# 28520 - Fundamentals of Mechanics of Machines T-1

## Learning outcomes

To provide the tools to understand the composition of machines and the principles of operation, the kinematic and static analysis, the dynamics of the rigid and flexible bodies machines, with the related problems.

## Course contents

1. Composition of machines
• 1.1. Introduction
• 1.2. Machines, mechanisms, members and classification
• 1.3. Degrees of freedom and constraints in plane and in space
• 1.4. Kinematic pairs
• 1.4.1. Rotoidal
• 1.4.2. Prismatic
• 1.4.3. Cylindrical
• 1.4.4. Helical
• 1.4.5. Spherical
• 1.4.6. Cylinder on cylinder
• 1.4.7. Cylinder in groove
• 1.4.8. Pair classification
• 1.5. Kinematic chains. Planar and spatial mechanisms
• 1.6. Degrees of freedom of a mechanism
• 1.6.1. Definition
• 1.6.2. Grubler and Kutzbach formulas
• 1.6.3. Application examples
• 1.6.4. Inessential dof
• 1.6.5. Repeated constraints
• 1.6.6. Mechanisms with 1 or more dof
2. Elements of rigid-body mechanics
• 2.1. Internal and external forces
• 2.2. Moments and transport theorem
• 2.3. Resultant vector and moment of a system of forces
• 2.4. Couples and generalized systems of forces
• 2.5. Reduction of a system in plane and in space. Resultant force.
• 2.6. Equilibrium of a mechanical system
• 2.7. Geometric relationships in equilibrated force systems
• 2.7.1. 2 forces
• 2.7.2. 3 forces
• 2.7.3. 4 forces
• 2.7.4.variants
3. Dissipative actions in machine components
• 3.1. Contact types in kinematic pairs
• 3.2. In/output members, motors/users
• 3.3. Kinetic friction.
• 3.4. Static friction. Friction cone.
• 3.5. Coulomb model
• 3.6. Causes of kinetic friction. Surface status. Lubrication.
• 3.7. Friction coefficient values. Influence of temperature and speed.
• 3.8. Rolling contact
• 3.8.1. Static friction
• 3.8.2. Hertz theory
• 3.8.3. Causes of rolling friction
• 3.8.4. Rolling friction parameter
• 3.8.5. Rolling friction coefficient
• 3.9. Wear
• 3.9.1. Adhesive and abrasive wear
• 3.9.2. Determination of the volume of removed material
• 3.9.3. Reye hypothesis
• 3.9.4. Wear for surface fatigue
• 3.10. Energy equations
• 3.11. Energy balance of a machine during direct motion. Periodic and absolute regime.
• 3.12. Efficiency in periodic and absolute regime: definitions
• 3.13. Instantaneous efficiency
• 3.14. Inverse motion. Energy balance of a machine during inverse motion.
• 3.15. Efficiency of inverse motion
• 3.15.1. Definitions
• 3.15.2. Relationship between direct and inverse motion efficiencies
• 3.15.3. Possibility of inverse motion
• 3.16. Efficiency of machines in series and in parallel
• 3.17. Inclined plane
• 3.17.1. Driving force: graphical and analytical solution
• 3.17.2. Efficiency of direct and inverse motion
• 3.17.3. Conditions for the inverse motion
• 3.17.4. Verification of the relationship between direct and inverse motion efficiencies
4. Static analysis of kinematic pairs
• 4.1. Rotoidal pair
• 4.1.1. Ideal and real constraint reaction
• 4.1.2. Friction circle
• 4.1.3. Equilibrium in ideal and real conditions
• 4.1.4. Friction moment
• 4.1.5. Efficiency: graphical and analytical method
• 4.2. Prismatic pair
• 4.2.1. Contact modes
• 4.2.2. Ideal and real constraint reaction
• 4.2.3. Equilibrium in ideal and real conditions
• 4.2.4. Stuck pair
• 4.3. Helical pair
• 4.3.1. Static analysis: ideal and real conditions
• 4.3.2. Equivalent fricion angle
• 4.3.3. Efficiency of direct and inverse motion
• 4.3.4. Efficiency variation with geometrical parameters
• 4.3.5. Direct and inverse motion condtions
• 4.3.6. Fastening screws
• 4.4. Cs Couple: Ideal, Real, Real with Rolling, Pure Rolling reactions
• 4.5. Wheels: static analysis of the driven, motor, braked wheel
• 4.6. Bearings
• 4.6.1. Hypotheses
• 4.6.2. Analysis of contact forces
• 4.6.3. Friction moment
• 4.6.4. Comparison with the rotoidal pair
• 4.7. Slewing revolute joint
• 4.7.1. Contact pressure distribution 4.7.2. Friction moment
• 4.8. Drum brake
• 4.8.1. Contact pressure distribution 4.8.2. Friction moment
5. Kinetostatic analysis of the mechanisms
• 5.1. Independent constraint reactions. Examples: rotoidal pair, prismatic, Cs.
• 5.2. Equilibrium of multi-body systems
• 5.3. Kinetastic analysis of mechanisms
• 5.3.1. Definition and characteristics
• 5.3.2. Number of unknowns
• 5.3.3. Systematic (or global) method
• 5.3.4. Direct method
• 5.3.5. Energy methods
• 5.3.6. Graphical methods
• 5.4. Kinetostatic analysis of the quadrilateral linkage (QL) with systematic, direct, energetic, graphical method. Matrix form.
6. Elements of kinematics of rigid bodies
• 6.1. Angular velocity
• 6.2. Fundamental law of rigid body kinematics
• 6.3. Particular instantaneous and finite motions: translation, rotation, planar motion
• 6.4. Instantaneous helical axis, instantaneous rotational axis, instantaneous rotation centre
• 6.5. Centroids
• 6.6. IRC of kinematic pairs
• 6.7. IRC of relative motions
• 6.8. Aronhold-Kennedy Theorem
• 6.9. Acceleration of the points of a rigid body
• 6.10. Example of the wheel
7. Kinematics analysis of mechanisms
• 7.1. Position, velocity, acceleration analysis
• 7.2. Systematic method for position analysis
• 7.3. Application to the QL: solution, multiplicity of solutions, closure configurations
• 7.4. Direct method with application to the QL
