Dynamics of Structures

Transcription

1 UNION Dyamis of Stutues Pat Zbigiew Wójii Jae Gosel Pojet o-fiae by Euopea Uio withi Euopea Soial Fu

2 UNION Sigle-egee-of-feeom systems Uampe systems The euatio of motio fo a uampe sigle-egee-of-feeom system a be witte i fom m m (t) This is the seo oe iffeetial euatio with ostat oeffiiets. The solutio of this euatio a be fou with the help of the system Mathematia. The vaiable eofmotio otais the euatio of motio without ampig (I[]), solut otais the solutio of this euatio with iitial oitios fo time t= (iitial isplaemet o, iitial veloity po). The symbol eotes a eivative, so you a plot isplaemet [t], veloity [t] (fist eivative), a aele-atio [t] (seo eivative). The ew futio [t] is efie (I[4]). Pojet o-fiae by Euopea Uio withi Euopea Soial Fu

3 UNION By evaluatig [] a [], you obtai the iitial oitios: The patile solutio a be obtaie fom the geeal solutio by etemiig the iitial oitios o po. The omma Blo i Mathematia maes it possible to ahieve the patiula solutio fo the values: m= g, = N/m, o= m, po=, m/s. Usig the optio PlotStyle, you a hage the popeties of the plot. I the eample below, the isplaemet is plotte i bla, the veloity i blue a the aeleatio i e. Pojet o-fiae by Euopea Uio withi Euopea Soial Fu

4 UNION By usig the omma Maipulate with the omma Plot este, you a obseve how the solutio hages with hagig paametes. Comma Plot ae followe by paametes (hee m,, o, po) with ages a optioally iitial values. Pojet o-fiae by Euopea Uio withi Euopea Soial Fu

5 UNION By movig the slies to the left o ight, you a obseve hages i the plot. By pushig the + butto, you ativate the sub-meu You a also plot isplaemet, veloity a aeleatio by applyig the omma Maipulate Pojet o-fiae by Euopea Uio withi Euopea Soial Fu

6 UNION Tas: Fi whih paametes eease the yami espose, a whih oes iease it. Dampe system The euatio of motio fo a ampe sigle-egee-of-feeom system a be witte i fom m Pojet o-fiae by Euopea Uio withi Euopea Soial Fu

7 UNION Pojet o-fiae by Euopea Uio withi Euopea Soial Fu

8 UNION Pojet o-fiae by Euopea Uio withi Euopea Soial Fu

9 UNION Tas: Fi whih paametes eease the yami espose, a whih oes iease it. Fi whih paametes eease the vibatio peio, a whih oes iease it Hamoially foe vibatio The euatio of motio fo foe vibatio of a sigle-egee-of-feeom system (SDOF) a be witte i fom m Ft The vibatio ouig whe the eitatio foe is eual to Ft F sipt F os pt hamoially foe vibatio. is alle The euatio of motio fo a system with oe egee of feeom is elae as the vaiable eofmotio3. The system is hamoially eite with the agula feuey p. s This euatio has bee solve usig the omma DSolve. The iitial oitio was tae as eual to zeo. Pojet o-fiae by Euopea Uio withi Euopea Soial Fu

10 UNION Futio 3a[t] was elae as the solutio solut3a of the euatio of motio eofmotio3. The iitial oitios ae hee below. The figue below shows the fist s of vibatio of a SDOF system. The SDOF system has the paametes: m= g, = 8 Ns/m, = 9 N/m, F s = N = N, p = a/s. The atual agula feuey fo this system euals m 9 3, a/s the atual feuey is f 7,8,477 Hz the peio of uampe fee vibatios is T,9 s f the fatio of itial ampig (ampig atio) is m 9 9,5,5% 3,,5 3,,999887, 999 Pojet o-fiae by Euopea Uio withi Euopea Soial Fu

11 UNION f,999, 477 f the peio of ampe fee vibatios is T T f f,9s a the peio of foe vibatios is T,34 p s The hamoially foe vibatios of a SDOF system ae esibe by a iffeetial ohomogeous euatio of motio. The solutio of suh a type of euatio is the sum of a geeal itegal a a patiula itegal. The geeal itegal a be itepete as fee vibatio of the system, whih ous with the atual feuey a faes ue to ampig. The time uig whih the fee vibatio pesists is alle the tasiet poess. The patiula itegal epesets foe vibatio with a ostat amplitue a a feuey eual to the feuey of the loa. The poit i time whe the geeal itegal is patially eual to zeo, a oly the patiula itegal emais, is the begiig of the steay-state poess. The figue above shows the tasiet poess, i whih we a see the eay of the vibatio. These vibatios ae a sum of fee vibatio with the peio.9s a foe vibatio whose peio is.34s. The motio oespoig to the fee vibatio is peomiat. As the vibatio eays, afte s, the fee vibatio eas zeo, a oly the foe vibatio emais. I the et figue we see the steay state vibatio oly with the peio.34s. Pojet o-fiae by Euopea Uio withi Euopea Soial Fu

