( ) ( ) ( ) 0. dt dt dt ME203 PROBLEM SET #6. 1. Text Section 4.5
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1 ME PROBLEM SET #6 T Sion 45 d w 6 dw 4 5 w d d Solion: Fis mlil his qaion b (whih w an do sin > o ansfom i ino h Cah- El qaion givn b w ( 6w ( 4 Thn b making h sbsiion (and sing qaion (7 on ag 88 of h, w ansfom h Cah-El qaion ino an qaion wih onsan offiins givn b d w dw dw 6 4w d d d o d w dw 5 4w ( d d This is a lina qaion wih onsan offiins and has h assoiad ailia qaion 5 4 whih has oos, 4 Thfo, a gnal solion o qaion ( is ( ( 4 4 To hang his qaion bak ino on wih h indndn vaiabl, w again s h sbsiion Thfo h solion boms 4 w ( T Sion 46 ( ( 6( Solion: Making h sbsiion (and sing qaion (7 on ag 88 of h, w ansfom h Cah-El qaion ino an qaion wih onsan offiins givn b d d d 6 d d d o d d 4 6 ( d d This is a lina qaion wih onsan offiins and has h assoiad ailia qaion 4 6 whih has oos ± i Thfo, a gnal solion o qaion ( is ( os sin ( ( ( os( ( sin( To hang his qaion bak ino on wih h indndn vaiabl, w again s h sbsiion o, solving his ssion fo, Thfo h solion boms ( os sin T Sion 6 ( ( 6 7 Solion: Th ailia qaion fo his oblm is 6 7 B insion w s ha is a oo o his qaion and so w an fao i as follows 6 7 ( (6 ( ( ( Ths, w s ha h oos o h ailia qaion a,, Ths oos a al and non-aing Thfo, a gnal solion o his oblm is giv b ( (4 4 4 Solion: Th ailia qaion fo his oblm is This an b faod as (
2 Thfo, his qaion has oos i, i, i, i, whih w s a ad and oml Thfo, a gnal solion o his oblm is giv b ( os( os( sin sin 4 T Sion 48 ( ( 6 8 Solion: Aoding o Tabl 4 on ag 8 of h, his non-homognos m is of T I Ths, w wan a aila solion of his diffnial qaion o hav h fom ( A A A A Thfo,, and a givn b ( A A A A A ( A A, and ( 6A A Sbsiing hs ssions ino h oiginal diffnial qaion ilds 6A A ( A A A ( A A A A A ( A A ( 6A A A ( A A A 8 B qaing offiins, w obain A A 6 A A A A 4 A A A A 8 A A A Thfo, a aila solion of h nonhomognos diffnial qaion 8 is givn b ( 7 9 sin Solion: Aoding o Tabl 4 on ag 8 of h, his non-homognos m is of T III (wih a, b, and β Ths, w wan a aila solion of his diffnial qaion o hav h fom ( Aos B sin Thfo,, and a givn b ( Aos B sin ( Asin B os, and ( 9Aos 9B sin Sbsiing hs ssions ino h oiginal diffnial qaion ilds 9 9B sin ( Asin B os ( Aos B sin 9Aos 9 Asin B os sin B qaing offiins, w obain A A B B Thfo, a aila solion of h nonhomognos diffnial qaion 9 sin is givn b ( os d θ dθ 9 5 6θ d d Solion: Fo his oblm, h osonding homognos qaion is θ 5 θ 6θ whih has h assoiad ailia qaion ρ 5ρ 6 This ailia qaion has oos ρ, Ths, a gnal solion of his homognos qaion is givn b θ ( C C h
3 Th non-homognos m of h oiginal diffnial qaion is Aoding o Tabl 4 on ag 8 of h, his nonhomognos m is of T IV Ths, w wan a aila solion of his diffnial qaion s o hav h fom θ ( ( A A Sin nih no a solions of h osonding homognos qaion, w l s Thfo, θ, θ and θ a givn b θ ( A A θ ( A ( A ( A A ( A A, and θ Sbsiing hs ssions ino h oiginal diffnial qaion ilds θ 5θ 6θ A A [ ] ( A A ( A A 5 A ( A A 6 (A A B qaing offiins, w obain A A A A A 4 Thfo, a aila solion of h nonhomognos diffnial qaion θ 5 θ 6θ is givn b θ ( 4 7 ( ( ( sin Solion: Th osonding homognos qaion is whih has h assoiad ailia qaion This ailia qaion has oos, Ths, a gnal