teacher profile teaching materials
Bachelor Degree L-9 Mechanical Engineering
TOPOLOGY (dimensionless quantities)
Link (kinematic element) and kinematic pair; degree of freedom of a pair, mobility, degree of constraint. Commonly used pairs (revolute, prismatic, helical, cylindrical, spherical, planar) with determination of the corresponding degrees of freedom and of constraint. Reuleaux classification; higher and lower pairs; force-closed and form-closed pairs. Mechanism and kinematic chain, with examples.
Representation of mechanisms and kinematic chains: functional diagram, polygonal representation, Watt and Stephenson chains, outline of graph theory and of Franke's notation.
Topological formulas for the degrees of freedom: Grubler's formula in space and in the plane, Euler's formula for the number of independent loops, Kutzbach's formula.
KINEMATICS (space and time, planar rigid-body motions)
Infinitesimal motions; velocity and acceleration of a link in the intrinsic (path) formulation and of two links belonging to the same rigid body; fundamental formula of kinematics. Relative motions with respect to two reference frames; instantaneous state of motion (velocity state).
Velocity centre C and instantaneous centre of rotation P0; velocity field. First-order centrodes: fixed centrode from the paths of two points, transfer method for the moving centrode, pure rolling between the centrodes, the centrodes of the slider-crank mechanism.
Acceleration centre K: existence and uniqueness, acceleration field, velocity of K, acceleration of P0, inflection circle, orthogonality of the centrodes to the acceleration of P0. Euler-Savary equation (first and second form). Centre of curvature of the path of a point given the centrodes (graphical construction and its justification); circle of stationary curvature.
Mechanisms with higher pairs: generation of the conjugate profile from a given one, the centrodes being known (envelope method, method of the normals); simultaneous generation of the conjugate profiles; epicycle method with auxiliary curve and with point path; centres of curvature of conjugate profiles; equivalent mechanisms. Aronhold-Kennedy theorem; sliding velocity.
Linkages and Grashof's rule for the four-bar linkage. Transmission couplings: rigid and flexible couplings (outline), articulated couplings, Oldham coupling, Cardan (universal) joint and its transmission ratio, double Cardan joint.
Exercises on oscillating and rolling levers by means of the relative-motion theorem and of the equivalent mechanism. Examples: parallelogram and antiparallelogram linkage, drafting machine, pantograph, Hart's and Peaucellier's inversors.
STATICS (forces, equilibrium under steady conditions)
Cardinal equations; graphical methods and elementary cases of a body subjected to 2, 3 and 4 forces only. Principle of dismemberment and dismemberment of complex subassemblies. Principle of virtual work applied to ideal mechanisms.
TRIBOLOGY (surface contacts with sliding and normal loads)
Roughness: nominal profile, maximum surface height Rt, average roughness, roughness parameter Rq, outline of the standards ASME B46.1 and ISO 4287 and 13565.
Coulomb friction coefficient: static and sliding friction. Hertz formulas: point contacts, mean radius of curvature, radius of the contact spot, maximum pressure, mutual approach; line contacts (optional).
Friction coefficient under the assumption of adhesive wear; other wear mechanisms (abrasion, erosion, fretting and corrosion, surface fatigue); phenomenological classification (scuffing, scoring, spalling, case crushing, pitting, galling); Reye's energy model, Archard's model and specific wear rate.
Friction in the journal (load-carrying) revolute pair (friction circle) and static equilibrium of a lever with friction; thrust revolute pair (mean radius). Rolling friction due to hysteresis (elastic lag), with the cases of the unloaded towed wheel and of the driving wheel; definition of the rolling friction coefficient; rolling friction due to impacts.
Rolling bearings: static rating of radial bearings, Stribeck's formula for balls and rollers, fatigue life calculation.
LUBRICATION
Lubricants and additives; viscosity, Petroff's law, Newtonian fluids, dependence of viscosity on temperature and pressure, Barus' formula and Reynolds' formula, viscosity index V.I.
