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Extension of the polygonal contact model for flexible multibody systems
Abstract and Figures
The Polygonal Contact Model or PCM is originally an algorithm for contact analysis between rigid bodies with complicated geometries in multibody dynamics. In applications where the flexibility of contacting bodies is not negligible, rigid body contact modeling can not achieve realistic results and elastic body contact modeling should be utilized instead. Therefore, due to the importance and necessity of considering elastic deformations, the extension of PCM for elastic bodies was required and is described in this paper. With the described modifications and extensions PCM is also applicable for contact modeling between flexible bodies with reasonable efficiency. For this purpose, geometric and kinematic information is updated repeatedly during the simulation and different options are implemented and compared.
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... PCM has been implemented in several academic and commercial multibody tools, partly with substantial extensions like surface height correction, flexible body support and self contact [3][4][5]. Its areal approach and polygonal discretisation have been taken over in other contact models [6][7][8]. ...
The polygonal contact model is an established contact algorithm for multibody dynamics based on polygonal surfaces. Some guidance for its practical use is presented. In particular, assignment and discretization of the surface meshes are discussed with regard to correct results and optimal efficiency. Moreover, notes on stiffness and damping parameters are given.
An improved implementation of the polygonal contact model is introduced. The new algorithm goes without quite complicated intersection construction steps. As a consequence, it is less complex, more robust and usually more efficient than the classic implementation.
Application examples of varied complexity demonstrate both notes on usage and improved implementation of the polygonal contact model.
... These papers reflect the feasibility and efficiency of the PCM model. A detailed introduction to the PCM method is in [34]. The specific integration of the PCM algorithm into the DEM frame has been summarized by Zeng et al. [35,36], which is not the focus of this paper. ...
The discrete element method (DEM) is usually applied in analyzing the scientifical origin/evolution of the asteroids and the landing/sampling of the regolith. In order to manage the contact between the non-spherical granules, the Polygonal Contact Model (PCM) has been introduced into the DEM method. This paper applies four different contact force models in the newly-proposed DEM algorithm to analyze their difference and implication. The four contact force models include one linear model and three nonlinear models derived from the complete Mindlin–Deresiewicz equations. By considering the macroscopical results and calculation efficiency, the single-collision and multiple-collision cases are analyzed by comparing the four contact models. Specifically, the restitution coefficient, the angular velocity, the rebound angle, and the kinetic energy are applied as indicators for the single collision. The multiple-collision case is studied under the Brazil nut effect with ellipsoidal granules. Additionally, the softening feasibility is also discussed by decreasing the Young’s modulus of the material, mainly analyzing the outgoing results and the calculation efficiency.
... The general formulation of contact impact problems in a multi-body system requires consideration contact detection mechanisms [258][259][260][261][262] and the non-smooth response of the contacting bodies [263]. Here, we briefly review the latter phenomenon. ...
The impact-induced fracture analysis has a wide range of engineering and defence applications, including aerospace, manufacturing and construction. An accurate simulation of impact events often requires modelling large-scale complex geometries along with dynamic stress waves and damage propagation. To perform such simulations in a timely manner, a highly efficient and scalable computational framework is necessary.
This thesis aims to develop a high-performance computational framework for analysing large-scale structural problems pertaining to impact-induced fracture events. A hierarchical grid-based mesh containing octree cells is utilised for discretising the problem domain. The scaled boundary finite element method (SBFEM) is employed, which can efficiently handle the octree cells by eliminating the hanging node issues.
The octree-mesh is used in balanced form with a limited number of octree cell patterns. The master element matrices of each pattern are pre-computed while the storage of the individual element matrices is avoided leading to a significant reduction in memory requirements, especially for large-scale models. Further, the advantages of octree cells are leveraged by automatic mesh generation and local refinement process, which enables efficient pre-processing of models with complex geometries.
To handle the matrix operations associated with large-scale simulation, a pattern-by-pattern (PBP) approach is proposed. In this technique, the octree-patterns are exploited to recast a majority of the computational work into pattern-level dense matrix operations. This avoids global matrix assembly, allows better cache utilisation, and aids the associated memory-bandwidth limited computations, resulting in significant performance gains in matrix operations.
The PBP approach also supports large-scale parallelism. In this work, the parallel computation is carried out using the mesh-partitioning strategy and implemented using the message passing technique. It is shown that the developed solvers can simulate large-scale and complex structural problems, e.g. delamination/fracture in sandwich panels with approximately a billion unknowns (or DOFs). A massive scaling can be achieved with more than ten thousand cores in a distributed computing environment, which reduces the computation time from months (on a single core) to a few minutes.
