{"title":"Reducing the Constrained Multibody Dynamics Problem to the Solution of a System of ODEs via Velocity Partitioning and Lie Group Integration","authors":"Alexandra Kissel, Luning Bakke, D. Negrut","doi":"10.1115/1.4065254","DOIUrl":null,"url":null,"abstract":"\n In multibody dynamics, formulating the equations of motion in absolute Cartesian coordinates results in a set of index-3 differential algebraic equations (DAEs). In this work, we present an approach that bypasses the DAE problem by partitioning the velocities in the system into dependent and independent coordinates, thereby reducing the task of producing the time evolution of the mechanical system to one of solving a set of ordinary differential equations (ODEs). In this approach, the independent coordinates are integrated directly, while the dependent coordinates are recovered through the kinematic constraint equations at the position and velocity levels. Notably, Lie group integration is employed to directly obtain the orientation matrix A at each time step of the simulation. This eliminates the need to choose generalized coordinates to capture the orientation of a body, as the matrix A is a byproduct of the solution algorithm. The approach is akin to the method presented in [1], which combines coordinate partitioning with an Euler parameter formulation. Herein, we outline the new approach and demonstrate it in conjunction with four mechanisms: a single pendulum, a double pendulum, a four-link mechanism, and a slider crank. We report on the convergence order behavior of the proposed method and a performance comparison with the solution introduced in [1]. The Python code developed to generate the reported results is open-source and available in a public repository for reproducibility studies [2].","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":"61 6","pages":""},"PeriodicalIF":16.4000,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"5","ListUrlMain":"https://doi.org/10.1115/1.4065254","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
In multibody dynamics, formulating the equations of motion in absolute Cartesian coordinates results in a set of index-3 differential algebraic equations (DAEs). In this work, we present an approach that bypasses the DAE problem by partitioning the velocities in the system into dependent and independent coordinates, thereby reducing the task of producing the time evolution of the mechanical system to one of solving a set of ordinary differential equations (ODEs). In this approach, the independent coordinates are integrated directly, while the dependent coordinates are recovered through the kinematic constraint equations at the position and velocity levels. Notably, Lie group integration is employed to directly obtain the orientation matrix A at each time step of the simulation. This eliminates the need to choose generalized coordinates to capture the orientation of a body, as the matrix A is a byproduct of the solution algorithm. The approach is akin to the method presented in [1], which combines coordinate partitioning with an Euler parameter formulation. Herein, we outline the new approach and demonstrate it in conjunction with four mechanisms: a single pendulum, a double pendulum, a four-link mechanism, and a slider crank. We report on the convergence order behavior of the proposed method and a performance comparison with the solution introduced in [1]. The Python code developed to generate the reported results is open-source and available in a public repository for reproducibility studies [2].
期刊介绍:
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.