Multibody Dynamics

Multibody Dynamics

ME41056 2023/2024

Syllabus

Course Description

ME41056: Multibody Dynamics
Faculty of Mechanical Engineering, Delft University of Technology
Year: 2023/2024, EC: 5
Study Guide: https://studiegids.tudelft.nl/a101_displayCourse.do?course_id=67061

After the completion of this class you will have developed the skills to model, interpret, simulate, and analyze multibody systems, i.e. systems which are made up of interconnected rigid bodies with arbitrary constraints and applied loads. Mathematical models of multibody systems are typically very useful at predicting the motion of macro scale objects. Newton's laws of motion are the foundation of developing predictive models of these systems. Examples of systems you will be able to model are: spacecraft trajectories, human/animal biomechanics, vehicle motion, robot motion, etc.

Learning Objectives

Students will be able to:

  • derive the equations of motion for a simple planar system, draw free body diagrams, set up constraint equations, and demonstrate the uniqueness of the solution
  • derive the equations of motion for a system of interconnected rigid bodies by means of a systematic approach
  • derive the equations of motion in terms of generalized independent coordinates
  • apply the techniques from above to systems having holonomic and nonholonomic constraints
  • perform various numerical integration schemes on the equations of motion, and predict the stability and accuracy of the applied methods
  • perform numerical integration on a coupled system of differential and algebraic equations (DAEs) with minimal constraint drift
  • derive the equations of motion for a general rigid three-dimensional body system bound by constraints
  • identify the need to describe orientation in 3D space by means of: Euler angles and Tait-Bryan angles and derive the angular velocity and accelerations in terms of these parameters and their time derivatives
  • create accurate, efficient, and documented computer programs that construct and simulate multibody systems
  • analyze and interpret the resulting motion from a multibody simulation

Prerequisites

We highly recommend taking TU Delft's "WB2630-T1: Rigid-Body Dynamics" prior to this course. Otherwise any introductory undergraduate level dynamics course should suffice. If you haven't had a dynamics course, you can probably get by if you are motivated and you already know:

  • Linear algebra
  • Vector calculus
  • Calculus based physics
  • Statics
  • Introductory numerical methods
  • Introductory scientific computing

You should also be proficient with at least one scientific programming language. You will be learning and using Python for the class assignments.

Instructors

Primary Instructor Co-Instructor Teaching Assistants
Dr. Jason K. Moore
Assistant Professor
BioMechanical Engineering Department
j.k.moore@tudelft.nl
Office Room #: 34 F-1-140
Pier de Jong, p.h.dejong@tudelft.nl
Karien ter Welle, C.S.terWelle@student.tudelft.nl
Niels Stienen, n.l.stienen@student.tudelft.nl
Riccardo Di Girolamo, R.DiGirolamo@student.tudelft.nl

Time and Location

Lecture videos will be posted online for viewing each Monday, starting February 12, 2024. You will be expected to watch the videos and read the online notes before the lecture and work sessions on Thursdays. We will have a brief in-person lecture followed by homework work sessions each week on Thursdays [*]:

  • Quarter 3:
    • Lecture: Thursdays 10:45-11:15 in CEG-Lecture Hall C (23.HG.0.53)
    • Work Sessions: Thursdays 11:15-13:15 in Pulse-Hall 10 (33.A2.600) & Pulse-Hall 6 (33.A1.500)
  • Quarter 4:
    • Lecture: Thursdays 10:45-11:15 in ME-Lecture Hall A - Leonardo da Vinci (34.A-0-820)
    • Work Sessions: Thursdays in AS-Classroom 12 (22.F.104) & AS-Classroom 4 (22.A.207) & AS-Classroom 6 (22.A.251) & AS-Classroom 7 (22.A.257) & ME-Hall I (34.D-1-200)

Planned lecture topics are shown on the schedule page.

[*]One lecture and work session is on Monday May 6 due to Acension Day being on a Thursday.

Course Text and Materials

The course will be based primarily on chapters from this online book:

Jason K. Moore. Learn Multibody Dynamics, 2023. https://moorepants.github.io/learn-multibody-dynamics

Chapters will be updated and added each week, so always check for the latest version when the associated lecture videos are released.

The primary principles and notation are based on this freely available book:

Thomas R. Kane, and David A. Levinson. Dynamics, Theory and Application. McGraw Hill, 1985. http://hdl.handle.net/1813/638.

Note that the book is out of print, but you can download a PDF copy from Cornell's eCommons digital repository for personal use.

Additionally, some topics will be derived from the following books:

  • Heike Vallery and Arend L. Schwab, "Advanced Dynamics", 3rd edition, Delft University of Technology, 2020, ISBN/EAN 978-90-8309-060-3
  • Thomas R. Kane, Peter W. Likins, and David A. Levinson. Spacecraft Dynamics. McGraw Hill, 1983. http://hdl.handle.net/1813/637.
  • Stephen H. Crandall, Dean C. Karnopp, Edward F. Kurtz, Jr., and David C. Pridmore-Brown, "Dynamics of Mechanical and Electromechanical Systems", 1968.

Assignments & Grades

The average of your best 10 of 12 homeworks will be counted for 40% of the course grade and the exam will count for 60% of the course grade. You must score at least a 5 of 10 (50%) on the exam to pass the course when the grade is in combination with your homework score. If the exam grade is better than the average homework grade, then the course grade is 100% from the exam. Homework scores can only be used to supplement your exam grade if the homeworks were completed in the same academic year as the exam is given. The rounding rules and grade calculations will follow the TU Delft exam regulations.

