A Comparison Case Study For Dynamics Analysis Methods In Applied .
2006-1859: A COMPARISON CASE STUDY FOR DYNAMICS ANALYSISMETHODS IN APPLIED MULTIBODY DYNAMICSShanzhong (Shawn) Duan, South Dakota State UniversityShanzhong (Shawn) Duan received his Ph.D. from Rensselaer Polytechnic Institute in 1999. Hehas been working as a software engineer at Autodesk for five years before he became an assistantprofessor at South Dakota State University in 2004. His current research interests include virtualprototyping, mechanical design and CAD/CAE/CAM.Page 11.27.1 American Society for Engineering Education, 2006
A Comparison Case Study for Dynamics AnalysisMethods in Applied Multibody DynamicsAbstractThis paper discusses how a simple comparison case study has been utilized in an appliedmultibody dynamics (AMD) course to enhance students’ learning of dynamic analysis methodsto set up equations of motion for multibody systems. The comparison case used is a planar rigidbody double pendulum with a pin joint connection between two bodies. This simple case hashelped students directly understand and see advantages and disadvantages of each dynamicanalysis method used to set up equations of motion. Based on what they have learned from thiscase study, students have a better understanding of targeted dynamic analysis methods and canmore efficiently choose a proper method to analyze the motion behaviors of their designapplications than they could previously.IntroductionAn applied multibody dynamics course is usually offered to mechanical engineeringundergraduates in their senior year and to graduates in their first year. It is an advanced topic andrequires that students have a background in linear algebra, vector-matrix operations, dynamics,numerical analysis, and fundamentals of computer science, as well as in basic programmingskills. The specific contents of multibody dynamics may vary from school to school. Butgenerally speaking, they may contain but are not limited to the following: (1) Multibodykinematics: coordinate transformation matrixes and direction cosines, kinematical formulas,partial velocities, partial angular velocities, Euler angles, Euler parameters and kinematicaldifferential equations, and so on; (2) Inertia: rotation of coordinate axes for inertia matrices andprincipal moments of inertia; (3) Multibody kinetics: various dynamic analysis methods forequations of motion. (4) Numerical issues in applied multibody dynamics6, 11, 12.In practice, many dynamics analysis methods are available for formulation of equations ofmotion of a multibody system. Newton-Euler equations, Lagrange’s equations, principles ofvirtual work, Hamilton’s principle, Gauss’s principle, Jordan’s principle, Kane’s method, andeven finite element methods have been used by researchers in various applications1. ThreePage 11.27.2
commonly-used methods are Newton-Euler equations, Lagrange’s equations, and Kane’smethod1, 5, 15.However, students may easily feel lost at such extremely mathematically-orientated methodswhen they need to select a proper dynamic analysis method to set up the equations of motion fortheir designs. Because they have difficulty in understanding methods, they eventually loseconfidence when they have to select a proper method for their applications.To facilitate students’ understanding of these three methods, case study methodology, aninstructional approach widely used in various subject areas, has been utilized in the appliedmultibody dynamics to help them learn how to select a proper method for virtual prototyping oftheir design applications.Applied Multibody Dynamics and Background of Students at SDSUThe dual-number course ME 592-03/492-03 applied multibody dynamics is a three-credittechnical elective course offered in the mechanical engineering program at South Dakota StateUniversity (SDSU) to students majoring in mechanical engineering and other engineeringdisciplines.Generally, applied multibody dynamics can be structured and organized in numbers of ways. Thefollowing are three common instructional approaches:(1) Introducing functions, commands, user interfaces, and a user manual of commercial virtualprototyping software without having a minimal knowledge of its theoretical bases.(2) Introducing multibody kinematics, multibody kinetics, and dynamic analysis methods forequations of motion and constraint equations but without proper use of commercial virtualprototyping computer software.(3) Introducing both multibody dynamics theory and computer software functions in anintegrated 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 AMDEmphasis on eory/ softwarecombinedLevel of courseWorkshop to trainsoftware userPh.D. level graduatecourseCollege level coursefor undergraduates &1st year graduatesTimeconstraintHigh% of use ofsoftwareHighDifficulty ofcourseLowLowLowHighMiddleMiddleMiddlePage 11.27.3During fall 2005, undergraduates and graduates enrolling in ME492-03/592-03 came from one ofthe following two groups:(1) They had taken EM 215 dynamics, MATH 471 numerical analysis, and CSC 150computer science I, but had not taken any advanced dynamics course yet. So they had nobackground in advanced dynamical analysis methods such as Lagrangian equations.