One-dimensional Reynolds theory: flow rate in a film with parallel flat faces and the need for a film of variable thickness; pressure gradient in a converging plane film; section of zero pressure gradient; pressure distribution, load-carrying force, tangential force, line of action, mean friction coefficient. Stepwise constant film (step bearings) and linearly varying film thickness.
Hydrodynamically lubricated thrust bearings; Michell (tilting-pad) bearings, direct and inverse problem. Hydrostatic lubrication of the thrust revolute pair: total flow rate and pressure, outline of hydrostatic compensation, resistance of the recess and of the restrictor.
Hydrodynamically lubricated journal bearing pair: film geometry, radial clearance, eccentricity, film thickness, pressure gradient and shear stresses. Full bearing according to Sommerfeld's theory: position of zero pressure gradient, side thrust (zero), load-carrying force, equivalent tangential force, mean friction coefficient and its physical meaning. Half-bearing model; conditions for determining the eccentricity and the attitude angle.
MECHANICAL EFFICIENCY
Work and energy; energy balance equation of a machine; absolute and periodic steady regime; ideal and actual operation. Mechanical efficiency; efficiency of mechanisms in series and in parallel; efficiency in systems under absolute steady regime. Backward (reverse) motion and self-locking, with the relevant conditions.
DYNAMICS OF SYSTEMS OF RIGID BODIES
Classification of forces: internal and external, driving and resisting, active and constraint forces. Dynamics of the link and of the rigid body: general cardinal equations, inertia loads and reformulation of the cardinal equations, principle of virtual work extended to dynamics. Direct and inverse dynamic problem: four-bar linkage (inverse) and compound pendulum (direct).
GEARS
Friction wheels: centre distance, transmission ratio, diameter of the pitch circles. Involute profiles: straight-line epicycle method with point path and with auxiliary curve, method of the normals, base circle as the evolute of the tooth profile. Characteristic (pressure) angle, pitch, module and modular proportioning, tooth thickness and space width; arc of approach and arc of recess, arc of action, contact ratio.
Interference in gears and means to avoid it: increase of the characteristic angle, stub profiles, profile-shifted (corrected) profiles. Sliding velocity and instantaneous efficiency; minimum number of teeth in the pinion-rack pair. Gear geometry by means of Computer Aided Design programs, simulation of tooth cutting, undercutting, specific slidings.
ANALYTICAL AND NUMERICAL METHODS OF KINEMATIC ANALYSIS
Complex-number method: Euler's formula, 90-degree rotation operator, velocities and accelerations of two points of the same rigid body.
Configurations of a four-bar linkage by means of the loop-closure equation and its solution by substitutions; kinematic characteristics of a generic coupler point; differentiation of the loop-closure equation for angular velocities and accelerations. Applications to the four-bar linkage by means of a symbolic manipulation program, with analytical methods and with constraint equations (link angles, natural coordinates).
Constraint-equation method: degrees of freedom, independence of the equations, position analysis by the Newton-Raphson numerical method, velocity and acceleration analysis with and without partitioning of the matrices. Critical (singular) configurations by means of Dini's theorem; critical configurations of the slider-crank mechanism with the crank as driving link.
DYNAMIC ANALYSIS BY LAGRANGE MULTIPLIERS (IN THE PLANE)
Lagrangian coordinates; mass matrix of the whole system; vectors of the accelerations, of the external forces and of the constraint reactions. Partitioning of the Jacobian matrix and subdivision of the coordinates into dependent and independent ones; assumption of vanishing work of the reaction forces (ideal case). Reaction vector as a function of the Lagrange multipliers; iterative solution of the direct dynamic problem with determination of the law of motion. Example by means of a symbolic manipulation program: double pendulum.
MECHANICAL OSCILLATIONS IN ELASTIC SYSTEMS REDUCIBLE TO LUMPED-PARAMETER SYSTEMS
Masses, springs, viscous dampers; complex-number method and Euler's formula; springs and dampers in series and in parallel.
Undamped free harmonic oscillator: equation of dynamic equilibrium, response for given initial conditions, Rayleigh's method and its applications.
