TRITA-MEK ISSN 0348-467X ISRN KTH/MEK/TR--99/06--SE Efficient Multibody Dynamics by Anders Lennartsson June 1999 Technical Reports from Royal Institute of Technology Department of Mechanics S-100 44 Stockholm, Sweden Typsatt i LATEX med Anders thesis-stil. Akademisk avhandling som med tillst˚andav Kungl Tekniska H¨ogskolan framl¨ag- ges till offentlig granskning f¨oravl¨aggandeav teknisk doktorsexamen fredagen den 4 juni 1999 kl 10.15 i Kollegiesalen, Administrationsbyggnaden, Kungl Tek- niska H¨ogskolan, Valhallav¨agen79, Stockholm c Anders Lennartsson 1999 Norstedts Tryckeri AB, Stockholm 1999 Errata Compared to the printed version the following corrections have been made: Whole document The hyperref LATEX-package is used to make this pdf-file clickable. In addition, the LATEX-code of all instances of \i.e.",\e.g.", and \etc." have been corrected to create better word spacing. Beginning This errata inserted. Page vii First line of second paragraph, \and and" changed to \and". Page vii End of fourth paragraph, Piirionen changed to Piiroinen. Page 11 `Equation (2.4) is not changed, and this file does not have a strange spacing which is in the printed version and was caused by a bug in the type setters postscript interpretor. The same applies to equation (2.12) on page 13. Page 21 \it's" changed to \its" on second line of third paragraph. 2 Page 24 Last term of equation (2.41), denominator corrected from @xj to @xk@xj. Page 25 Appropriate use of @ in the denominator of the third order term of equation (2.52). Page 69 Caption to figure 5.3, second line, \a a" changed to \a". i i Page 85 Last line, fj changed to f;j. Page 99 Caption to figure 6.4, second line, \given the text" changed to \given in the text". Page 134 End of paragraph on Table, \methodof" changed to \method of". Bibliography All URL:s changed by use of hyperref url-command, some mod- ified to end in /. Efficient Multibody Dynamics Anders Lennartsson 1999 Department of Mechanics, Royal Institute of Technology S-100 44 Stockholm, Sweden Abstract The dynamics of macroscopic objects connected to each other by joints is of fundamental importance in physics, engineering and everyday life. Such systems are commonly referred to as multibody systems and are well known to be non- linear systems. Multibody systems consisting of particles and rigid bodies are considered in this thesis. Understanding their dynamics is essential for design, optimization, and control of current and new technology involving these types of systems. For a long time, investigators were constrained by the lack of computational power. With the invention of the digital computer this changed abruptly. Today software for modeling and investigating properties of multibody systems are common engineering tools. The main purpose of this study is to enhance the present methods and tools used in the area. Both the derivation of equations of motion and the analysis of given equations are investigated. Symbolic and numeric computations are utilized where appropriate. Efficient algorithms for vector algebra and calculus using software for symbolic computations are proposed. Use of these methods significantly reduce the efforts needed for practical analysis of some common types of mechanisms. Numerical analysis of dynamical systems often involve simulation, computing stability of fixed points and periodic motion. Analytical stability calculations depend on the Jacobian and the variational equations of the dynamical system. This work presents a unified environment for the actual generation of efficient code for computing such properties in combination with the differential equations of the base system. Discontinuous systems can be analyzed if suitable integration methods are used. Several versions of the equations of motion may be necessary due to different behaviour before and after impacts. On the theoretical side, kinematic transformations are examined for possible ways of reducing the computational effort necessary to evaluate the equations of motion. A power equation for systems with and without nonholonomic con- straints is derived. The derivation includes a short and illustrative argument for the relationship between Kane's equations and various forms of the virtual power formulations. Descriptors: multibody systems, nonlinear dynamics, power equations, vari- ational equations, code-generation, kinematic transformations, diagonalization, bicycle, symbolic computations. TRITA-MEK • ISSN 0348-467X • ISRN KTH/MEK/TR--99/06--SE Preface This monograph concerns modeling and analyzing of multibody systems, i.e. mechanical systems consisting of bodies connected with joints. It consists of seven chapters, of which the the first two serves as introduction and background material. The following five chapters deal with different aspects