12 UNION The geate the ampig, the shote the time uig whih geeal itegal peomiates. I the figue below, the tasiet poess lasts fo the fist s of vibatio oly, as the ampig oeffiiet is ouble. Whe = Ns/m, the tasiet poess lasts appoimately s, as show i the figue below. To obseve oly the patiula itegal, apply the followig Mathematia omma Pojet o-fiae by Euopea Uio withi Euopea Soial Fu

13 UNION I egieeig appliatios, the steay-state poess is moe impotat. A iffeet fom of witig ow the above fomula may also be use, as pesete below. p ( t) Fs si( pt ) F os( pt ) The amplitue of the yami espose is eual to am p s F F Fo st whih meas that the yami espose is times geate o smalle tha the stati espose st. is alle a yami magifiatio fato am p st ( ) ( ) ata p feuey atio the fatio of itial ampig (ampig atio). m Pojet o-fiae by Euopea Uio withi Euopea Soial Fu

14 UNION The followig thee figues show the iagam of. The fist oe is aw o two liea aes, the seo oe o a vetial logaithmi ais a the last oe o two logaithmi aes. The uve with statig paamete =, is aw i bla. Tas: Fi the age of whee the yami magifiatio fato patially oes ot epe o the itial ampig fato. Pojet o-fiae by Euopea Uio withi Euopea Soial Fu

15 UNION The amplitue of espose epes ot oly o the yami magifiatio fato but also o the stiffess of the system. The fomula fo am p oul be ewitte i tems of system paametes, m,. am p Fo F ( ) o ( ) p substitutig a m you obtai am p Fo p F o ( mp ) Pojet o-fiae by Euopea Uio withi Euopea Soial Fu

16 UNION Usig followig ommas you a obseve amplitue of oigial stutue (bla lie) a moifie stutue i.e. with hage m, o (e lie). You a obseve that thee ae egio i whih espose of moifie stutue eeases but thee ae also egio whee the espose ieases. Tas Fi: how oes the espose hage with hage of m,,, F, what is the best way to suppess the yami espose, egios whee the espose is geate eve if you eease the maimum espose. Poblem Afte five peios, the amplitue of the fee vibatios hages fom A to,8a. Calulate the amplitue afte te peios. Afte how may peios will the amplitue beome smalle tha,5 A? Solutio: I systems with visous ampig, vibatios eays logaithmially. The logaithmi eemet is efie as follows: l ( t) ( t T T ) whee is the umbe of the obseve peios. Fo the osiee system, the logaithmi eemet is eual to Pojet o-fiae by Euopea Uio withi Euopea Soial Fu

17 UNION ( t) A l l.44. ( t T ) 5.8 A Fo a lightly ampe system, the followig fomula is tue hee / %. Usig the efiitio of the logaithmi eemet, you a alulate the amplitue afte peios ( t) l ( t T ) ( t T ) ( t) ep the amplitue afte peios is A A. 4 A ep.44. Afte how may peios will the amplitue beome smalle tha.5 A? A A l l 5,5,5 A,5 A. Afte the th peio, the amplitue of motio beome smalle tha 5% of the iitial amplitue Poblem Fi the atual vibatios peio of a wate towe with a tipo. The towe s apaity fo stoig wate is V= m 3. The aial efomatio a the mass of the stutue is egligible. How will the atual feuey hage i ase of the stutue without the tipo. Data: ρ= g/m 3, E = GPa, J = m 4, h = m, h =5 m. Pojet o-fiae by Euopea Uio withi Euopea Soial Fu

18 UNION Solutio: As the aial efomatio may be eglete, followig sheme oul be aepte m h h e m As the mass of the stutue may be eglete, m is eual to the mass of wate, m = V ρ = 3 = 5 g. The euivalet stiffess a be alulate fom fomula = Q/, whee isplaemet ause by the foe Q. This stutue is statially ietemiate, so the isplaemet a by alulate by ay metho suitable fo statially ietemiate stutues (fleibility metho, isplaemet metho), o with the use egieeig tables. Q h h Fo the above sheme Qh 3h 4h E J Q h E J 3,84 N/m 3h 4h Pojet o-fiae by Euopea Uio withi Euopea Soial Fu

