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. (11=1=1kgm”, b=1.8Nms/rad, k=4Nm/rad) b ez O: OBROTUD T

Answer 1

Page Date I d² 1 +b, do k (0-2)=0 dt² at T = I2 d²02 + b2 d 2 4 K CO2-6) dt² dt Laploce transfom of equation () ( ² It sb, tk) o, - KO2=0 3 TCS) - (S?I 2 + 5b2 + k) 02 - Ke -4 From equation 3 Kez s²I, tsb, tk +5)= (s? I2 + 5b2+ ) 02 - K20 2 s²1 it sbitk T(S) = [CS2 12 +562+k) (525, +sbitk) - K2 ]oz(s) s²I, tsbitk O2(5) g²I, tsb, tk ICS) 841, 12 + (12 b, +116.953+(K 12 + b2b2+ KI, 15+ C kb 2 tkbi)s +6²-k² 02 (5) S’+, +5b.tk t(s) 5 [sº1,12 +(12b2+1, 6,95?+CkIzt bibzt ki,) + Stkb2tkb, from eg 3) 82 +1, + 5 bitke, . k 1(5): (5412 + 5b2+k) ( $21, tsb, tk) -K) ,(s) 541,12+ (12 bz t II b)s8+ (KI 2 +61 + bxt ki, )s*+(kbz+kb.)+s+k+ 103) 0,(S) 13 K @05). 105) STC5%I, 12+ (Izby+1, b.) 52+k12 tb,b2+k1,)s +kCbitb]

Put I = I 2 = 1, b = b 2 =0.9 ,K=4 Bi(5) T(5) 4 S[ 53+1.85?+ 8.81 5 + 1-2 O2 (s) TCS) s²+0.95 +4 8 53 +1.833+ 8.815+7.2] 0.3 OCS) TCS) 4 s(3+ 1-88?+ 8.815 +7.2) No Zero's Poles at 8=0 S - O a s =-0.45 + 12.7924

1 Re -2 2