Damped free harmonic oscillator: equation of dynamic equilibrium, overdamped (hypercritical), critical and underdamped (subcritical) damping factor, logarithmic decrement.
Damped forced vibrations: forcing function, solution in the complex field, magnitude and phase of the response, vector diagrams for the quasi-static, resonance and seismographic regions, transmissibility coefficient.
Flexural vibrations: flexural critical speeds with one degree of freedom, bending stiffness of a shaft, deflection at the section where the load is applied, second moment of area of the circular cross-section. Rotating shafts: self-centring, case of zero eccentricity; rotordynamics and the Jeffcott rotor.
Outline of systems with n degrees of freedom: synchronous solution in the complex field, mass and stiffness matrices.
Torsional natural frequencies: torsional stiffness of a shaft, polar second moment of area, torsional pendulum, natural frequencies of two flywheels with free ends, trivial solution and its interpretation, elastic line (mode shape).
REGULATING DEVICES: FLYWHEEL SIZING
Reduction of forces and masses; balancing of a slider-crank mechanism; energy balance equation; periodic irregularity (coefficient of speed fluctuation); maximum peripheral speed; flywheel sizing.
BRAKES
The braking problem and the reference equations. Particular cases: stopping from rectilinear travel, maintaining constant speed with excess driving force. Brakes with rigid approach: jamming. Brakes with free approach: partialization of the shoe.
CAMS
Dynamics (follower jump problem) and tribology (lubrication and wear). Lift diagram; equivalent translating plate with knife-edge follower; envelope construction for the offset roller follower; constant-acceleration cam.
HOISTS
Lifting devices: direct-pull and inverted-pull hoists, applications, operation in the ideal case, static force analysis. Optional: Witt's hoist and differential pulley block.
NUMERICAL INTEGRATION METHODS
Representation scales of planar vectors and length scales, with an example of representation of mechanisms. Graphical methods: pole method and polar diagram method. Numerical integration: Bezout's and Cavalieri-Simpson formulas. Use of the spreadsheet; solution of simple differential equations by the step-by-step method; outline of the finite-difference method.
CLASSROOM EXERCISES
No.1 Kinematic analysis of a planar four-bar linkage by a graphical method based on points of the same rigid body.
No.2 Centrodes of the motion of the coupler relative to the frame in an in-line slider-crank mechanism (graphical method).
No.3 Kinematic analysis of mechanisms with slotted links, higher pairs and short coupler links: Fairbairn guide and cam-follower mechanism.
No.4 Kinematic analysis by a graphical method based on first- and second-order poles: four-bar linkage and slider-crank mechanism.
No.5 Involute and cycloid: path of a tracing point for given centrodes of the motion (graphical method).
No.6 Equilibrium of mechanisms by graphical statics in the ideal case: free-body graphical method for the ideal reactions; virtual work for the driving actions under a load Q.
No.7 Calculation of the instantaneous efficiency of a mechanism.
No.8 Inverse dynamic problem by the free-body method.
No.9 Gear geometry and interference in gears.
COMPUTER LAB ACTIVITIES
PCLAB 1. Elementary direct dynamic problem: rectilinear motion of an accelerated mass (analytical method and numerical methods).
PCLAB 2. Newton-Raphson method for the configuration of a planar four-bar linkage and of an in-line slider-crank mechanism.
PCLAB 3. Involute and cycloid: path of a tracing point for given centrodes of the motion (analytical method).
PCLAB 4. Numerical method based on constraint equations for the kinematic analysis of mechanisms.
PCLAB 5. Hydrodynamic lubrication: inverse and direct problem.
PCLAB 6. Multi Body System dynamics (MBS Dynamics): direct dynamic problem.
PCLAB 7. Free mass-spring-damper system: direct dynamic problem, analytical and numerical solution.