(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 andno experience with virtual prototyping software.Based on the technical background of the students, the approach of combining theory with theuse 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 theinstructional approach was determined, other teaching materials were chosen as follows:(1) Textbook and reference booksa) Thomas R. Kane/David A. Levinson, Dynamics Online: Theory and Implementation withAutolev, Online Dynamics, Inc., 2000b) Ahmed A. Shabana, Computational Dynamics, 2nd edition, Wiley, 2001c) Jerry H. Ginsberg, Advanced Engineering Dynamics, 2nd edition, Cambridge UniversityPress, 1998(2) Computer software: Autolev and Matlab.(3) Course length: Forty lectures were delivered during fall semester of 2005: three fifty-minutelectures each week.However, how can students be motivated to learn theory? More specifically, how can students bemotivated to proactively learn and understand various dynamic analysis methods to set upequations of motion for their applications? In order to encourage students’ learning theory, theAMD class exploited case study methodology in teaching three commonly-used dynamicanalysis methods: the Newton-Euler approach, the Lagrange approach and Kane’s method.Case Study Methodology for Teaching and LearningCase study methodology has been widely exploited as an instructional approach in varioussubject areas such as medicine, law, business, education, engineering, technology, and science.Use of this teaching method has been extensively discussed in the literature8, 9, 10, 16.The case study method promotes team-based activities, active learning and the ability to handleopen-ended problems10. Case study methodology also fosters the development of higher-levelcognitive skills8, 9. Shapiro13 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 limitationof techniques”; case methodology promotes the “development of philosophies, approaches andskills”.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 al14 in “Importance ofEthical and Business Issues in Making Engineering Design Decision.” They concluded that theuse of the case study methodology to deal with real-world examples is highly motivating andincreases understanding of the importance of ethical issues in making engineering designdecisions.Page 11.27.4
Jensen discussed the merits of case study methodology for teaching freshman engineeringcourses4. The range of engineering disciplines and contents covered were engineering analysis,design methods, engineering calculations, technical communications and ethics. The approachhas improved students’ involvement, motivation, and interest. The outcomes of the study arepositive and promising.Beheler et al3 specifically applied a case study approach to teaching engineering technology.Their experiences showed that it is a viable teaching method to enhance educational outcomesand provide students with a more meaningful and relevant academic experience. Graduatingstudents develop and obtain the skills and knowledge that corporate employers have reported tobe essentials to improving job seekers’ employability. Also, their experience indicated that theapproach provides a valid way to enhance problem-solving, critical-thinking, communication,and documentation skills.General merits of the case study approach in Barrott2 are summarized as follows:a) Providing students with a link to the real worldb) Developing students’ critical-thinking and problem-solving skillsc) Developing students’ communication skillsd) Involving students in a cooperative learning activityApplication of the Case Study Method to Teaching and Learning Dynamic AnalysisMethods in AMD1. Selection of the caseA planar rigid body double pendulum connected by a pin joint was selected as the case. Thependulum as shown in Figure 1 has joint axes at points O and P parallel to the unit vector n̂3 .Bodies A and B are slender uniform rods with mass mA and mB , and length LA and LBrespectively. A torsional spring with the spring constant K A acts between body A and the ground.q1 and q2 are generalized coordinates. The basis vectors âi , b̂i , and n̂i (i 1,2,3) are fixed onbody A, body B and the ground respectively. A force FQ is applied to point Q in the direction b̂1 .Page 11.27.5Figure 1: A Simple Case Study – the Rigid Body Double PendulumThough the double pendulum case is simple, it contains basic features that are necessary todiscuss the principles of the targeted dynamic analysis methods. For example, its generalized
