Aerospace Engineering - Meccanica Aerospaziale

Full exam

Meccanica Aerospaziale Prova scritta del 22 Gennaio 2020Tempo a disposizione:1500Il punteggio massimo attribuibile a ciascuna domanda e indicato in parentesi quadrata Esercizio 1 Nel sistema schematizzato nella Figura 1, una piattaforma circolare, omogenea, di raggioRe massaM e posta in un pianoorizzontale, avendo il proprio centroA ssato a terra mediante una cerniera ideale. Volendo controllare la rotazione della piattaforma senza intervento esterno, un disco omogeneo di raggio re massamviene collegato alla piattaforma mediante una cerniera ideale posta nel suo centroB, a distanzaR=2 daA. Tramite un motoreinterno al sistema, sul disco agisce una coppia di momento C(t)k, essendokun versore ortogonale al piano su cui si muove il sistema. Utilizzando le coordinate libereeindicate nella Figura 1: 1. Scrivere l'espressione del lavoro virtuale della sollecitazione attiva sul sistema. [2] 2. Scrivere l'espressione dell'energia cinetica del sistema. [3] 3. Scrivere le equazioni di Lagrange in forma non conservativa. [2] 4. DeterminareC(t) in modo che la rotazione della piattaforma segua la legge oraria(t) =2 sin2 (!t), che consente alla piattaforma, nell'intervallo di tempo [0;2 !], di partire da ferma a t= 0 ed arrestarsi dopo aver eettuato un quarto di giro attorno al proprio asse. [2] 5. Disegnare il diagramma di corpo libero del sistema, mettendo in evidenza la sollecitazione attiva ereattiva agente su ciascun corpo.[3] 6. Nelle condizioni di moto del punto 4, determinare la reazione vincolare sulla cernieraAin funzione del tempo. [4]Figura 1: Esercizio 1 1AB M,Rθ m,r φ Esercizio 2 Nel sistema della Figura 2, un'asta omogenea di lunghezza 2Le massaMha il proprio punto medioO ssato al pianoverticale, identi cato dai versorijek, mediante una cerniera piana ed ideale, che consente la rotazione dell'asta nel piano, intorno all'assei. All'estremitaAdell'asta e saldato un contrappesoP di massa 2m, mentre all'estremitaBl'asta e collegata ad un'elica mediante una seconda cerniera piana ed ideale che vincola la rotazione relativa dell'elica rispetto all'asta nel piano ortogonale adAB. L'elica e costituita da due aste identiche, di massame lunghezzaL, saldate nei rispettivi punti medi, in modo da essere mutuamente ortogonali. Utilizzando gli angolie, indicati nella Figura 2, che descrivono rispettivamente la rotazione diABintorno all'asseie la rotazione propria dell'elica attorno all'asseAB: 1. Scrivere le espressioni delle velocita angolari dell'asta e dell'elica. [2] 2. Scrivere l'energia cinetica del sistema. [5] 3. Scrivere le equazioni di Lagrange del sistema e determinarne il moto con le condizioni iniziali(0) = 0;_ (0) =_ 0; (0) = 0;_ (0) =_ 0. [3] 4. In corrispondenza del moto trovato al punto 2, determinare il momentoM0 della coppia vincolare che la cerniera inBesercita sull'elica. [6]Figura 2: Esercizio 2 2i 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