A Comparison Case Study For Dynamics Analysis Methods In Applied Multibody Dynamics

This paper discusses how a simple comparison case study has been utilized in an applied multibody dynamics (AMD) course to enhance students’ learning of dynamic analysis methods to set up equations of motion for multibody systems. The comparison case used is a planar rigid body double pendulum with a pin joint connection between two bodies. This simple case has helped students directly understand and see advantages and disadvantages of each dynamic analysis method used to set up equations of motion. Based on what they have learned from this case study, students have a better understanding of targeted dynamic analysis methods and can more efficiently choose a proper method to analyze the motion behaviors of their design applications than they could previously. Introduction An applied multibody dynamics course is usually offered to mechanical engineering undergraduates in their senior year and to graduates in their first year. It is an advanced topic and requires that students have a background in linear algebra, vector-matrix operations, dynamics, numerical analysis, and fundamentals of computer science, as well as in basic programming skills. The specific contents of multibody dynamics may vary from school to school. But generally speaking, they may contain but are not limited to the following: (1) Multibody kinematics: coordinate transformation matrixes and direction cosines, kinematical formulas, partial velocities, partial angular velocities, Euler angles, Euler parameters and kinematical differential equations, and so on; (2) Inertia: rotation of coordinate axes for inertia matrices and principal moments of inertia; (3) Multibody kinetics: various dynamic analysis methods for equations of motion. (4) Numerical issues in applied multibody dynamics 6, 11, 12 . In practice, many dynamics analysis methods are available for formulation of equations of motion of a multibody system. Newton-Euler equations, Lagrange’s equations, principles of virtual work, Hamilton’s principle, Gauss’s principle, Jordan’s principle, Kane’s method, and even finite element methods have been used by researchers in various applications 1 . Three P ge 1.27.2 commonly-used methods are Newton-Euler equations, Lagrange’s equations, and Kane’s method 1, 5, 15 . However, students may easily feel lost at such extremely mathematically-orientated methods when they need to select a proper dynamic analysis method to set up the equations of motion for their designs. Because they have difficulty in understanding methods, they eventually lose confidence when they have to select a proper method for their applications. To facilitate students’ understanding of these three methods, case study methodology, an instructional approach widely used in various subject areas, has been utilized in the applied multibody dynamics to help them learn how to select a proper method for virtual prototyping of their design applications. Applied Multibody Dynamics and Background of Students at SDSU The dual-number course ME 592-03/492-03 applied multibody dynamics is a three-credit technical elective course offered in the mechanical engineering program at South Dakota State University (SDSU) to students majoring in mechanical engineering and other engineering disciplines. Generally, applied multibody dynamics can be structured and organized in numbers of ways. The following are three common instructional approaches: (1) Introducing functions, commands, user interfaces, and a user manual of commercial virtual prototyping software without having a minimal knowledge of its theoretical bases. (2) Introducing multibody kinematics, multibody kinetics, and dynamic analysis methods for equations of motion and constraint equations but without proper use of commercial virtual prototyping computer software. (3) Introducing both multibody dynamics theory and computer software functions in an integrated way. Each way has its strengths and weaknesses. The following table shows a brief comparison: Table 1: A Brief Comparison of Three Different Ways to Organize AMD Emphasis on course contents Level of course Time constraint % of use of software Difficulty of course Softwareorientated Workshop to train software user High High Low Theoryorientated Ph.D. level graduate course Low Low High Theory/ software combined College level course for undergraduates & 1 st year graduates Middle Middle Middle During fall 2005, undergraduates and graduates enrolling in ME492-03/592-03 came from one of the following two groups: (1) They had taken EM 215 dynamics, MATH 471 numerical analysis, and CSC 150 computer science I, but had not taken any advanced dynamics course yet. So they had no background in advanced dynamical analysis methods such as Lagrangian equations. P ge 1.27.3 (2) They had taken an advanced dynamics course and at least knew Lagrangian equations. All students in these two groups had little or no background in applied multibody dynamics and