coordinate can be linear or angular, and absolute or relative. It contains two-level coordinatetransformations. The formulas for velocity and acceleration of two points fixed on a rigid bodyand of one point moving on a rigid body can be applied to the same case respectively so thatstudents can do a cross check for their derivation and simulation. The low complexity of the casealso permits it fit into our forty-class schedule. Its simplicity brings benefits to teaching andlearning of the targeted dynamic analysis methods. In short, this case makes comparison of threetargeted dynamic analysis methods clear with less effort.The double pendulum as a case study has been utilized for teaching and research in varioussubject areas. Newberry17 used a double pendulum for students to learn and understandHamilton’s principle. Gulley found that a double pendulum was a useful case in learning the Sfunction of Matlab18. Swisher et al19 mentioned to use a double pendulum as a case study in anintegrated vibrations and system simulation course. Romano20 applied a double pendulum toresearching a modular modeling methodology in real-time multi-body vehicle dynamics.2. Use of the case in ME 592-30/492-03 AMDIn the fall of 2005, the double pendulum case was repeatedly used in teaching and learningAMD. The case and its variation were integrated with various teaching and learning scenarios.The first use of the case was in a student homework assignment. Students were asked to deriveequations of motion out of the case shown in Figure 1 according to the Newton-Euler equations.The purpose was to help students review what they had learned from the basic rigid-bodydynamics course. At the same time the case helped them apply new vector-matrix notations andcoordinate transformation matrix techniques in advanced dynamics to what they had learned.The second use of the case was in class lectures when Lagrange equations were introduced. Theequations of motion in form of energy for the case were derived by the instructor to show ananalytical way to obtain them. The derivation was compared with the students’ derivation forthe same case in their homework using the Newton-Euler method.Then the original case shown in Figure 1 was reduced into a double pendulum of two particlesconnected by massless rigid links A and B as shown in Figure 2. The lectures about this reducedcase illustrated Kane’s method in detail. The key to Kane’s method lies in use of the conceptsn̂1gOn̂3Aq1n̂2â1â2â3mP PBq2b̂1b̂2b̂3mQ QPage 11.27.6Figure 2: A Variation of the Case Study – a Double Pendulum of Two Particles
of generalized speeds, partial angular velocities and partial velocities. These concepts werediscussed in detail in terms of the reduced case. Solution and simulation procedures usingKane’ equations and the Autolev package were summarized for students as shown in the flowcharts of Figure 3. The simulation results produced by Autolev and Matlab for the givengeometric data and initial conditions for the reduced case are presented in Figure 4.Kinematical analysis forvelocities, angular velocities,accelerations, angularaccelerations, partial velocitiesand partial angular velocitiesSelect generalizedcoordinates &generalized speedAutolev producesmotion simulationcodes in Matlab or Cbased on givengeometric data &initial conditionsEquationsof motion1Kinetic analysis forgeneralized activeforces & generalizedinertial forcesCompile & executeMatlab code or Ccode to producesimulation data1InterpretsimulationresultsFigure 3: Solution Procedure Using Kane’s Equation and AutolevGeneralized Coordinate Q2 vs. Time TGeneralized Coordinate Q1 vs. Time T102081564Q2 (degree/sec)Q1 (degree/sec)1050-520-2-4-10-6-15-8-20-1002468T (sec)1012140162468T (sec)10121416Generalized Speed U1 vs. Time TGeneralized Speed U2 vs. Time T0.80.80.60.60.40.4U2 (degree/sec)U1 102468T (sec)10121416-0.80246810121416T (sec)Figure 4: Simulation Results of the Reduced Case Produced by Autolev & Kane’s MethodPage 11.27.7
