Question

4. Suppose two identical pendulums are coupled by means of a spring with constant k. See the figure. When the displacement angles 01(t) and 02(t) are small the system of linear differential equations describing the motion is k (01 02) + m k (002) 02+02 m Use the Laplace transform to solve the system when the initial conditions are 01(0) - 00,0{(0) = 0,02(0) = -0,02 (0) = 0. Can you discuss the motion for this case? NNWNANNNNA m

Answer 1

Aruaar page yatem of diferenta equations that modlel the mobion of coupled Conaytants Conider That below is a pendulums K and m are o11 ,t0,- (-92) 2 m th folloing Can reuite the ystem Uhing Subitrutions for the const ants 9 and k K- -KCO,-0,) 17 intial condiftons Aloo haur thetolloaing e co) e co)-0 Co) P and Conytants are Such that and apply the equations TO solve tha syrte m Laplace trans form te the system Cam ue Of L foy2f0} =K.Lica-0,13

Cbing the aplace tramtoms fr darivatives and tetfing Lfo3 - e) and LiGj= 6,(s) apresent the laplace trans forms we have secs)-se, (o)-0' (0)te,-ec)-,) S6,(s)-S0,(0)-0 co)te,(s)K(ecs) -e () in the inihal Cond. tions, the system beoes s'e(s)-s-0toe, Cs) = (e,ca)- 0,cs) ) ce,c)-S-0tae,s) (e, c)-e,cs) Smptfying, the sytem i s s'e,(5)-stwe, s) Ke,c)-, ) the first euation in the ystem in cam Solr for e, ) Uising we s) + kO,cs) = *e, Cs) se,c)-s, -S

Tate the above eguahon and plong it into Pag te second eguahion of the system in euation e,) Ciuke S = Ke, co)- k -S K Rpitying the right hand side, e s) tk-s3.00-40t 0c) cue hare K = KO,C)-Ce, caC404 k) - S.) Naw we multiply out all the ferms on both ndes Oo -S CSe) O - K,C)-OpCo4+) + s 0 on Noww twe can oak the termu with ,u) tefe hand side and multiply by K the +0,C) K K se K

PageC . Ko,t stot skyo+sao S factoring e ca) term, ur get he S KOt tsKp tsu o, eR04)K")- SKOts,tsky, + swo, eiCacsuk-)4 u't Ktk -skotto,+sktsue 3 finally isd late the e(s) - (an term ec)Ketots Kso For the above ewabon frachin decomporihion betre ue an taplace trantfam skosOt sktswo k Mead to Ute partial pply the invere As+ B Cst D tar S4w 2K muliplying both ades of the euation by the den aminator of the teft hand Ade and aulthiplyhg out the ight hand Ade we have

pages skogtstsksde, - Ace)(4ze)tD) (h) P0,+ (KO,tkp,to )s = Ata's+2 FAs+ Bs4s +2 KB tCs3+Cs DstDo the abovr eation we an Uing tems to sove foY A,8,C, and D. Looting at the Compare Cite Coutaut tofms we hane Burt2KBt Du Cast2K) B+Du CCs42kB-Du -s42k = D LoOking at the s terms and -C2) C B=D we han s BSt Ds at the S3 terms so ae haue that D- Looti CWe have Os=Atc

at the s fermu and A- 0-C Page C) hare Looei ng we CKOst ko toO,)s -Aust 2KAS + Csa KOst tvo tao A t 2)+C KOtEPotovos (-cat 2k) +c KOD teptuo. = 0,+20,-cu-2kctCa Sanplity tho akoue equation to Aolve for c. KOotKt o = 8,+2k-2KC Kyo-Ko=2c So A is A - - 0o otCo-0a) plugging faund into egatio the const anti we we hare V stw star2k the aboe into eetin pugging Laplace trans form is (2 wK have fhat The

Page stut2k Leplacs transfora right hond side is he fiad the inverse Nao we cam of the above eguation Latlace traniforas of conne The L statzk 2. 2. 2 te diferential e4ation fram he yetem Uning Cue hare 01M.4ao o,-Q o,/uYK (c Jae the abwe to solve fir ()y pussing CAuation fov 0,ce) and C). Fnt u find darivahur of eaton he second

Pogz& Pluzzing nto esuatin 6 6,- tocaut-- aJot ue haire Cos K 2 be hare Bechy adang 2 K 2k 2k Next, ue maulhply Dut the terms 2K 2K -2k cos V42Kt t Co,tKasut 2K 2K 2k Now we can cancel terms G--8440t-(- astt 2 4 (0tylacast t 2K +Co,at + ) caett 2K

Poge 6f the term in the above zen out Some Cos-) core2rt and Such as 2K wcasw SO we hare Ceaut-Jonyot 2 2. to lution to the Syate of diterentia eqjlations is guen by aguationand 8,Ct)= Coto cosat t(,-)o4kt 2 houng the intiel Conditions O,co)=0,,0,C0)-0, 0,0 - and co) - 0 Aauing &,Col-,8, ) -0, 8,0)- i jially we Cucre 1 and 0,to)= 0 fleld roeplaa o, pong tht Bayic ly equeahionbec Cue omes, o,-0) Cos t S, CE)=(t00)casut- ces t 2

,lwe hane O,CE)-200 Casut + (o)cas a2Kt 2 e,l cout- Co) o a4t So the copled penduluns Fytu beloas &CE)= OCasest Ppenited ty are Pthe Snce the equothos Pendulumi move the same that maans are the same The two uay at the hevially Sama ne Thuy n cnison are oigidly given the inihal Conaltions aue aere Tiven 0,(0)=00,6,-0, 0l0) -0, Now we are and co) jhould replace , - 0 Dolng Bariclly tteqaatior becomes e,cE)-0-00csut ,to) we Varelt 0)caxut -(t) coakt cos 6,CE)- 2

pageT epl&rg O, ce)= Jue have C2 Cas e2t Coswt t Cos Eo te coupled pendulums The System be loas 0,CE)-GocOsVast 2kt eprinted y are e, ce) co42Kt The qutons are the sama bul have they oparte FnThet auay Move means at the same time emd then touards cach Other at the ame time