LAPLACE TRANSFORM TABLE Th Laplac afom of am mpl fuco a gv h Tabl. Fuco U mpul U Sp U Ramp Expoal Rpad Roo S Co Polyomal Dampd Dampd co f δ u -a -a co,,... -a -a co F / / /a /a / /!/ /a a/a
Thom : Shf Tm Th Laplac afom of f dlayd by m T qual o h Laplac afom f mulpld by -T ; ha, [ T u T ] F L f T Thom : Complx Shfg Th Laplac afom of f mulpld by ±α, wh α a coa qual o h Laplac afom F, wh placd by ±α; ha, [ mα ] f F ± α L
Exampl: Fd h Laplac afom of h gv fuco L[ 4 ] f 4 4 [ ] L [ ] L L[ ] [ ] L [ ] L f 4
Iv Laplac Tafom: I a majoy of cool ym, h valuao of h v Laplac afom do o ly o h u of vo gal. Rah, h v Laplac afom opao volvg aoal fuco ca b cad ou ug a Laplac afom abl ad paal-faco xpao. Th la ad h lookg up of h v afom h afom abl ca b coducd by compu pogam. X N 7 0 D 5
Paal-Faco Expao: Wh h Laplac abfom oluo of a dffal qua a aoal fuco, ca b w a G N D wh N ad Q a polyomal of. I aumd ha h od of N ga ha ha of Q. Th polyomal D may b w D a... a a wh a 0, a,..., a - a al coffc. 0
Th mhod of paal-faco xpao wll ow b gv fo h ca of mpl pol, mulpl-od pol, ad complx cojuga pol of G. G ha mpl pol al pol: If all h pol of G a mpl ad al, h fuco G ca b w a G N D N... wh... Applyg h paal-faco xpao, h G w G...
Cod h fuco N D 5 G Th coffc -, -, ad - a dmd a follow: 5 5 7 5 6
G 7 6 Takg h v Laplac afom by ug h Tabl g 7 6 0.7 0.6 0.5 0.4 g 0. 0. 0. 0-0. 0 4 5 6 Tm
G ha mulpl-od pol If of h pol of G a dcal, o w ay ha h pol a - of mulplcy, G w N D N G...,,...,-. Th G ca b xpadd a A A A G...... - m of mpl pol m of pad pol Th - coffc,,,..., -, whch copod o mpl pol, may b valuad by h mhod dcbd abov. Th dmao of h coffc ha copod o h mulpl-od pol dcbd a follow:
[ ] [ ] [ ] [ ] G d d A G d d A G d d A G A!...!
Cod h fuco G 0 A A A G Th coffc copodg o h mpl pol a [ ] 0 0 G [ ] G Ad ho of h hd-od pol a
[ ] [ ] [ ]! 0 d d G d d A d d G d d A G A Th compld paal faco xpao 0.5 0.5 G ad v Laplac afom g 0 0.5 0.5 0.5
g 0.5 0 0.5 0.5 0.5 0.45 0.4 0.5 0. g 0.5 0. 0.5 0. 0.05 0 0 4 6 8 0 Tm
G ha mpl complx-cojuga pol Suppo ha G coa a pa of complx pol α ad α Th copodg coffc of h pol a α α α G α α G α Cod h fuco G ξ L u aum ha h valu of ξ ad a uch ha h pol of G a complx. Th G xpadd a follow:
G wh α α α α α ξ ad Th coffc a dmd a ξ α α α α G G α α Th compl paal-faco xpao of G G α α
g α Th v afom co co g ξ α co co g α g ξ ξ ξ 0
ad/ ad fo dff ξ dampg ao valu 0.8 0.6 0.4 0. ξ0 ξ0. ξ0.4 ξ0.6 g 0-0. ξ0.8-0.4 ξ. -0.6-0.8-0 4 6 8 0 Tm
If Laplac afom ha h fom G ξ Th h v afom ca b w a g ξ ξ ξ θ wh θco - ξ, ξ<
Applcao of Laplac Tafom o h Soluo of La Oday Dffal Equao La oday dffal quao ca b olvd by h Laplac afom mhod wh h ad of h hom o Laplac afom. Th pocdu ould a follow:. Tafom h dffal quao o h -doma by Laplac afom ug h Laplac afom abl.. Mapula h afomd algbac quao ad olv fo h oupu vaabl.. Pfom paal-faco xpao o h afomd algbac quao. 4. Oba h v Laplac afom fom h Laplac afom abl.
Exampl: 5 u y d dy d y d Wh u h u-p fuco. All h al codo a zo. Y Y Y 5 [ ].5 5.5 0 5 5 0 Y Y Y y.5 5.5
y.5 5.5.5.5 y 0.5 0 0 4 5 6 7 8 Tm
f m00 kg c00 N/m k5000 N/m m f50 N p fuco x f 50 k N/m c N/m c F 50
f kx cx x m & && [ ] 5000 00 00 50 k c m F X F k X c m 5.5 0.5 X 5.5 00 5 5.5 0.5 00 00 X 0.0 X ξ
x 0.0 ξ ξ ξ θ 5 ξ. 5 5ad / ξ0.5 θ co ξ co 0.5.40 ad x 0.0 0.00 0.75 4.944.40
x 0.0 0.00 0.75 4.944.40 0.08 0.06 x max 0.04 x max -x x x 0.0 0.005 0.0095 0.0 0.008 0.5*x 0.006 0.004 0.00 0 0 4 5 6 7 8 Tm d max
max π. ξ π ξ 0<ξ<0.69 4. 5ξ ξ>0.69 Maxmum Ovhoo x max x πξ ξ d 0.7ξ. 0.5ξ 0.469ξ Slg m, d Dlay m
>> mulk Soluo by MATLAB-Smulk
50 N 0.08 0.06 0.04 0.0 0.0 x 0.008 0.006 0.004 0.00 0 0 4 5 6 7 8 Tm