19 UNION the atual agula feuey is m 3,84 5 4,38 a/s the atual feuey is f 4,38,7 Hz a the peio of uampe fee vibatios is T,43 s f Whe a stutue without a tipo is osiee, the followig sheme will be vali Q h + h I this ase Q h h 3 E J 3 Q 3E J 3,35 N/m 3 h h the atual agula feuey is m,35 5,33 a/s the atual feuey is f,33, Hz a the peio of uampe fee vibatios is T 4,7 s f Pojet o-fiae by Euopea Uio withi Euopea Soial Fu

20 UNION Poblem Choose a geealize ooiate a fomulate a euatio of motio. Fi the paametes of motio a amplitue of vibatio i poit A. Bas ae uefomable. Data: = g/m, = Ns/m, = N/m, p= a/s, P = N. D E B C A P o 3 m 3 m 3 m 3 m The system osist of two igi boies, i.e. o-efomable elemets (ba A-D a D-B), a five ostaits (two i pi suppot B, two i hige D a oe i pi-olle suppot A). As a igi boy o a plae has thee egees of feeom, two boies have si egees of feeom. The ostaits i this system, howeve, leave oly oe egee of feeom. It is theefoe eessay to hoose a geealize ooiate, whih will esibe ay possible (vitual) isplaemet i the system, e.g. a vetial isplaemet of poit A () o a otatio aou suppot B (ooiate ). D u D' D u D u D ue E E' B C u C C' P o P o P o si pt os pt A' A u A Pojet o-fiae by Euopea Uio withi Euopea Soial Fu

21 UNION The figue above shows the sheme of isplaemet. Evey vitual isplaemet oul be efie as the geealize ooiate. The euatio of motio will fist be witte ow usig ooiate, a the usig ooiate. u A = u C =u D = u E = ½ u A u C u E u D u D =u D os 45 = / u D u D Epessios fo eegies will be establishe both fo a eal system a fo a yami moel (show i the figue beeath). The popeties of the yami moel m e, e, e, F(t) - oul be alulate fom the euality of eegies i both systems. F(t) me (t) e e Kieti eegy D u B u A Pojet o-fiae by Euopea Uio withi Euopea Soial Fu

22 Pojet o-fiae by Euopea Uio withi Euopea Soial Fu UNION The loal isplaemet o veloity of a vey small pat of ba oul be witte ow as a futio of o u u u u Kieti eegy of a small pat of ba AD a ba DB,, 3 m u E m u E DB AD The total ieti eegy: ,, ) ( E E E AD l DB DB l AD The total ieti eegy fo the euivalet system is eual to m E e The system is a euivalet oe if the eegy has the same value as i the eal system, so g 8 ) (8 e e m m Rayleigh Dissipatio Futio The issipatio futio is eual to 4 4 u E Fo the euivalet system, the issipatio futio is eual to e The issipatio futio shoul be the same i both fomulatios e e Ns/m Potetial eegy The potetial eegy fo a sigle spig elemet is eual to u

23 UNION whee u is the hage i legth of the spig elemet. I the osiee system the spig elemet has oe e fie, so the hage i spig legth will be eual to that pat of the isplaemet of the othe e (poit D) whih is paallel to the spig ais. The total isplaemet of poit D is u D, the pat of this isplaemet paallel to the spig ais is u D, a the pat of total isplaemet pepeiula to the spig ais is u D. The potetial eegy fo the system is E p u D The potetial eegy fo the euivalet system is E p e Fom the euality of these fomulas e e 5 5 N/m Wo of atig foes You will obtai the same esults if you osie the vetial a hoizotal ompoets of the otatig foe P istea of the whole foe. The vetial ompoet of the foe is P si pt a the hoizotal ompoet is P os pt. The wo of the atig foe is eual to L P os pt u C, hoizotal P os pt P sipt u P sipt u C C, vetial P si pt u C P sipt The wo of the atig foe fo the euivalet system is L Ft hee the euivalet yami foe is F t P si pt = si ( t) [N] Euatio of motio m e ( t) ( t) ( t) F( t) e e Substitutig the obtaie values 8 ( t) ( t) ( t) P si pt 4 5 ( t) 5 ( t) 5 ( t) sit Paametes of motio the atual agula feuey is m e e 5 5 7,8 a/s the atual feuey is Pojet o-fiae by Euopea Uio withi Euopea Soial Fu

