Question

Problem B-4-1 Find the transfer function X (s)/X/(s) of the mechanical system shown in Figure 4-50. The displacements x; and X, are measured from their respective equilibrium positions. Obtain the displacement xo(t) when the input x;(t) is a step displacement of magnitude X; occurring at t = 0. Assume that x.(0-) = 0. T Figure 4-50 Mechanical system.

Answer 1

ANSlater The Mechanical s ustem : The framfer function Xo (s)/xe(s) of the mechanical system. ki The of the system an, equation of the motion K. (Wi-y)= bicy-xo) — o bicy-so) - Ky do o

Apply Laplace framform to fiorst ear of motion ki (Xi(s) - YCs)) = 6,(sy(s) - sxo(s)) k, Y(s) + 6,5YCS) = 6,5 xo(s) + k,*;(s) yes) : k, Rils) + bis to(s) ki + bis Apply Laplace tramform to and earn of motion. .bi(sycs) - sxo(s)). kz xo(s). Substitute Kixi(s) + bus xo(s) - for ycs). kit bis lo (has ), c) *. by CCsCkyxi (s) +bisxols-(boys+ki)s xocs)) = (bys +ki) ks * bik, sx;(s)= (6,5+k.)K, XO(S)+(6x5+ki) bys to (s)-b*s xols bik,st; (s) = (bi ky s + kike k,615) Xo(s). bikis Cobitis Xi() $) (biky + kibi)s + kiko Tron, the tramfer fonction of the systemin,

Xo(s) bikis x;(s) (biky +kibi)s+kike The Input xi(t) Pn a step din placement et magnitude xi occuring at too. The Laplace transform of nect) in, - L[up ct)] = x xi(s) = - Substitute for X, (s) in the transfer function. Xo(s) bikis Whatsay. (holky tk, bi )s trike bikis Xo(s) =3 / bike+kibi) st Keka Xabiki xols): - (bika+k,61/s + kike

nue Invente Apply 190 Laplace transform kika X; bik, - bakat kibit Holt) = 1 bik, tkibi Thon, the displacement noct) Pn, xo bike a XoCt)= (tiky +kibile - Kik₂ b2kz+kib, t.