of the modeling and analysis process. In chapter three velocity transformations are examined for possible findings of efficient methods and theoretical gains. A power equation is derived in chapter four together with compact expressions for the terms in this equation. Chapter five deals with complexity issues of algorithms for describing geometry and motion of multibody systems. Efficient tools for numerical investi- gations is the subject of chapter six. In the last chapter, two systems are studied as application examples of the methods discussed in the previous chapters. Some investigation of their dynamics is performed. v Acknowledgments I am glad to have the opportunity to publicly thank several people who have made my time at the Department of Mechanics interesting and most enjoyable. First and foremost I thank my advisor Professor Martin Lesser for his guidance, constant encouragement, and patience. I am very grateful to him for sharing his knowledge in mechanics and related topics, and allowing me to pursue my own ideas. I am deeply indebted to Dr. Hanno Ess´enfor his support and his valuable comments on the material. Dr. Ess´enand Dr. Arne Nordmark have provided much knowledge and insight in the field of dynamics. In addition, Arne has always taken the time to answer my questions on computational issues and com- puter related problems, for which I thank him sincerely. I consider their influence an important part of my doctoral training. In a joint project with Dr. Annika Stensson and Lars Drugge at the division of Computer Aided Design, Lule˚aTechnical University, I have had the opportunity to visit Lule˚aon several occasions. I thank them for providing a memorable and productive environment during my visits. It has been a pleasure to work with my fellow students and racing car fans Dr. Mats Fredriksson and Jesper Adolfsson. Another member of our research group, Dr. Harry Dankowicz, I thank for constructive criticism and discussions on any subject. In the early part of studies I worked with Claes Tisell at the Department of Machine elements and I value discussions with him highly. While not a member of the group, my room-mate Per Olsson is a jolly fellow. I wish him and the younger students in the group, Gitte Ekdahl and Petri Piirionen the best of luck in their future careers. The collegues and friends at the department are the ones that provide the friendly environment. Many thanks to all of you! Finally, I thank my family, Ulrika and our son Jacob, for their endless support. Financial support from the Swedish Research Council for Engineering Sci- ences (TFR) and Banverket is gratefully acknowledged. vii Contents 1 Introduction1 2 Methods of multibody dynamics7 2.1 Mechanism configuration and kinematics..............7 2.1.1 Reference frames.......................8 2.1.2 Coordinates..........................9 2.1.3 Constraints.......................... 10 2.1.4 Kinematics.......................... 11 2.2 Dynamics............................... 11 2.2.1 Equations of motion..................... 11 2.2.2 Formulations......................... 12 2.3 Analysis tools............................. 19 2.3.1 Differential geometry..................... 19 2.3.2 Dynamical systems...................... 22 2.3.3 Numerical integration.................... 28 2.3.4 Computer algebra systems.................. 29 3 Kinematic transformations 31 3.1 Formulation.............................. 32 3.1.1 Matrix factorization..................... 33 3.1.2 Gram-Schmidt transformations............... 35 3.2 Physical interpretation........................ 37 3.3 Pair of swinging legs......................... 38 3.4 Numerical efficiency......................... 43 3.4.1 Numerical performance................... 45 3.5 Discussion............................... 45 4 Power 47 4.1 Introduction.............................. 47 4.2 Derivations.............................. 48 4.3 Results................................. 52 4.4 Example................................ 53 4.5 Conclusions.............................. 56 ix 5 Delayed component evaluation 59 5.1 Introduction.............................. 59 5.2 Implementation............................ 61 5.2.1 Relative orientations..................... 62 5.2.2 Vector algebra........................ 63 5.2.3 Vector calculus........................ 64 5.2.4 Utilizing angular velocity.................. 66 5.2.5 Function family for partitioned vectors........... 67 5.3 Examples............................... 68 5.4 Performance evaluation....................... 69 5.4.1 The n-pendulum case.................... 69 5.4.2 The n-pendulum with Sophia................ 70 5.5 Results................................. 73 5.6 Discussion............................... 73 6 Processing of ODE 75 6.1 Introduction.............................