24 UNION f 7,8,8 Hz the peio of uampe fee vibatio is T,355 s f the fatio of itial ampig (ampig atio) is m 5 5 5,44 4,4% 7,8,44 7,8,999 7, a/s f 7,,8 Hz the peio of ampe fee vibatios is T,35 s f a the logaithmi eemet of ampig is T,77 Steay-state solutio of euatio of motio The euatio of motio fo foe vibatio is m e e e F t F sipt F os pt The steay-state (patiula) solutio of this euatio is ( t) si pt os pt p s p ( t) Fs si( pt ) F os( pt ) e e the amplitue of this motio is eual to s am p e s F F Fo e st whee the yami magifiatio fato p,, 3 ( ) ( ) 7,8 Pojet o-fiae by Euopea Uio withi Euopea Soial Fu

25 UNION (, 3 ) (,44, 3) 3,4 whih meas that the espose of the system is ove thee times geate tha the espose of the same system whe the same foe is applie statially. I this poblem F s = P, F =, a the amplitue of motio is eual to Fs P am p 3,4 3,4, 5 5 e e 3,8 m The amplitue of vibatio of poit A is eual to the amplitue of p, am u A =,8 m. Poblem Solve the pevious poblem hoosig a iffeet geealize ooiate. Solutio This time let us hoose the otatio aou suppot B as the yami ooiate. D u D' D u D u D ue E E' B C u C C' P o P o P o si pt os pt A A' u A The sheme of isplaemet is show i the figue above. Evey possible isplaemet oul be efie by the geealize ooiate. ua = uc =ud = m ue = 3m u u u A C 3 u E D Pojet o-fiae by Euopea Uio withi Euopea Soial Fu

26 Pojet o-fiae by Euopea Uio withi Euopea Soial Fu UNION ud =ud os 45 = / ud u D 3 Kieti eegy A D u u B The loal isplaemet a veloity oul be witte ow as a futio of a u u u u The ieti eegy of a small pat of ba AD a ba DB is,, 3 3 m u E m u E DB AD The total ieti eegy is ,, ) ( E E E AD l DB DB l AD

27 UNION The total ieti eegy fo the euivalet system is eual to E m e A system is euivalet if the eegy has the same value as i the eal system, so /a [g m ] m ( 88) g m e m e Rayleigh Dissipatio Futio The issipatio futio is eual to 3 9 u E 9 Fo the euivalet system, the issipatio futio is eual to e The issipatio futio shoul be the same i both fomulatios 9 e e 9 9 N m s/a [N m s] Potetial eegy Applyig the same metho as i the pevious solutio E p D 8 u 8 Potetial eegy fo the euivalet system E p e Fom the euality of these fomulas 8 e e 8 8 Nm/a [Nm] Wo of atig foes The wo of the atig foe is eual to L P os pt u C, hoizotal P os pt P si pt u P si pt u C C, vetial P si pt u The wo of the atig foe fo the euivalet system is L Ft hee the euivalet yami foe is F t P sipt = si ( t) [Nm] Euatio of motio C P si pt P si pt Pojet o-fiae by Euopea Uio withi Euopea Soial Fu

28 UNION m e ( t) ( t) ( t) F( t) e e Afte substitutig the evaluate values, we ahieve 88 ( t) 9 ( t) 8 ( t) P si pt 57 ( t) 9 ( t) 8 ( t) sit Paametes of motio The atual agula feuey is m e e ,8 a/s the atual feuey is f 7,8,8 Hz the peio of uampe fee vibatios is T,355 s f the fatio of itial ampig (ampig atio) is 9,44 4,4% m ,8,44 7,8,999 7, a/s f 7,,8 Hz the peio of ampe fee vibatios is T,35 s f a the logaithmi eemet of ampig is T,77. Colusios: As i this euatio the geealize ooiate is assume to be a uit agula isplaemet, the paametes m e, e, e, F(t) of the euatio ae iffeet tha the oes fo the euatio obtaie with the use of uit taslatioal isplaemet (show above). Solutio of euatio of motio Pojet o-fiae by Euopea Uio withi Euopea Soial Fu

29 UNION The amplitue of steay-state motio is eual to am p e s F F Fo e st whee the yami magifiatio fato p,, 3 ( ) ( ) 7,8 3,4. (, 3 ) (,44, 3) I this poblem, F s = P, F =, a the amplitue of motio is eual to am p Fs e 3 P 3 3,4 3,4 3,33 8 e,3 a The amplitue of vibatio of poit A is eual to u A amu am,3,8m A p Colusio: the esults o ot epe o the hoie of the ooiate. Pojet o-fiae by Euopea Uio withi Euopea Soial Fu