NUMERICAL EVALUATION of DYNAMIC RESPONSE

Transcription

1 NUMERICAL EVALUATION of YNAMIC RESPONSE Aalycal solo of he eqao of oo for a sgle degree of freedo syse s sally o ossble f he excao aled force or grod accelerao ü g -vares arbrarly h e or f he syse s olear. Sch robles ca be acled by ercal e-seg ehods for egrao of dffereal eqaos. Maheacal Model SOF Syse Free Body agra

2 A e sa, he dyac eqlbr ll be, FI F FS F Sce he force level chage by e, he he dyac eqlbr ad herefore F I, F, F S ad F ll chage. Eve hogh [M] ad [C] are cosa F I ad F, [K] deeds o he force level F S. ad a shor e laer, he dyac eqlbr ll be, F F F F I S Sbracg Eq. fro Eq. resls he dffereal eqao of oo ers of crees, F F F F 3 I here he creeal forces hs eqao are, S

3 S S S I I I F F F F F F F F F F F F f e asse ha he dag force s a fco of he velocy ad he srg force a fco of dslacee as sho Fg., hle he eral force reas rooroal o he accelerao, 6 here c ad are he dag ad sffess a he e. F c F F S I c

4 The coeffce s defed as he crre evalao for he dervave of he srg force h resec o he dslacee, 8 7 The coeffce c s defed as he crre evalao for he dervave of he dag force h resec o he velocy, S d df d df c 9

5 The sbso of Eqs. 6 o Eq. 3 resls a covee for for he creeal eqao, 0 If e asse c as cosa, F c F c

6 Three ora reqrees for a ercal rocedre are:. coverge- as he e se decreases, he ercal solo shold aroach he axac solo. sably- he ercal solo shold be sable he resece of ercal rod-off errors 3. accracy- he ercal rocedre shold rovde resls ha are close eogh o he exac solo Three yes of e-seg rocedres ll be reseed:. Mehods based o erolao of he excao fco. Mehods based o fe dfferece exressos of velocy ad accelerao 3. Mehods based o assed varao of accelerao

7 Se by Se Mehods for he Solo of yac Excaos. Mehods Based o Ierolao of Excao A hghly effce ercal rocedre ca be develoed for lear syses by erolag he excao over each e erval ad develog he exac solo. Fg. Shos ha over he e erval +, he excao fco s gve by here he e varable vares fro 0 o.

8 The eqao of oo of a SOF syse ho dag s, The resose over he erval 0 s he s of hree ars:. Free vbrao de o al dslacee ad a =0. Resose o se force 3. Resose o ra force / Adag he avalable solos for hese hree cases, gves 3 cos cos s cos 4a 4b cos s cos s

9 Evalag hese eqaos a = gves he dslacee ad velocy a e +: s ] cos [ s cos cos s cos s 5a 5b These eqaos ca be rere as recrrece forlas: ' ' ' ' C B A C B A 6 Ths ercal rocedre s esecally sefl he he excao s defed a closely saced e ervals-as for earhqae grod accelerao-so ha he lear erolao s esseally erfec. If he e se s cosa, he coeffces A, B,., ` eed o be coed oly oce.

10 Recrrece forlas for he daed case: e A cos s e B s e C cos s e cos s

11 Recrrece forlas for he daed case: e A s e C cos s e B s cos e cos s

12 Exale: A SOF syse has he follog roeres =0.045 Nsec /c, =.7858 N/c, T = sec, =0.05. eere he resose of hs syse o P defed by he half-cycle se lse, sg ecese lear erolao of P h =0. sec.

13 Aly he recrrece forlas. A=0.88 B= C=0.069 A`=-3.58 B `= C `=0.957 A A' B B' C C' ' = `=.0483 The reslg coaos are sarzed he follog ables.

