OGLEDNI PRIMJER ZADAAK Odredte dnamčke karakterstke odzv armranobetonskog okvra C-C prkazanog na slc s prpadajućom tlorsnom površnom, na zadanu uzbudu tjekom prve tr sekunde, ako je konstrukcja prje djelovanja sle bla u stanju mrovanja (materjal: beton C 4/5, prgušenje: %). Opterećenje na nvou međukatnh konstrukcja: stalno g9,kn/m, promjenjvo p4,kn/m. Pr proračuna masa, promjenjvo opterećenje uzet u znosu od 5%. C tlors presjek B A B A 5 5 5 5 4 C 6 5 5, F, F,5 F 5 kn F(t) F,5F,5,,5, t(s)
Dnamčke karakterstke odzv konstrukcje određujemo na zamjenjujućem sustavu s tr stupnja slobode (promatramo samo horzontalne pomake). Masu konstrukcje svodmo na koncentrrane mase na nvoma međukatnh konstrukcja. 4 5 4 6 5 4 4 5 4 4 I I s s Is m m m 4 5 5 A) POČENI (PRIPREMNI) PRORAČUNI ) Matrca masa [m] Mase koncentrramo na nvoma međukatnh konstrukcja preko kojh je zadano svo opterećenje. S obzrom da promatramo damo horzontalne pomake, matrca je masa djagonalna. Dakle, velčnu masa određujemo na temelju zadanog opterećenja prpadajuće tlorsne površne. B A C C 6 B A 5 5 5 5 m ( g + γp ) A / a g m (9+,5 4, ) (4,,) / 9,8 49, t m m 49, 49, [ m] 49, [ t]
) Matrca popustljvost [a] Elemente matrce popustljvost dobjemo, sukladno defncj, određvanjem pomaka pojednh čvorova na djelovanje jednčnh koncentrranh sla na nvoma čvorova (F,kN), ručno l pomoću nekog programskog paketa (npr. SAP, ROBO, SRUDL td.).,4,497,49,497,757 [] a,49,499,47 [ m],47 ) Vlastte kružne frekvencje (ω), vlastt perod () frekvencje (f) Vlastt (prrodn) oblc Dnamčke karakterstke sustava određujemo rješavanjem problema vlastth vrjednost metodom karakterstčnog polnoma l karakterstčne determnante. [ m]{ u&& } λ[ k]{} u { } / [ k] [ a], ([ D] λ[ I] ){ u} { } gdje je λ ω Dnamčka matrca [D] [ D ] [ a][ m],4,49,497,49,499,47,497 49,,47,757 49, 49, [ D],56,975,,975,7,7,,7,5777
det det [ D] λ[ I ] ( d λ) d d ( d λ) d d ( d λ) λ,669987λ +,5λ 7,766 d d Korjen su jednadžbe: λ,586 λ,659 λ,8487 ω,6 rad/s,48 s, f,79hz ω 9,57 rad/s,6 s, f 6,5Hz ω 7,547 rad/s,85 s, f,76hz Karakterstčne tj. vlastte l prrodne vektore oblka (modove) vbrranja, određujemo z uvjeta λ I u. ([ D ] [ ]){ } { } a) I. vektor oblka {u} za λλ ( d λ ) u + du + du ( ) du + ( d λ ) u + du () d u + d u + ( d λ ) u () Za npr. pretpostavljen u,, z jednadžb npr. () () dobje se: {} u,4455,75794, b) II. vektor oblka {u} za λλ ( d λ ) u + du + du ( ) du + ( d λ ) u + du () d u + d u + ( d λ ) u () Za npr. pretpostavljen u,, z jednadžb npr. () () dobje se: {} u,865,645,
