FORCED VIBRATION of MDOF SYSTEMS

Transcription

1 FORCED VIBRAION of DOF SSES he respose of a N DOF sysem s govered by he marx equao of moo: ] u C] u K] u 1 h al codos u u0 ad u u 0. hs marx equao of moo represes a sysem of N smulaeous equaos u ad s me dervaves. For lear sysems, he orhogoaly properes of he mode shapes ca be used o sgfcaly smplfy he geeral equaos of moo. hs smplfcao volves expressg he equao of moo erms of he so-called ormal coordaes sead of geomerc coordaes. hs rasformao ucouples he N smulaeous equaos o gve N depede equaos ha ca be veed as goverg N SDOF sysems.

2 FORCED VIBRAION of DOF SSES he exac dyamc respose he orgal geomerc coordaes ca be obaed by superposg he calculaed respose of each ormal coordae. For he purpose of dyamc-respose aalyss, s ofe advaegous o express he dsplaced posos of N-DOF lear sysems erms of he free-vbrao mode shapes. hese shapes cosue N depede dsplaceme paers, he ampludes of hch may serve as geeralzed coordaes o express ay se of dsplacemes. Normal Coordaes Normal vbrao mode shapes defe a se of coordaes ha are o dyamcally lked,.e., for a N DOF sysem, hese modes are N depede dsplaceme shapes. he ampludes of hese modal shapes ca herefore be used as geeralzed coordaes ha ould characerze ay deformed shape. hese coordaes are called modal coordaes or ormal coordaes. Ay dsplaceme vecor u ca be obaed by superposo of he mode shapes, each scaled by a specfc modal amplude.

3 Normal Coordaes he procedure s llusraed Fg. u ] u 1 1 y1 u y u 3 3 y3 Ay dsplaceme vecor u for hs srucure ca be developed by superposg suable ampludes of he ormal modes as sho. he dsplacemes of modal compoe u are gve by he produc of he modal amplude y ad he mode shape, ha s u y

4 Normal Coordaes he oal dsplaceme s equal o he sum of he modal compoes N 1 y1 y... N yn y 1 u 3 hch ca be expressed as N u y ] 4 1 here s he ormal or geeralzed coordaes vecor ad ] s he mode shapes marx. arx ] s used o rasform he geeralzed coordaes o geomerc coordaes u. Iversely, ay modal coordae ca be easly calculaed usg he orhogoaly properes of mode shapes. If e premulply Eq. 4 by ], e ge ] u ] ]

5 Normal Coordaes Sce he mode shapes are orhogoal h respec o he mass marx, all erms of he precedg seres are equal o zero, excep for he erm correspodg o. We herefore have from hch If vecor u s me depede, coordaes ll also be me depede. akg he me dervave of hs equao, e ge 5a u ] ] u ] ] u ]... ]... ] ] ] 1 1 u ] ] 5b

6 Ucoupled Equaos of oo: Udamped he orhogoaly properes of he ormal modes ll be used o smlfy he equaos of moo of he DOF sysem. ] u K] u u ] u ] 6 If subsued o Eq. 6, ] ] K] ] Afer premulplyg Eq. 6 by leads o ] ] K] ] Cosderg he orhogoaly codos, ] 0 K] 0 r k r k r k 7

7 Ucoupled Equaos of oo: Udamped all erms excep he h mode ll vash because of he mode-shape orhogoaly properes; hece he resul s Ne symbols ll be defed as follos: So, Eq. 8 ll be, hch s a SDOF equao of moo for mode. he procedure descrbed above ca be used o oba a depede SDOF equao for each mode of vbrao of he udamped srucure. K ] ] K K ] ] K Geeralzed mass Geeralzed sffess for ode Geeralzed load 8 9

8 Ucoupled Equaos of oo: Udamped hs equao may be expressed alerave form: 10 hus, he use of he ormal coordaes serves o rasform he equaos of moo from a se of N smulaeous dffereal equaos, hch are coupled by he off-dagoal erms he mass ad sffess marces, o a se of N depede ormal-coordae equaos. he dyamc respose herefore ca be obaed by solvg separeely for he respose of each ormal modal coordae ad he superposg hese by Eq. 4 o oba he respose he orgal geomerc coordaes. hs procedure s called he mode-superposo mehod.

