Formulation Of Multibody Systems Research Articles

Third-order geometric stiffness formulation for improved mesh convergence of thin and wide spatial beams in the generalized strain beam formulation

AbstractThis paper presents the stiffness formulation of a beam element with the relevant third-order nonlinear geometric effects for relatively wide and thin rectangular beams, in particular when loaded in the plane and simultaneously deformed out of the plane. The element is initially straight in its undeformed configuration. The formulation is based on Timoshenko beam theory with nonuniform torsion and Wagner effects. The derivation is carried out by means of the Hellinger–Reissner variational principle with custom interpolation functions. The element is incorporated into the generalized strain beam formulation for multibody systems. Numerical simulations of precision flexure mechanisms show that the use of a single third-order element per flexible member can already yield adequate performance, at a significant reduction of the necessary degrees of freedom and the computation time, compared with using multiple second-order elements in the generalized strain beam formulation.

Computational efficiency of multi-body systems dynamic models

AbstractFor several decades, simulation and analysis of mechanisms have been performed with dedicated computer-aided engineering software that implements general dynamic formulations, known in the literature as multi-body systems (MBS) formulations. The MBS name is related to the structure of the mechanism, which is often considered to be a collection of bodies interconnected by mechanical joints (pairs). Nevertheless, only a few formulations are really based on a true multi-body mechanical model, while many others instantiate mathematically quite different mechanical concepts. This paper aims to identify and discuss the mechanical models that fundament the main multi-body mathematical formulations known in the literature. The main features of each model are outlined, based on a detailed presentation of the structure and equations of motion, together with their link with the kinematic and dynamic formulations. A comparative study related to computational efficiency is then presented for the identified main models, based on a test mechanism. Comparative advantages and disadvantages are discussed at the end, considering the identified mechanical models.

A three-dimensional approach for contact detection between realistic wheel and rail surfaces for improved railway dynamic analysis

The wheel-rail contact modeling problem assumes a preponderant role on the dynamic analysis of railway systems using multibody systems formulations. The accurate and efficient evaluation of both location and magnitude of the wheel-rail contact forces is fundamental for the development of reliable computational tools. The wheel concave zone might be a source of numerical difficulties when searching the contact points, which has been neglected in several works. Here, it is demonstrated that the minimum distance method does not always converge when the wheel surface is not fully convex, being an alternative methodology proposed to perform the contact detection. This approach examines independently the contact between each wheel strip and the rail, where the maximum virtual penetration is determined and associated with the location of the contact point. Then, an Hertzian-based force model is considered for both normal and tangential forces. The results obtained from dynamic simulations show that the minimum distance method and the proposed methodology provide a similar response for simplified wheel profiles. However, the new approach described here is reliable in the identification of the contact point when realistic wheel profiles are considered, which is not the case with the minimum distance method.

A new version of the Riccati transfer matrix method for multibody systems consisting of chain and branch bodies

Computational speed and stability are two important aspects in the dynamics analysis of large-scale complex multibody systems. In order to improve both in the context of the multibody system transfer matrix method, a new version of the Riccati transfer matrix method is presented. Based on the new version of the general transfer matrix method for multibody system simulation, recursive formulae are developed which not only retain all advantages of the transfer matrix method, but also reduce the truncation error. As a result, the computational speed, accuracy and efficiency are improved. Numerical computation results obtained by the proposed method and an ordinary multibody system formulation show good agreement. The successful computation for a spatial branch system with more than 100000 degrees of freedom validates that the proposed method is also working for huge systems.

Floating Frame of Reference formulation for modeling flexible multi-body systems in premise operational conditions

The Floating Frame of Reference (FFR) formulation is a well-established and reliable method for modeling flexible multi-body systems. The FFR formulation has been adopted for dynamic systems that are characterized by large rotations with relatively small deformations. Many scientific papers have pointed out these characteristics with the scrutiny of the simulation results and comparisons with other modeling techniques. However, the FFR is still enclosed in the theoretical aspects and simulation work and faces difficulties when being applied to practical systems. The crucial point in these difficulties centers on coordinate reduction and the associated mapping between nodal and modal coordinates. The process of selecting the necessary modes may be theoretically simple, but the situation is different when applied in a real operational environment. The strategy developed in this work combines the Operational Modal Analysis, specifically the Frequency Domain Decomposition (FDD) approach, and the FFR formulation to build a suitable model of practical multi-body systems. The output model has been validated experimentally. The results show that the proposed FFR-FDD method can be efficiently used to construct multi-body models for those systems that work in premise operational conditions.

