The course aims to provide students with up-to-date and innovative skills, abilities and professionalism in the area of functional mechanical design and dynamic simulation of mechanical systems and robots, which will be used in product development and service applications and plant development. In this regard, mechanical systems for industrial, service, automation and automotive applications will first be explained through the study of their kinematic structure and function. Innovative mechanism design methods will then be discussed, in particular, for industrial automation, human centered applications and automotive, proposing, among others, kinematic synthesis methods for infinitesimal and finite motions, function-generating mechanisms, rigid body guidance and trajectory generator. Tribological implications will also be considered. Next, the dynamic simulation of multibody systems in space and vehicle dynamics will be modeled in SE(3). Mechanical design and dynamic simulation will be applied to the following topics: industrial robotics for automation and service; micro and nano (MEMS and NEMS) systems; navigation systems based on inertial sensors; wearable systems; powertrains, planetary gearboxes, automatic transmissions, differential, cam systems, clutches, special mechanisms.
Curriculum
teacher profile teaching materials
Master Degree LM-33 Mechanical Engineering
Second edition, academic year 2025/26
Introduction
Aims of the course and relationship between theory, applications and exercises; history of the discipline; overview of the variety of topics covered; teaching methods adopted and examination arrangements.
Topology of mechanisms
Topological analysis and synthesis of mechanisms. Franke's notation. Graph-mechanism correspondence; enumeration of kinematic chains; isomorphism. Atlases of kinematic chains; loop-closure equations and generalized kinematic analysis. Spanning tree, graph drawing and planarity; automatic representation of kinematic chains and mechanisms. Atlases of mechanisms: classification and the atlases of Artobolewsky, Jones, Jensen, Chironis and Sclater.
Kinematic analysis of planar mechanisms by analytical methods
Bresse circles F and S; circle of zero normal jerk and circle of zero tangential jerk; velocity centre P1, acceleration centre P2 and jerk centre P3, with application to the slider-crank mechanism. Geometric invariants for the kinematic analysis and synthesis of mechanisms; determination of the centrodes of the slider-crank mechanism by means of the geometric invariants; tangency of the centrodes by complex numbers; proof of the Euler-Savary equation; Aronhold-Kennedy theorem and its applications to the generation of conjugate profiles; kinematic analysis method based on polar coordinates. Outline of performance indices, pressure angle and mechanical advantage.
Kinematic synthesis for infinitesimal displacements
Classical Burmester theory developed by means of the geometric invariants; Burmester points; examples of application to the four-bar linkage. Outline of the quartic of the curvature derivative; curve C and curve C. Evolute of a curve; generalized Burmester theory, with examples and exercises.
Kinematic synthesis for finite displacements
Displacement matrices in homogeneous coordinates in SE(2). Outline of the graphical methods. Rigid body guidance mechanisms and the Suh and Radcliffe method. Function generator and path generator mechanisms; method based on Freudenstein's equation. Outline of affine four-bar linkages.
Kinematics of spatial motions and robotics
Attitude matrix [A] in SO(3) and its properties; parametrization by Euler and Cardan angles; kinematic analysis of spatial motions in Cartesian coordinates. Introduction to industrial robots: history and classification; industrial robotics for automation and service robotics; elements of robot mechanics; robotic wrists of gyroscopic complexity. Outline of displacement matrices in homogeneous coordinates for spatial displacements in SE(3), of experimental methods for measuring the orientation (attitude) of a body in space, and of self-driving cars.
Dynamics of spatial multibody systems
Newton-Euler equations in space in matrix notation. Inertia matrix in SO(3). Axis-angle notation for the description of a displacement in SO(3) and Rodrigues' formula; development of the equations of dynamics and of the constraint equations in non-minimal notation. Euler parameters and quaternions; properties of the vector of the Euler parameters and of its derivatives; constraint and dynamic equations in compact matrix form. Assembly of the overall system of constraint and dynamic equations of a multibody system. Solution of the direct dynamic problem in SE(3) by Lagrange multipliers and step-by-step integration procedure; computer example in MatLab for SE(2) and in Simscape for SE(3): amphibious rover. Outline of the principle of inertia concordance.
