation through Comparison between Measured and Calculated Stresses on a Prototype Car

Numerical methods for the dynamics of flexible multi-body systems havebeen developed by the related scientific and technical community duringthe last two or three decades. However, most examples which can be foundin the bibliography are either very simple and/or have not beenconfronted with the results of real tests. The solution of realistic andcomplex problems along with their corresponding experimental validation,will be useful to evaluate the different formulations available, andwill make them more valuable for the industrial world.

Application of Plasticity Theory and Absolute Nodal Coordinate Formulation to Flexible Multibody System Dynamics

The objective of this investigation is to develop a nonlinear finite element formulation for the elastic-plastic analysis of flexible multibody systems. The Lagrangian plasticity theory based on J2 flow theory is used to account for the effect of plasticity in flexible multibody dynamics. It is demonstrated that the principle of objectivity that is an issue when existing finite element formulations using rate-type constitutive equations are used is automatically satisfied when the stress and strain rate are directly calculated in the Lagrangian descriptions using the absolute nodal coordinate formulation employed in this investigation. This is attributed to the fact that, in the finite element absolute nodal coordinate formulation, the position vector gradients can completely define the state of rotation and deformation within the element. As a consequence, the numerical algorithm used to determine the plastic deformations such as the radial return algorithm becomes much simpler when the absolute nodal coordinate formulation is used as compared to existing finite element formulations that employ incrementally objective algorithms. Several numerical examples are presented in order to demonstrate the use of the formulations presented in the paper.

Non-linear dynamics of flexible multibody systems

In this work we set to examine several important issues pertinent to currently very active research area of the finite element modeling of flexible multibody system dynamics. To that end, we first briefly introduce three different model problems in non-linear dynamics of flexible 3D solid, a rigid body and 3D geometrically exact beam, which covers the vast majority of representative models for the particular components of a multibody system. The finite element semi-discretization for these models is presented along with the time-discretization performed by the mid-point scheme. In extending the proposed methodology to modeling of flexible multibody systems, we also present how to build a systematic representation of any kind of joint connecting two multibody components, a typical case of holonomic contraint, as a linear superposition of elementary constraints. We also indicate by a chosen model of rolling contact, an example of non-holonomic constraint, that the latter can also be included within the proposed framework. An important aspect regarding the reduction of computational cost while retaining the consistency of the model is also addressed in terms of systematic use of the rigid component hypothesis, mass lumping and the appropriate application of the explicit–implicit time-integration scheme to the problem on hand. Several numerical simulations dealing with non-linear dynamics of flexible multibody systems undergoing large overall motion are presented to further illustrate the potential of presented methodology. Closing remarks are given to summarize the recent achievements and point out several directions for future research.

Dynamics of Flexible Multibody Systems with Non-Holonomic Constraints: A Finite Element Approach

In this article it is shown how non-holonomic constraints can beincluded in the formulation of the dynamic equations of flexiblemultibody systems. The equations are given in state space formwith the degrees of freedom, their derivatives and the kinematiccoordinates as state variables, which circumvents the use ofLagrangian multipliers. With these independent state variables forthe system the derivation of the linearized equations of motion isstraightforward. The incorporation of the method in a finiteelement based program for flexible multibody systems is discussed.The method is illustrated by three examples, which show, amongother things, how the linearized equations can be used to analysethe stability of a nominal steady motion.

Modeling and Control of Elastic Bodies. 1st Report. Proposals the Extended Reduced Order Physical Model.

This paper proposes a new modeling method of elastic body to represent the rigid motion and elastic deformation. This model can be used to control flexible multibody systems like flexible manipulators, vehicles, and space structures etc. The model consists of mass particles, springs, and rigid bodies and the behavior of mass particles represents rigid motion and deformation of elastic bodies. The mass and stiffness matrices are calculated using the modal mass, natural frequencies, and mode shapes. The efficacy of the model is verified by considering an example of an elastic car body supported by a rectangular plate. This modeling procedure is also extended to represent the dynamics of flexible multibody systems.

Dynamics and control of flexible multibody systems: Part II: simulation code and parametric studies with nonlinear control

This paper describes a rather general numerical simulation code newly developed for studying dynamics and control of flexible multibody systems. It accounts for an arbitrary level of topological branching, character of the structural members, slewing, deployment, orbital perturbations, etc. The program is based on a recursive order n formulation whose details are given in Part I. Versatility and validity of the code are illustrated through parametric studies of several distinctively different systems. The dynamical analysis indicates that critical situations can lead to an undesirable system response. To establish specified performance, feasibility of a nonlinear control strategy based on the feedback linearization technique (FLT) in conjunction with the order n algorithm is explored. The FLT can control both the rigid and flexible degrees of freedom. The effectiveness of the control is assessed through numerical simulation.

On rigid components and joint constraints in nonlinear dynamics of flexible multibody systems employing 3D geometrically exact beam model

In this work, we discuss the finite element implementation of the internal constraints in a three-dimensional (3D) geometrically exact beam model which can be formulated as holonomic constraint relationships. Model problems chosen for a more detailed consideration include a general joint constraint between beams and the beam connected to a rigid component. Appropriate modifications of the standard form of the geometrically exact beam element arrays are carried out in order to impose explicitly these kinds of constraints into time-integration schemes for nonlinear dynamics. Consequently, only the minimum number of unknowns is retained for the global set of nonlinear equations to be solved, thus avoiding the use of the extra variables (the Lagrange multipliers) and the pertinent difficulties in integrating the system of differential–algebraic equations. A number of numerical simulations considering dynamic analysis of multibody systems with rigid–flexible components and joint constraints are presented in order to illustrate versatility of the proposed procedure.