Homework
There will be 12 computational homework assignments (HW00 is not graded). Homeworks will be made available via Brightspace-Vocareum one week before they are due. You may turn in homework as a pair or as an individual. To submit as a pair, you must invite your partner within the Vocareum interface for each homework. All homework submissions should be the unique work of the individual or the pair. You must provide a contribution statement for each homework explaining any help you have received and any copyright licenses for materials you have used. See the schedule page for homework deadlines. No homework will be accepted late.
Exam
The exam will have a 3 hour duration. Effective use of the computational tools taught in class will give you the best chance at succeeding, but they are not necessarily required to succeed. You will be able to bring reference materials to the exam. No help from other people during the exam is permitted. Exact exam rules will be shared in Q4.

Brightspace

We will be using several features in Brightspace:

Announcements
This will be the instructor's primary communication avenue to you. These announcements can be forwarded to your TU Delft email address. You are expected to read these when shared.
Content -> Vocareum (Jupyter Notebook Server)
You will access the homework Jupyter notebook assignments here. You can edit and execute the notebooks in the Vocareum interface that is linked via each assignment. The "Sandbox" assignment gives access to a Vocareum Jupyter instance where you can practice and explore the software.
Collaboration -> Discussions
All questions for the instructors (or fellow classmates) that are not of a private nature should be asked in Brightspace discussions. If you need to discuss something of a private nature with the instructor(s), use email or talk in person.
Grades
Homework grades will be posted to Vocareum and/or Brightspace throughout the duration of the course.

Software

We will be making extensive use of the computer aided algebra software SymPy along with NumPy and SciPy to model and simulate multibody systems. These packages are written in the open source Python programming language and leverage the scientific Python ecosystem of scientific and engineering computing tools. You will have access to these through Vocareum in Brightspace. You may also install the software on your own computer. It is recommended that you bring your laptop to the work sessions. See the software page on this website for more information.

Academic Integrity

Academic dishonesty will not be tolerated. All homework assignments turned in for a grade must be your (or you and your partner's) unique work. You will have to include a contribution explanation with each homework submission. This contribution explanation should explain the contributions each of the partners made and any help you received from people other than the instructors.

If you make use of code found in other sources that you did not write yourself, either directly or in a modified form, you must follow the copyright licenses associated with that material. If there is no copyright license present, then you must obtain a written and signed permission from the author of the materials and provide that with your assignment submission. If there is a copyright license present in the materials you use (e.g. GPL, MIT, BSD, CC-BY), then you must follow the terms of that license. Most licenses, at minimum, require you to include the license with your work submission. This mirrors what you will have to do, by law, in your future work.

All code and written answers will be checked for plagiarism amongst student submissions and against external materials. Unattributed plagiarized materials will be marked with a 0 grade. Multiple offenses will result in no grade for the course.

Homework Contribution Statements

A single contribution statement with explanations for both sections are required for each homework whether you submit individually or as a pair. Homeworks will be graded with a 0 if a sufficient contribution statement is not included or no statement is included.

The contribution statement consists of two parts:

  1. Descriptions of your or you and your partner's contributions to the work and any contributions from other non-instructors to the solution.
  2. Copyright permission from the creators of code, text, images, etc. that were copied or copied and modified for the solution.

Section 1

If you worked as an individual, then state that here.

If you work in a pair, both partners are expected to make intellectual and coding contributions to the code written for the solutions. Describe who wrote what code and how each partner contributed to the formulation of the solution.

For solo and partner submissions you may obtain gain help from others, but you must explain how any non-instructor contributed to your solution. This includes help derived from any living or non-living (e.g. AI generated) source.

We expect that the submitters (you and/or your partner) formulate, write, and execute the submitted version of the code.

Section 2

If you make use of materials (code, text, images, etc.) that you did not create yourself, either directly or in a modified form, you must follow the copyright licenses associated with that material. If there is no copyright license present, then you must obtain a written and signed permission from the author of the materials and provide that with your assignment submission. If there is a copyright license that allows reuse present in the materials you use (e.g. GPL, MIT, BSD, CC-BY), then you must follow the terms of that license. Most licenses, at minimum, require you to include the license with your work submission. This mirrors what you will have to do, by law, in your future work. For online materials, include URLs to the materials you used and URLs to their licenses. For other materials, include the creator's permission or their licenses. If you did not use any other code or materials, then say so.

Example contribution statement:

  1. Moses Dinkle and Sandra Dee worked on this homework together as partners. We each did problems 1-3 independently first and then compared answers. We reworked our solutions together and Sandra typed the final combined answers into the notebook. For problem 4, Moses typed the solution while Sandra discussed and suggested what to do and made the necessary sketches. For problem 5, we struggled with the problem and our classmate Rutger Hauer helped talk us through the errors we were making. With Rutger's explanation we then typed up the solution the solution together.
  2. All solutions were our original work, except for problem 2 and 5. For problem 2 we found an example on Stackoverflow that was similar. We copied the Stackoverflow code and then reworked it to solve Problem 2. Here is the Stackoverflow post https://stackoverflow.com/questions/8739227/how-to-solve-a-pair-of-nonlinear-equations-using-python and the copyright license is CC-BY-SA 4.0 which is shown in tiny font at the very bottom right of the Stackoverflow page. We even used modified versions of two lines from Rutger's code that he showed us and he gave his permission to use those lines in our work.

Previous Year Materials