ENGI 9420 Engineering Analysis Assignment 2 Solutions

Transcription

1 ENGI 940 Engineering Analysis Assignmen Soluions 0 Fall [Second order ODEs, Laplace ransforms; Secions.0-.09]. Use Laplace ransforms o solve he iniial value problem [0] dy y, y( 0) 4 d + [This was Quesion on Assignmen ] { } Le Y( s) y( ) L, hen he Laplace ransform of he iniial value problem is + 4s ( sy 4) + Y ( s+ ) Y + 4 s s s 4s+ a b Y + s s+ s s+ By he cover-up rule, 0+ a 0 0+ and 8+ b + Y + L + L s s+ { } { e } + e y

2 ENGI 940 Assignmen Soluions age of. Find he complee soluion of he iniial value problem d y dy y 6 e, d d y( 0), y ( 0) (a) wihou using Laplace ransforms; and [8] (b) using Laplace ransforms. [8] A.E.: λ + 7λ λ+ 5 λ+ 0 λ 5 or (a) 5 C.F.: yc Ae + Be aricular soluion by he mehod of undeermined coefficiens: The righ side funcion r 6e is a simple muliple of one of he basis funcions of he complemenary funcion. ry ( ) ( 4 4) y ce y c e y c e OR Subsiue his rial paricular soluion ino he inhomogeneous ODE: y + 7 y + 0y c e ce 6e c y e aricular soluion by he mehod of variaion of parameers: 5 y e, y e, r 6e y y 5 e e 7 W e y 5 y 5e e 0 y 4 W yr e ( 6e ) 6e r y y W + yr e ( 6e ) 6e y r W 4 6e u e u e W 7 e W 7 6e v v W 7 e 5 y u y + vy e e + e e e Bu any consan muliple of e is par of he complemenary funcion and can be absorbed ino i. he paricular soluion is y e.

3 ENGI 940 Assignmen Soluions age of (a) (coninued) The general soluion is C ( ) 5 y y + y Ae + + B e 5 y 5Ae + 4 B e Applying he iniial condiions, y 0 A + B B A ( 0) 5 + ( ) ( ) y A B Subsiue equaion () ino equaion (): 5A + + A A A A B 0 The complee soluion is herefore 5 + y e e [I is edious bu sraighforward o verify ha his complee soluion does saisfy he iniial value problem.] { } (b) Le Y( s) y( ) L, hen he Laplace ransform of he iniial value problem d y d is dy y 6 e, y( 0), y ( 0) d s Y s sy + 0Y s + 6 s + 7s+ 0 Y + s + 7 s s + 6s+ 8 ( s+ )( s+ 5) Y + s + 4 s+ s+ s + 6s+ 4 a b c Y + + s+ s+ 5 s+ s+ s+ 5 6 By he cover-up rule, c ( 5 ) ( 5 5) s s a s s b s s s b+ 0 b Maching coefficiens of s : a + 0+ a 0

4 ENGI 940 Assignmen Soluions age 4 of (b) (coninued) ( s + ) 5 { } { } Y + L e + L e s y e e. An underdamped mass-spring sysem, wih an oscillaing force applied, is modelled by he ordinary differenial equaion d d + d d + 0 sin Find he general soluion (a) wihou using Laplace ransforms; and [6] (b) using Laplace ransforms. [0] (a) ± 4 8 A.E.: λ + λ + 0 λ ± C.F.: e Acos + Bsin C j aricular soluion by he mehod of undeermined coefficiens: The righ side funcion r 0sin is independen of he complemenary funcion. [0 sin is no a muliple of eiher e sin or e ry ccos + d sin csin + d cos cos.] ccos d sin Subsiue his rial paricular soluion ino he inhomogeneous ODE: + + c+ d + c cos + d c+ d sin 0 sin d + c 0 and d c 0 5d 0 d and c 4 sin 4 cos OR

