LAPLACE TRANSFORM TABLE

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1 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

2 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

3 Exampl: Fd h Laplac afom of h gv fuco L[ 4 ] f 4 4 [ ] L [ ] L L[ ] [ ] L [ ] L f 4

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

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

6 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...

7 Cod h fuco N D 5 G Th coffc -, -, ad - a dmd a follow:

8 G 7 6 Takg h v Laplac afom by ug h Tabl g g Tm

9 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:

10 [ ] [ ] [ ] [ ] G d d A G d d A G d d A G A!...!

11 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

12 [ ] [ ] [ ]! 0 d d G d d A d d G d d A G A Th compld paal faco xpao G ad v Laplac afom g

13 g g Tm

14 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:

15 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

16 g α Th v afom co co g ξ α co co g α g ξ ξ ξ 0

17 ad/ ad fo dff ξ dampg ao valu ξ0 ξ0. ξ0.4 ξ0.6 g 0-0. ξ ξ Tm

18 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<

19 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.

20 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 [ ] Y Y Y y.5 5.5

21 y y Tm

22 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

23 f kx cx x m & && [ ] k c m F X F k X c m X X 0.0 X ξ

24 x 0.0 ξ ξ ξ θ 5 ξ. 5 5ad / ξ0.5 θ co ξ co ad x

25 x x max 0.04 x max -x x x *x Tm d max

26 max π. ξ π ξ 0<ξ< ξ ξ>0.69 Maxmum Ovhoo x max x πξ ξ d 0.7ξ. 0.5ξ 0.469ξ Slg m, d Dlay m

27 >> mulk Soluo by MATLAB-Smulk

28 50 N x Tm

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