Multibody System Dynamics
From WikIRI
Master's Degree in Mechanical Engineering
Department of Mechanical Engineering - UPC-BarcelonaTech
Contents |
Course data
- Prerequisites: Ideally, students should have knowledge on rigid body kinematics and dynamics (the one imparted in the "Mechanics" course from ETSEIB). However, these topics will be introduced as needed so as to build up the Lagrange formulation. Students should also know MATLAB and have an English level allowing them to understand textbooks and oral presentations.
- Academic year: 2023-2024
- Semester: February-May.
- ECTS Credits: 3.
- Teaching language: Catalan (Spanish or English if needed).
- Instructor: Lluís Ros.
- Coordinator: Rosa Pàmies.
- Previous instructors: Rosa Pàmies, Albert Peiret, Josep M. Font.
- Classroom: G.1 for theory sessions (how to get there). Lab practices will be in Pavelló E, upper floor.
- Schedule: Wednesday 17:00 - 19:00.
- Start date: 21 march.
- Consultation hours: Each Wednesday after class, but please email the teacher to arrange them beforehand.
Objective
This course deals with the modelling and simulation of multibody systems composed of rigid bodies. On completion of the course, students should be able to model a mechanical multibody system, and conduct kinematic and dynamic analyses by means of computational simulation methods. Connections with control theory will also be stressed out during the course. The lab practices are designed to take the student from the construction of the dynamics model of a system, to its use on designing control laws in fully actuated robots.
Methodology
The subject will be worked out in theory and problem sessions, and by means of lab sessions with personal computers. Please install Matlab in your laptop (instructions below), and be ready to bring it to class with enough battery.
Theory slides and their notes will be published before each class, so the student can study them or print them for annotation.
Depending on the course evolution, some seminar sessions might be programmed as well. In such sessions, the active participation of the assistants is a fundamental aspect, and the teacher's task is, essentially, to direct the session, presenting and setting the topics in context, and coordinating the discussion among participants.
Grading
Global course grade NFIN will be based on the following partial grades:
- NLAB = Lab assignments grade (0 to 10)
- NEXAM = Final exam grade (0 to 10)
The final mark will be
- NFIN = 0.5 NLAB + 0.5 NEXAM
A student will pass the course if NFIN ≥ 5.
Clarity of presentation is a must. Poorly presented answers or reports may reduce the score up to 30% of the maximum one achievable.
Calendar
Here's a calendar for the course. We may change it on the fly depending on the course evolution. Any changes will be announced.
Theory and exercises
I'm grateful to Dr. Albert Peiret for sharing his lecture material with me. The slides below contain part of his material and new slides I've created myself. I’m also grateful to Dr. Rosa Pàmies and Dr. Ana Barjau for their help, constructive ideas, and encouraging comments at all times. Any errors or omissions are my sole responsibility.
Unit 0 "Introduction"
After studying this material, you can proceed to the Lab1 assignment.
For those needing more background on numerical methods to solve ODEs you can read this document. From the beginning to the trapezoidal rule is sufficient (but knowledge of the Runge-Kutta methods is useful in engineering).
Unit 1 "Rigid body kinematics"
Unit 2 "Rigid body dynamics"
Unit 3 "Introduction to multibody kinematics"
Unit 4 "Lagrange's equations"
- Here's a list of exercises which you can try on your own (more to be posted). Soon we will solve them in class.
Lab assignments
I will keep polishing the assignments until the corresponding session, so wait a bit before downloading them. Lab practices 1 and 2 are ready for download.
Lab 1: Simulation of a driven pendulum
Preparation: Study the theory of Unit 0 before attempting to solve this lab practice. Also understand the practice assignment and the program pendulum.m in detail. The following videos may be of help to these ends:
Lab 2: Simulation of an acrobot
Preparation: Read the practice assignment and the document "Preparation for Lab practice 2. The latter gives a piece of theory needed to understand the concepts of mass matrix, Coriolis matrix, and gravity term. You will also need to understand the theory of Units 1 and 2 regarding the formulation of velocities and kinetic/potential energies. After that, proceed to solve the practice assignment. You'll need to have the symbolic math toolbox of Matlab installed in your laptop. Type "ver" in the Matlab command window. If the toolbox is listed, it means you already have it installed.
Update: we have finally devoted two sessions to this practice, and I've given you this material:
- Blackboards of the 1st Lab2 session.
- Flow diagram of model.mlx and acrobot.m.
- Help for points (I) and (J).
Curiosities: motivated by a question by Weike Liao, here's an animation that shows the effect of chaos on a triple pendulum. The animation shows the simulation of many triple pendula whose initial conditions are almost the same (they just differ very slightly). Initially the motions coincide, but soon they diverge because of chaos (high sensitivity of the evolution of the system to its initial conditions). A similar effect arises in the double pendulum of this practice and could be reproduced by applying some edits in acrobot.m. Despite chaos arises in many systems, the time it manifests itself is very variable. For example, the solar system is chaotic, but for reasonably close initial conditions its simulations almost coincide during time horizons of millions of years.
