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2006-1859: A COMPARISON CASE STUDY FOR DYNAMICS ANALYSISMETHODS IN APPLIED MULTIBODY DYNAMICS
Shanzhong (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.
© American Society for Engineering Education, 2006
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A Comparison Case Study for Dynamics Analysis
Methods in Applied Multibody Dynamics
Abstract
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 dynamics6, 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 applications1. Three P
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commonly-used methods are Newton-Euler equations, Lagrange’s equations, and Kane’s
method1, 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
Software-
orientated
Workshop to train
software user
High High Low
Theory-
orientated
Ph.D. level graduate
course
Low Low High
Theory/ software
combined
College level course
for undergraduates &
1st 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.
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(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, 2nd edition, Wiley, 2001 c) Jerry H. Ginsberg, Advanced Engineering Dynamics, 2nd 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 literature8, 9, 10, 16
.
The case study method promotes team-based activities, active learning and the ability to handle
open-ended problems10
. Case study methodology also fosters the development of higher-level
cognitive 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 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 al14
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. Page 11.27.4
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Jensen discussed the merits of case study methodology for teaching freshman engineering
courses4. 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 al3 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 Barrott2 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 3n̂ .
Bodies A and B are slender uniform rods with mass Am and Bm , and length AL and BL
respectively. A torsional spring with the spring constant AK acts between body A and the ground.
1q and 2q are generalized coordinates. The basis vectors iâ , ib̂ , and in̂ )3,2,1( =i are fixed on
body A, body B and the ground respectively. A force QF is applied to point Q in the direction 1b̂ .
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
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coordinate can be linear or angular, and absolute or relative. It contains two-level coordinate
transformations. The formulas for velocity and acceleration of two points fixed on a rigid body
and of one point moving on a rigid body can be applied to the same case respectively so that
students can do a cross check for their derivation and simulation. The low complexity of the case
also permits it fit into our forty-class schedule. Its simplicity brings benefits to teaching and
learning of the targeted dynamic analysis methods. In short, this case makes comparison of three
targeted dynamic analysis methods clear with less effort.
The double pendulum as a case study has been utilized for teaching and research in various
subject areas. Newberry17
used a double pendulum for students to learn and understand
Hamilton’s principle. Gulley found that a double pendulum was a useful case in learning the S-
function of Matlab18
. Swisher et al19
mentioned to use a double pendulum as a case study in an
integrated vibrations and system simulation course. Romano20
applied a double pendulum to
researching a modular modeling methodology in real-time multi-body vehicle dynamics.
2. Use of the case in ME 592-30/492-03 AMD
In the fall of 2005, the double pendulum case was repeatedly used in teaching and learning
AMD. 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 derive
equations 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-body
dynamics course. At the same time the case helped them apply new vector-matrix notations and
coordinate 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. The
equations of motion in form of energy for the case were derived by the instructor to show an
analytical way to obtain them. The derivation was compared with the students’ derivation for
the 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 particles
connected by massless rigid links A and B as shown in Figure 2. The lectures about this reduced
case illustrated Kane’s method in detail. The key to Kane’s method lies in use of the concepts
Figure 2: A Variation of the Case Study – a Double Pendulum of Two Particles
1̂a
2â
1b̂ 2b̂
1̂n
2n̂3n̂
3b̂
3â
Qm
Pm
1q
2q
A
B
Q
P
Og
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of generalized speeds, partial angular velocities and partial velocities. These concepts were
discussed in detail in terms of the reduced case. Solution and simulation procedures using
Kane’ equations and the Autolev package were summarized for students as shown in the flow
charts of Figure 3. The simulation results produced by Autolev and Matlab for the given
geometric data and initial conditions for the reduced case are presented in Figure 4.
Figure 3: Solution Procedure Using Kane’s Equation and Autolev
-20
-15
-10
-5
0
5
10
15
20
0 2 4 6 8 10 12 14 16
Q1
(d
egre
e/se
c)
T (sec)
Generalized Coordinate Q1 vs. Time T
-10
-8
-6
-4
-2
0
2
4
6
8
10
0 2 4 6 8 10 12 14 16
Q2
(d
egre
e/se
c)
T (sec)
Generalized Coordinate Q2 vs. Time T
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12 14 16
U1
(d
egre
e/se
c)
T (sec)
Generalized Speed U1 vs. Time T
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 2 4 6 8 10 12 14 16
U2 (
deg
ree/
sec)
T (sec)
Generalized Speed U2 vs. Time T
Figure 4: Simulation Results of the Reduced Case Produced by Autolev & Kane’s Method
Select generalized
coordinates &
generalized speed
Kinematical analysis for
velocities, angular velocities,
accelerations, angular
accelerations, partial velocities
and partial angular velocities
Kinetic analysis for
generalized active
forces & generalized
inertial forces 1
1 Equations
of motion
Autolev produces
motion simulation
codes in Matlab or C
based on given
geometric data &
initial conditions
Compile & execute
Matlab code or C
code to produce
simulation data
Interpret
simulation
results
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Finally, the original case was assigned to students to derive the equations of motion by Kane’s
method and produce simulation data using the Autolev/Matlab software packages. Through four
repeated cycles of teaching and learning of the selected case, the students’ understanding and
learning 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 in
corresponding research areas. Further reading of journal papers and technical articles was
suggested. The appendix, which has been scanned from the AMD class materials, shows a brief
side 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 cycles
In multibody dynamics, most dynamical formulations fall into either State Space Form (SSF) or
DeScriptor (DSF) Form as shown in equations (1) and (2) respectively:
↓→↑
=
=
),,(),(IIIIII
III
qqtRHSqqtM
qq
�
�
)1(
)1(
b
a
°↓
°→
↑
=Φ
=−−
=−
0),(
0),,(),(),(
0
I
IIII
T
III
III
qt
qqtRHSqtAqqtM
qq
λ�
�
)2(
)2(
)2(
c
b
a
In equations (1) and (2), I
q and II
q are position and velocity state variables. Matrix M is the
system mass matrix, and matrix RSH is the right hand side of the equation of motion that
contains all of external loads, body loads and inertia forces associated with centripetal and
Coriolis accelerations. Due to constraints in the equation (2c), the Jacobi constraint matrix A
and Lagrange multiplier λ appear in the equation (2b).
