The course is taught in English.
The course provides the students with the basic skills required for model building and simulation of electromechanical systems. This means the theoretical and practical capabilities that will be used in order to model mechatronic components, equipment, and systems. The student will acquire the capability to understand the interactions between electromechanical structures and their actuations of various nature (such as magnetic, capacitive, piezoelectric. Unifying methodologies (state coordinates, work, potential and kinetic energy and co-energy, dissipation) and modeling approaches (Lagrange equations, Bond-Graph approach) constitute the core topics of the course.

The student acquires the following knowledge
- How to describe the structure of a system; various types of analytical representation (linear, non-linear, time invariant, time-varying, etc.) and their representations based on physical principles
- Vectors, matrices, reference systems: how to use this concepts to represent motions of solid bodies in space
- Rigid displacements in space: translations and rotation matrices , homogeneous representation of the body pose
- Kinetic and potential energy: the building block for modeling approaches based on Lagrange equations.
- The concept of state variables.
- From Lagrange equations to state variable equations.
- The concepts of flows and efforts; power and the bond-graph methodology;
The student develops the following abilities
- Ability to describe a rigid body in space and its motion using vectors and matrices (rotation and homogeneous)
- Ability to compute analytically and numerically the kinetic and potential energy of a system that includes many bodies (multi-body analysis)
- Ability to write the Lagrange equation, to transform in state variable equations and use for model simulation
- Ability to write the Bond-Graph model, to transform in state variable equations and use for model simulation
- Ability to use the above methods for modeling mechanical, electromagnetic and electromechanical systems such that actuators and transducer (electromagnets, capacitive, moving-coil transducers, piezoelectric material, etc.).

The course requires the knowledge of the basic elements of physics, in particular mechanics and electromagnetism, and the capacity to compute with vectors and matrices; elements of system theory (states, inputs, outputs, transfer functions) elements of automatic control (simple PID control networks) are not mandatory, but beneficial.

1. Introduction, examples of mechanical, electrical, hydraulic, electromechanical equipment, definition of power, energy and co-energy. (8 hours)
2. Analytical models: understanding of the system structure, of the analytical representation (linear, nonlinear, time invariant, time varying, etc.) and its representation based on physical principles. (10 hours)
3. Definition of the effort and flux variables in mechanical, electrical and hydraulic systems. (8 hours)
4. State functions: kinetic co-energy and potential energy. (10 hours)
5. Mathematical language needed in the course: vectors and matrices, rigid body kinematics and dynamics, generalized variables, degrees of freedom, constraints, holonomicity. Roto-translations of rigid bodies. general properties of the body dynamics: inertia, dissipation, elasticity. (8 hours)
6. Lagrange Equations. (8 hours)
7. Bond-graph modelling based on power exchange between subsystems. Power variables, reversibility, construction rules, relation between bond-graphs and block diagrams. (12 hours)
8. Configuration space and state equations. The dynamic system as a unifying mathematical representation. Linear and nonlinear models, Fundamental properties: stability and passivity. (8 hours)
9. How to write the equations that describe the interactions inside electromechanical actuators and transducer, as electromagnets, capacitive transducers, moving coil transducers, piezoelectric materials, synchronous motors. (10 hours)
10. Mechanical models of the rigid body, finite elements models. Finite elements models of lumped parameters electrical and electronic systems. Modal reduction techniques and bond-graph representation. (10 hours)
11. Examples and implementation of simulated systems. (8 hours)

Practical exercises in class will be devoted to analytically model several simple systems, using the different approaches introduced in the course.

For the Lagrange approach: B. Bona, "Dynamic Modelling of mechatronic Systems", CELID, 2013
For the Bond-graph approach : Dean C. Karnopp, Donald L. Margolis, Ronald C. Rosenberg, "System dynamics: modeling and simulation of mechatronic systems", New York, Wiley, 2000.
Slides, notes and other written material is available through the course web page

The exam is constituted by a written test (no oral exam), and it is aimed at evaluating the competencies of the candidate with reference to all the topics of the course program.
It consists of two parts:
Part 1 – (LAG approach): one to three exercises (usually two); the exam lasts approximately 1 hour and 10 min; books and notes CAN be used. Examples are given during the course.
Part 2 – (BG approach): one to three exercises (usually two); the exam lasts approximately 1 hour and 10 min; books and notes CAN be used. Examples are given during the course.
The two parts can be taken together in the same exam session or separately at different times during the year. Each part is evaluated with a separate mark, computed as the weighted mean of the scores of the proposed questions, and expressed in 1/30. The final mark is unique and is the arithmetical mean of the marks of the two parts.
Each part must get at least a mark > 9/30; if the mark is lower, that part must be repeated.