Fondamenti di meccanica applicata alle macchine
Terza edizione | 2024 | Casa Editrice Ambrosiana. Distribuzione esclusiva Zanichelli
Programme
SUMMARY OF THE SYLLABUS OF THE COURSE OF APPLIED MECHANICS (MECHANICS OF MACHINES) - 9 CFUBachelor Degree L-9 Mechanical Engineering
TOPOLOGY (dimensionless quantities)
Link (kinematic element) and kinematic pair; degree of freedom of a pair, mobility, degree of constraint. Commonly used pairs (revolute, prismatic, helical, cylindrical, spherical, planar) with determination of the corresponding degrees of freedom and of constraint. Reuleaux classification; higher and lower pairs; force-closed and form-closed pairs. Mechanism and kinematic chain, with examples.
Representation of mechanisms and kinematic chains: functional diagram, polygonal representation, Watt and Stephenson chains, outline of graph theory and of Franke's notation.
Topological formulas for the degrees of freedom: Grubler's formula in space and in the plane, Euler's formula for the number of independent loops, Kutzbach's formula.
KINEMATICS (space and time, planar rigid-body motions)
Infinitesimal motions; velocity and acceleration of a link in the intrinsic (path) formulation and of two links belonging to the same rigid body; fundamental formula of kinematics. Relative motions with respect to two reference frames; instantaneous state of motion (velocity state).
Velocity centre C and instantaneous centre of rotation P0; velocity field. First-order centrodes: fixed centrode from the paths of two points, transfer method for the moving centrode, pure rolling between the centrodes, the centrodes of the slider-crank mechanism.
Acceleration centre K: existence and uniqueness, acceleration field, velocity of K, acceleration of P0, inflection circle, orthogonality of the centrodes to the acceleration of P0. Euler-Savary equation (first and second form). Centre of curvature of the path of a point given the centrodes (graphical construction and its justification); circle of stationary curvature.
Mechanisms with higher pairs: generation of the conjugate profile from a given one, the centrodes being known (envelope method, method of the normals); simultaneous generation of the conjugate profiles; epicycle method with auxiliary curve and with point path; centres of curvature of conjugate profiles; equivalent mechanisms. Aronhold-Kennedy theorem; sliding velocity.
Linkages and Grashof's rule for the four-bar linkage. Transmission couplings: rigid and flexible couplings (outline), articulated couplings, Oldham coupling, Cardan (universal) joint and its transmission ratio, double Cardan joint.
Exercises on oscillating and rolling levers by means of the relative-motion theorem and of the equivalent mechanism. Examples: parallelogram and antiparallelogram linkage, drafting machine, pantograph, Hart's and Peaucellier's inversors.
STATICS (forces, equilibrium under steady conditions)
Cardinal equations; graphical methods and elementary cases of a body subjected to 2, 3 and 4 forces only. Principle of dismemberment and dismemberment of complex subassemblies. Principle of virtual work applied to ideal mechanisms.
TRIBOLOGY (surface contacts with sliding and normal loads)
Roughness: nominal profile, maximum surface height Rt, average roughness, roughness parameter Rq, outline of the standards ASME B46.1 and ISO 4287 and 13565.
Coulomb friction coefficient: static and sliding friction. Hertz formulas: point contacts, mean radius of curvature, radius of the contact spot, maximum pressure, mutual approach; line contacts (optional).
Friction coefficient under the assumption of adhesive wear; other wear mechanisms (abrasion, erosion, fretting and corrosion, surface fatigue); phenomenological classification (scuffing, scoring, spalling, case crushing, pitting, galling); Reye's energy model, Archard's model and specific wear rate.
Friction in the journal (load-carrying) revolute pair (friction circle) and static equilibrium of a lever with friction; thrust revolute pair (mean radius). Rolling friction due to hysteresis (elastic lag), with the cases of the unloaded towed wheel and of the driving wheel; definition of the rolling friction coefficient; rolling friction due to impacts.
Rolling bearings: static rating of radial bearings, Stribeck's formula for balls and rollers, fatigue life calculation.
LUBRICATION
Lubricants and additives; viscosity, Petroff's law, Newtonian fluids, dependence of viscosity on temperature and pressure, Barus' formula and Reynolds' formula, viscosity index V.I.