Compliant mechanisms, MEMS and NEMS
Introduction to compliant mechanisms; deformations; pseudo rigid body equivalent mechanism; design methods. Determination of the centre C of the relative rotation between two adjacent members and drift of C as the loads change; application of the stability circle to the design of a microgripper based on MEMS technology. Introduction to the mechanical design of micro-electro-mechanical systems MEMS and NEMS; fabrication methods, in particular D-RIE on SOI wafers and EBL lithography.
Epicyclic gear trains and automotive transmissions
Willis' formula for epicyclic gear trains; computation of the geometric invariants for cycloidal motions and in epicyclic and Cardan gear trains; outline of square and hexagonal turning. Elementary epicyclic gear train; epicyclic reduction gears and automatic computation of the overall transmission ratio; automatic gearbox, characteristics and constraints of epicyclic reduction gears; automotive differential. Outline of the functional design of automotive components: automatic gearboxes, differentials, suspensions and dampers, steering mechanisms. Outline of clutches and couplings, Geneva mechanisms and ratchets.
Bevel gears, conjugate surfaces and the helical pair
Bevel gears in machinery; general methods for the generation of conjugate surfaces in space. Efficiency of the inclined plane by analytical treatment; introduction to the helical pair in machinery; fastening screws and power screws; computation of the efficiency of a screw; screw and nut pair with translating nut; recirculating ball screw. Outline of transmissions between parallel, intersecting and skew axes, and of helical and hyperboloidal gears.
Lubrication
Outline of elastohydrodynamic lubrication EHD and of the computation of the entraining velocity.
Functional design for ocean engineering and for energy
Outline of the functional design of components for ocean engineering systems and for WEC systems for wave energy conversion; outline of the functional design of components for the development of energy sources.
Computational methods and methods for innovation in design
Outline of computational intelligence: optimization algorithms for mechanisms, neural networks, genetic algorithms. Lateral thinking in design; TRIZ method.
Exercises and computer activities
Introductory elements of the Wolfram Mathematica symbolic manipulation environment and programming in its language.
Derivation of the Bresse circles, of the jerk circles and of the centres P1, P2 and P3 with Wolfram Mathematica.
Computation of the geometric invariants and determination of the centrodes of the slider-crank mechanism.
Kinematic synthesis by means of geometric invariants developed in Wolfram Mathematica.
Computation of the geometric invariants in epicyclic and Cardan gear trains.
Determination of the centre of the finite rotation and application to compliant mechanisms.
Multibody dynamic simulation in MatLab for SE(2) and in Simscape for SE(3).
Fondamenti di meccanica applicata alle macchine
Terza edizione | 2024 | Casa Editrice Ambrosiana. Distribuzione esclusiva Zanichelli
Lecture Notes from the Professor
Programme
Syllabus of the course of Mechanics and Dynamics of Machinery - 9 CFUMaster Degree LM-33 Mechanical Engineering
Second edition, academic year 2025/26
Introduction
Aims of the course and relationship between theory, applications and exercises; history of the discipline; overview of the variety of topics covered; teaching methods adopted and examination arrangements.
Topology of mechanisms
Topological analysis and synthesis of mechanisms. Franke's notation. Graph-mechanism correspondence; enumeration of kinematic chains; isomorphism. Atlases of kinematic chains; loop-closure equations and generalized kinematic analysis. Spanning tree, graph drawing and planarity; automatic representation of kinematic chains and mechanisms. Atlases of mechanisms: classification and the atlases of Artobolewsky, Jones, Jensen, Chironis and Sclater.
Kinematic analysis of planar mechanisms by analytical methods
Bresse circles F and S; circle of zero normal jerk and circle of zero tangential jerk; velocity centre P1, acceleration centre P2 and jerk centre P3, with application to the slider-crank mechanism. Geometric invariants for the kinematic analysis and synthesis of mechanisms; determination of the centrodes of the slider-crank mechanism by means of the geometric invariants; tangency of the centrodes by complex numbers; proof of the Euler-Savary equation; Aronhold-Kennedy theorem and its applications to the generation of conjugate profiles; kinematic analysis method based on polar coordinates. Outline of performance indices, pressure angle and mechanical advantage.