c) III. vektor oblka {u} za λλ ( d λ ) u + du + du ( ) du + ( d λ ) u + du () d u + d u + ( d λ ) u () Za npr. pretpostavljen u,, z jednadžb npr. () () dobje se: {} u,684,6665786, 4) Modalna matrca [Φ] Ortonormranje vektora oblka j {}[ u m]{} u j j u α u {} {} α {}[ u m]{} u α,7699, α,8598, α,897, te su ortonormran vektor oblka, redom,47558,8 u,86 u,5768 u,7699,8598,785,897 {} {} {},8657. Prema tome je modalna matrca oblka,47558,8,785 [ Φ],86,5768,8657,7699,8598,897 5) Odabr vremenskog koraka teracje t t n Za ovaj školsk prmjer odabermo za n, te je stoga,48 t,48. Odabrano: t, s
B) IERAIVNI POSUPAK ) ransformacja sustava u sustav nezavsnh dferencjalnh jednadžb tzv. modalnh jednadžb [ m]{} && x + [ c]{ x& } + [ k]{} x { F( t) } / [ Φ] / [ Φ][ Φ] [ Φ] [ m][ Φ][ Φ] {} && x + [ Φ] [ c][ Φ][ Φ] {} x& + [ Φ] [ k][ Φ][ Φ] {} x [ Φ] { F() t } []{} I && η + [ ξω]{} & η + [ ω ]{} η f () t Svaku jednadžbu zasebno rješavamo jednom od teratvnh metoda, npr. metodom korak po korak s nterpolacjom uzbudne sle. Rješenjem određujemo modalne tj. projcrane brzne pomake. ) Rješenje modalnh jednadžb a) Modalna jednadžba & η & + ξωη + ωη f ( t) t,s ξ, ω,6rad/s ω D,58rad/s k 7,5897 kn/m n,6 A,96695 A' -,555648 B,96777 B',9559674 C,99 C',9766 D 6,6666E-5 D',998764 t (t) F() x() x(+) v() v(+),, 5,65745,984697,94945,,96 4,65446955,984697,5489,94945,5544547,4,9 4,486665,5489,477445,5544547,765676,6,88,4676,477445,4584,765676,96597 4,8,84,86686,4584,65997,96597,969595 5,,8,5796,65997,79764,969595,958689 6,,76,64556,79764,986977,958689,87468 7,4,7,99856,986977,4848,87468,769647 8,6,68,8497,4848,68648,769647,49655 9,8,64 9,76964668,68648,6796,49655,698569,,6 9,59447,6796,55969,698569 -,47996,,56 8,5484457,55969,79 -,47996 -,878664,4,5 7,9787674,79,477 -,878664 -,656959,6,48 7,74776,477,785 -,656959 -,86975 4,8,44 6,766878,785,8857 -,86975 -,5649498 5,,4 6,6898,8857,6546447 -,5649498 -,954676
6,,6 5,495468,6546447,486749 -,954676 -,5665 7,4, 4,884884,486749,579 -,5665 -,487648 8,6,8 4,7486,579 -,8574 -,487648 -,6775 9,8,4,666788 -,8574 -,56799 -,6775 -,4597,4,,5449 -,56799 -,4969 -,4597 -,898478,4,6,44459 -,4969 -,6869 -,898478 -,577849,44,,888694 -,6869 -,7985 -,577849 -,9967556,46,8,5796 -,7985 -,754458 -,9967556 -,67885 4,48,4,66898 -,754458 -,757676 -,67885,95486 5,5, -,757676 -,666587,95486,4668996 6,5,4,66898 -,666587 -,5587,4668996,6878748 7,54,8,5796 -,5587 -,9588,6878748,8757 8,56,,888694 -,9588 -,445969,8757,7879 9,58,6,44459 -,445969,5948,7879,865477,6,,5449,5948,5869,865477,6865,6,4,666788,5869,44796,6865,56769,64,8 4,7486,44796,644499,56769,959576,66, 4,884884,644499,897,959576,795984 