Lab 3: Control of a Scara robot
Final exam
Date/place
Friday 14 June 15:00-18:30. Room L-LS.4.
What you can/cannot bring
You can bring:
- Any printed or handwritten material you want, in paper.
- A scientific calculator.
- Ball pen, pencil, rubber.
You cannot bring any device with the ability to communicate with others. In particular, cell phones must be shut down during the exam. Computers will not be allowed as a substitute of a scientific calculator.
Priority topics
Unit 0: the whole unit, and in special: the concept of state, how to transform a 2nd order ODE into a 1st order one, numerical integration through Euler's method, and the concept of initial value problem and its geometric interpretation.
Unit 1: the whole unit with an emphasis on velocities, not accelerations. You have to master: the formulation of velocities of points by any means (taking derivatives of position vectors in fixed or moving bases, and using composition or propagation of velocities), the formulation of angular velocities, the concept of instantaneous center of rotation and how to formulate velocities of other points using such a center.
Unit 2: Moment of inertia. Steiner's theorem. How to formulate spring and damper forces. How to express the kinetic and potential energy functions of a multibody system in terms of q and qdot (potential energy may be due to gravity and spring forces). How to compute the attractive/repulsive force of a spring in an equilibrium configuration of your multibody system (this may require the application of statics conditions on bodies of your system, so you may need to identify constraint forces/torques transmitted through joints).
Unit 3: Distinction between constrained and unconstrained system. You must know which kind of Lagrange equation applies to each type of system (ordinary or with multipliers).
Unit 4: Formulate Lagrange's equation for systems with or without nonconservative forces. How to formulate the generalized force F^nc* (right hand side of Lagrange's equation). Also how to rewrite Lagrange's equation in the standard form M·qddot + C·qdot + G = F^nc* (for simple systems, departing from the traditional Lagrange equations). How to convert M·qddot + C·qdot + G = F^nc* into a 1st order ODE in order to solve initial value problems numerically.
Lab practices: all concepts of the practices may be subject to evaluation, but a computer will not be needed. I will not ask you to implement any Matlab program.
Help
Vocabulary table
We list the concepts in order of appearance in the course:
| English | Catalan | Spanish |
|---|---|---|
| Revolute joint | Enllaç de revolució | Enlace de revolución |
| Reference frame | Referència | Referencia |
| Coordinate system | Sistema de coordenades | Sistema de coordenadas |
| Trihedral | Tríedre | Trihedro |
| Pose (position+orientation) | Posa (posició+orientació) | Pose (posición+orientación) |
| Skew symmetric matrix | Matriu antisimètrica | Matriz antisimétrica |
| Transportation velocity | Velocitat d'arrossegament | Velocidad de arrastre |
| Transportation acceleration | Acceleració d'arrossegament | Aceleración de arrastre |
| Sliding | Lliscament | Deslizamiento |
| Instantaneous axis of rotation | Eix instantani de rotació | Eje instantáneo de rotación |
| Torque (or couple) | Parell | Par |
| Moment | Moment | Momento |
| Constraint force | Força d'enllaç | Fuerza de enlace |
| Wrench | Torsor de forces | Torsor de fuerzas |
| Spring | Molla | Muelle |
| Damper | Amortidor | Amortiguador |
| Linear momentum | Quantitat de moviment | Cantidad de movimiento |
| Angular momentum | Moment cinètic (o angular) | Momento cinético (o angular) |
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Solutions for other browsers are unknown.
How to install MATLAB
If you have specific instructions given to you by UPC, please follow them.
Otherwise, try the UPC instructions in this page. The web portal they refer to is this one.
You will need the login credentials of the Mathworks account associated with your university license. In principle, they coincide with those you use to access the UPC intranet.
I recommend to install all toolboxes available.
Bibliography
On rigid body mechanics
A helpful web is:
The course books of reference are:
- J. Agulló i Batlle, A. Barjau Condomines, Rigid body kinematics, Cambridge University Press, 2020.
- J. Agulló i Batlle, A. Barjau Condomines, Rigid body Dynamics, Cambridge University Press, 2022.
Catalan versions of these books are:
- J. Agulló i Batlle, Mecànica de la partícula i del sòlid rígid, Publicacions OK punt, 2002.
- J. Agulló i Batlle, Introducció a la mecánica analítica, percussiva i vibratòria, Publicacions OK punt, 1998. Link.
Further material is available in Prof. Agulló's page, including Spanish versions of the books and collections of solved exercises.
On Matlab
An excellent book introducing Matlab is Higham:
- D. J. Higham and N. J. Higham. Matlab Guide (3rd edition). SIAM 2017.
A MATLAB cheat sheet from MIT.
Trick: expand all or collapse all in Matlab file.
Links