One-dimensional Reynolds theory: flow rate in a film with parallel flat faces and the need for a film of variable thickness; pressure gradient in a converging plane film; section of zero pressure gradient; pressure distribution, load-carrying force, tangential force, line of action, mean friction coefficient. Stepwise constant film (step bearings) and linearly varying film thickness.
Hydrodynamically lubricated thrust bearings; Michell (tilting-pad) bearings, direct and inverse problem. Hydrostatic lubrication of the thrust revolute pair: total flow rate and pressure, outline of hydrostatic compensation, resistance of the recess and of the restrictor.
Hydrodynamically lubricated journal bearing pair: film geometry, radial clearance, eccentricity, film thickness, pressure gradient and shear stresses. Full bearing according to Sommerfeld's theory: position of zero pressure gradient, side thrust (zero), load-carrying force, equivalent tangential force, mean friction coefficient and its physical meaning. Half-bearing model; conditions for determining the eccentricity and the attitude angle.
MECHANICAL EFFICIENCY
Work and energy; energy balance equation of a machine; absolute and periodic steady regime; ideal and actual operation. Mechanical efficiency; efficiency of mechanisms in series and in parallel; efficiency in systems under absolute steady regime. Backward (reverse) motion and self-locking, with the relevant conditions.
DYNAMICS OF SYSTEMS OF RIGID BODIES
Classification of forces: internal and external, driving and resisting, active and constraint forces. Dynamics of the link and of the rigid body: general cardinal equations, inertia loads and reformulation of the cardinal equations, principle of virtual work extended to dynamics. Direct and inverse dynamic problem: four-bar linkage (inverse) and compound pendulum (direct).
GEARS
Friction wheels: centre distance, transmission ratio, diameter of the pitch circles. Involute profiles: straight-line epicycle method with point path and with auxiliary curve, method of the normals, base circle as the evolute of the tooth profile. Characteristic (pressure) angle, pitch, module and modular proportioning, tooth thickness and space width; arc of approach and arc of recess, arc of action, contact ratio.
Interference in gears and means to avoid it: increase of the characteristic angle, stub profiles, profile-shifted (corrected) profiles. Sliding velocity and instantaneous efficiency; minimum number of teeth in the pinion-rack pair. Gear geometry by means of Computer Aided Design programs, simulation of tooth cutting, undercutting, specific slidings.
ANALYTICAL AND NUMERICAL METHODS OF KINEMATIC ANALYSIS
Complex-number method: Euler's formula, 90-degree rotation operator, velocities and accelerations of two points of the same rigid body.
Configurations of a four-bar linkage by means of the loop-closure equation and its solution by substitutions; kinematic characteristics of a generic coupler point; differentiation of the loop-closure equation for angular velocities and accelerations. Applications to the four-bar linkage by means of a symbolic manipulation program, with analytical methods and with constraint equations (link angles, natural coordinates).
Constraint-equation method: degrees of freedom, independence of the equations, position analysis by the Newton-Raphson numerical method, velocity and acceleration analysis with and without partitioning of the matrices. Critical (singular) configurations by means of Dini's theorem; critical configurations of the slider-crank mechanism with the crank as driving link.
DYNAMIC ANALYSIS BY LAGRANGE MULTIPLIERS (IN THE PLANE)
Lagrangian coordinates; mass matrix of the whole system; vectors of the accelerations, of the external forces and of the constraint reactions. Partitioning of the Jacobian matrix and subdivision of the coordinates into dependent and independent ones; assumption of vanishing work of the reaction forces (ideal case). Reaction vector as a function of the Lagrange multipliers; iterative solution of the direct dynamic problem with determination of the law of motion. Example by means of a symbolic manipulation program: double pendulum.
MECHANICAL OSCILLATIONS IN ELASTIC SYSTEMS REDUCIBLE TO LUMPED-PARAMETER SYSTEMS
Masses, springs, viscous dampers; complex-number method and Euler's formula; springs and dampers in series and in parallel.
Undamped free harmonic oscillator: equation of dynamic equilibrium, response for given initial conditions, Rayleigh's method and its applications.