Kinematic synthesis for infinitesimal displacements
Classical Burmester theory developed by means of the geometric invariants; Burmester points; examples of application to the four-bar linkage. Outline of the quartic of the curvature derivative; curve C and curve C. Evolute of a curve; generalized Burmester theory, with examples and exercises.
Kinematic synthesis for finite displacements
Displacement matrices in homogeneous coordinates in SE(2). Outline of the graphical methods. Rigid body guidance mechanisms and the Suh and Radcliffe method. Function generator and path generator mechanisms; method based on Freudenstein's equation. Outline of affine four-bar linkages.
Kinematics of spatial motions and robotics
Attitude matrix [A] in SO(3) and its properties; parametrization by Euler and Cardan angles; kinematic analysis of spatial motions in Cartesian coordinates. Introduction to industrial robots: history and classification; industrial robotics for automation and service robotics; elements of robot mechanics; robotic wrists of gyroscopic complexity. Outline of displacement matrices in homogeneous coordinates for spatial displacements in SE(3), of experimental methods for measuring the orientation (attitude) of a body in space, and of self-driving cars.
Dynamics of spatial multibody systems
Newton-Euler equations in space in matrix notation. Inertia matrix in SO(3). Axis-angle notation for the description of a displacement in SO(3) and Rodrigues' formula; development of the equations of dynamics and of the constraint equations in non-minimal notation. Euler parameters and quaternions; properties of the vector of the Euler parameters and of its derivatives; constraint and dynamic equations in compact matrix form. Assembly of the overall system of constraint and dynamic equations of a multibody system. Solution of the direct dynamic problem in SE(3) by Lagrange multipliers and step-by-step integration procedure; computer example in MatLab for SE(2) and in Simscape for SE(3): amphibious rover. Outline of the principle of inertia concordance.
Compliant mechanisms, MEMS and NEMS
Introduction to compliant mechanisms; deformations; pseudo rigid body equivalent mechanism; design methods. Determination of the centre C of the relative rotation between two adjacent members and drift of C as the loads change; application of the stability circle to the design of a microgripper based on MEMS technology. Introduction to the mechanical design of micro-electro-mechanical systems MEMS and NEMS; fabrication methods, in particular D-RIE on SOI wafers and EBL lithography.
Epicyclic gear trains and automotive transmissions
Willis' formula for epicyclic gear trains; computation of the geometric invariants for cycloidal motions and in epicyclic and Cardan gear trains; outline of square and hexagonal turning. Elementary epicyclic gear train; epicyclic reduction gears and automatic computation of the overall transmission ratio; automatic gearbox, characteristics and constraints of epicyclic reduction gears; automotive differential. Outline of the functional design of automotive components: automatic gearboxes, differentials, suspensions and dampers, steering mechanisms. Outline of clutches and couplings, Geneva mechanisms and ratchets.
Bevel gears, conjugate surfaces and the helical pair
Bevel gears in machinery; general methods for the generation of conjugate surfaces in space. Efficiency of the inclined plane by analytical treatment; introduction to the helical pair in machinery; fastening screws and power screws; computation of the efficiency of a screw; screw and nut pair with translating nut; recirculating ball screw. Outline of transmissions between parallel, intersecting and skew axes, and of helical and hyperboloidal gears.
Lubrication
Outline of elastohydrodynamic lubrication EHD and of the computation of the entraining velocity.
Functional design for ocean engineering and for energy
Outline of the functional design of components for ocean engineering systems and for WEC systems for wave energy conversion; outline of the functional design of components for the development of energy sources.
Computational methods and methods for innovation in design
Outline of computational intelligence: optimization algorithms for mechanisms, neural networks, genetic algorithms. Lateral thinking in design; TRIZ method.
Exercises and computer activities
Introductory elements of the Wolfram Mathematica symbolic manipulation environment and programming in its language.
Derivation of the Bresse circles, of the jerk circles and of the centres P1, P2 and P3 with Wolfram Mathematica.
Computation of the geometric invariants and determination of the centrodes of the slider-crank mechanism.
Kinematic synthesis by means of geometric invariants developed in Wolfram Mathematica.
Computation of the geometric invariants in epicyclic and Cardan gear trains.
Determination of the centre of the finite rotation and application to compliant mechanisms.
Multibody dynamic simulation in MatLab for SE(2) and in Simscape for SE(3).
