MACHINE DESIGN – II (9 cfu)

Objectives and organization

This course is aimed at delivering the main concepts of finite elements and of structural dynamics in the design of engines, machines and their mechanical components. The familiarization with modeling issues is also promoted, together with the implementation of the above concepts to practical cases.

Class exercises will be organized for practising with both self-written computer programs as well as commercial f.e. codes.



CONTENT OF THE COURSE

1) Matrix method and Finite Elements for structural calculation:

Matrix method for monodimensional plane structures - Stiffness matrix for an element - Global /local reference rotation matrix - Structure stiffness matrix - Permutation of dof and stiffness partitioning - Nodal displacements and reaction loads - Internal and external constraints - Element geometry, nodal displacement approximating polynomials - Shape functions - Differentiation and strains - Stress-load relationship - Principle of virtual work, Rayleigh Ritz method - Truss elements - Euler beam - 4 nodes plane elements (plane stress / plane strain membrane, Kirchoff plate, axysimmetric solid) - 8 noded esahedric element - Isoparametric formulation - Gauss integration - Matrix Equations for dynamic equilibrium - Consistency and direct lumping for mass matrices - Free undamped oscillatory solution - Rayleigh damping - State equations for damped oscillatory motion - Transitory and regimen damped oscillations.

References:


“Belingardi”, Parte 1 cap. 1; Parte 2 capp. 1, 2, 3; Parte 3 chapt. 1, 2,3, 4, 6, 7, 9, 11; Appendice 2.
“Zienkiewicz-Taylor” chapt. 17, Dispensa Gavin x matrici




2) Rotors and rotating discs:

Fundamental equations - constant thickness disc - effect of shaft and blade supporting ring - Hyperbolic, conical and uniform stress profile discs - Grammel method for arbitrary shape discs - Thermal stresses in discs with arbirtary profile – Centrifugal stress due to radial blades - Stress concentation around holes

References:

Giovannozzi –II” chapt. 10.



3) Critical speed of shafts:

Shaft with a single lumped mass - Critical speed, eccentricity, oscillations - Effect of transverse inertia - Effect of axial stress - Shaft with multiple lumped masses - Equivalent shaft - Vibration modes, eigenvalues and eigenvectors.

References:

“Giovannozzi –II” chapt. 11.



4) Torsional vibrations of shafts:

Introduction to torsional vibrations - Equivalent system for a torsionally vibrating shaft - Free and forced solutions for torsional vibrations of a shaft - Frequencies of series and parallel shafts systems - Inertia reduction to crank axis for traslating and rotating components of piston engines - Armonic decomposition of torque compinents induced by pressure and inertia - Main and secondary armonics in multi cylinder engines - Resonating armonics - Vibration amplitude of forced and resonating armonics - dynamic stress on crankshafts

References:

“Giovannozzi –II” chapt. 12.



5) Main components of piston engines:

Typical cranshaft configurations - Pulsating and rotating forces and moments of 1st and 2nd order - Loads on crankcase and engine mounts - Approximations for calculating critical speed of crankshafts - Design of piston pins - Design of connecting rod arm and eyes - Design of piston rings – Dynamics of engine distribution system - Valve opening, speed and acceleration - Cam profile calculation - Loads on camshafts, springs, valves and seats.

References:

“Giovannozzi –II” chapt. 17, 19.



6) Transmission gears:

Straight tooths conical gears - Tredgold model - Minimum number of teeth - Selection of gear parameters - Loads on conic gears - Helicoidal gears - front profile and normal profile - Minimum number of teeth - Helicoidal stub tooth profile - Loads on cylindrical gears with helicoidal teeth - Skew axes transmission - Arc of action and effective width - Loads and efficiency – Loads and efficiency – Conical gears with helicoidal teeth – Tipical shapes of inclined teeth – Planar/conical equivalent gear – Radial, tangential, axial loads.

References:

“Giovannozzi –II” chapt. 2, 3.



Textbooks

See next section.

Giovannozzi R., “Costruzione di Macchine Vol. II”, Patron (in Italian);
Belingardi G., “Il Metodo degli Elementi Finiti nella Progettazione Meccanica”, Levrotto & Bella (in Italian);
Zienkiewicz & Taylor, The finite elements method V. 1, Butterworth & Heinemann; (Cap. 17);
Cophra A., “Dynamics of Structures Theory and Application”, Prentice Hall;
Software Manuals (MATLAB, MSC MARC) ;