SD 652: Dynamics of Multibody Systems

Essentially, this course shows how an engineer can efficiently model and simulate the motion of complex

multibody systems such as robots, humans, vehicles, and mechanisms. Systematic methods, including conventional

approaches and linear graph theory, are used to generate kinematic and dynamic models for 2-d and

3-d systems of bodies connected by joints, springs, dampers, and actuators. The extension of these methods

to biomechanics, vehicles, mechatronic systems and flexible robot manipulators is also presented. Numerical

solutions for the dynamic equations provide a simulation of the system response, which will be obtained using

commercial software packages (ADAMS and MapleSim). The course concepts are demonstrated through

applications to the kinematic and dynamic analysis of mechanisms, biomechatronic systems, serial and parallel

robot manipulators, autonomous and hybrid vehicles, and other industrial multibody systems. Whenever

possible, physical prototypes are brought to the lectures..


Course Organization:

  1. Introduction

1.1  Overview of multibody dynamics

1.2  Review of kinematics 

1.2.1       degrees of freedom

1.2.2       vector kinematics

1.2.3       applications to particles

1.2.4       applications to multibody systems

1.2.5       rotation transformations

1.3  Review of dynamics

1.3.1       vectorial mechanics (particles, rigid bodies)

1.3.2       analytical mechanics (multibody systems)


The reviews of kinematics and dynamics are best supported by the books of Hibbeler and Ginsberg (see below).


2.    Conventional Multibody Dynamics

2.1   Planar systems

2.1.1       absolute coordinates and constraints

2.1.2       kinematic analysis

2.1.3       dynamic analysis

2.1.4       numerical computer implementation:   MSC.Adams

2.2  Spatial systems

2.2.1       rotational coordinates and equations

2.2.2       kinematics

2.2.3       dynamics


The material on planar systems is covered in the first half of the Haug textbook [1], which is available on Learn.  The material on spatial systems is supported by the second half of the Haug textbook.


3.     Graph-Theoretic (G-T) Dynamics of Multibody Systems


3.1. Basics of G-T modelling

3.2. One-dimensional multibody systems

3.3. Multi-dimensional multibody systems [Stanford manipulator graph and plots]

3.4. Symbolic computer implementation:  MapleSim

Section 3 is supported by the SD 652 course notes and the slides on GT modelling.


4.     Advanced Topics

4.1. Mechatronic systems

4.2. Flexible bodies

4.3. Contact dynamics

4.4. Tires in vehicle dynamics




[1].  Computer-Aided Kinematics and Dynamics of Mechanical Systems, E.J. Haug, Allyn and Bacon, 1989.

[2].  Dynamics of Multibody Systems: Conventional and Graph-Theoretic Approaches, J. McPhee, SD652 course notes, 2004.
[3].  Engineering Mechanics, R. Hibbeler, Prentice-Hall, 2004.
[4].  Vector Mechanics for Engineers, F. Beer and R. Johnston, McGraw-Hill, 2003.
[5].  Advanced Engineering Dynamics, 2nd ed., J. Ginsberg, Cambridge University Press, 1998.

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