Question

Consider the mechanical dynamics of a 2DOF rotary motion system shown below, where the torque is applied to the right shaft but the angular position of the left shaft is to be controlled, k is the stiffness of the linear rotary spring and b is the viscous friction coefficient of the ball bearing that supports the right shaft and acts as a linear viscous damper with rotary motion. The left shaft is only supported by the right shaft, so there is no friction between the left shaft and the ground. PLEASE show all mathematical steps explicitly. (1,=Iz=1kgm'. b=1.8Nms/rad, k=4Nm/rad) b ( MOULDS 1) Find the dynamic model of the system. 2) Find the transfer functions 0;(s)/T/s) and 0,45)/T(s).

Answer 1

1) mm 22) I I FB Ds ) I loc Fs=ke,-) cos ou is spring force acting ou I, 1, , = Fs -> I, 0,= K (02-0) 1 5

-> I, Ö, +K@, -KQ, = 0 ---O Fa K Fs I2 -30 hehe Combining Fd is Damping Force Igöz = T-Fs - Fd = T-K(0-0) -6 (0) => I, Ö, +66,-KQ, +KO, - T ---(2) get 118 }+[ Lille } Ek ;} {i} => [I]{ ö} + [Cefe] {} [Keff]{0}-{m} 1 & 2 we I O In 2 2 5 t

& 2) Consider system initially at rest mean position : 0, 6020,(0) = 0,0) = 0,0)=0 Laplace Traus form of eq. O (T6, 1K9 - K9,) - Ý (6) => 321, 0,(s) tk 0,(s) - K O2 (s) = 0 [:2 (0,(t)) = 50,45) -59,6os = $20,(3) => (s) K 3 O, (s) 8² I tk 2 Laplace Traus form of eq 2 (12ő, +60, - k, +KO,)-(TC+)) -> $I20,(s) + bs Oz(s) – K 0,(s) +K 0,(5) T(s) 3 5

value of ין k? => 0,05) [8?I_ tbs+k] - K 0,05) = T (s) Putt itting O, (s) from eq. 3 => O₂ (s) [s²Iz+bstk] -kok O, (s) = T(s) s²I, tk => O₂ (s) 1 T(s) s²I₂+bstk S²I, tk s’I, +k (s?I, +k) (521,+bstk) -k? S²I, tk s^T, I2 + Ks? Iz +65I, tkbs +s?I,K => O2(s) s²I, tk T(s) s“I, I, + 831,6 +5K (I, + I) +8 Kb 2 Bunu tek at, rhy) v Kb 0265) = T(s) 82 + 4 S"+1.83 +85² +7.2s 4 5 Aus.

Combining 3 & 0,(s) 0,(s) O2(5) 0,(s) T(s) T(3) k S²I tk s“I, Iz +5% 1, b +sk (I+12 7+sKb S²I, tk , TK K s”I, I₂ + s² Ibt + s’I, 6 +s?K (I, +I) +skb 4 s4 +1.853 +85² +7.2s 0,6) T(s) - Aus It You LIKE THE Sol. Please Give Thumbs UPI 01 5