5 ENGI 940 Assignmen Soluions age 5 of (a) (coninued) aricular soluion by he mehod of variaion of parameers: e cos, e sin, r 0sin W e cos e sin e e ( cos sin ) ( cos sin ) ( cos cos sin cos sin sin ) e + + e 0 W r e e sin 0sin 0 sin r 0 W + r e cos 0sin 0e cos sin r We shall need he double angle formulae: sin sin cos and cos cos sin W 0e sin u 0e sin 5e cos W e Afer a edious inegraion by pars, we find u e ( sin + cos 5) W 0e cos sin v 0e cos sin 5e sin W e Afer anoher edious inegraion by pars, we find v e ( sin cos ) u + v e sin + cos 5 e cos + e sin cos e sin 4sin cos + cos sin cos 5cos + sin cos + sin 4sin cos Clearly he mehod of undeermined coefficiens is much faser in his case! The general soluion is e A cos + B sin 4 cos + sin [I is edious bu sraighforward o verify ha his general soluion does saisfy he ordinary differenial equaion.]

6 ENGI 940 Assignmen Soluions age 6 of { } (b) Le X ( s) ( ) L, hen he Laplace ransform of he ordinary differenial equaion d d sin d d 0 is ( s X as b) + ( sx a) + X s +, where a ( 0) and b ( 0). 0 as + b+ a s + as + b + a+ 0 s + s+ X + as + b + a s + s + as + ( b+ a) s + as + ( b + a+ 0) cs + d e( s+ ) + f X + s + s+ + s + s + + as + b+ a s + as + b + a+ cs + d s+ + + e s+ + f s + 0 Maching coefficiens: s : a c + e (A) s : 0+ b+ a c+ d + e+ f (B) s : a + 0 c+ d + e (C) s 0 : b+ a+ 0 d + e+ f (D) (C) (A) c+ d 0 c d (D) (B) d c 0 d + 4d 0 d 0 5 c 4 (A) e a + 4 (B) f b+ a+ 8 a 4 a+ b+ e and f are epressed in erms of he unknown consans a, b. e and f are herefore arbirary consans - relabel hem as A, B. ( ) ( s ) 4s + A s+ + B X + L + + Ae + Be s he general soluion is { 4 cos sin cos sin } e A cos + B sin 4 cos + sin

7 ENGI 940 Assignmen Soluions age 7 of 4. Find he general soluion of he ordinary differenial equaion [0] d y π + y sec, 0 < d A.E.: λ + 0 λ ± j C.F.: yc Acos + Bsin aricular soluion: The righ side (sec ) is no of a form for which he paricular soluion can be found by he mehod of undeermined coefficiens. There is no choice bu o find he paricular soluion by he mehod of variaion of parameers. y cos, y sin, r sec cos y y cos sin W cos + sin y y sin cos 0 y W yr sin an r y cos y 0 W + yr cos y r cos W an sin sin u u d ln cos W cos cos W v v W y uy + vy ( ln cos ) cos + sin The general soluion is herefore ( + ) + ( + ) y A ln cos cos B sin Laplace ransforms canno be used in his quesion, because L { sec } does no eis in erms of elemenary funcions.

8 ENGI 940 Assignmen Soluions age 8 of 4 (coninued) Verificaion of his soluion (no required): ( + ) + ( + ) y A ln cos cos B sin sin y ( ) cos cos A+ ln cos sin + sin + ( + B)cos sin y ( ) sin ( A+ ln cos ) cos + cos ( + B) sin cos sin y ( ) + + cos y( ) cos sin + cos y + y sec cos cos 5. A mass-spring sysem is a res unil i is sruck by a hammer a ime 4 (seconds). [0] The response () is modelled by he iniial value problem d where ( a) d δ ( 4 ), ( 0) ( 0) 0 d d δ is he Dirac dela funcion. Use Laplace ransforms o find he complee soluion of his iniial value problem. { } s Le X ( s) ( ) L, hen he Laplace ransform of he iniial value problem is 4 4s s X sx 0 + 5X e s X e ( s ) 4s { } e X ( s) L e sin 7 e Using he second shif heorem, he complee soluion is ( ) 4 e sin 7 4 H 4 This mehod for he soluion is much faser han he mehod of variaion of parameers! 4s