Core Documentation
Nicola Pio Belfiore, Augusto Di Benedetto, Ettore PennestrìFondamenti di meccanica applicata alle macchine
Terza edizione | 2024 | Casa Editrice Ambrosiana. Distribuzione esclusiva Zanichelli
Lecture Notes from the Professor
Attendance
Attendance is strongly recommended.Type of evaluation
Traditional examination consisting of a written test and an oral interview, with possible discussion of exercises, PC lab activities, and projects. teacher profile teaching materials
Master Degree LM-33 Mechanical Engineering
Second edition, academic year 2025/26
Introduction
Aims of the course and relationship between theory, applications and exercises; history of the discipline; overview of the variety of topics covered; teaching methods adopted and examination arrangements.
Topology of mechanisms
Topological analysis and synthesis of mechanisms. Franke's notation. Graph-mechanism correspondence; enumeration of kinematic chains; isomorphism. Atlases of kinematic chains; loop-closure equations and generalized kinematic analysis. Spanning tree, graph drawing and planarity; automatic representation of kinematic chains and mechanisms. Atlases of mechanisms: classification and the atlases of Artobolewsky, Jones, Jensen, Chironis and Sclater.
Kinematic analysis of planar mechanisms by analytical methods
Bresse circles F and S; circle of zero normal jerk and circle of zero tangential jerk; velocity centre P1, acceleration centre P2 and jerk centre P3, with application to the slider-crank mechanism. Geometric invariants for the kinematic analysis and synthesis of mechanisms; determination of the centrodes of the slider-crank mechanism by means of the geometric invariants; tangency of the centrodes by complex numbers; proof of the Euler-Savary equation; Aronhold-Kennedy theorem and its applications to the generation of conjugate profiles; kinematic analysis method based on polar coordinates. Outline of performance indices, pressure angle and mechanical advantage.
Kinematic synthesis for infinitesimal displacements
Classical Burmester theory developed by means of the geometric invariants; Burmester points; examples of application to the four-bar linkage. Outline of the quartic of the curvature derivative; curve C and curve C. Evolute of a curve; generalized Burmester theory, with examples and exercises.
Kinematic synthesis for finite displacements
Displacement matrices in homogeneous coordinates in SE(2). Outline of the graphical methods. Rigid body guidance mechanisms and the Suh and Radcliffe method. Function generator and path generator mechanisms; method based on Freudenstein's equation. Outline of affine four-bar linkages.
Kinematics of spatial motions and robotics
Attitude matrix [A] in SO(3) and its properties; parametrization by Euler and Cardan angles; kinematic analysis of spatial motions in Cartesian coordinates. Introduction to industrial robots: history and classification; industrial robotics for automation and service robotics; elements of robot mechanics; robotic wrists of gyroscopic complexity. Outline of displacement matrices in homogeneous coordinates for spatial displacements in SE(3), of experimental methods for measuring the orientation (attitude) of a body in space, and of self-driving cars.
Dynamics of spatial multibody systems
Newton-Euler equations in space in matrix notation. Inertia matrix in SO(3). Axis-angle notation for the description of a displacement in SO(3) and Rodrigues' formula; development of the equations of dynamics and of the constraint equations in non-minimal notation. Euler parameters and quaternions; properties of the vector of the Euler parameters and of its derivatives; constraint and dynamic equations in compact matrix form. Assembly of the overall system of constraint and dynamic equations of a multibody system. Solution of the direct dynamic problem in SE(3) by Lagrange multipliers and step-by-step integration procedure; computer example in MatLab for SE(2) and in Simscape for SE(3): amphibious rover. Outline of the principle of inertia concordance.
Compliant mechanisms, MEMS and NEMS
Introduction to compliant mechanisms; deformations; pseudo rigid body equivalent mechanism; design methods. Determination of the centre C of the relative rotation between two adjacent members and drift of C as the loads change; application of the stability circle to the design of a microgripper based on MEMS technology. Introduction to the mechanical design of micro-electro-mechanical systems MEMS and NEMS; fabrication methods, in particular D-RIE on SOI wafers and EBL lithography.