Modeling and analysis of multiple impacts in multibody systems under unilateral and bilateral constrains based on linear projection operators

This paper presents a unifying dynamics formulation for nonsmooth multibody systems (MBSs) subject to changing topology and multiple impacts based on a linear projection operator. An oblique projection matrix ubiquitously yields all characteristic variables of such systems as follows: (i) the constrained acceleration before jump discontinuity from the projection of unconstrained acceleration, (ii) post-impact velocity from the projection of pre-impact velocity, (iii) impulse during impact from the projection of pre-impact momentum, (iv) generalized constraint force from the projection of generalized input force, and (v) post-impact kinetic energy from pre-impact kinetic energy based on projected inertia matrix. All solutions are presented in closed-form with elegant geometrical interpretations. The formulation is general enough to be applicable to MBSs subject to simultaneous multiple impacts with nonidentical restitution coefficients, changing topology, i.e., unilateral constraints becoming inactive or vice versa, or even when the overall constraint Jacobian becomes singular. Not only do the solutions always exist regardless of the constraint condition, but also the condition number for a generalized constraint inertia matrix is minimized in order to reduce numerical sensitivity in computation of the projection matrix to roundoff errors. The model is proven to be energetically consistent if a global restitution coefficient is assumed. In the case of nonidentical restitution coefficients, the set of energetically consistent restitution matrices is characterized by using a linear matrix inequality (LMI).

Efficient embedding of empirically-derived constraints in the ODE formulation of multibody systems: Application to the human body musculoskeletal system

We present a novel method for deriving the governing equations of the musculoskeletal system, a new class of multibody systems in which the constituent components are connected together via anatomical joints which behave differently compared with traditional mechanical joints. In such systems, the kinematics of the joints and the corresponding constraints are characterized experimentally. We generate the equations of motion of these complex systems in which the homogeneous transformation matrices become matrix-valued functions of the generalized coordinate vector due to the empirical expression of body coordinates as smooth functions of generalized coordinates. The detailed mathematical procedure is provided to derive each term of the equations of motion using the novel calculus for the efficient evaluation of the partial derivatives of matrix-valued functions with respect to a vector. The governing equations obtained using the presented technique are expressed with ordinary differential equations rather than algebraic differential equations while not suffering from any simplification in experimental data describing the kinematics of the system. We then apply this method to derive the equations of motion of the “Andrews’ squeezer mechanism” for the validation. Furthermore, we successfully use this technique to model the shoulder rhythm with empirically-derived constraints in a trajectory tracking problem.

A projection-based approach for the derivation of the floating frame of reference formulation for multibody systems

The reduction in the number of coordinates for flexible multibody systems is necessary in order to achieve acceptable simulation times of real-life structures and machines. The conventional model order reduction technique for flexible multibody systems is based on the floating frame of reference formulation (FFRF), using a rigid body frame and superimposed small flexible deformations. The FFRF leads to strongly coupled terms in rigid body and flexible coordinates as well as to a non-constant mass matrix. As an alternative to the FFRF, a formulation based on absolute coordinates has been proposed which uses a co-rotational strain. In this way, a constant mass matrix and a co-rotational stiffness matrix are obtained. In order to perform a reduction in the number of coordinates, by means of the component mode synthesis, e.g., the number of modes needs to be increased, such that all modes are represented in every possible rotated configuration. This approach leads to the method of generalized component mode synthesis (GCMS). The present paper shows in detail how the equations of motion of the FFRF evolve from the ones of the GCMS by considering rigid body constraint conditions and subsequently eliminating them via an appropriate null-space projection. This approach allows a straightforward, term-by-term interpretation of the FFRF mass matrix and of the generalized gyroscopic forces, which, to the same extent, cannot be deduced from former publications on the FFRF. From a practical point of view, the resulting expressions allow to calculate all inertia coefficients from the constant finite element mass matrix together with standard input data of the finite element model in the course of a preprocessing step. Then, the repeated updates of the FFRF mass matrix and of the gyroscopic forces in the course of time integration involve only simple vector matrix operations of low dimensions. In contrast to previous implementations of the FFRF, no evaluations of extra inertia integrals are required. Consequently, the present formulation can be implemented entirely independent of the related finite element code.

A method for analyzing the revolute joint wear of planar multibody systems and its application in unbalance loading working condition

This paper develops an efficient method to describe the evolution of the revolute joint wear during the process of the mechanism movement. In the framework of multibody systems formulation, the prediction method of the revolute joint wear in different motion time interval is established by considering the wear ranges between the pin and the bushing on the basis of the Archard’s wear model. Then a typical non-planar mechanism with unbalancing loads acting on single non-ideal revolute joint is used to illustrate the application of the method. Moreover, the corresponding experimental mechanism is tested to validate the predicting results of the proposed method. The analysis shows that the method proposed is of high efficiency and can guarantee enough predicting accuracy when the predicted wear depth is not big. Therefore, the method is practical to a certain degree in engineering.

