Question

GA Problem-5: The system given in the following figure consists of a pulley with inertia ) and radius r, and a hanging object with mass m. The input of this system is the torque t applied on the pulley and angle is measured along CCW direction in radians. Assume that the effect of gravity is balanced by the springs before the torque is applied. a) Obtain a parametric mathematical model for this system, b) Find the input to output transfer function H(s) = X(s)/T(s). c) Assuming a unit step input torque T = u(t), find the position output x(t) as t+ 0. m kis 2004

please can you solve clearly and step by step

Answer 1

ave - a To- From the given figure we can say That the force acceleration relation for the mass and the Prile m : - Vị 2 + ki(3-4) - 5 % 10 (2) K2 (YX) Y and o are related by 6) Y sro substi tu ting from (3) to (1) and (2). mii = -6,56 + Kz (n-ro) - h (5) Jö ET-K2 (70-2) terms in (4) and (5). m + bät (ky-k2 ) 2 + kyro = 0 (+) jö +K2 ro - K221 > T The set of equations (6) and (7) into s domain, MCS? xis) -sly 0) - X'COM)] + Bf5 xrs ) – X (a*)] (s) + (ki - Kq) x B) + K RO (5) -K2 X () + 7 (2005) - 50 (*) - 0°C0+) ] + K, RO (3) T(S) - Rearranging the - 06) -19). le arranging terms (8) and (9) M52 + B 5+ ki -ky) x (5+ K2 RO (8) = ( Ms + B) x 0 +mot'0 :) (10) -K2 X () + JA? + Ky dois) = 702 + Iso (0) + JO'(0+) (11).

in mairine form (10) and (1) can be writren al ܬܐ ܕܬܵܐ 7162 (13) IMs ² A Botki-K2 - K2 [M JS2 + K2 R. LO Solving (13) for x( ), K2 R XS) 1 (5) Is ² + Kar K2 R Mis2 + 13 st Ko-K₂ -K2 Js2 + K₂ R -T2 KŻ R (MS? + Be + k_kz) (Js² + K2R) + K2 K; R (14). H(S): -K2 R (M52 +85 # Ko - K2).(Is? + KąR) + k, K2 R ] (15). to output frownsfer THIS) is the input function, of input step (C) The unit Laplate Frawformaron TH) = ult)e is TS); - 5 → (16) substituting the equation in in ('6). K2 R --(17) XIS ) = stimet Ms2 + Be + K - K2) (J5 + kq R + ki kz R]

"The Final value the meorens says that lim 21) lim X(5) 118), 26) 5-10 Appuing (18) in (17) lim 2 (7) lim -SK2 R 70 100 [cms ² + Bs + ki-K2) (J5 + K₂ R)t k, kz R] K2 - 2K, Co K₂ -2K, is the steady state as tend to infinity.
If you have any query comment down below i will surely help you as soon as possible.Please upvote my answer.Thanks in advance...!