Finally, the original case was assigned to students to derive the equations of motion by Kane’smethod and produce simulation data using the Autolev/Matlab software packages. Through fourrepeated cycles of teaching and learning of the selected case, the students’ understanding andlearning of three targeted dynamic analysis methods were enhanced. At the end of four cycles,the students and faculty generalized and summarized about the use of each method incorresponding research areas. Further reading of journal papers and technical articles wassuggested. The appendix, which has been scanned from the AMD class materials, shows a briefside by side comparison between Lagrange’s method and Kane’s method as used in this case.3. Summary of the targeted methods after the repeated case study cyclesIn multibody dynamics, most dynamical formulations fall into either State Space Form (SSF) orDeScriptor (DSF) Form as shown in equations (1) and (2) respectively:q I q II (1a ) M (t , q )q RHS (t , q , q )(1b)IIIIII q I q II 0(2a) T M (t , q I )q II A (t , q I )λ RHS (t , q I , q II ) 0 (2b) Φ (t , q I ) 0( 2c ) In equations (1) and (2), q I and q II are position and velocity state variables. Matrix M is thesystem mass matrix, and matrix RSH is the right hand side of the equation of motion thatcontains all of external loads, body loads and inertia forces associated with centripetal andCoriolis accelerations. Due to constraints in the equation (2c), the Jacobi constraint matrix Aand Lagrange multiplier λ appear in the equation (2b).Page 11.27.8Of the three targeted dynamic analysis methods, generally the Newton-Euler method treats eachbody separately, which results in a large but sparse system mass matrix and a simple formulation.But if not used wisely, this method may result in order n to the fourth power, O(n4),computational complexity with respect to n number of degree-freedom of a multibody system. Inthe Newton-Euler method, much effort is required to eliminate workless constraint forces.Lagrange’s equation can automatically eliminate workless constraint forces. But this benefit canbe offset by complicated derivatives of Lagrangians, which often results in a phenomenon ofintermediate ‘swell’ and complex formulation. Generally speaking, Lagrange’s method is anO(n3) method. Kane’s method can avoid these disadvantages and keep the advantages of bothNewton-Euler and Lagrange. It has the first order form of equation and an O(n3) computationcomplexity. A comparison of labor involved in deriving the equations of motion via differentmethods may be found in the reference7. The following table shows a brief comparison.Table 2: A Brief Comparison of Three Dynamics Analysis MethodsMethodsWorklessComputationalGeneralized Complexity ofconstraints complexitycoordinates formulationNewton-Euler YesO(n4)Nolow3LagrangeEliminatedO(n )Yeshigh3KaneEliminatedO(n )Yeslow
Assessment of the Case Study Method for Teaching and Learning AMDHomework problems, computing assignments, quizzes and exams were used to assess students’learning and the effectiveness of the teaching of dynamic analysis methods through the casestudy. In addition, team-based course projects were used to evaluate teaching and learning. Eachproject team was formed by three students. The project topic was a component or subsystem ofsenior design project, Mini-Baja project, or a real dynamic system that all team members wereinterested in modeling, designing, analyzing and simulating. Then they would further apply whatthey had learned from this case study to select a proper analysis method for their applications,derive kinematical and force equations, set up equations of motion, and eventually producesimulation results. Figure 5 shows the selected examples of team projects.Figure 5: Selected Team Project Titles in AMDEvaluation of teaching and learning was conducted anonymously. Twelve graduate students andeight senior students took part in the survey. Table 3 shows the percentage of students whostrongly agree or agree with questions listed in the survey about course outcomes.Table 3: Course Outcomes from the Student Survey for AMD% ofstudentsMy Learningincreased inthis course76%I made progresstowards achievingcourse objectives75%My interestin subjectincreased75%Course helps me tothink independentlyabout subject73%I involved inwhat I amlearning77%Page 11.27.9Since in fall 2005 ME 492-03/592-03 applied multibody dynamics was offered for the first timein the mechanical program, no baseline data were available to conduct a direct comparisonbetween teaching AMD with the selected double pendulum case and without the case. As arelative comparison, Table 4 shows the same survey questions answered by the students whotook EM 215 dynamics, in which the rigid body double pendulum was not used as a case study atall.
