B2376 - Mechanics and Dynamics of Machines

Learning outcomes

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.

Course contents

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: classification of Cams and followers; Displacement diagram; Kinematic coefficients; Pressure angle; Radius of curvature of cam profile; Cam size; examples. Kinematic analysis: equivalent linkages. Graphical synthesis of cam profile: examples with translating cam; examples with rotating disc cam. Analytical synthesis of the cam profile with the envelope method. Analytical synthesis of cam profile and determination of other design parameters: translating radial follower (knife-edge; roller; flat-faced), and oscillating roller follower. Prevention of profile undercutting.

3. Gears. Introduction: general concepts and definitions; Types of gears; Ordinary and Epicyclic gear trains; Transmission ratio and gear ratio; the fundamental Law of gearing; tracing pairs of conjugate profiles; the involute curve. The involute gears: the geometry of involute gears; base circle and pitch circle; the advantage of involute gears; contact line and pressure angle; the involute rack. Spur involute gears. 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.

 

Readings/Bibliography

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, 2002.

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 (available on https://virtuale.unibo.it/).

Teaching methods

The lectures will focus on the theoretical foundations of the course topics and will be complemented by guided problem-solving sessions.

Although attendance is not mandatory, regular participation is strongly encouraged, as it is a key factor in achieving the intended learning outcomes.

The progressive development of concepts, the discussion of methodological aspects, and the use of examples during class provide essential support for understanding the subject matter and acquiring the skills required for the examination.

Students who attend lectures regularly are therefore expected to benefit significantly in their learning process.

Assessment methods

The final examination consists of a series of questions designed to assess both the acquisition of the knowledge covered by the course syllabus and the achievement of the intended learning outcomes:

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

- ability to address modelling issues in mechanical systems, with reference to multibody systems and mechanical vibrations.

The questions are aimed at assessing both the understanding of theoretical concepts and the ability to apply analytical methods to engineering problems.

The examination consists of a written test comprising multiple-choice questions, numerical-response questions requiring the calculation of specific quantities, together with an exercise involving the analysis of a two-degree-of-freedom mass–spring system.

The examination takes place in a computer laboratory and is conducted using the Examination On Line (EOL) platform.

During the examination, the use of textbooks, notes, or other written materials is not permitted.
Access to online learning resources is prohibited.

The use of Artificial Intelligence (AI) tools during the examination is strictly forbidden in any form. Any such use constitutes a violation of academic integrity.

Students are advised to use a non-programmable pocket calculator.

To be admitted to the examination, students must submit an Exercise Workbook to the Examining Committee a few days before the scheduled examination date, in accordance with the instructions provided on https://virtuale.unibo.it/ .
An email containing the submission instructions will be sent to students a few day before the scheduled examination date.

In accordance with the University Didactic Regulations, students may request the cancellation of a positive grade and retake the examination. The following rules apply:

• a positive grade may be cancelled only on the day of the examination;
• a cancelled grade cannot be reinstated under any circumstances;
• each student may request grade cancellation only once during the course.

Further information on the examination structure, passing requirements, and grading criteria is available on https://virtuale.unibo.it/ .

 

Teaching tools

Blackboard, PC.

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

Office hours

See the website of Alessandro Rivola