Dynamics of Mechanical Systems


The objective of this course is to convey the fundamental concepts of dynamics of multiple body systems, systematizing their equations of motion with the Cartesian formulation and solving them with numerical methods, thus allowing the development and analysis of mechanical systems used in different industrial applications. from mechanisms in general to biomechanics, robotics and vehicles. The aim is thus to instruct students about the potentialities and limitations of numerical models constructed from the programming of equations of motion with Cartesian or other formulations, either by the use of commercial codes, promoting their rational and appropriate use with regard to representation of real systems and their components.

General characterization





Responsible teacher

António Paulo Vale Urgueira, Marta Isabel Pimenta Verdete da Silva Carvalho


Weekly - 4

Total - 66

Teaching language



Basic knowledge of programming in MATLAB language.


1.   Computer Aided Analysis of Mechanical Systems: Parviz Nikravesh, 1988 Prentice – Hall Publishers. Englewod Cliffs, New Jersey.

2.   Lecture Notes. Dynamics of Mechanical Systems: M. S. Pereira 1992 Lecture notes, COMMETT courses

Teaching method

Classes are theoretical and practical, in a classroom equipped with computers and appropriate software. The concepts are taught through the theoretical exposition of the formulation and practical resolution of some type problems, by the computational implementation in code developed by the student or by the use of existing commercial software to compare or to implement more complex engineering cases.

Evaluation method

The evaluation is made through the appreciation of four practical assignments to be done individually by the students.

Score = 0.2 (A1 + A2) + 0.3 (A3 + FA)

Subject matter

1. Introduction: Concepts of mechanism, kinematic joint, rigid body and flexible body. Two-dimensional and three-dimensional analysis.

2. Kinematic analysis: Relative coordinates. Equations of constraints, velocities and accelerations. Coordinate Partition Method. Motor constraints.

3. Planar kinematics with Cartesian coordinates: Coordinates, constraints and joints. Equations of position, velocity and acceleration. Kinematic joints. Applications.

4. Dynamic analysis in the plane: Equations of motion. Forces vector. Spring-damper-translation and rotation actuator. Reactions due to constraints. Lagrange Multipliers. System of equations of motion. Applications.

5. Numerical methods for solving ordinary differential equations.

6. Contact and impact of mechanical systems. Applications to motor vehicle collisions.

7. Other formulations used for dynamic multibody analysis.


Programs where the course is taught: