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,-1,-1kgm', b=1.8Nms/rad. k=4Nm/rad) b O2 ma 12 T

Answer 1

+ K d 12. dt atz + b2d Oz (2 db (3) we know that I, 120 bi (0,- 02) = 0 O Again T = Iz do, 2+k(01-02) = 6 Now, Laplace transformation (D) (5%, +sbitk) 0,- K0, = 0 and OM, 02 KO2 ($?1, + sbit) Agam Laplace transfoomation of ego 210 © wwe get. T(S) = (s' Tot 5b94x) Oy – K.q2 31, 1sbitk T(S)= (s'],+ Sb2FkX's in + sbi 4K) 69 - RE, s'], +Sbi + k T(5) 02? (SIsusb,tw)(SI, #Shytk) -ky s²1, + sbitk Og(s) (51 tsbitk) {sºn 19 + 5*1961 + $?yk + sºb2314 $t1b24 sbyk + k$si + ksbi + x32x3] SI + Sbitk ${"sör,22 +3 ( Isb2+ Sibi)+ (Kfzt bib 24 KS)s + Kbytkbi Now, (s? + Sb1+ k) 0,- K02 20 (Slitsbitk) o 02- TO (Sa, + Sb1 tk 2015 (8197 Sb2+x)(så + sort k) - K?} T159 SI, A sbitk. son K. K

(s?1, 4 Sby+ k) 0, x (5 To 1 $b2+ x) ($9, +361 + x) 267 +195 + Is} R S's, +5+) 11821, +Sb2+k){ $+5b1+x) 11 K. 0,13) к K 0,19 TIS.) s[539119 +52 (Igbyt-Iyba) + s (K7 et bibz+ KIJ) + 61 62 Giren 11 212 2 Ingm?; 6548 Nes/rad; Kayn/ red birb2=0:1 be birba 4 81 182 T(S) Lons [ 3571852-18-515+4-27 02652 T(S) skaitsbitk s[582.g+ s4( 19b2 fibi) + (KI9Hby b94 kJ) s + kbytk.biz s²10195 tk s [ +8.815 + 7:27 Ans s3 + 1887