Finite Element Method in Dynamics of Flexible Multibody Systems: Modeling of Holonomic Constraints and Energy Conserving Integration Schemes

In this work we discuss an application of the finite elementmethod to modeling of flexible multibody systems employing geometricallyexact structural elements. Two different approaches to handleconstraints, one based on the Lagrange multiplier procedure and anotherbased on the use of release degrees of freedom, are examined in detail.The energy conserving time stepping scheme, which is proved to be wellsuited for integrating stiff differential equations, gouverning themotion of a single flexible link is appropriately modified and extendedto nonlinear dynamics of multibody systems.

Conserving Properties in Constrained Dynamics of Flexible Multibody Systems

The context of this work is the non-linear dynamics ofmultibody systems (MBS). The approach followed for parametrisation ofrigid bodies is the use of inertial coordinates, forming a dependent setof parameters. This approach mixes naturally with nodal coordinates in adisplacement-based finite element discretisation of flexible bodies,allowing an efficient simulation for MBS dynamics. An energy-momentumtime integration algorithm is developed within the context of MBSconstraints enforced through penalty methods. The approach follows theconcept of a discrete derivative for Hamiltonian systems proposed byGonzalez, achieving exact preservation of energy and momentum. Thealgorithm displays considerable stability, overcoming the traditionaldrawback of the penalty method, namely numerical ill-conditioning thatleads to stiff equation systems. Additionally, excellent performance isachieved in long-term simulations with rather large time-steps.

Flexible Multibody Dynamic Analysis Using Multirate Integration Method

A Nordsick form opf the multirate integration scheme has been proposed for flexible multibody dynamic systems. It is assumed that vibrational modal coordinates in the equations of motion are treated as fast variables, whereas the relative joint coordinates are treated as slow variables. In the multirate integration, the fast variables are integrated with small step-size, and the slow variables are integrated with larger step-size. The proposed multirate integration method is based on the Adams-Bashforth-Moulton predictor-corrector method and implemented in the Nordsieck vector form. The Nordsieck form of multrate integration method provides effective step-size control and at the same time, inherits the efficiency from the Adams integration method. Simulations of a flexible gun and turret system of the military tank have been carried out to show the effectiveness and efficiency of the proposed method.

Modeling and analysis of a coupled rigid-flexible system

Correct predictions of the behavior of flexible bodies undergoing large rigid-body motions and small elastic vibrations is a subject of major concern in the field of flexible multibody system dynamics. Because of failing to account for the effects of dynamic stiffening, conventional methods based on the linear theories can lead to erroneous results in many practical applications. In this paper, the idea of “centrifugal potential field”, which induced by large overall rotation is introduced, and the motion equation of a coupled rigid-flexible system by employing Hamilton's principle is established. Based on this equation, first it is proved that the elastic motion of the system has periodic property, then by using Frobenius' method its exact solution is obtained. The influences of large overall rigid motion on the elastic vibration mode shape and frequency are analysed through the numerical examples.

Dynamics of Flexible Multibody Systems having Rolling Contact: Application of the Wheel Element to the Dynamics of Road Vehicles

SUMMARYThis paper presents the wheel on a surface as a special finite element for the description of the geometry of the rolling contact. The two-dimensional wheel and the general case of a thin disc on an arbitrary spatial surface are treated; they are suitable wheel models to investigate approximately the rolling contact phenomena in road vehicles. A procedure for formulating the equations of motion for flexible non-holonomic systems and their linearization is presented as well. The method is illustrated by the analysis of shimmy behaviour of a swivel wheel. As a road vehicle example the IAVSD benchmark problem of the Bombardier ILTIS vehicle is solved and some results are shown.

Analysis of joint reaction forces of flexible multibody system with closed loops

The analysis of dynamic forces is essential to the design of systems, stress analysis, or life prediction of part of machine. Calculation of dynamic forces has very close relations with multibody dynamics algorithm. In this paper, an algorithm which calculates joint reaction force/moment of flexible multibody dynamic systems is proposed by using inverse dynamic algorithm and velocity transformation technique.

An efficient and accurate dynamic stress computation by flexible multibody dynamic system simulation and reanalysis

This paper presents an efficient and accurate method for dynamic stress computation based on reanalysis in flexible multibody dynamic system simulation. The mode acceleration concept that is widely used in linear strucural dynamics was utilized for accuracy improvement. A mode-acceleration equation for each flexible body is defined and the load term in the right hand side of the equation is represented as a combination of space-dependent and time-dependent terms so that efficient computations of dynamic stresses can be achieved. The load term is obtained from dynamic simulation of a flexible multibody system and a finite element method is used to compute stresses by quasi-static analyses. A numerical example of a flexible four-bar mechanism shows effectiveness of the proposed method for flexible multibody dynamic systems such as linkages and vehicle systems.