A flexible natural coordinates formulation (FNCF) for the efficient simulation of small‐deformation multibody systems

SummaryThis paper introduces the novel flexible natural coordinates formulation to model small‐deformation multibody dynamics. The main contribution of this work is its resulting constant mass matrix and quadratic constraint equations devoid of any other nonlinearities. These properties are similar to those of a natural coordinates formulation for rigid multibody systems with the addition of constant damping and stiffness matrices to model the flexibility under the assumption of small deformations. As such, it is a straightforward extension to natural coordinates while maintaining its beneficial properties. The main concept of the current approach is to introduce ample redundancy in the set of generalized coordinates to simplify the kinematics ensuring the aforementioned properties and the similarity to a natural coordinates approach. This is not achievable by standard techniques that introduce redundancy. Not only does this offer a very simple equation structure but also interesting properties toward the development of system‐level model order reduction techniques for flexible multibody systems as well as a straightforward parameter gradient extraction. The formulation accuracy is validated with a floating frame of reference implementation.

Nonlinear dynamics of slender structures: a new object-oriented framework

With this work, we present a new object-oriented framework to study the nonlinear dynamics of slender structures made of composite multilayer and hyperelastic materials, which combines finite element method and multibody system formalism with a robust integration scheme. Each mechanical system under consideration is represented as a collection of infinitely stiff components, such as rigid bodies, and flexible components like geometrically exact beams and solid-degenerate shells, which are spatially discretized into finite elements. The semi-discrete equations are temporally discretized for a fixed time increment with a momentum-preserving, energy-preserving/dissipative method, which allows the systematic annihilation of unresolved high-frequency content. As usual in multibody system dynamics, kinematic constraints are employed to render supports, joints and structural connections. The presented ideas are implemented following the object-oriented programming philosophy. The approach, which is perfectly suitable for wind energy or aeronautic applications, is finally tested and its potential is illustrated by means of numerical examples.

On the use of absolute interface coordinates in the floating frame of reference formulation for flexible multibody dynamics

In this work a new formulation for flexible multibody systems is presented based on the floating frame formulation. In this method, the absolute interface coordinates are used as degrees of freedom. To this end, a coordinate transformation is established from the absolute floating frame coordinates and the local interface coordinates to the absolute interface coordinates. This is done by assuming linear theory of elasticity for a body’s local elastic deformation and by using the Craig–Bampton interface modes as local shape functions. In order to put this new method into perspective, relevant relations between inertial frame, corotational frame and floating frame formulations are explained. As such, this work provides a clear overview of how these three well-known and apparently different flexible multibody methods are related. An advantage of the method presented in this work is that the resulting equations of motion are of the differential rather than the differential-algebraic type. At the same time, it is possible to use well-developed model order reduction techniques on the flexible bodies locally. Hence, the method can be employed to construct superelements from arbitrarily shaped three dimensional elastic bodies, which can be used in a flexible multibody dynamics simulation. The method is validated by simulating the static and dynamic behavior of a number of flexible systems.

A Study on the Dynamics of Spatial Mechanisms With Frictional Spherical Clearance Joints

An investigation on the dynamic modeling and analysis of spatial mechanisms with spherical clearance joints including friction is presented. For this purpose, the ball and the socket, which compose a spherical joint, are modeled as two individual colliding components. The normal contact-impact forces that develop at the spherical clearance joint are determined by using a continuous force model. A continuous analysis approach is used here with a Hertzian-based contact force model, which includes a dissipative term representing the energy dissipation during the contact process. The pseudopenetration that occurs between the potential contact points of the ball and the socket surface, as well as the indentation rate play a crucial role in the evaluation of the normal contact forces. In addition, several different friction force models based on the Coulomb's law are revisited in this work. The friction models utilized here can accommodate the various friction regimens and phenomena that take place at the contact interface between the ball and the socket. Both the normal and tangential contact forces are evaluated and included into the systems' dynamics equation of motion, developed under the framework of multibody systems formulations. A spatial four-bar mechanism, which includes a spherical joint with clearance, is used as an application example to examine and quantify the effects of various friction force models, clearance sizes, and the friction coefficients.

