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Academic year
Didactic period
Secondo Semestre

Training objectives

The course aims at providing the basic tools for the analysis and synthesis of planar mechanisms and to present in a systematic way the main issues involved in the functional study of machines.


Basic knowledge of Mathematics, Geometry, Physics and Rational Mechanics.

Course programme

The course lasts 120 hours, including lessons and exercises, and it consists of two integrated modules: Mechanics of Machines A (held by prof. Raffaele Di Gregorio, 60 hours), Mechanics of Machines B (held by prof. Giorgio Dalpiaz, 60 hours). The topics of the integrated course are as follows:
PRELIMINARY CONCEPTS AND DEFINITIONS (5 hours): Machines and mechanisms. Kinematic pairs. Degrees of freedom.
KINEMATICS REVIEW (5 hours): Rigid body’s kinematics. Instantaneous center of rotation. Polodes. Acceleration. Relative motions.
MECHANISM KINEMATICS (12 hours): Concept of loop and mechanism classification (parallel, serial and hybrid). Kinematic analysis. Four-bar linkage. Slider-crank linkage.
FORCES APPLIED TO THE MACHINES AND MACHINE KINETOSTATICS (15 hours): Classification of force systems. Friction work. Definition of efficiency. Efficiency of in-series and in-parallel machines. Retrograde motion. Reaction forces in the ideal kinematic pairs. Free-body diagrams. Kinetostatic analysis.
TRIBOLOGY (10 hours): Friction and wear in sliding and rolling contacts. Friction in kinematic pairs. Distribution of contact pressures (Clutches, Brakes). Roller bearings. Equilibrium of a vehicle in rectilinear motion.
LUBRICATION (18 hours): Lubricant viscosity and properties. Elementary theory of hydrodynamic lubrication. technical applications. Journal bearings: theory and design (use of Raimondi-Boyd diagrams). Hydrostatic lubrication. technical applications. Performances and applications of bearings.
CAM MECHANISMS (5 hours): Law of motion. Determination of the cam profile. Equivalent linkage. Kinematic analysis. Pressure angle. Undercutting. Elementary laws of motion.
LINKAGE SYNTHESIS (15 hours): Synthesis problems (number synthesis, type synthesis, path generation, body guidance, function generation). Analytic methods for the synthesis of planar linkages. Planar four-bar linkage (4bl) and its applications: transformation of a continuous rotatory motion into an oscillating rotatory motion; static analysis; the parallelogram linkage. 4bl synthesis with graphical methods. Roberts’ theorem. Use of the polodes: path generation, Euler-Savary equation, inflection circle and its applications (harbor crane). Use of the four-bar coupler curve atlas in the linkage synthesis: examples.
GEARS (25 hours): Spur gears: properties of the involute, velocity ratio, static analysis, manufacturing processes, contact ratio and interference, dimension standards, non-standard gear teeth, gear measurements. Helical gears. Bevel gears. Worm gears.
GEAR TRAINS (5 hours): Ordinary gear trains. Planetary gear trains. Differential gear systems. Torque relationships in gear systems.
VIBRATIONS OF SINGLE DEGREE OF FREEDOM SYSTEMS (5 hours): Free vibrations. Forced vibrations. Vibration isolation.


Didactic methods

During the course, the subjects are addressed both from a theoretical and from a practical point of view. In particular, for each topic, a number of exercises, whose solution is partly shown in class and partly left to the student as a tool for self-training, are proposed.

Learning assessment procedures

The students of the mechanical engineering school must take the examinations of the courses "Mechanics of Machines A" and "Mechanics of Machines B" in a unique session which tests the acquired skills and knowledge on the topics addressed in both the modules.

The examination of "Mechanics of Machines A” evaluates the skills in solving problems of kinematics and kinetostatics analysis of planar mechanisms, the organic knowledge of the topics listed in the course program and the exercise book of the student.

In practice, the student, over the presentation of his exercise book, must solve a problem of kinematic analysis and of kinetostatic analysis of a planar mechanism and answer to a number (two/three) of questions about the topics listed in the course program.

Reference texts

About all the topics addressed during the lectures are reported in the textbook:

[1] Funaioli, E., Maggiore, A. e Meneghetti, U., "Lezioni di MECCANICA APPLICATA ALLE MACCHINE – prima parte: fondamenti di meccanica delle macchine ", Ed. Patron, Bologna, 2005.

or in the alternative textbook

[2] Uicker,J.J., Pennock, G.R., Shigley,J.E., "Theory of Machines and Mechanisms", third edition 2003, Oxford University Press (, ISBN:0-19-515598-X

For the topics not reprted in [1] (or [2]), lecture notes freely downloadable from the section "Dispense" of the course website are available.

In the library, the following textbooks are also available:

[3] N. P. Belfiore, A. Di Benedetto, E. Pennestrì, "Fondamenti di Meccanica Applicata alle macchine", Casa Editrice Ambrosiana, Milano, 1a Ed. 2005 oppure 2a Ed. 2011;

[4] A. Zanarini, “Analisi cinetostatica grafica di meccanismi piani – applicazioni per la meccanica delle macchine”, Società Editrice Esculapio, 2012;

[5] Doughty, S., "Mechanics of Machines", John-Wiley & Sons, 1988;

[6] Paul, B., "Kinematics and dynamics of planar machinery", Prentice-Hall, 1979

[7] Waldrom K.J., Kinzel G.L., “Kinematics, Dynamics, and Design of Machinery”, John-Wiley & Sons, 1999

[8] Mabie H., Reinholtz C., “Mechanisms and dynamics of machinery,” John-Wiley & Sons, 1987.

Also, from the website ( of the Cornell University, a number of freely downloadable textbooks are available, among which the following:

[9] Hartenberg R.S., Denavit J, “Kinematic Synthesis of Linkages,” Mc.Graw-Hill, Inc., 1964

[10] Bickford J.H, “Mechanisms for Intermittent Motion,” Industrial Press, Inc, 1972

Eventually,it is worth stressing that, for the background concepts of rational mechanics,

- from Prof. Alberto Strumia's website (, the following textbook is freely downloadable:

[11] A. Strumia, “Meccanica Razionale”, Casa Editrice Nautilus, 2a Ed. 1996;