• 7.5. Velocity analysis. Application to the QL: transmission ratios.
• 7.6. Singularities: effects on position, velocity, forces, precision. Methods to avoid singularities.
• 7.8. Acceleration analysis
• 7.9. Definition of position, velocity, acceleration of generic points. Application to the QL.
• 7.10. Graphical method for the velocity analysis: vector method, IRC method.
• 8.1. QL: applications, Grashof rule
• 8.2. Parallelogram: applications, antiparallelogram
• 8.3. Slider-crank: applications, kinematic analysis at order I and II
• 8.4. Inverted slider-crank: applications, quick return mechanisms, Maltese cross
• 8.5. Double slider: IRC and centroids, Scott Russell mechanism, Oldham coupling, inertial forces
• 8.6. Spatial QL
• 8.7. Cardan joint: features, transmission ratio
• 8.8. Double Cardan joint: transmission ratio, homokinetic joint
9. Gears
• 9.1. Centroids with constant τ
• 9.2. Friction wheels: force transmission, limitations
• 9.3. Conjugate curves: envelope
• 9.4. Spur gears
• 9.4.1. Tooth profiles
• 9.4.2. Geometrical parameters
• 9.4.2. Pitch and base circle
• 9.4.3. Pressure angle
• 9.4.4. Force transmission (no friction)
• 9.5. Helical gears
• 9.5.1. Tooth profiles
• 9.5.2. Tooth pressure
• 9.5.3. Force transmission (no friction)
• 9.5.4. Axial force compensation
• 9.5.5. Rack
• 9.6. Motion transmission between intersecting axes with constant τ: centroids and determination of τ
• 9.7. Conical friction wheels and bevel gears
• 9.8. Gears for skewed axis transmission: common solutions (worm and hypoid gears), problems
10. Gear trains
• 10.1. Definitions and features
• 10.2. Determination of τ in a standard gear train. Examples. Idler gears.
• 10.3. Efficiency. Motor torque.
• 10.4. Division of τ between different gears
• 10.5. Epicyclic gearing. Determination of τ. Willis formula: fixed wheel 1, fixed wheel n, two degrees of freedom. Examples.
• 10.6. Differential for automobiles: velocity and moment. Differential blocking.
11. Cams
• 11.1. Definitions, features and classifications
• 11.2. Force and shape contact
• 11.3. Motion laws: characteristics, triangular diagram, complete and adimensional forms, common laws
• 11.4. Definition of cam profiles
• 11.5. Equivalent mechanisms
• 11.6. Kinetostatic analysis and pressure angle
• 11.7. Mechanical and electronical cams
12. Flexible organs
• 12.1. Definitions and typologies
• 12.2. Elastic and anelastic flexional stiffness
• 12.3. Work done and geometrical configuration during winding and unwinding: elastic, anelastic and combined conditions. Total work.
• 12.4. Lost work (dissipation) computation. Equivalent δ parameter and stiffness model.
• 12.5. Static analysis of the fixed and moving pulley and of the hoist:
• 12.5.1. K parameter
• 12.5.2. Motor force
• 12.5.3. Efficiency
• 12.5.4. Lifting velocity
• 12.6. Belt transmission:
• 12.6.1. Features
• 12.6.2. Moment
• 12.6.3. Belt velocity
• 12.6.4. Ideal and real τ ratio
• 12.6.5. Efficiency
• 12.6.6. Adherence and slippage angles
• 12.7. Eithelwein equation:
• 12.7.1. Tension variation
• 12.7.3. Tension in belt
• 11.7.3. Maximum moment
• 12.8. Belt brake
13. Elements of dynamics of rigid bodies
• 13.1. Fundamental equation of dynamics
• 13.2. Resultant and resultand moment of inertia forces with respect to the center of gravity. Inertia tensor. Reduction of the force system.
• 13.3. Kinetic energy
• 13.4. Special cases: principal axes of inertia, planar motion, translation
• 13.5. Substitution masses: spatial and planar case
14. Dynamic analysis of machines
• 14.1. Total and partial balancing of mechanisms
• 14.2. Direct dynamics of the slider-crank mechanism. Forces to the ground: couples, rotating forces, alternating forces.
• 14.3. Balancing of the forces to the ground: example on the single slider-crank mechanism
• 14.4. Kinetic energy of the slider-crank mechanism. Energy of rotating and alternating masses.
• 14.5. Reduced moment of inertia. Reduced moment of rotating and alternating masses.
• 14.6. Mass-spring-damper model: importance, definitions, vibrations with 1 dof
• 14.7. Free vibrations:
• 14.7.1. Motion of the mass
• 14.7.2. Anharmonic, damped harmonic, harmonic motion
• 14.7.3. Initial conditions
• 14.8. Forced vibrations:
• 14.8.1. Force model
• 14.8.2. Transient and steady-state solution
• 14.8.3. Graphical solution with rotating vectors
• 14.8.4. Amplitude and phase with and without damping. Discussions of results.
• 14.8.5. Amplitude and phase resonance
• 14.8.6. Result discussion with exciting force amplitude proportional to ω2
• 14.9. Reduction of the force to the ground:
• 14.9.1. Graphical solution with rotating vectors
• 14.9.2. Transmissibility
• 14.9.3. Suspension optimization
• 14.10. Reduction of vibrations generated by a vibrating ground:
• 14.10.1. Relative displacement
• 14.10.2. Suspension optimization
• 14.11. Rotor dynamics:
• 14.11.1. Ideal and real rotors
• 14.11.2. Static and dynamic imbalance
• 14.11.3. Inertia forces
• 14.12. Flexional vibrations in rotors:
• 14.12.1. Jeffcott rotor
• 14.12.2. Motion: transient and steady-state solution, with and without damping
• 14.12.3. Critical valocity, alignment and self-centering
• 14.13. Rotor balancing on two planes:
• 14.13.1. Balancing forces
• 14.13.2. Correction masses
• 14.13.3. Balancing machines