System-Based Approaches for Structural Optimization of Flexible Mechanisms

This paper reviews the state-of-the-art methods to perform structural optimization of flexible mechanisms. These methods are based on a system-based approach, i.e. the formulation of the design problem incorporates the time response of the mechanism that is obtained from a dynamic simulation of the flexible multibody system. The system-based approach aims at considering as precisely as possible the effects of nonlinear dynamic loading under various operating conditions. Also, the optimization process enhances most existing studies which are limited to (quasi-) static or frequency domain loading conditions. This paper briefly introduces flexible multibody system dynamics and structural optimization techniques. Afterwards, the two main methods, named the weakly and the fully coupled methods, that couple both disciplines are presented in details and the influence of the multibody system formalism is analyzed. The advantages and drawbacks of both methods are discussed and future possible research areas are mentioned.

Dynamic response multiobjective optimization of road vehicle ride quality—A computational multibody system approach

Computational multibody system (MBS) method is a practical technique utilized for modeling, simulation, and optimization of mechanical systems. In the methodology of computational multibody system, equations of motion are derived, formulated, and solved through a systematic, generalized, and well-structured computational-mathematical approach. In this paper, the computational multibody system formulation, based on the appended Lagrangian method, is implemented to establish the governing equations of ride dynamics for a nonlinear ride model that represents a versatile half-car two-track model of a road vehicle. The input to the system is a simulated road surface model based on the ISO road surface classification. The solution of the equation of motion is obtained using the direct integration approach along with constraint violation elimination and control techniques. Following the simulation, a time-domain multiobjective design optimization procedure is performed to improve the ride quality of the model. The ride quality comprises both ride comfort and ride safety. The optimization considers relevant objective functions including vibration isolation, suspension travel, road holding, and force index. The results of work show that the proposed method could acceptably estimate the optimal values of design variables for specific road classes and vehicle driving speeds. The simulation-based ride quality optimization performed here could facilitate improvement of suspension and tyre.

Contact force control in multibody pantograph/catenary systems

In this paper, a new continuum-based pantograph/catenary model based on the absolute nodal coordinate formulation (ANCF) is proposed and used to develop an effective method to control the contact force which arises from the pantograph/catenary interaction. In the proposed new model, only one ANCF gradient vector is used in the formulation of the pantograph/catenary contact conditions, thereby allowing for using the proposed approach for both fully parameterized and gradient-deficient ANCF finite elements. The proposed contact formulation can also be considered as a more general sliding joint formulation that allows for the use of the more efficient gradient-deficient ANCF finite elements in modeling very flexible cables. A three-dimensional multibody system (MBS) model of a pantograph mounted on a train is developed using a nonlinear augmented MBS formulation. In order to take into account the catenary large deformation, ANCF finite elements are used. The contact between the pantograph and the catenary system is ensured using a sliding joint constraint whereas the contact between the rail vehicle wheels and the train track is modelled using an elastic contact formulation. In addition to the use of the new MBS approach to model the pantograph/catenary interaction, the contact force between the pantograph and the catenary is computed using a simpler lumped parameter model which describes the pan-head and the plunger subsystem dynamics. In order to reduce the standard deviation of the contact force without affecting its mean value, a control actuator is used between the pan-head and the plunger. To this end, three types of control laws for the control action are designed to improve the contact quality both in the transient phase and in the steady state phase of the pantograph/catenary interaction. The first control law proposed features a feedback structure whereas the second and the third control strategies employ a feedback plus feed-forward architecture. In order to demonstrate the effectiveness of the proposed method, the results of a set of numerical simulations with and without the controllers are presented.

Static form-finding analysis of a railway catenary using a dynamic equilibrium method based on flexible multibody system formulation with absolute nodal coordinates and controls

This paper proposes a dynamic equilibrium method for finding the initial equilibrium configuration of a railway catenary. In the proposed method, the catenary composed of flexible wires is modeled using two-node cable elements with absolute nodal coordinates based on a flexible multibody system formulation. Dynamic conditions that characterize the initial equilibrium configuration of the catenary are given and addressed as control processes in the form-finding procedure. The key feature of the proposed method is that the catenary configuration is continually evolved by dynamic simulation until characterization conditions are attained and an equivalent configuration of the centenary at static equilibrium is thus computed. It is validated using two examples and applied to the form-finding analysis of a two dimensional simple railway catenary. The obtained results are analyzed and discussed. It is general and can be applied to catenaries with complex configurations.

