
WB141304: 
Multibody Dynamics B 
ECTS: 
4 
Responsible Instructor: 
M. Wisse 
Instructor: 
Dr.ir. A.L. Schwab 
Contact Hours / Week x/x/x/x: 
0/0/2/2 
Education Period: 
3, 4 
Start Education: 
3 
Exam Period: 
none, Different, to be announced 
Course Language: 
English 
Expected prior knowledge: 
wb1113wb, wb1216 
Course Contents: 
In this course we will cover a systematic approach to the generation and solution of equations of motion for mechanical systems consisting of multiple interconnected rigid bodies, the socalled Multibody Systems. This course differs from "Advanced Dynamics", which mostly covers theoretical results about classes of idealized systems (e.g. Hamiltonian systems), in that the goal here is to find the motions of relatively realistic models of systems (including, for example, motors, dissipation and contact constraints). Topics covered are: NewtonEuler equations of motion for a simple planar system, free body diagrams, constraint equations and constraint forces, uniqueness of the solution. Systematic approach for a system of interconnected rigid bodies, virtual power method and Lagrangian multipliers. transformation of the equations of motion in terms of generalized independent coordinates, and lagrange equations. Nonholonomic constraints as in rolling without slipping, degrees of freedom and kinematic coordinates. Unilateral constraints as in contact problems. Numerical integration of the equations of motion, stability and accuracy of the applied methods. Numerical integration of a coupled differential and algebraic system of equations (DAE"s), Baumgarte stabilisation, projection method and independent coordinates. NewtonEuler equations of motion for a rigid threedimensional body, the need to describe orientation in space, Euler angles, Cardan angles, Euler parameters and Quaternions. Equations of motion for flexible multibody systems, introduction to Finite Element Method approach, Linearised equations of motion. Upon request and if time and ability of the instructor allows, related topics are open for discussion. 
Study Goals: 
The student is able to find the motions of linked rigid body systems in two and three dimensions including systems with various kinematic constraints, like there are: sliding, hinges and rolling, and closed kinematic chains. More specifically, the student must able to: 1. derive the NewtonEuler equations of motion for a simple planar system, draw free body diagrams, setup constraint equations and introduce constraint forces, and demonstrate the uniqueness of the solution 2. derive the equations of motion for a system of interconnected rigid bodies by means of a systematic approach: virtual power method and Lagrangian multipliers 3. transform the equations of motion in terms of generalized independent coordinates, and derive and apply the Lagrange equations of motion 4. apply the techniques from above to systems having nonholonomic constraints as in rolling without slipping, degrees of freedom and kinematic coordinates 5. apply the techniques from above to systems having unilateral constraints as in contact problems 6. perform various numerical integration schemes on the equations of motion, and predict the stability and accuracy of the applied methods 7. perform numerical integration on a coupled system of differential and algebraic equations (DAE"s), apply Baumgarte stabilization, the coordinate projection method and transformation to independent coordinates 8. derive the NewtonEuler equations of motion for a general rigid threedimensional body system connected by constraints, identify the need to describe orientation in space describe the orientation in 3D space of a rigid body by means of: Euler angles, Cardan angles, Euler parameters and Quaternions, derive the angular velocity and accelerations in terms of these parameters and their time derivatives, and their inverse 9. derive the equations of motion for flexible multibody systems by means of a Finite Element Method approach, and extend this to linearised equations of motion 
Education Method: 
Lectures (2 hours per week) 
Computer Use: 
The course is computeroriented. In doing the assignments you will be using Matlab, Maple or related computer software. 
Literature and Study Materials: 
Course material: Arend L. Schwab, `Lecture Notes on Multibody Dynamics", Delft, 2003 References from literature: A.A.Shabana, " Dynamics of multibody systems", Wiley, New York, 1998. E.J.Haug, " Computer aided kinematics and dynamics of mechanical systems, Volume I: Basic methods", Allyn and Bacon, Boston, 1989. P.E.Nikravesh, " Computeraided analysis of mechanical systems", PrenticeHall, Englewood Cliffs, 1988. M. Géradin, A. Cardano, " Flexible multibody dynamics: A finite element approach", J. Wiley, Chichester, New York, 2001. 
Assessment: 
Final Project 
Remarks: 
There will be weekly assignments and a final project. You have to make a report on the final project. In doing the assignments I strongly encourage you to work together. The final project is individual. Check out the uptodate web page at http://bicycle.tudelft.nl/schwab/ 