9 Ucoupled Equaos of oo: Vscous Dampg ode shapes are geerally o orhogoal h respec o he dampg marx C]. I ha case, he equaos of moo of a damped sysem cao be ucoupled. Hoever, f he orhogoaly codos are appled o he dampg coeffces, he orhogoaly properes are applcable o he dampg marx. he equao of moo marx form for a damped sysem s From Eq. 4, If subsued o Eq. 11, ] u C] u K] u 11 u ] u ] u ] ] ] C] ] K] ]

10 Ucoupled Equaos of oo: Vscous Dampg Afer premulplyg Eq. 11 by leads o ] ] C] ] K] ] Cosderg he orhogoaly codos, ] 0 K] 0 m m m 1 causes all compoes excep h mode erm he mass ad sffess expressos. A smlar reduco ll aplly o he dampg expresso f s assumed ha he correspodg orhogoaly codo apples o he dampg marx; ha s, assume ha C] m 0 m

11 Ucoupled Equaos of oo: Vscous Dampg I hs case, Eq. 1 may be re here If Eq. 13 s dvded by he geeralzed mass, hs modal equao of moo may be expressed alerave form: here represes a defo of he modal vscous dampg rao, Jus as each mode has a assocaed frequecy, also has a assocaed dampg rao,. K C 13 C C K K ] ] ] 14 C

12 Respose Aalyss by ode-superposo ehod he oal respose of he DOF sysem ca be obaed by solvg he N ucoupled modal equaos ad superposg her effecs. I he ode-superposo ehod, he Duhamel Iegral or drec umercal egrao ca be used he me doma, hle he Fourer rasform s used he frequecy doma. Sce he dvdual resposes are superposed, hs mehod s oly applcable o lear sysems. 1 Drec Numercal Iegrao Dffere umercal egrao echques ca be used o solve ucoupled modal equaos Eqs. 10 ad 14. Duhamel Iegral he soluo of each equao Eq. 10 or Eq. 14 ca be obaed h he Duhamel egral, hch s expressed as follos for a damped sysem:

13 Respose Aalyss by ode-superposo ehod hch also may be re sadard covoluo egral form, hch, s he u-mpulse respose fuco. If he al codos are dffere from zero, d e D D s 1 0 d h D 0 1 s 1 D D e h Vbrao Forced D D Vbrao Free D D D d e e s 1 s cos 0 15

14 Respose Aalyss by ode-superposo ehod here, K C D 1 Ial codos 0, 0 are deermed from Eqs. 5 he forms, 0 ] u0 0 ] u 0 Havg geeraed he oal respose for each mode, he dsplacemes expressed he geomerc coordaes ca be obaed as, u ] y y... y

15 Respose Aalyss by ode-superposo ehod For he udamped sysem he soluo of Eq. 10 ll be he, cos s d s 17 0 Free Vbrao Forced Vbrao I should be oed ha for mos ypes of loadgs he dsplaceme corbuos geerally are greaes for he loer modes ad ed o decrease for he hgher modes. Cosequely, usually s o ecessary o clude all he hgher modes of vbrao he superposo process.

16 Respose Aalyss by ode-superposo ehod he dsplaceme me-hsores vecor u may be cosdered o be he basc measure of a srucure s overall respose o dyamc loadg. I geeral, oher repose parameers such as sresses or forces developed varous srucural compoes ca be evalaued drecly from he dsplacemes. For example, he elasc forces f S hch ress he deformao of he srucure are gve drecly by Wrg hs equao erms of he modal corbuos, ad subsug, he above equao leads o, K] u K] ] f S f S 3 K] 1 1 K] K] 3... ] K] f S 1 ] 1 1 ] 3 ]

17 Respose Aalyss by ode-superposo ehod rg hs seres marx form gves, f S ] ] here represes a vecor of modal ampludes each mulpled by he equare of s modal frequecy. Because each modal corbuo s mulpled by he square of he modal frequecy, s evde ha he hgher modes are of greaer sgfcace defg he forces he srucure ha hey are he dsplacemes. Cosequely, ll be ecessary o clude more modal compoes o defe he forces o ay desred degree of accuracy ha o defe he dsplacemes. 19