Anno Accademico 2023/2024

  • Docente: Alessandro Rivola
  • Crediti formativi:: 9
  • SSD: ING-IND/13
  • Lingua di insegnamento: Inglese
  • Moduli: Alessandro Rivola (Modulo 1) Alessandro Rivola (Modulo 2)
  • Modalità didattica: Convenzionale - Lezioni in presenza (Modulo 1) Convenzionale - Lezioni in presenza (Modulo 2)
  • Campus: Forli
  • Corso: Laurea Magistrale in Mechanical Engineering for Sustainability (cod. 5980)

Conoscenze e abilità da conseguire

The student acquires advanced concepts and methods for the functional design of the machines and is able to develop models of mechanical systems, with reference to multibody systems and mechanical vibrations.


Recall of basic concepts. Machine and Mechanism. Analysis versus synthesis. Links and Kinematic Pairs. Linkages. Kinematic chain. Mobility analysis. Degrees of Freedom of a mechanism. Kinematics of a rigid body in a plane. The Instantaneous center of velocity. Kennedy’s Theorem. Center of curvature. Centrodes. Conjugate profiles. Kinematic analysis of planar linkages.

1. Kinematics of linkages. Analysis versus synthesis. Modular approach to kinematic analysis. Assur groups. Kinematic Synthesis. Tasks of Kinematic Synthesis. The Four-Bar linkage. Grashof rule. Graphical synthesis. Synthesis of a crank-rocker four-bar linkage. Synthesis of a double rocker four-bar linkage. Motion generation: two and three prescribed positions. Path generation: Euler-Savary equation; inflection circle; center of curvature; Robert's theorem; example and applications. Analytical synthesis. The Dyad. The four-bar linkage: motion generation; path generation; function generation. Order synthesis.

2. Cam mechanism design. Introduction. Cam and follower types. Pressure angle. Kinematic analysis: equivalent linkages. Graphical cam profile synthesis. Analytical cam profile synthesis. Translating radial follower: point; roller; flat-face. Oscillating roller follower. Prevention of profile undercutting.

3. Gears. Conjugated profiles. Spur involute gears. Geometry of involute gears. Generation of involute curves by tools. Tooth element proportions. Rack-Cutter. Undercutting. Wildhaber’s concept. Modified involute Gears. Involute Helical Gears with parallel axes. Bevel Gears. Worm gearing.

4. Dynamics of Machines and Fundamental of mechanical vibrations. Dynamics of machines: inertia forces and moments; kinetic energy, D'Alembert's principle; principle of virtual works; principle of conservation of energy; Lagrange's equations; equivalent mass, inertia, forces and moments. 
Mechanical vibrations: discrete and continuous systems; spring elements; damping elements (viscous damping; Coulomb or Dry-Friction damping; hysteretic damping); harmonic motion.

5. Single Degree of Freedom Systems. Free vibrations: undamped system; viscous damping; Coulomb damping; hysteretic damping. Phase-plane representation. Logarithmic decrement method. Energy method: introduction to Rayleigh's method. Excited vibrations: harmonic excitation; Frequency Response Function (FRF); half-power bandwidth method; response of a damped system under rotating unbalance; forced vibration with hysteresis damping; impulse response function; response under a non-periodic force. 

6. Two Degrees of Freedom Systems. Equations of motion: choice of coordinates; static and dynamic coupling. Free vibrations: characteristic equation; natural frequencies; modes of vibration; initial conditions; rigid-body motion. Forced vibration analysis: impedance matrix; example; the mass damper system.

7. Multi-degree of Freedom systems. Equations of motion of undamped systems in matrix form. Eigenvalue problem: eigenvalues and eigenvectors; orthogonality; modal matrix; uncoupled equations; rigid-body motions. Forced vibration of viscously damped systems: proportional damping; modal and pseudo-modal methods.

8. Continuous Systems. Transverse vibration of a String. Longitudinal vibration of a bar. Orthogonality of mode shapes. Torsional vibration of a shaft. Lateral vibration of beams. Approximate methods: Rayleigh’s method; Rayleigh-Ritz method. Examples of excited systems.

9. Vibration measurements and modal analysis. Vibration measurement scheme. Signal analysis: time and frequency domain. Data sampling: Shannon's theorem; Aliasing. Discrete Fourier Transform. Choice of acquisition parameters. Experimental modal analysis: basic concepts; Transfer Function and Frequency Response Function (FRF); experimental measurements of FRFs.

10. Elastodynamic modelling. Lumped parameter modeling: longitudinal vibration of a bar; lateral vibration of a beam; simplified vehicle; forging hammer; mechanism with backlash; mechanisms for reciprocating motion.



1. Erdman A.G., Sandor G.N., Kota S., Mechanism Design: Analysis and Synthesis, 4th edition, Prentice Hall, 2001.

2. Norton R.L., Cam Design and Manufacturing Handbook, Industrial Press.

3. Uicker J. J., Pennock G. R., Shigley J. E., Theory of Machines and Mechanisms, 5th edition, Oxford University Press, 2016.

4. Vullo V., Gears: Geometric and Kinematic Design, vol. 1, Springer Nature, 2020.

5. Rao S.S., Mechanical vibrations, Sixth edition, Pearson Education, 2018.

6. Inman D.J., Engineering Vibration, 4th edition, Pearson Education, 2014.

7. Slides and notes from the lessons.

Metodi didattici

The lectures will focus on theoretical aspects of the course items. The lectures will be supplemented with practical exercises.

Class attendance, although not mandatory, plays a fundamental role in the learning and evaluation process.

Modalità di verifica e valutazione dell'apprendimento

The final examination consists of three questions that aim to ensure the acquisition of knowledge expected by the course program and to assess the achievement of learning objectives:

- knowledge of advanced methods for the functional design of the machines;

- ability to deal with issues concerning the modelling of mechanical systems, with reference to multibody systems and mechanical vibration.

Depending on the number of students registered for the exam, questions may be asked in written form through exercises.

The ability of the student to articulate and correctly explain the answers to the questions, has a fundamental weight in the attribution of the final marks.

To be admitted to the exam, students must submit an Exercise Workbook to the Examining Committee, as specified in https://virtuale.unibo.it/ .

In accordance with the Art. 16 of the University Didactic Regulations, if a positive grade does not meet the expectations of a student, the latter may ask its cancellation and the repetition of the examination. The instructors of the course comply with the following rules:

  • a student may request the cancellation of a positive grade only within the date of the examination;
  • a cancelled positive grade can in no way be recovered;
  • a student may request the cancellation of a positive grade at most once.

Strumenti a supporto della didattica

Blackboard, PC.

On the E-learning Platform (https://virtuale.unibo.it/ ), students may find slides of the course lectures.

Orario di ricevimento

Consulta il sito web di Alessandro Rivola


Istruzione di qualità Imprese innovazione e infrastrutture

L'insegnamento contribuisce al perseguimento degli Obiettivi di Sviluppo Sostenibile dell'Agenda 2030 dell'ONU.