An Efficient Dynamic Formulation for Solving Rigid and Flexible Multibody Systems Based on Semirecursive Method and Implicit Integration

This paper presents a method for solving the dynamic equations of multibody systems containing both rigid and flexible bodies. The proposed method uses independent coordinates and projects the dynamic equations on the constraint tangent manifold by means of a velocity transformation matrix. It can be used with a wide variety of integration formulae, considering both fixed and variable stepsizes. Topological semirecursive methods are used to take advantage of the relatively small number of parameters needed. An in depth implementation analysis is performed in order to evaluate the terms involved in the integration process. Numerical and stability issues are also discussed.

Co-simulation method for solver coupling with algebraic constraints incorporating relaxation techniques

Based on the stabilized index-2 formulation for multibody systems, a semi-implicit co-simulation approach for solver coupling with algebraic constraints has been presented by Schweizer and Lu (Multibody Syst. Dyn., 2014) for the case that constant approximation is used for extrapolating/interpolating the coupling variables. In the manuscript at hand, this method is generalized to the case that higher-order approximation is employed. Direct application of higher-order polynomials for extrapolating/interpolating the coupling variables fails. Using linear approximation polynomials, artificial oscillations in the Lagrange multipliers of the kinematical differential equations are observed. For quadratic and higher-order polynomials, the co-simulation becomes unstable. In this work, the key idea to obtain stable solutions without artificial oscillations is to apply a relaxation technique. A detailed stability and convergence analysis is presented in the paper for the case of higher-order approximation. In this context, the influence of the relaxation parameter on the stability and convergence behavior is investigated. Applicability and robustness of the stabilized index-2 co-simulation method incorporating higher-order approximation polynomials is demonstrated with different numerical examples. Using piecewise constant approximation polynomials for the coupling variables produces discontinuous accelerations and reaction forces in the subsystems at the macrotime points, which may entail problems for the subsystem integrator. With higher-order approximation polynomials, the coupling variables and in consequence the accelerations and reaction forces in the subsystems become continuous.

Guest Editorial

Multibody system (MBS) dynamics is a very open discipline in the sense that different authors propose drastically different approaches. Several steps are required to address the simulation of a multibody system: modeling, coordinates selection, formulation of the equations of motion, integration of the equations of motion, and computer implementation of the final algorithm. Many alternatives are possible for each of them. Moreover, these steps are tightly related such that a decision made for one step usually affects some of the others. And, in addition, there can be issues often just tiny details not even described in papers, that must be taken into account in order to have a successful implementation of a MBS formulation. In multibody dynamics, no formulation can be considered as the best for all problems, and different approaches can lead to very different results for a given problem, both from the accuracy and efficiency points of view.In today's multibody dynamics discipline, flexibility and contact are also essential ingredients, whose consideration may also affect some aspects of the formulation. Another topic that has a big impact on formulations is parallelization: some methods that were suitable for sequential computing are not adaptable to parallel environments, while other algorithms become more efficient when several processors are used. There are other topics, such as control, optimization, or multiphysics cosimulation that may also affect the way a formulation is developed, or applications having peculiarities that require special adaptations such as rotor or human body dynamics.For these reasons, this special issue on multibody dynamics formulations has intended to gather papers documenting recent and most powerful MBS formulations and approaches by some representative groups of the international multibody community. The thirteen papers and one technical brief are organized as follows.The first four papers describe full multibody formulations along with related considerations and applications: implicit Newton–Euler formulation with constraints in algebraic form, unified mixed coordinate formulation on the special Euclidean group, and divide-and-conquer algorithm (two connected papers).The fifth and sixth papers point out particular aspects or applications of already known formulations: adaptation of the operational space formulation to wheeled vehicles, and determination of reactions from nonholonomic constraints in an index-3 augmented Lagrangian formulation with projections of velocity and acceleration.The seventh paper is fully devoted to parallelization in multibody dynamics. The eighth and ninth papers deal with issues that find applicability in control and optimization, respectively, (although not only in them): underactuated systems with specified in time outputs, and graph theory for sensitivity equations obtained by direct differentiation.The tenth paper is about contact, presenting a general and efficient method for the contact detection problem, while the last three papers and the technical brief address flexibility issues: description of the beam section to avoid errors due to simplified theories, plasticity and hyperelasticity in a multibody formulation, consideration of flexibility in symbolic codes, and derivation of velocity-dependent inertia terms in the floating frame of reference formulation irrespective of the rigid-body rotational parameterization.I wish to thank the authors, reviewers and journal staff for their commitment and effort, which made possible to complete this special issue on time. Finally, my gratitude to Professor Ahmed Shabana who, as Editor of the journal, offered me the possibility of compiling this set of excellent works and always provided his wise advice and kind support.