solion of his homognos qaion is givn b h ( C C Th non-homognos m of h oiginal diffnial qaion is sin Aoding o Tabl 4 on ag 8 of h, his nonhomognos m is of T VI wih α, β, a, and b Ths, a aila solion of h non-homognos diffnial qaion will hav h fom s ( ( Aos Bsin Sin nih os no sin is a solion of h osonding homognos qaion, w l s Thfo,, and a givn b ( ( Aos Bsin ( ( Aos B sin ( ( Asin B os [( A B os ( A B sin ] [( A B os ( A B sin ] [ ( A B sin ( A B os] [ B os Asin ] Sbsiing hs ssions ino h oiginal diffnial qaion ilds ( B os Asin [( A B os ( A B sin ] { } [ ( Aos B sin ] ( B A os ( A B sin sin B qaing offiins, w obain A B and A B B and A Thfo, a aila solion o h nonhomognos diffnial qaion sin is givn b os sin ( Ths, a gnal solion of h oiginal, nonhomognos diffnial qaion is
4 ( h C ( C ( θ os sin ( θ ( θ sinθ ( ( Solion: W fis solv h assoiad homognos qaion and obain as gnal solion θ θ h ( θ C C N w will s h sosiion inil and onsid saal h qaions sinθ ( and θ (4 Fo qaion (, aoding o Tabl 4 on ag 8 of h, his non-homognos m is of T III (wih a, b, and β Ths, w wan a aila solion of qaion ( o hav h fom ( θ Aosθ sinθ B Thfo, and a givn b ( θ Aosθ Bsinθ ( θ Aosθ Bsinθ Sbsiing hs ssions ino qaion ( ilds ( Aosθ Bsinθ Aosθ Bsinθ Aosθ Bsinθ sinθ B qaing offiins, w obain A A B B Thfo, a aila solion of h nonhomognos diffnial qaion ( is sinθ ( θ Fo qaion (4, aoding o Tabl 4 on ag 8 of h, his non-homognos m is of T II (wih a and α Ths, w wan a aila solion of qaion (4 o hav h fom θ ( θ D Thfo, and a givn b θ ( θ D θ ( θ 4D Sbsiing hs ssions ino qaion (4 ilds θ θ 4D D θ D θ B qaing offiins, w obain D D Thfo, a aila solion of h nonhomognos diffnial qaion (4 is θ ( θ I follows fom h sosiion inil ha a gnal solion o h oiginal qaion is givn b ( θ h ( θ ( θ ( θ θ θ θ sinθ C C Ths w hav θ θ θ osθ ( θ C C Th iniial ondiions giv ( C C ( C C 7 Ths w hav C and C 4 Thfo h solion is θ θ θ 7 sinθ ( θ 4 5 T Sion 49
5 s Solion: S solv omlimna qaion ± 4(( ±i C os C sin os sin S Gnal fom of solion f ( d, f ( d,, os sin sin os os ( sin sin s d an d os os s d d os os sin Ths h gnal solion is os os sin C os C sin s d os s d sin 6 Solion: S solv omlimna qaion ± C 4(( C S Gnal fom of solion f ( d, f ( d,, ( ( d d d d Ths h gnal solion is C C 7 s Solion: S solv omlimna qaion ± 4(( ±i C os C sin os sin S Gnal fom of solion W will s h sosiion inil and onsid saal h qaions s (5 and (6 Fo qaion (5 f ( d, f ( d,
6 , os sin sin os os ( sin sin s d an d os os s d d os os sin W an s h mhod of ndmind offiins o solv qaion (6 Aoding o Tabl 4 on ag 8 of h, his non-homognos m is of T I Ths, w wan a aila solion of his diffnial qaion o hav h fom ( A A A Thfo, and a givn b ( A A A ( A Sbsiing hs ssions ino qaion (6 ilds A A ( A A A A A B qaing offiins, w obain A A A A A Thfo, a aila solion of qaion (6 is givn b ( Ths h gnal solion is C os C sin A sin os os Solion: Th givn diffnial qaion is a Cah El qaion W mak h sbsiion o ansfom h Cah-El qaion ino an qaion wih onsan offiins givn b S solv omlimna qaion ± C 4(( C S Gnal fom of solion Ths, f ( d, f ( d,, ( ( d d d ( d
7 ( Ths h gnal solion is ( C C C C Wih (so ha, C and C, h gnal solion an b ssd as follows ( ( ( [ ]
( ) ( ) + = ( ) + ( )
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