... This increases the degree of discretization and causes the computing time to rise dramatically. A proper detection of the initial instant of contact is critical when dealing with these models [50,55,56]. ...
In the present work, different algorithms for contact detection in multibody systems based on smooth contact modelling approaches are presented. Beginning with the simplest ones, some difficult interactions are subsequently introduced. In addition, a brief overview on the different kinds of contact/impact modelling is provided and an underlining of the advantages and the drawbacks of each of them is determined. Finally, some practical examples of each interaction are presented and analyzed and an outline of the issues arisen during the design process and how they have been solved in order to obtain stable and accurate results is given. The main goal of this paper is to provide a resource for the early-stage researchers in the field that serves as an introduction to the modelling of simple contact/impact events in the context of multibody system dynamics.
... This forces to increase the discretization of the dynamic analysis, making the computing time to skyrocket. In these models, the proper detection of the initial instant of contact is critical [88,[101][102][103]. ...
In the present work, an introduction to the contact phenomena in multibody systems is made. The different existing approaches are described, together with their most distinctive features. Then, the term of coefficient of restitution is emphasized as a tool to characterize impact events and the algorithm for calculating the relative indentation between two convex-shaped bodies is developed. Subsequently, the main penalty contact models developed in the last decades are presented and developed, analysing their advantages and drawbacks, as well as their respective applications. Furthermore, some models with specific peculiarities that could be useful to the reader are included. The aim of this work is to provide a resource to the novice researcher in the field to facilitate the choice of the appropriate contact model for their work.
... In this approach, integrating the equations of motion is not halted when the contact happens [12]. Contact procedures are investigated in the literature as in [16][17][18][19]. ...
Clearances are necessary in assemblage of mechanisms to allow the relative motion between the members. This clearance is due to machining tolerances, wear, material deformations, and imperfections, and it can worsen mechanism performance such as precision and vibration. As a new study in this topic, the effect of joint stiffness on the variation of instantaneous natural frequencies and mode shapes of a flexible four-bar mechanism with a clearance between coupler and follower is studied in this paper. To model the clearance, the continuous contact approach is used. The Lankarani’s and Nikravesh’s continuous contact force model is used to model the contact force arising from contact between journal and bearing. Finite element method is used to determine the instantaneous natural frequencies and their corresponding mode shapes. The stiffness of the clearance is modeled as a linear spring added to the assembled stiffness matrix. To validate the clearance model in rigid mechanism, the dynamic response is compared with the results in the literature. To show the validity of the formulation which calculates the instantaneous natural frequencies, two methods are used and compared with each other in the case no clearance exists. The results show that taking the joint stiffness into account has a considerable effect on the instantaneous natural frequencies and their corresponding mode shapes of a flexible multibody system.
... In this approach, integrating the equations of motion is not stopped when the contact happens. Contact procedures are investigated e.g. in [2][3][4][5]. ...
In this paper, the method of multiple scales is used in an innovative way to conduct a vibration analysis of a mechanical system such as a sliding pendulum with multiple clearances. Analytical approaches can give more insight into such problems due to the nonlinearity involved. At first, it is assumed that a single clearance exists in the joint between the slider and the guide. Then, a second clearance is assumed in the joint position between the pendulum and the slider. The horizontal vibration of the system is ignored. For the first case, the clearance is modeled just as a nonlinear spring and in the second one a nonlinear damper is also considered which gives a more realistic model. The Lankarani-Nikravesh contact force model is used to represent the contact force between joined bodies. Primary resonance and internal resonance are discussed in detail. It is shown that the pendulum angle is not so affected by the clearance. The natural frequency corresponding to the oscillation of the pendulum is far from the one corresponding to the vertical motion of the system. However, the vertical motion of the system components can have influence on each other.
... This algorithm, referred to as polygonal contact model, is based on representation of the body surfaces by polygon meshes and the contact force evaluation is done by using an elastic foundation model. This approach has been used by other researchers (Ebrahimi et al. 2005;Ebrahimi and Eberhard 2006). He et al. (2007) presented a multigrid contact detection method, where the multigrid idea was integrated with contact detection problems. ...
In this book, several compliant contact force models are analyzed within the context of multibody dynamics, in which the main issues associated with the fundamental contact mechanics are also revisited. In particular, various con-tact force models from linear to nonlinear, purely elastic to dissipative con-tact force models are presented and their parameters are described. The different numerical methods and algorithms dealing with contact problems in multibody systems are presented and discussed. In present work, the gross motion of multibody systems is described by using a two-dimensional formulation based on the absolute coordinates and the contact-impact events are represented by different contact models. Results for some planar multi-body mechanical systems are presented and utilized to discuss the main assumptions and procedures adopted throughout this work. The material provided here indicates that the prediction of the dynamic behavior of mechanical systems involving contact-impact strongly depends on the selection of the contact force model. Overall, this book is aimed to provide a collective source for the multibody dynamics community and beyond on modeling contact forces and dynamics of mechanical systems undergoing contact-impact events.
