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3100 BME Test I Name: t(Closed Book) October 2, 20 (20 points) 1. Suppose a. +8000i + 12 x 10`y= 624 x 10`' with initial conditions y(0) = 96 and y(0) 48 x 10'. (a). Write the characteristic equation 2 F (; 5 ,^ aoo i2HO U S, (b). Write the MATLAB commands necessary to solve for the roots of the characteristic equation (c). Given the roots are -6000 and - 2000, write the natural solution +^Le ( d). Find the forced response. ^f = K3 )2 O V3 62q>110 6 K3. 52 (e). Write the MATLAB commands necessary to solve for the unknown coefficients in the natural solution. b) r, t- 5Z 6004, _Z0C-c0 1 ZV0U k2- 6 >> K r 2 (20 points) 2. Suppose ij + 4^ + 4y fi(t) with initial conditions t/(0) = 0 and ij(0) = 0. (a). Write the characteristic equation and write the MATLAB commands necessary to solve for the roots = c) (b). Given the roots are -2 , -2, write the natural solution ( c). Find the forced response to a unit step input. ^-^, (c3 K3 - y (d). Write the MATLAB commands necessary to solve for the unknown coefficients in the natural solution with a unit step input. ^- (^1u^a^= rJ'12 +y J , ^ Cu - -2 e ( ^,-^r ^a t K e K= 4\c (e). Find the impulse response if the unit step response is y (t ) e 21 -1 t- 1 + 1 u(t) 244 ^ ^ (10 :-- y) - Le-2-1 >LLC-O -L ^^ S ^f) y t=u 13 6 Uit) 3 (20 points) 3. Suppose ij + 16000' + 108 t/ = 6 x 10" (u (t) it (t y(O) 300 5)) with initial conditions t/(0) = -45x 10 and (a). Write the characteristic equation and write the MATLAB commands necessary to solve for the zt roots Cl l^aov leg]" r tr(^^ (b). Given the roots are -8000 +j6000, write the natural solution Vn (c). Find the forced response to, 6 x 10`' a (t) . f 1C ^ 6 tOgK3= C 0 0 (d). Write the MATLAB commands necessary to solve for the unknown coefficients the in natural solution with a 6 x W it (t) input and zero initial conditions. Sovo7! k = e ( ^ ^, GCS 6 G^o^ -^. (off ^ ^ -^ ` vG -` O - -0600 S,n bOQ ) 4`, U6 4- e -(- 6600 let 51 A 60 qq Q. -^Uoa ^p0^` KL 6Ut) (e). Find the response to 6 x 10' (u (t) - u (t - 5)) with initial conditions y(O) - -45 x 10 s and y(0)= 300, if the response to 6xWu(t) is y (t)= (e 800 ' (-0.O6cos 6OOOt-O.O8sin 6OOOt)+8)u(t) and the natural solution to the initial conditions with 0 input is y;1(t)= e soonr (-0.O45cos6000t-0.01sin6000t))u(t) + ^ - ^ U-- 5 -= (e r (^ v -SSDov f t016 DUO s IT- cj^ ulF) Lk + yu5 1,0 d 6.6k 1 e- 8000Ck 7^-O6 elo 6 Wt uC^ 60 ^ `5) 4 (20 points) 4. Suppose i^ +7tt + 10y = 6e ;'u(t) with initial conditions y(0) 3 and Y(O) = 3 (a). Write the characteristic equation and write the MATLAB commands necessary to solve for the roots S -+ -75 t- t0 _ v p-z C1 tom c 7 1&1 (b). Given the roots are -5 and -2, write the natural solution ^n I^ e -St + (c). Find the forced response. ^_^^ K3e 16 K" e ^ _ 3 s A + 7 K3 e _ s^ - 0 V3 = (d). Write the MATLAB commands necessary to solve for the unknown coefficients in the natural solution. K, e s-t + KZ (f - 2 ^( _ 2 - e K, ^. C i/3 ' H/3L 5 (20 points) 5. Given y, _ y, + /2 +3y, Y2 = 2y, -5y2 +2y, -5Y2 - 'h+y. (a) Using the D-Operator matrix approach, write the Mathab commands to find the determinant and the adjoint syrns (b) If the determinant is D^3+7*D^2-10*D-88 and the adjoint is [ DA2+6*D-5, D+16, 17+3*D] [ 2`(DA3+7*DA2-10*D-88)/(D^2+3*D-22), (DA3+7*DA2-1O*D-88)*(D-2)/(DA2+3*D-22), 2*(D^3+7*DA2-1O*D-88)/(DA2+3*D-22)] [ 3`D+25, 5*D+8, D^2+6*D+3] Write a single differential equation involving only y3 and the input. c4 (Dik){ -4^ c^ Co+^D2-ioo-^^^Y 1 II _, a 3 - ^^ ^^ ^ ^5 +6ys t3

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