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plugin-hints - Math245 Final Project - Part II – Hints...

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Unformatted text preview: Math245 Final Project - Part II – Hints Analytical solutions to Problem 1 & 2 are given as follows: 1. y = f K u(t − t0 )e−a(t−t0 ) sin b(t − t0 ) + u(t − t1 )e−a(t−t1 ) sin b(t − t1 ). b b 2πa b 2. K = −f e− and t1 = t0 + 2π . b In this project students are required to work out the complete solutions to derive these answers. An example covered in class as attached in following pages and a Chapter 5 homework problem (§ 5.7 #13) are essentially the same as the present problems, and therefore, are expected to be helpful for completing the solutions. Above solutions can be directly used to perform the simulation tasks specified in subsequent problems in the project. N on ') t---. \l I'au l,-\ J , fua + 4r[rn) s-r*Ct-nl. 4ltt-n1 Sl/trtt-lr.) l I ol Tv = = f,t?nj # * e"'f,frr'! # +Zusfu ,h I .E*s t^d Fl3 tn"$ g-.? HW This example below gives you some idea to solve the first two problems in the final project-2. This example has been covered in class, and it is very similar to the homework problem in Chapter 5 (#13 in Sec. 5.7). In the project you need to solve the similar equation in terms of parameters a, b, ... etc. The next page shows how to determine the strength of the second impulse (K) and its timing (t0, in this example) to bring the response to rest after exactly onecycle of oscillation. J l l + X a t ! ' + 5 3 = 6 ft-t)+ K Sft*t;, J6rcl^t f ,'co;6, t > 2 $co)=o, tct-r) ( t ct-t ) . f f,lq-3 ' s'Vo- s,y/r-yr 2 {M,-S! + 5 \(r = a "* K et' + o O (s-+zs 5 )Vl : 6 * + Ke t tt + 'ro . J--. . 5 V rl : € " S'+zS-t- + K a 6 St+e-stS 2 !--a/^._J 6 4rt)= f -'{ H 'l Hct)= Htsl , V)= €ttFt,r,+KeosHur J.. + Jo f to=f J YqJ= 4)k-2)lntt'-z) K l rt*-+l?ct-t"; . I (L+aSt S - tz. - z (s+l)L+22 4 f ,eo) * a*s,'rzt ? ^V: ltfr ti be ^ exact.fuJtt s1^t,X&ft osct/I*do^". got hn* l=o J rr t zta K. l y a t6*s+u6 5tt-r-) Ttnn,', fr-d l.r \ ra.0*sJk * oC t " K Kt tt-r) rra,{r J =o f r t }t", C,rl.rl c,s'"sldl t l"c.$ -0i^ f tr t zt z tT J A .b . trt = +,€t-t-v"'?.:"-z)"* k e*).;zct-t ). = t t-z) + At.-t { s -n^2 + K erf* ( rrt-.r- gI) I T *K ( f"v t }t" ) . 5^q- T = o ^r-pari.l, L T=2T.. I / /, - t f n= 2 -rr4 rs I 5 r = * , a '*'\ m ect-z) r * K n t} " !!, t ? 1", H r elpr'r'trt{ "'t-o, t i 'e, Smc--u ^q n r4-d J =o t * a-rk€T=o + Cbwrvs,iqr-: u.ritht k ^z) t t t-z) K=-e'=-€nfu V 6tt-a; ff w ith I K t tt-t; K*-- ...
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