This work reviews the main techniques to model dynamical systems with contact-impact events. Regularized and non-smooth formulations are considered, wherein the fundamental features associated with each approach are analyzed. A brief description of contact dynamics is presented, and an overview of the state-of-the-art of the main aspects related to the contact dynamics discipline is provided. This paper ends by identifying gaps in the current techniques and prospects for future research in the field of contact mechanics in multibody dynamics.
Purpose
The purpose of this paper is to investigate the effect of joint clearance on the behavior of a needle driver mechanism (a slider-crank linkage) of a typical sewing machine with an imperfect joint between the coupler and the slider (including needle).
Design/methodology/approach
In order to model the clearance, the momentum exchange approach is used. The Lankarani’s and Nikravesh’s continuous contact force model is used to model the contact force and the modified Coulomb’s friction law represents the friction between sliding members. The penetration force applied on the needle by fabric is chosen based on an experimental data in the literature. The dynamic response is validated for the existing properties in the literature without considering the penetration force.
Findings
It is shown that the clearance joint made considerable effect on the dynamic response of the system. The rough changes of the needle acceleration and jerk are obvious. The base reaction force changed roughly and did not vary as smooth as that of the mechanism with ideal joint. So, clearance joint in the mechanism could lead to an undesirable vibration in the system. Furthermore, the crank driver must provide a non-smooth moment on the crank to keep the crank rotational velocity constant. Moreover, reducing the clearance size sufficiently could make the dynamic response closer to that of the mechanism with ideal joint. In addition, smoother crank moment could be required if the clearance size is reduced sufficiently. Furthermore, the rough change of the base reaction force which can represent the vibration caused by the mechanism on the fixed frame could be reduced if the clearance size is small enough.
Originality/value
Lockstitch sewing machine is one of the most common apparel industrial machines. The needle driver mechanism of a sewing machine could have an important role for proper stitch forming. On the other hand, clearances are inevitable in assemblage of mechanisms to allow the relative motion between the members. This clearance is due to machining tolerances, wear, material deformations, and imperfections, and it can worsen mechanism performance such as precision, dynamic behavior and vibration. Unfortunately, despite the importance of the dynamic behavior of the needle driver mechanism from practical point of view, very little publications have focused especially on the investigation of the effect of clearance joint on the dynamic behavior of the sewing machine which could lead to undesired vibration of the system and shorter lifetime as a result. In this paper, the dynamic behavior of the system including, needle velocity and acceleration, crank moment and base reaction force was compared with that of the ideal mechanism. Finally, the effect of clearance size on the dynamic behavior of the system was investigated.
This is the second edition of the valuable reference source for numerical simulations of contact mechanics suitable for many fields like civil engineering, car design, aeronautics, metal forming, or biomechanics. Boundary value problems involving contact are of great importance in industrial applications in engineering such as bearings, metal forming processes, rubber seals, drilling problems, crash analysis of cars, rolling contact between car tires and the road, cooling of electronic devices... Other applications are related to biomechanical engineering design where human joints, implants or teeth are of consideration. Due to this variety, contact problems are today combined either with large elastic or inelastic deformations including time dependent responses. Thermal coupling also might have to be considered. Even stability behaviour has to be linked to contact. The topic of computational contact is described in depth providing an up-to-date treatment of different formulations, algorithms and discretisation techniques for contact problems which are established in the geometrically linear and nonlinear range. This book provides the necessary continuum mechanics background which includes the derivation of the contact constraints. Constitutive equations stemming from tribology which are valid at the contact interface are discussed in detail. Discretization schemes for small and finite deformations are discussed in depth. Solid and beam contact is considered as well as contact of unstable systems and thermomechanical contact. The algorithmic side covers a broad range of solution methods. Additionally adaptive discretisation techniques for contact analysis are presented as a modern tool for engineering design simulations. The present text book is written for graduate, Masters and PhD students, but also for engineers in industry, who have to simulate contact problems in practical application and wish to understand the theoretical and algorithmic background of contact treatment in modern finite element systems. For this second edition, illustrative simplified examples and new discretisation schemes as well as adaptive procedures for coupled problems are added. © Springer-Verlag Berlin Heidelberg 2006. All rights are reserved.