9 ENGI 940 Assignmen Soluions age 9 of 6. Find he Laplace ransform F(s) of [0] cos f e Quoing he wo sandard ideniies { e a cos } s + ω a s+ a + ω L and L { f ( ) } L f ( ) ds d { } { cos } { cos } d { }: F s L f L e L e ds ( ( s ) ) ( s ) ( ( s ) 0 d s ) ds ( s ) + + (( s + ) + ) ( )( s + ) 9 s + 4s 5 (( s + ) + 9) ( s + 4s+ ) F s s + 4s 5 ( s + 4s+ ) OR Applying he firs shif heorem o L { ω} L{ cos } L{ cos } F s e cos s ω : s s+ ( s + ω ) ( s + ) (( s + ) + ) F s s + 4s 5 ( s + 4s+ )

10 ENGI 940 Assignmen Soluions age 0 of 7. Find he funcion f () whose Laplace ransform is [] F s 48 ( s+ )( s + 4s+ 0) Soluion by parial fracions: 48 a b( s+ ) + 4c F( s) + s+ s + 4s+ 0 s + s a( s s ) b( s ) c( s ) s 48 a a [or use he cover-up rule o find a ] Coefficiens of s : 0 + b + 0 b ( s + ) ( s ) s c c c 0 6 { } { } F( s) L e L e cos 4 s ( cos 4) e f or f 6e sin Soluion by firs shif heorem and inegraion propery: F s s+ s + 4s s s + 6 ( s ) ( s ) s s+ By he firs shif heorem, he inverse Laplace ransform of F (s) is 4 f ( ) L e s( s + 4 ) Using he inegraion propery of Laplace ransforms: 4 f ( ) L dτ e 4 sin 4τ dτ e s f cos 4τ e f cos 4 e 0

11 ENGI 940 Assignmen Soluions age of 7. (coninued) Soluion by convoluion: 48 4 F( s) s+ s + 4s+ 0 s + s { e } { e } L L sin 4 τ τ f ( ) e ( e sin 4) e e sin 4τ dτ 0 e 4sin 4 d e cos 4 e cos 4 + τ τ τ τ 0 τ 0 ( cos 4) e f

12 ENGI 940 Assignmen Soluions age of 8. Use he inegraion propery of Laplace ransforms, ω F( s) { F( s) } L L s 0 sinω wice, in order o esablish L [8] s ( s + ω ) ω and confirm his resul using parial fracions. [8] dτ Using he inegraion propery: sinω L s + ω ω sinωτ cosωτ cosω + L d τ s s + ω 0 ω ω 0 ω cos sin sin 0 L ωτ ωτ ωτ ω ω dτ s s + ω 0 ω ω 0 ω Using parial fracions: a b cs+ dω + + s s + ω s s s + ω ω sinω L s ( s + ω ) ω as s + ω + b s + ω + cs + dωs Maching coefficiens of powers of s: s : 0 + bω b ω s : 0 aω a 0 s : dω d ω ω s : c + 0 c 0 ω ω ωl L s ( s + ω ) ω s s + ω ω 0 ω sinω L s ( s + ω ) ω ( { } { sinω} )

13 ENGI 940 Assignmen Soluions age of 8. (coninued) Also noe ha his inverse can be found via convoluion: sinω sinω L{ } L L s s + ω ω ω sin ω 0 L τ sinωτ dτ s ( s + ω ) ω ω Afer a wo-sep inegraion by pars, his inegral evaluaes o τ ω τ cosωτ + sinωτ ω τ 0 ω 0 + sinω ω 0 ( ) ω sinω L s ( s + ω ) ω Reurn o he inde of assignmens