Question

Problem 2: (30 points) A mode ontrolled DC motor is she l or is shown below. The em 2: (30 points) A model of an armature contro aller, here modeled as a rigid body with mass load attached to the rotor of the motor is a prope is transmitted to the propeller through a shaft moment of inertia IL. The rotation of the rotor with torsional damping constant br. The proper eller motion generate a further load TL due to the drag with air. The motor has a torque const tant Kr and a back-emf constant K, such that the torque T = K L and the hackam . Kew. The rotor has a mass moment of inertia respect to its rotational axis equal to Ir. (a) (4 points) Write the Kirchhoff's voltage laws laws (KVL) for the armature circuit. (b) (4 points) Draw the F-B-D of the rotor. (c) (4 points) Draw the F-B-D of the propeller. (d) (4 points) Write the equations of motion of the rotor (e) (4 points) Write the equations of motion of the propei (1) (8 points) Find the transfer function that relate the angular speed of the propeller to the voltage applied at the armature circuit. (8) (2 points) Find the steady-state response of the propeller when vs = A ug (step function of amplitude A) FELD CRCUIT rotor load ARMATURE CIRCUIT

Answer 1

1.) The epuvalant concuit of animadame is, Va To ) VB. Applying KVL to abovo circuit, Vo = Tolot Lood To tvo - dt cohero, Ves = asumature Voltage Bo - amtaro - prasistance Loraxmaturo inductance Ve - Back Emkom To imateerd awurant. - b). The a mass rotor and a system dampor can be represented as bolow, with Hot Free body diagram » T = Jd²e + B do . tadt In our problem, J. IR and B = b . IT = IR do ti bj w . / dit shore i do :w

Hoos Since propeller is modelled as ou sugid body with Tabs somerdinandia, IL TL = Il de d) Equation of motion of roth is T = Udo B do TE- as atz , dt. Slydes and load. ano takon as o..complete cincit e dt. A L cohore TI - Il do Ž Equation of motion of propeller from 0 Vac Bow Tals) La ST()+ V2 (8) aller applying Laplate. Translam to co 0 NB = KRW → Vore) = KB 2016) Vece- (Bast&la) 10+ KRWCB. T = KT Ja (6) Ia (3) = T (8) 100 = atsla) TⓇ+ Kg wc.). - © T8) = Tyscoce) + buy cs] + TL 8 Wee) -

Put 6 in 6, we get Vacs) - (Ratelo) (Ips + by to IL) wted + Kere (619) = (Ratela) (Ip+&+I +67) + KB) core) _L KT Vals) 1: Wie) KT Barsta) (18+811+by) + KEK) g) of Ve = Va = A Us(t) - Vals)- A + ( tiring laplace transform. ce) = KTA sfrastela) CIR+2TL +b) + Kokt y step suspense coith amplitude À