4,68,6 5,495468,897,95974,795984,599698 5,7,4 6,6898,95974,57676,599698,768644 6,7,44 6,766878,57676,97587,768644,4498 7,74,48 7,74776,97587,5444,4498 -,856799 8,76,5 7,9787674,5444,774877 -,856799 -,9874 9,78,56 8,5484457,774877,8 -,9874 -,4676664 4,8,6 9,59447,8,89946 -,4676664 -,5964447 4,8,64 9,76964668,89946,7659 -,5964447 -,6759559 4,84,68,8497,7659,68947 -,6759559 -,68854967 4,86,7,99856,68947,497965 -,68854967 -,64677884 44,88,76,64556,497965,7749 -,64677884 -,54977956 45,9,8,5796,7749,77 -,54977956 -,44467 46,9,84,86686,77,495 -,44467 -,565 47,94,88,4676,495,96985 -,565 -,66 48,96,9 4,486665,96985,94976 -,66,594 49,98,96 4,65446955,94976,76,594,484575 5,, 5,65745,76,759865,484575,684599 5,,98 4,95977,759865,5476,684599,74647 5,4,96 4,65446955,5476,67685,74647,8686 5,6,94 4,4968,67685,88884,8686,84454987 54,8,9 4,486665,88884,4777,84454987,79949976 55,,9,78565,4777,54567,79949976,69456 56,,88,4676,54567,784757,69456,575844 57,4,86,796,784757,66766,575844,9799 58,6,84,86686,66766,45569,9799,5465 59,8,8,57594,45569,479,5465 -,4776 6,,8,5796,479,64669 -,4776 -,5876 6,,78,967565,64669,7748 -,5876 -,564665 6,4,76,64556,7748,47 -,564665 -,7467 6,6,74,9656,47,979755 -,7467 -,874446 64,8,7,99856,979755,79677865 -,874446 -,9486 65,,7,685557,79677865,648 -,9486 -,966986 66,,68,8497,648,49546 -,966986 -,9595785 67,4,66,749478,49546,758567 -,9595785 -,888845 68,6,64 9,76964668,758567,8566779 -,888845 -,6895 69,8,6 9,4644499,8566779 -,76 -,6895 -,49849
7,4,6 9,59447 -,76 -,984 -,49849 -,884475 7,4,58 8,8574 -,984 -,47559 -,884475 -,66899 7,44,56 8,5484457 -,47559 -,8968 -,66899,57646 7,46,54 8,49 -,8968 -,884898,57646,4986949 74,48,5 7,9787674 -,884898 -,456,4986949,575568 75,5,5 7,6565 -,456,55668,575568,6494865 76,5,48 7,74776,55668,557,6494865,75968 77,54,46 7,97,557,987,75968,7486464 78,56,44 6,766878,987,544646,7486464,6979688 79,58,4 6,449,544646,6794,6979688,6864 8,6,4 6,6898,6794,7866448,6864,4746769 8,6,8 5,8775,7866448,86489,4746769,999744 8,64,6 5,495468,86489,9744,999744,575 8,66,4 5,946,9744,9945,575 -,47567 84,68, 4,884884,9945,859445 -,47567 -,94884 85,7, 4,579575,859445,77865 -,94884 -,49747 86,7,8 4,7486,77865,665748 -,49747 -,65565 87,74,6,9689887,665748,5888865 -,65565 -,77695 88,76,4,666788,5888865,5644 -,77695 -,8445759 89,78,,58599,5644,8458 -,8445759 -,868468 9,8,,5449,8458,49 -,868468 -,8658 9,8,8,74774,49 -,4448 -,8658 -,74777559 9,84,6,44459 -,4448 -,89977 -,74777559 -,6948 9,86,4,74 -,89977 -,9649 -,6948 -,46454 94,88,,888694 -,9649 -,4645576 -,46454 -,7596848 