Epicyclic gear trains and automotive transmissions
Willis' formula for epicyclic gear trains; computation of the geometric invariants for cycloidal motions and in epicyclic and Cardan gear trains; outline of square and hexagonal turning. Elementary epicyclic gear train; epicyclic reduction gears and automatic computation of the overall transmission ratio; automatic gearbox, characteristics and constraints of epicyclic reduction gears; automotive differential. Outline of the functional design of automotive components: automatic gearboxes, differentials, suspensions and dampers, steering mechanisms. Outline of clutches and couplings, Geneva mechanisms and ratchets.
Bevel gears, conjugate surfaces and the helical pair
Bevel gears in machinery; general methods for the generation of conjugate surfaces in space. Efficiency of the inclined plane by analytical treatment; introduction to the helical pair in machinery; fastening screws and power screws; computation of the efficiency of a screw; screw and nut pair with translating nut; recirculating ball screw. Outline of transmissions between parallel, intersecting and skew axes, and of helical and hyperboloidal gears.
Lubrication
Outline of elastohydrodynamic lubrication EHD and of the computation of the entraining velocity.
Functional design for ocean engineering and for energy
Outline of the functional design of components for ocean engineering systems and for WEC systems for wave energy conversion; outline of the functional design of components for the development of energy sources.
Computational methods and methods for innovation in design
Outline of computational intelligence: optimization algorithms for mechanisms, neural networks, genetic algorithms. Lateral thinking in design; TRIZ method.
Exercises and computer activities
Introductory elements of the Wolfram Mathematica symbolic manipulation environment and programming in its language.
Derivation of the Bresse circles, of the jerk circles and of the centres P1, P2 and P3 with Wolfram Mathematica.
Computation of the geometric invariants and determination of the centrodes of the slider-crank mechanism.
Kinematic synthesis by means of geometric invariants developed in Wolfram Mathematica.
Computation of the geometric invariants in epicyclic and Cardan gear trains.
Determination of the centre of the finite rotation and application to compliant mechanisms.
Multibody dynamic simulation in MatLab for SE(2) and in Simscape for SE(3).
Fondamenti di meccanica applicata alle macchine
Terza edizione | 2024 | Casa Editrice Ambrosiana. Distribuzione esclusiva Zanichelli
Lecture Notes from the Professor
Programme
Syllabus of the course of Mechanics and Dynamics of Machinery - 9 CFUMaster Degree LM-33 Mechanical Engineering
Second edition, academic year 2025/26
Introduction
Aims of the course and relationship between theory, applications and exercises; history of the discipline; overview of the variety of topics covered; teaching methods adopted and examination arrangements.
Topology of mechanisms
Topological analysis and synthesis of mechanisms. Franke's notation. Graph-mechanism correspondence; enumeration of kinematic chains; isomorphism. Atlases of kinematic chains; loop-closure equations and generalized kinematic analysis. Spanning tree, graph drawing and planarity; automatic representation of kinematic chains and mechanisms. Atlases of mechanisms: classification and the atlases of Artobolewsky, Jones, Jensen, Chironis and Sclater.
Kinematic analysis of planar mechanisms by analytical methods
Bresse circles F and S; circle of zero normal jerk and circle of zero tangential jerk; velocity centre P1, acceleration centre P2 and jerk centre P3, with application to the slider-crank mechanism. Geometric invariants for the kinematic analysis and synthesis of mechanisms; determination of the centrodes of the slider-crank mechanism by means of the geometric invariants; tangency of the centrodes by complex numbers; proof of the Euler-Savary equation; Aronhold-Kennedy theorem and its applications to the generation of conjugate profiles; kinematic analysis method based on polar coordinates. Outline of performance indices, pressure angle and mechanical advantage.
Kinematic synthesis for infinitesimal displacements
Classical Burmester theory developed by means of the geometric invariants; Burmester points; examples of application to the four-bar linkage. Outline of the quartic of the curvature derivative; curve C and curve C. Evolute of a curve; generalized Burmester theory, with examples and exercises.
Kinematic synthesis for finite displacements
Displacement matrices in homogeneous coordinates in SE(2). Outline of the graphical methods. Rigid body guidance mechanisms and the Suh and Radcliffe method. Function generator and path generator mechanisms; method based on Freudenstein's equation. Outline of affine four-bar linkages.