RECOMMENDED BOOKS

E. Funaioli, A. Maggiore, U. Meneghetti, “Lezioni di Meccanica Applicata alle Macchine – Prima parte – Fondamenti di Meccanica delle Macchine”, ed. Pàtron, Bologna.

OTHER MATERIAL

Slides provided by Professor, published on "Insegnamenti online".

OTHER BOOKS

M. Callegari, P. Fanghella, F. Pellicano, “Meccanica applicata alle macchine”, ed. CittàStudi.

A. Zanarini, “Analisi cinetostatica grafica di meccanismi piani”, ed. Esculapio.

EXERCISES

Besides Professor's slides, exercises and previous tests are published on "Insegnamenti online".

## Teaching methods

The course consists of theoretical lessons and classroom exercises.

As for the lessons, the theoretical and analytical aspects of typical problems of Applied Mechanism are analysed. Starting from the prior knowledge of physics and analytical mechanics (recalled and discussed in the course), new theoretical tools for modeling and functional analysis of machines and mechanisms are provided, with particular emphasis on kinematic, static and dynamic problems.

Exercises provide examples (including some particular cases) of some methods for the computation of degrees of freedom and for the kinematic and kinetostatic analysis of mechanisms.

Since suggested readings cover all course contents, lesson attendance is not mandatory. However, it is strongly suggested to attend lessons in order to better understand the theoretical derivations of models and methods presented in the course.

Module 2 (Mechanics of Material) for Electrical Energy Engineering is given in parallel to Module 1 and is the subject of a separate exam.

## Assessment methods

EXAM

Written + Oral

WRITTEN TEST

It is an exercise of graphical kinematic and kinetostatic analysis.

ORAL EXAM

It consists of two theoretical questions on the course contents. For Electrical Energy Engineering, questions are related to Module 1 alone, Module 2 being the subject of a separate exam.

Mathematical derivations could be asked. Candidates must explain mathematical steps, to clarify the logic behind them.

EXAM

The written test lasts 60 minutes. At the end of the written test, the oral exam begins. Candidates will be called in list order to:

• correct the exercise of the written test;

Each question is assigned a maximum score of 30. To continue the oral test, the written test has to be sufficient. Overall, the exam is passed if both of the following conditions are met:

• Average rating ≥18;
• Two theoretical questions ≥18, OR one theoretical question ≥18 and one theoretical question ≥12, OR two theoretical questions ≥15.

Depending on the number of candidates, oral exams may last several days.

MARK

Automation Engineering: arithmetic mean of the single marks of the written test and theoretical questions.

Electrical Energy Engineering: if M1 and M2 are the marks of Module 1 and 2 respectively, the final mark is computed as 2/3 M1 + 1/3 M2.

OTHER RULES

Six possible dates a year to pass the exam, in general 3 in january-february, 2 in june-july, 1 in september.

Candidates can reject the vote only once.

In case of exam failure, the candidate can attend next date.

In case of no-show, the candidate cannot attend next date.

Notes cannot be checked during the exam.

## Teaching tools

During the course some presentations will be used to support theoretical topics. The material is made available on the "Insegnamenti online" website.

## Office hours

See the website of Nicola Sancisi