95,9,,565745 -,4645576 -,55 -,7596848 -,886645 96,9,8,5796 -,55 -,4977885 -,886645,57499 97,94,6,959447 -,4977885 -,456748,57499,8844546 98,96,4,66898 -,456748 -,8586,8844546,47968774 99,98,,5449 -,8586 -,8496,47968774,55787,, -,8496 -,665478,55787,677876,, -,665478 -,87,677876,6497987,4, -,87,9795,6497987,69545,6,,9795,464,69545,5755484 4,8,,464,8456,5755484,47888878 5,,,8456,488567,47888878,5994 6,,,488567,45766,5994,5947 7,4,,45766,489,5947,8947 8,6,,489,47586,8947 -,49677 9,8,,47586,475 -,49677 -,7797899,,,475,6757 -,7797899 -,4697655,,,6757,6975 -,4697655 -,54667,4,,6975,59477 -,54667 -,5844454,6,,59477,95789 -,5844454 -,699757 4,8,,95789 -,8848 -,699757 -,5964599 5,, -,8848 -,9645 -,5964599 -,546769 6,, -,9645 -,964587 -,546769 -,4584459 7,4, -,964587 -,75487 -,4584459 -,745 8,6, -,75487 -,48645 -,745 -,9686 9,8, -,48645 -,45789 -,9686 -,444685,4, -,45789 -,44446654 -,444685,958,4, -,44446654 -,4757,958,578,44, -,4757 -,48866,578,8449,46, -,48866 -,5679,8449,486746
4,48, -,5679 -,587644,486746,5455 5,5, -,587644 -,4454,5455,576 6,5, -,4454,756658,576,56967 7,54,,756658,8479,56967,5575 8,56,,8479,759,5575,489757 9,58,,759,5784,489757,664447,6,,5784,464,664447,8497789,6,,464,4488,8497789,494589,64,,4488,4799758,494589 -,9544,66,,4799758,884 -,9544 -,77986 4,68,,884,4765 -,77986 -,56947 5,7,,4765,48769 -,56947 -,44988 6,7,,48769,4656 -,44988 -,5854455 7,74,,4656,498 -,5854455 -,57546 8,76,,498 -,66559 -,57546 -,575986 9,78, -,66559 -,67574 -,575986 -,488544 4,8, -,67574 -,567566 -,488544 -,446988 4,8, -,567566 -,768 -,446988 -,745 4,84, -,768 -,757546 -,745 -,7748967 4,86, -,757546 -,97968 -,7748967 -,4578864 44,88, -,97968 -,9869 -,4578864,9884 45,9, -,9869 -,67949,9884,9754 46,9, -,67949 -,66884,9754,4846 47,94, -,66884 -,779,4846,4958 48,96, -,779 -,489,4958,478574 49,98, -,489 -,459,478574,5454 5,, -,459,59588,5454 #REF! b) Modalna jednadžba & η & + ξωη + ωη f ( t) t,s ξ, ω 9,57rad/s ω D 9,49rad/s k 5,7649 kn/m n,784 A,76886 A' -7,9745 B,77875 B',689578 C,88 C',869 D 6,459E-5 D',94855
c) Modalna jednadžba & η & + ξωη + ωη f ( t) t,s ξ, ω 7,547rad/s ω D 7,5rad/s k 549,69 kn/m n,4794 A,6444 A' -7,7665 B,946 B',7775799 C,47 C',49756 D 5,89777E-5 D',86759 ) Određvanje stvarnh pomaka brzna Inverznom transformacjom nodalnh pomaka, brzna ubrzanja dobvamo stvarne generalzrane pomake, brzne ubrzanja. [ Φ] {} && x {} && η {&& x} [ Φ]{ && η} [ Φ] {} x& {} η& {} x& [ Φ]{} η& [ Φ] {} x {} η {} x [ Φ]{} η
SVARNI POMACI (m),,5 pomac x(m),,5,,,5,,5,,5, -,5 -, vrjeme t(s) pomac I.kata pomac II.kata pomac III.kata