Damped free harmonic oscillator: equation of dynamic equilibrium, overdamped (hypercritical), critical and underdamped (subcritical) damping factor, logarithmic decrement.
Damped forced vibrations: forcing function, solution in the complex field, magnitude and phase of the response, vector diagrams for the quasi-static, resonance and seismographic regions, transmissibility coefficient.
Flexural vibrations: flexural critical speeds with one degree of freedom, bending stiffness of a shaft, deflection at the section where the load is applied, second moment of area of the circular cross-section. Rotating shafts: self-centring, case of zero eccentricity; rotordynamics and the Jeffcott rotor.
Outline of systems with n degrees of freedom: synchronous solution in the complex field, mass and stiffness matrices.
Torsional natural frequencies: torsional stiffness of a shaft, polar second moment of area, torsional pendulum, natural frequencies of two flywheels with free ends, trivial solution and its interpretation, elastic line (mode shape).
REGULATING DEVICES: FLYWHEEL SIZING
Reduction of forces and masses; balancing of a slider-crank mechanism; energy balance equation; periodic irregularity (coefficient of speed fluctuation); maximum peripheral speed; flywheel sizing.
BRAKES
The braking problem and the reference equations. Particular cases: stopping from rectilinear travel, maintaining constant speed with excess driving force. Brakes with rigid approach: jamming. Brakes with free approach: partialization of the shoe.
CAMS
Dynamics (follower jump problem) and tribology (lubrication and wear). Lift diagram; equivalent translating plate with knife-edge follower; envelope construction for the offset roller follower; constant-acceleration cam.
HOISTS
Lifting devices: direct-pull and inverted-pull hoists, applications, operation in the ideal case, static force analysis. Optional: Witt's hoist and differential pulley block.
NUMERICAL INTEGRATION METHODS
Representation scales of planar vectors and length scales, with an example of representation of mechanisms. Graphical methods: pole method and polar diagram method. Numerical integration: Bezout's and Cavalieri-Simpson formulas. Use of the spreadsheet; solution of simple differential equations by the step-by-step method; outline of the finite-difference method.
CLASSROOM EXERCISES
No.1 Kinematic analysis of a planar four-bar linkage by a graphical method based on points of the same rigid body.
No.2 Centrodes of the motion of the coupler relative to the frame in an in-line slider-crank mechanism (graphical method).
No.3 Kinematic analysis of mechanisms with slotted links, higher pairs and short coupler links: Fairbairn guide and cam-follower mechanism.
No.4 Kinematic analysis by a graphical method based on first- and second-order poles: four-bar linkage and slider-crank mechanism.
No.5 Involute and cycloid: path of a tracing point for given centrodes of the motion (graphical method).
No.6 Equilibrium of mechanisms by graphical statics in the ideal case: free-body graphical method for the ideal reactions; virtual work for the driving actions under a load Q.
No.7 Calculation of the instantaneous efficiency of a mechanism.
No.8 Inverse dynamic problem by the free-body method.
No.9 Gear geometry and interference in gears.
COMPUTER LAB ACTIVITIES
PCLAB 1. Elementary direct dynamic problem: rectilinear motion of an accelerated mass (analytical method and numerical methods).
PCLAB 2. Newton-Raphson method for the configuration of a planar four-bar linkage and of an in-line slider-crank mechanism.
PCLAB 3. Involute and cycloid: path of a tracing point for given centrodes of the motion (analytical method).
PCLAB 4. Numerical method based on constraint equations for the kinematic analysis of mechanisms.
PCLAB 5. Hydrodynamic lubrication: inverse and direct problem.
PCLAB 6. Multi Body System dynamics (MBS Dynamics): direct dynamic problem.
PCLAB 7. Free mass-spring-damper system: direct dynamic problem, analytical and numerical solution.
Core Documentation
Nicola Pio Belfiore, Augusto Di Benedetto, Ettore PennestrìFondamenti di meccanica applicata alle macchine
Terza edizione | 2024 | Casa Editrice Ambrosiana. Distribuzione esclusiva Zanichelli
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.