no experience with virtual prototyping software. Based on the technical background of the students, the approach of combining theory with the use of software was utilized to deliver the AMD course. Such an approach has several benefits. One obvious benefit is that students are usually attracted by the use of simulation tools. After the instructional approach was determined, other teaching materials were chosen as follows: (1) Textbook and reference books a) Thomas R. Kane/David A. Levinson, Dynamics Online: Theory and Implementation with Autolev, Online Dynamics, Inc., 2000 b) Ahmed A. Shabana, Computational Dynamics, 2 edition, Wiley, 2001 c) Jerry H. Ginsberg, Advanced Engineering Dynamics, 2 edition, Cambridge University Press, 1998 (2) Computer software: Autolev and Matlab. (3) Course length: Forty lectures were delivered during fall semester of 2005: three fifty-minute lectures each week. However, how can students be motivated to learn theory? More specifically, how can students be motivated to proactively learn and understand various dynamic analysis methods to set up equations of motion for their applications? In order to encourage students’ learning theory, the AMD class exploited case study methodology in teaching three commonly-used dynamic analysis methods: the Newton-Euler approach, the Lagrange approach and Kane’s method. Case Study Methodology for Teaching and Learning Case study methodology has been widely exploited as an instructional approach in various subject areas such as medicine, law, business, education, engineering, technology, and science. Use of this teaching method has been extensively discussed in the literature 8, 9, 10, 16 . The case study method promotes team-based activities, active learning and the ability to handle open-ended problems 10 . Case study methodology also fosters the development of higher-level cognitive skills 8, 9 . Shapiro 13 summarizes several teaching and learning approaches as follow: lectures and readings facilitate “acquiring knowledge and becoming informed about techniques”; exercises and problem sets provide “the initial tools for exploring the applications and limitation of techniques”; case methodology promotes the “development of philosophies, approaches and skills”. Case study methodology has been widely used in teaching and learning of engineering subjects. Advantages of case study methods have been presented by Sankar et al 14 in “Importance of Ethical and Business Issues in Making Engineering Design Decision.” They concluded that the use of the case study methodology to deal with real-world examples is highly motivating and increases understanding of the importance of ethical issues in making engineering design decisions. P ge 1.27.4 Jensen discussed the merits of case study methodology for teaching freshman engineering courses 4 . The range of engineering disciplines and contents covered were engineering analysis, design methods, engineering calculations, technical communications and ethics. The approach has improved students’ involvement, motivation, and interest. The outcomes of the study are positive and promising. Beheler et al 3 specifically applied a case study approach to teaching engineering technology. Their experiences showed that it is a viable teaching method to enhance educational outcomes and provide students with a more meaningful and relevant academic experience. Graduating students develop and obtain the skills and knowledge that corporate employers have reported to be essentials to improving job seekers’ employability. Also, their experience indicated that the approach provides a valid way to enhance problem-solving, critical-thinking, communication, and documentation skills. General merits of the case study approach in Barrott 2 are summarized as follows: a) Providing students with a link to the real world b) Developing students’ critical-thinking and problem-solving skills c) Developing students’ communication skills d) Involving students in a cooperative learning activity Application of the Case Study Method to Teaching and Learning Dynamic Analysis Methods in AMD 1. Selection of the case A planar rigid body double pendulum connected by a pin joint was selected as the case. The pendulum as shown in Figure 1 has joint axes at points O and P parallel to the unit vector 3 n̂ . Bodies A and B are slender uniform rods with mass A m and B m , and length A L and B L respectively. A torsional spring with the spring constant A K acts between body A and the ground. 1 q and 2 q are generalized coordinates. The basis vectors i â , i b̂ , and i n̂ ) 3 , 2 , 1 ( = i are fixed on body A, body B and the ground respectively. A force Q F is applied to point Q in the direction 1 b̂ . Figure 1: A Simple Case Study – the Rigid Body Double Pendulum Though the double pendulum case is simple, it contains basic features that are necessary to discuss the principles of the targeted dynamic analysis methods. For example, its generalized P ge 1.27.5 coordinate can be linear or angular, and absol