14 A B C B C + A

15 A' B' C' ' C` ` + A` B '

16

17 . Ceral fferece Mehod a b

18 . Ceral fferece Mehod By sbsrcg o sdes each oher, s he Eq. a By addg o sdes each oher, s he Eq. b

19 . Ceral fferece Mehod The eqao of oo of a SOF syse s, c P 3 A he o corresodg o he e, c P 4 here ü,, ad P are fco of, P P 5

20 . Ceral fferece Mehod Sbsg Eq. ad Eq. o Eq. 4,he Reorgazg Eq. 6, here 6 P c P c P c ˆ 7 8 ˆ ˆ P 9 ˆ c

21 . Ceral fferece Mehod If ad - are o, he + ll be calclaed fro Eq ˆ c P P 3 ˆ ˆ P Fro Eq. 8, Observe Eq. 30 ha o dslacees ad - are sed o coe ad +. Ths 0 ad - are reqred o deere ; he secfed al dslacee 0 s o. To deere -, e secalze Eqs. ad for =0 o oba,

22 . Ceral fferece Mehod Usg Eq., The al dslacee 0 ad al velocy eqao of oo a e 0 0 =0, 0 are gve, ad he 0 c0 0 P0 34 rovdes he accelerao a e 0: 0 0 c

23 . Ceral fferece Mehod Eq. 34 becoes o by he hel of Eq. 35. The ceral dfferece ehod ll blo, gvg eagless resls, he resece of ercal rod-off f he e se chose s o shor eogh. The secfc reqree for sably s, T Ths s ever a cosra for SF syses becase a ch saller e se shold be chose o oba resls ha are accrae. Tycally, / 0. o defe he resose adeqaely, ad os earhqae resose aalyses eve a shorer e se, ycally = 00.0 o 0.0 sec, s chose o defe he grod accelerao ü g accraely.

24 STEPS CALCULATION: A Ial Calclaos:. 0 = P 0 c 0 0. = = + c. a = c. b = 0 B Calclao for e se :. P = P a b. + = P /. If reqred: = + = + +

25 3. Near s Mehod Near develoed a faly a e-seg ehods based o he follog eqaos: The araeers β ad defe he varao of accelerao over a e se ad deere he sably ad accracy characerscs of he ehod. Eqs. 36 ad 37 have o be cobed h he eqao of oo of he dyac syse. c fs P c fs P There are o secal cases for he Near s Mehod.

26 a If he chage of accelerao fro se o + s assed as he average vale, s called o be he average accelerao ehod. b If he chage of accelerao fro se o + s assed o be lear, he he ehod s called he lear accelerao ehod.

27 Average Accelerao Mehod. for 4 4 for For =/ ad β=/4, Near s eqaos becoe he Average Accelerao Mehod.

28 Lear Accelerao Mehod. for for For =/ ad β=/6, Near s eqaos becoe he Lear Accelerao Mehod.

29 Icreeal Alcao of Near s Mehod Eqs. 40 ad 4 ca be re as, P P P

30 Sbsg Eq. 46 o Eq. 44 gves, 47 Eq. 6 ca be solved for, 46 a = The eqao of oo s: a = + The eqao of oo s: P fs c P fs c By sbracg he las o eqaos, e ge he creeal eqao of oo. Boh eqaos are secalzed o lear syses f s = ad f s + = +. c 48

31 If Eq. 49 s orgased, 50 By sbsg Eq. 7 ad Eq. 8 o Eq. 9 gves, c 49 Eq. 50 ca be re-exressed as, here c c c ˆ ˆ 5 c ˆ c c ˆ

32 The creeal dslacee s calclaed fro Eq. 5, ˆ ˆ 5 Fro Eqs. 46, 47 ad 5, Eqs. 4 ll be 53 The accelerao ca also be obaed fro he eqao of oo a e +: P c 54 Ths eqao s eeded o oba ü 0 o sar he coaos.

33 Near s ehod ll be sable f, T For =/ ad β=/4 hs codo becoes for average accelerao ehod T For =/ ad β=/6 hs codo becoes for lear accelerao ehod T 0.55

34 Near s Mehod: Average Accelerao Mehod =/, β=/4 Lear Accelerao Mehod =/, β=/6 A Ial Calclaos: P c Selec, /T ˆ c 4 a c b c

35 B Calclao for each e se: C Reea he ses B for + b a ˆ ˆ ˆ

36 Aalyss of Nolear Resose The dyac resose of a syse beyod s learly elasc rage s geerally o aeable o aalycal solo eve f he varao of he excao s descrbed by a sle fco. Nercal ehods are herefore esseal he aalyss of olear syses.