Table 4: Course Outcomes from the Student Survey for EM 215 Dynamics% ofstudentsMy Learningincreased inthis course71%I made progresstowards achievingcourse objectives69%My interestin subjectincreased66%Course helps me tothink independentlyabout subject69%I involved inwhat I amlearning64%Concluding remarksA rigid body double pendulum and its variation have been used repeatedly as a case study forteaching and learning through various phases of the AMD course. Though the case is simple, theintegration of the case with various educational activities provides many benefits for teachingand learning about three targeted dynamic analysis methods used for virtual prototyping ofmechanical systems. Student surveys have provided first-hand information for furtherimprovement and future investigations.Bibliographies1. Anderson, K. S. (1990). Recursive Derivation of Explicit Equation of Motion for Efficient Dynamic/ControlSimulation of Large Multibody Systems. Ph.D. Dissertation Stanford University. UMI, No. 91087782. Barrott, J. L. (2001). Why Should Case Studies be Integrated into the Engineering Technology Curriculum.Proceedings of the 2001 American Society for Engineering Education Annual Conference & Exposition.Albuquerque, NM.3. Beheler, A. and Jones, W. A. (2004). Using Case Studies to Teach Engineering Technology. Proceedings of the2004 American Society for Engineering Education Annual Conference & Exposition. Salt Lake City, UT.4. Jensen, J. N. (2003). A Case Study Approach to Freshman Engineering Courses. Proceedings of the 2003American Society for Engineering Education Annual Conference & Exposition. Nashville, TN.5. Hollerbach, J. M. (1980). A Recursive Lagrangian Formulation of Manipulator Dynamics and a ComparativeStudy of Dynamics Formulation Complexity. IEEE Trans. Systems, Man, and Cybernetics. Vol. SMC – 10, No. 11,November. pp. 730 – 736.6. Huston, R. L. (1990). Multibody Dynamics. Butterworth-Heinemann.7. Kane, T. R. and Levinson, D. A. (1980). Formulation of Equations of motion for Complex Spacecraft. Journal ofGuidance and Control, Vol. 3, No. 11. pp. 99-112.8. Kolodner, J. (1993). Case-Based Reasoning. Morgan Kaufman, San Manteo, CA.9. Leake, D. (1996). Case-Based Reasoning.: Experiences, Lessons, and Future Directions. AAAI Press/MIT Press,Cambridge, MA.Page 11.27.1010. Meyers, C. and Jones, T. B. (1993). Promoting Active Learning: Strategies for the College Classroom. NewYork, Wiley.
11. Roberson, R. E. and Schwertassek, R. (1988). Dynamics of multibody systems. New York: Springer-Verlag.12. Shabana, A. A. (1998). Dynamics of Multibody Systems. Cambridge. Cambridge University.13. Shapiro, B. P. (1984). Introduction to Cases. Harvard Business Online, Boston, MA. 9-584-097.14. Sankar, C. S. and Raju, P. K. (2001). Importance of Ethical and Business Issues in Making Engineering DeisgnDecisions: Teaching through Case Studies. Proceedings of the 2001 American Society for Engineering EducationAnnual Conference & Exposition. Albuquerque, NM.15. Walker, M. W., and Orin, D. E. (1982). Efficient Dynamic Computer Simulation of Robotic Mechanisms.Journal of Dynamic Systems, Measurements, and Control, Vol. 104, Sept. pp. 3363 – 3387.16. Wright, S. (1996). Case-based instruction: Linking theory to practice. Physical Educator. Vol. 53, Issue 4.17. Newberry, C. F. (2005). A Missile System Design Engineering Model Graduate Curriculum. Proceedings of the2005 American Society for Engineering Education Annual Conference & Exposition. Portland, OR.18. Gulley, N. (1993). PNDANTM2 S-function for Animating the motion of a double pendulum. The Math Works,Inc.19. Swisher, G. M. and Darvennes, C. M. (2001) An Integrated Vibrations and System Simulation Course. (2001).Proceedings of the 2001 American Society for Engineering Education Annual Conference & Exposition.Albuquerque, NM.20. Romano, R. (2003). Real-Time Multi-Body Vehicle Dynamics Using a Modular Modeling Methodology. SAETechnical Paper Series No. 2003-01-1286.Page 11.27.11
Appendix: A Brief Derivation Comparison between Kane’s Equation and Lagrange’sEquations for the Selected Case in Figure 1 (Scanned from the ME 492/592AMD Course Materials)From Figure 1, coordinate transformation matrixes between body A, body B & the ground are asfollows:Page 11.27.12
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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 nd edition, Wiley, 2001 c) Jerry H. Ginsberg, Advanced Engineering Dynamics, 2 nd edition, Cambridge University Press, 1998 (2) Computer software: Autolev and Matlab.
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