Based on a general hierarchical data structure, we present a fast algorithm for exact collision detection of arbitrary polygonal rigid objects. Objects consisting of hundreds of thousands of polygons can be checked for collision at interactive rates. The pre-computed hierarchy is a tree of discrete oriented polytopes (DOPs). An efficient way of realigning DOPs during the traversal of such trees allows us to use simple interval tests for determining the overlap between DOPs. The data structure is very efficient in terms of memory and construction time. Extensive experiments with synthetic and real-world CAD data have been carried out to analyze the performance and memory usage of the data structure. A comparison with oriented bounding box (OBB) trees indicates that DOP-trees are as efficient in terms of collision query time and more efficient in memory usage and construction time.
A new contact algorithm designed for multibody dynamics is presented. It is based on representation of the body surfaces by polygon meshes and contact force determination by the elastic foundation model. Areal discretisations of the contact patches are constructed using methods closely related to computer graphics, e.g. collision detection based on bounding vol- ume hierarchies and generation of subdivision surfaces by means of boundary representation data structures. Two examples prove the robustness of the method for complexly shaped bodies causing multiple and multiply bordered contact patches and conforming contacts.
We introduce the boxtree, a versatile data structure for representing triangulated or meshed surfaces in 3D. A boxtree is a hierarchical structure of nested boxes that supports efficient ray tracing and collision detection. It is simple and robust, and requires minimal space. In situations where storage is at a premium, boxtrees are effective alternatives to octrees and BSP trees. They are also more flexible and efficient than R-trees, and nearly as simple to implement.
Die vorliegende Arbeit beschreibt Grundlagen, Modellierung, Implementierung und Anwendung eines Verfahrens zur Analyse von Kontaktvorgängen zwischen komplex geformten Körpern im Rahmen der Mehrkörperdynamik. Nach einer kurzen Einführung werden einige Grundlagen der Simulation von Mehrkörpersystemen erläutert. Es folgt eine Übersicht der wichtigsten Methoden zur Kontaktanalyse, wobei zwischen Festkörper- und Oberflächenmodellen sowie eindimensionalen Modellen unterschieden wird. Das im Rahmen dieser Arbeit entwickelte Polygonflächen-Kontaktmodell basiert auf Körperoberflächen in Polygondarstellung. Die ausführlich erläuterten Methoden und Algorithmen zur Kollisionserkennung sowie zur Approximation und Diskretisierung der Kontaktgebiete weisen daher eine enge Verwandtschaft zur Computergrafik auf. Zur Bestimmung der Kontaktspannungen wird auf die Kontaktelemente ein um viskose Dämpfung erweitertes Randschichtmodell sowie eine regularisierte Version des Coulombschen Reibegesetzes angewendet. Vier Simulationsbeispiele unterschiedlicher Komplexität demonstrieren die Anwendung des Verfahrens. Mehrfache, mehrfach berandete und schmiegende Kontakte werden robust und effizient ausgewertet. Die Arbeit schließt mit einer Bewertung der Ergebnisse und einem Ausblick.
Impact-contact problems including frictional effects appear in many technical applications. For these problems a finite element formulation is presented which is based on a new frictional interface law and fully implicit algorithmic treatment for the integration of the constitutive relations and the dynamics. The first is performed via radial return schemes, which are well established in computational plasticity. As radial return methods are amenable to consistent linearizations, a full Newton-type algorithm can be constructed within a time step. This leads, however, to non-symmetric tangent matrices. Several examples compare this method with other formulations and shows its efficiency.
Dynamics of Multibody Systems introduces multibody dynamics, with an
emphasis on flexible body dynamics. Many common mechanisms such as
automobiles, space structures, robots, and micro machines have
mechanical and structural systems that consist of interconnected rigid
and deformable components. The dynamics of these large-scale, multibody
systems are highly nonlinear, presenting complex problems that in most
cases can only be solved with computer-based techniques. The book begins
with a review of the basic ideas of kinematics and the dynamics of rigid
and deformable bodies before moving on to more advanced topics and
computer implementation. This new edition includes important new
developments relating to the problem of large deformations and numerical
algorithms as applied to flexible multibody systems. The book's wealth
of examples and practical applications will be useful to graduate
students, researchers, and practicing engineers working on a wide
variety of flexible multibody systems.
Collision detection and response can make a virtual-reality application seem more believable. Unfortunately, existing collision-detection algorithms are too slow for interactive use. The authors present a new algorithm that is not only fast but also interruptible, allowing an application to trade quality for more speed. The algorithm uses simple four-dimensional geometry to approximate motion, and sets of spheres to approximate three-dimensional surfaces. The algorithm allows a sample application to run five to seven times faster than it runs with existing algorithms.< >