Dynamic Magnification Factor of SDOF Oscillators under. Harmonic Loading

Transcription

1 Dynmic Mgnificion Fcor of SDOF Oscillors under Hrmonic Loding Luis Mrí Gil-Mrín, Jun Frncisco Cronell-Márquez, Enrique Hernández-Mones 3, Mrk Aschheim 4 nd M. Psds-Fernández 5 Asrc The mgnificion fcor for he sedy-se response of SDOF sysem under hrmonic loding is descried in mny srucurl dynmics eooks; he well known nlyicl soluion is esily oined from he soluion o he dmped equion of moion for hrmonic loding. The complee nd sedy se soluions cn differ significnly. An nlyicl epression for he mimum response o he complee soluion (sedy se plus rnsien) remins elusive; however, simple nlyicl epression is idenified herein for he undmped cse. Differences in he mgnificion fcors oined for oh soluions re discussed. Keywords: Dynmic mgnificion fcor, sedy se, rnsien se. Associe Professor, Universiy of Grnd, Cmpus de Fuenenuev. 87 Grnd. Spin. mlgil@ugr.es. Ph.D. cndide. Universiy of Grnd, Cmpus de Fuenenuev. 87 Grnd. Spin. jfcronell@ugr.es. 3 Professor, Universiy of Grnd, Cmpus de Fuenenuev. 87 Grnd. Spin. emones@ugr.es 4 Professor, Sn Clr Universiy, 5 El Cmino Rel, Sn Clr, Cliforni. mschheim@scu.edu. 5 Professor, Universiy of Grnd, Cmpus de Fuenenuev. 87 Grnd. Spin. mpsds@ugr.es Pge

2 . Inroducion One of he fundmenl opics in mos if no ll srucurl dynmics ooks is he well sudied equion of moion for dmped hrmonic oscillor [-5], which for free virion is given s: m () + c () + k () () where m, k, nd c re he mss, he siffness nd he coefficien of viscous dmping, respecively. When sujeced o n eernl force p(), s presened in Figure, he equion of moion is: m () + c () + k () p() () where p() is he force pplied o he mss. k P m p() f S () P f I () m p() c f D () Figure. Oscillor: () idelized physicl configurion nd () forces cing on he mss for ccelerion >. Cusomrily, wo new prmeers re defined s funcions of m, c, nd k. The undmped nurl frequency,, is given y: k (3) m nd he dmping rio,, is given y: Pge

3 c m (4) Inroducing hese wo prmeers llows () o e rewrien s: p() () + () + () (5) m If he eciion is sinusoidl ( p() p ( ) sin Ω ), where p nd Ω re he mpliude nd frequency of he pplied force, respecively, he equion of moion cn e solved nlyiclly. The well-known soluion is: p Ωcos( Ω ) + ( Ω )sin( Ω) () e ce + ce + i i ( ) 4 4 m ( ) Trnsien + Ω +Ω Sedyse (6) where (7) is he dmped nurl frequency of he sysem nd c nd c re wo comple conjuged consns h cn e evlued once he oundry condiions hve een specified. As indiced in (6), he response is composed of rnsien erm h vnishes wih ime nd sedy-se erm h is hrmonic funcion of ime. This pper ddresses he soluion o (5) given y (6). The pek response (mimum of solue vlue) is sough nd is compred wih he mpliude of he sedy se porion of he response in (6). Resuls re considered s funcion of frequency rio, β, where β he rio of he frequency of he eciion ( Ω ) nd he undmped nurl frequency of he sysem ( ). As will e seen, priculrly for β >, he pek response is significnly greer hn he sedy-se pek response. Pge 3

4 . Tol response versus sedy-se response A res oundry condiions (i.e. () nd () ) re considered ecuse hose re he cul condiions under ny erhquke moion or ny oher dynmic moion in srucurl engineering. For hese oundry condiions, (6) ecomes: p k e ( ) β cos 4 [ ( ) β + β ] I β + φ rcn ; β φ rcn β II ( φ ) + cos( Ω φ ) (8) The rnsien nd sedy-se componens of (8) re idenified y I (mening imes I) nd II respecively. The mimum of he sedy-se componen is esily compued since his pr is consn ( ) muliplied y funcion ( II ) whose pek vlue is. Therefore, he pek sedy-se response is given y : m p (9) 4 ( ) β + β k sedy The erm of Eq. 9 h muliplies p /k hs een clled he sedy-se dynmic mgnificion fcor in numerous eooks. Figure shows he sedy-se dynmic mgnificion fcor s funcion of β for differen vlues of he dmping rio. Resonnce is esily pprecied in Figure. For he undmped cse ( ) he resonn frequency is equl o he undmped nurl frequency ( ). I will e ppren h he resonn frequencies re minined even for he complee soluion o (8), h is, where he rnsien porion is included in deermining he pek response mpliude. Pge 4

5 4. m /(p /k) β Figure. Vriion of he sedy-se dynmic mgnificion fcor wih nd β. Alhough simple, eplici, funcion descriing he pek response o (8) cnno e oined esily, he vlue of he pek response does no depend on he vlue of, s should e ppren upon closer inspecion of Eq. (8). Consider h he responses of wo sysems hving he sme β nd u differen undmped nurl frequencies nd would e descried y: ( ) p k 4 [ ( ) β + β ] e β cos φ + cos ( β φ ) () ( ) p k 4 [ ( ) β + β ] e β cos φ + cos ( β φ ) () Le us susiue for ime in Eq. () he modified ime : Pge 5