Kinematics of spatial motions and robotics
Attitude matrix [A] in SO(3) and its properties; parametrization by Euler and Cardan angles; kinematic analysis of spatial motions in Cartesian coordinates. Introduction to industrial robots: history and classification; industrial robotics for automation and service robotics; elements of robot mechanics; robotic wrists of gyroscopic complexity. Outline of displacement matrices in homogeneous coordinates for spatial displacements in SE(3), of experimental methods for measuring the orientation (attitude) of a body in space, and of self-driving cars.
Dynamics of spatial multibody systems
Newton-Euler equations in space in matrix notation. Inertia matrix in SO(3). Axis-angle notation for the description of a displacement in SO(3) and Rodrigues' formula; development of the equations of dynamics and of the constraint equations in non-minimal notation. Euler parameters and quaternions; properties of the vector of the Euler parameters and of its derivatives; constraint and dynamic equations in compact matrix form. Assembly of the overall system of constraint and dynamic equations of a multibody system. Solution of the direct dynamic problem in SE(3) by Lagrange multipliers and step-by-step integration procedure; computer example in MatLab for SE(2) and in Simscape for SE(3): amphibious rover. Outline of the principle of inertia concordance.
Compliant mechanisms, MEMS and NEMS
Introduction to compliant mechanisms; deformations; pseudo rigid body equivalent mechanism; design methods. Determination of the centre C of the relative rotation between two adjacent members and drift of C as the loads change; application of the stability circle to the design of a microgripper based on MEMS technology. Introduction to the mechanical design of micro-electro-mechanical systems MEMS and NEMS; fabrication methods, in particular D-RIE on SOI wafers and EBL lithography.
Epicyclic gear trains and automotive transmissions
Willis' formula for epicyclic gear trains; computation of the geometric invariants for cycloidal motions and in epicyclic and Cardan gear trains; outline of square and hexagonal turning. Elementary epicyclic gear train; epicyclic reduction gears and automatic computation of the overall transmission ratio; automatic gearbox, characteristics and constraints of epicyclic reduction gears; automotive differential. Outline of the functional design of automotive components: automatic gearboxes, differentials, suspensions and dampers, steering mechanisms. Outline of clutches and couplings, Geneva mechanisms and ratchets.
Bevel gears, conjugate surfaces and the helical pair
Bevel gears in machinery; general methods for the generation of conjugate surfaces in space. Efficiency of the inclined plane by analytical treatment; introduction to the helical pair in machinery; fastening screws and power screws; computation of the efficiency of a screw; screw and nut pair with translating nut; recirculating ball screw. Outline of transmissions between parallel, intersecting and skew axes, and of helical and hyperboloidal gears.
Lubrication
Outline of elastohydrodynamic lubrication EHD and of the computation of the entraining velocity.
Functional design for ocean engineering and for energy
Outline of the functional design of components for ocean engineering systems and for WEC systems for wave energy conversion; outline of the functional design of components for the development of energy sources.
Computational methods and methods for innovation in design
Outline of computational intelligence: optimization algorithms for mechanisms, neural networks, genetic algorithms. Lateral thinking in design; TRIZ method.
Exercises and computer activities
Introductory elements of the Wolfram Mathematica symbolic manipulation environment and programming in its language.
Derivation of the Bresse circles, of the jerk circles and of the centres P1, P2 and P3 with Wolfram Mathematica.
Computation of the geometric invariants and determination of the centrodes of the slider-crank mechanism.
Kinematic synthesis by means of geometric invariants developed in Wolfram Mathematica.
Computation of the geometric invariants in epicyclic and Cardan gear trains.
Determination of the centre of the finite rotation and application to compliant mechanisms.
Multibody dynamic simulation in MatLab for SE(2) and in Simscape for SE(3).
Core Documentation
Nicola Pio Belfiore, Augusto Di Benedetto, Ettore PennestrìFondamenti di meccanica applicata alle macchine
Terza edizione | 2024 | Casa Editrice Ambrosiana. Distribuzione esclusiva Zanichelli
Lecture Notes from the Professor
Attendance
Attendance is strongly recommended.Type of evaluation
Traditional examination consisting of a written test and an oral interview, with possible discussion of exercises, PC lab activities, and projects.