Of the three targeted dynamic analysis methods, generally the Newton-Euler method treats each
body 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. In
the Newton-Euler method, much effort is required to eliminate workless constraint forces.
Lagrange’s equation can automatically eliminate workless constraint forces. But this benefit can
be offset by complicated derivatives of Lagrangians, which often results in a phenomenon of
intermediate ‘swell’ and complex formulation. Generally speaking, Lagrange’s method is an
O(n3) method. Kane’s method can avoid these disadvantages and keep the advantages of both
Newton-Euler and Lagrange. It has the first order form of equation and an O(n3) computation
complexity. A comparison of labor involved in deriving the equations of motion via different
methods may be found in the reference7. The following table shows a brief comparison.
Table 2: A Brief Comparison of Three Dynamics Analysis Methods
Methods Workless
constraints
Computational
complexity
Generalized
coordinates
Complexity of
formulation
Newton-Euler Yes O(n4) No low
Lagrange Eliminated O(n3) Yes high
Kane Eliminated O(n3) Yes low
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Assessment of the Case Study Method for Teaching and Learning AMD
Homework problems, computing assignments, quizzes and exams were used to assess students’
learning and the effectiveness of the teaching of dynamic analysis methods through the case
study. In addition, team-based course projects were used to evaluate teaching and learning. Each
project team was formed by three students. The project topic was a component or subsystem of
senior design project, Mini-Baja project, or a real dynamic system that all team members were
interested in modeling, designing, analyzing and simulating. Then they would further apply what
they 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 produce
simulation results. Figure 5 shows the selected examples of team projects.
Figure 5: Selected Team Project Titles in AMD
Evaluation of teaching and learning was conducted anonymously. Twelve graduate students and
eight senior students took part in the survey. Table 3 shows the percentage of students who
strongly agree or agree with questions listed in the survey about course outcomes.
Table 3: Course Outcomes from the Student Survey for AMD My Learning
increased in
this course
I made progress
towards achieving
course objectives
My interest
in subject
increased
Course helps me to
think independently
about subject
I involved in
what I am
learning
% of
students
76% 75% 75% 73% 77%
Since in fall 2005 ME 492-03/592-03 applied multibody dynamics was offered for the first time
in the mechanical program, no baseline data were available to conduct a direct comparison
between teaching AMD with the selected double pendulum case and without the case. As a
relative comparison, Table 4 shows the same survey questions answered by the students who
took EM 215 dynamics, in which the rigid body double pendulum was not used as a case study at
all.
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Table 4: Course Outcomes from the Student Survey for EM 215 Dynamics My Learning
increased in
this course
I made progress
towards achieving
course objectives
My interest
in subject
increased
Course helps me to
think independently
about subject
I involved in
what I am
learning
% of
students
71% 69% 66% 69% 64%
Concluding remarks
A rigid body double pendulum and its variation have been used repeatedly as a case study for
teaching and learning through various phases of the AMD course. Though the case is simple, the
integration of the case with various educational activities provides many benefits for teaching
and learning about three targeted dynamic analysis methods used for virtual prototyping of
mechanical systems. Student surveys have provided first-hand information for further
improvement and future investigations.
Bibliographies
1. Anderson, K. S. (1990). Recursive Derivation of Explicit Equation of Motion for Efficient Dynamic/Control
Simulation of Large Multibody Systems. Ph.D. Dissertation Stanford University. UMI, No. 9108778
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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 the
2004 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 2003
American Society for Engineering Education Annual Conference & Exposition. Nashville, TN.
5. Hollerbach, J. M. (1980). A Recursive Lagrangian Formulation of Manipulator Dynamics and a Comparative
Study 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 of
Guidance 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.
10. Meyers, C. and Jones, T. B. (1993). Promoting Active Learning: Strategies for the College Classroom. New
York, Wiley.
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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 Deisgn
Decisions: Teaching through Case Studies. Proceedings of the 2001 American Society for Engineering Education
Annual 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 the
2005 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,
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Technical Paper Series No. 2003-01-1286.
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Appendix: A Brief Derivation Comparison between Kane’s Equation and Lagrange’s
Equations for the Selected Case in Figure 1 (Scanned from the ME 492/592
AMD Course Materials)
From Figure 1, coordinate transformation matrixes between body A, body B & the ground are as
follows:
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