6 Pge 6 ( ) [ ] ( ) ( ) e k p 4 cos cos φ β φ β β β + + Therefore, he pek response (over ll ime) is independen of, nd cn e oined using Eq. (8) for n rirry vlue of he undmped nurl frequency (e.g..hz ). Sysems hving oher vlues of will hve he sme pek, u occurring differen ime (see Figure 3). For sysem hving undmped nurl frequency s, he ime of he pek is delyed n moun s Δ (3) where is he ime corresponding o he mimum response for.hz, nd s is he undmped nurl frequency of he sysem eing nlyzed. ()

7 /(p /k).5 ( Hz ), ( Hz ), (s) s s M Δ s s Figure 3. Responses corresponding o., β. for differen nurl frequencies. The sme mimum occurs, u differen imes s shown. Pek vlues of he response epression cn e deermined using vrious mhemicl sofwre progrms. In he presen cse, he nonliner consrined glol opimizion pckge funcion FindMimum ws used wihin he Mhemic progrm. Plos of oh sedy-se nd complee mgnificion fcor (including oh rnsien nd sedy-se responses) for differen vlues of dmping rio re presened in Figure 4. Resuls were oined using he undmped nurl frequency se o Hz (.Hz ). Figure 4 shows h he complee mgnificion fcor is lwys greer hn or equl o he sedy-se mgnificion fcor. The closer β is o he smller he difference is eween oh mgnificion fcors, ecep for he cse. Also, he lower he vlue of he Pge 7

8 dmping rio ( ) he greer is he difference eween he complee nd sedy-se mimum responses. For insnce, in cse of, wih vlue of β.7, he complee mgnificion fcor is 3.3 while he sedy-se mgnificion fcor is.96. In his cse he sedy-se mgnificion fcor represens only 6% of he complee mimum response. For.5 nd β 3., Figure 4 shows h he sedy-se mgnificion fcor is round he 45% of he complee mgnificion fcor. m /(p /k) 4. Complee mgnificion fcor Sedy-se mgnificion fcor β Figure 4. Complee nd sedy-se mgnificion fcors for Hz. To furher illusre differences in he mgnificion fcors oined for he complee nd sedy-se soluions, he rio of hese mgnificion fcors is ploed in Figure 5 s funcion of β nd. Dshed lines in Figure 5 correspond o he rio eween pek response given y he complee soluion (oined from Eq. (8)) nd he pek response given y he sedy-se mgnificion fcors (oined from Eq. (9)) for differen vlues of he nd β. Vlues of β vry from o 3. in seps of.5. In cse h he dmping rio is equl o zero Pge 8

9 ( ) nd he frequency rio is equl o one (β, i.e. resonnce response) disconinuiy pper in Figure 5. 4 Complee Sedy Se β. ( nd β) β Figure 5. Complee o sedy-se mgnificion fcors rio. In cse h β nd œ (,) he rio of peks response (complee over sedy-se) is equl o one. This coincides wih he oservion given y Clough nd Penzien (993) in 3.3 of heir ook. For he cse of β nd, he rio of peks response presens disconinuiy nd wo vlues re possile, s cn e seen in Figure 5: m m complee sedyse m Lim m (, β ) (, β ) m Lim β m complee sedyse complee sedyse (, β ) (, β ) (, β ) (, β ) (4) Pge 9

10 Clough nd Penzien (993) pproched he soluion imposing β, so hey go he firs soluion given y epression (4). I is ppren h he mimum rios re oined for he highes vlue of β nd he lowes dmping rio,. Inspecion of he plos indices h for he undmped cse, he rio of he mgnificion fcors for he complee nd sedy-se soluions is lmos liner. Irregulriies in he rio of mgnificion fcors re ppren for.. In he undmped cse, funcion h epresses he rio of he complee nd he sedyse mgnificion fcor s funcion of β cn e simply djused. This funcion is given y (coninuous hick grey line in Figure 5): ( β) ( β) m, complee m, sedy β + (5) Thus, considering Eq. (9), n eplici epression for he complee mimum response for he undmped cse is given y: p p m (, β) ( β k k complee + ) (6) 4 β + β ( β ) wih he ecepion of of β nd, s i ws shown in Eq. (4). Furhermore, Figure 5 demonsres h Eq. (5) provides n upper ound o he rio of complee nd sedy-se mgnificion fcors. Thus, for ny dmping level ( ), he produc of Eqs. (9) nd (5) is n upper ound o he complee soluion. Pge

11 4. Conclusions The preceding demonsres he significnce of he rnsien pr of he response of SDOF sysem sujeced o hrmoniclly vrying lod of sine-wve form. Ner resonnce, differences eween pek rnsien nd pek full responses re negligile; hese differences ecome significn s he forcing frequency egins o devie significnly from he undmped frequency of virion of he oscillor. Trdiionlly, srucurl dynmics ooks define he mimum of he sedy-se response of SDOF sysems s he dynmic mgnificion fcor. A eer erm for he convenionl mgnificion fcor is he sedy-se mgnificion fcor, since i represens n imporn componen of he complee mgnificion fcor. A new eplici epression for he mimum response for he complee soluion for he undmped cse (i.e. for he mos unfvourle siuion) ws developed, nd is given y Eq. (6), wih he ecepion of he cse of β nd where wo vlues re possile. An upper ound on he complee soluion for cses wih dmping ws idenified. Pge

12 6. References.. Chopr, A.K. Dynmics of Srucures-Theory nd Applicions o Erhquke Engineering, 3 rd Ediion. Prenice Hll.. Clough, R.W, nd Penzien, J. Dynmics of Srucures, nd Ediion. McGrw-Hill Villverde, R. Fundmenl Conceps of Erhquke Engineering. CRC Press R. R. Crig Jr. nd A. J. Kurdil. Fundmenls of Srucurl Dynmics. Wiley, New York, Grcí-Reyes L.E. Dinámic Esrucurl Aplicd l Diseño Sísmico. Unindes. Bogoá Pge