DOF Systems & odal Aalyss
odal Aalyss (hese otes cover sectos from Ch. 0, Dyamcs of Structures, Al Chopra, Pretce Hall, 995). Refereces Dyamcs of Structures, Al K. Chopra, Pretce Hall, New Jersey, ISBN 0-3- 8554-
Procedure to Compute ode Shapes Start wth the equato of moto for a Lear ult-degree Of Freedom (DOF) system, wth base groud exctato: mu& + cu& + ku m& u& g wth tal codtos: u u u & u& (t0) (t0) 3
Upo completo of the forced vbrato phase 0 thereafter), the system cotues to oscllate a Free-Vbrato phase. he correspodg Free Vbrato Equato s (forget about dampg for ow): & u& g m u&& + ku 0 hs system wll oscllate a steady-state harmoc fasho, such that: u&& u ( t) + b cos( t) e.g. u a s gves && u - u 4
substtutg for u& &, we get: ( - m + k) u 0 or ( k - m) u 0 Equato he above equato represets a classc problem ath/physcs, kow as the Ege-value problem. he trval soluto of ths problem s u 0 (.e., othg s happeg, ad the system s at rest). 5
For a o-trval soluto (whch wll allow for computg the deformed shape the system exhbts durg free vbrato), k - m 0 k - λm 0 or where λ For a -DOF system, the above determat calculato wll result a quadratc equato the ukow term. If ths quadratc equato s solved (by had), two roots are foud ( λ ad λ ), whch defe ad (the atural resoat frequeces of ths -DOF system). For a geeral DOF system, atlab or smlar computer program ca be used to solve the determat equato (of order equal to the DOF system, thus defg DOF roots or DOF atural frequeces,,, ). Note: hese resoat (atural) frequeces,, are covetoally ordered lowest to hghest (e.g., 8 radas, 4 radas, ad so forth). DOF 6
Cotug wth the Ege-value problem soluto (aga, atlab does ths, or by had for a -dof system), for each we get a assocated mode shape. o do ths (for each detfed ), go ahead ad substtute ths for Eq. above. Upo ths substtuto, you ca solve for the correspodg vector u, the compoets of whch defes the mode shape.
Notes: ) Whe you substtute ay of the values to Eq., the determat of the matrx ( ) k - m automatcally becomes 0, sce ths s a root of the determat equato (.e., the matrx becomes sgular). he determat beg zero s a ecessary codto for obtag a vector u that s ot equal to zero all t s elemets to satsfy Equato (.e., a soluto other tha the trval soluto of u 0, deotg that o vbrato s happeg as we sad earler). he u vector obtaed whe s substtuted s what we call the mode shape. 8
. Ay mode shape oly defes relatve ampltudes of moto of the dfferet degrees of freedom the DOF system. For stace, f you re solvg a -dof system, you mght ed up wth (whe solvg for the frst mode): u u 0, oly defg a rato betwee ampltudes of u ad u (for stace, f u, the u 0.5, or f you choose u, the u, ad so forth). Geerally, go ahead ad make u ad solve for the other degrees of freedom the vector u, made of : u u u u mdof 9
Propertes of a) odes are orthogoal such that (for r) k r m r 0 (ot ) r 0 b) Also ca be ormalzed such that f m the dvde the elemets of by ad get m.0 c) k K ad f m.0 the k 0
Soluto by mode superposto Example of a -DOF system ( mode shapes ad ) m u m u
ultply by to get: Let kow as geeralzed coordates () or u Φq or q q u u [ ]q Φq Equato of oto: g u& & & & m ku u m + u& g & & & m kφ q mφ + q u& g & & & m Φ kφ Φ mφq Φ + q g u q q q q && && && + m m k k k k m m m m or Φ
Due to the orthogoalty property of mode shapes (see prevous slde), the matrx equato becomes u-coupled ad we get: && q && q + + q q L L && u g && u g or && q && q + + q q L L && u g && u g 3
Note that the orgal coupled matrx Eq. of moto, has ow become a set of u-coupled equatos. You ca solve each oe separately (as a SDOF system), ad compute hstores of q ad q ad ther tme dervatves. o compute the system respose, plug the q vector back to Equato ad get the u vector (ad the same for the tme dervatves to get velocty ad accelerato). he beauty here s that there s o matrx operatos volved, sce the matrx equato of moto has become a set of u-coupled equato, each cludg oly oe geeralzed coordate q. 4
For a dagoal mass matrx: NDOF m j j j NDOF L m j j K j... NDOF L L he terms ad are kow as modal partcpato factors. hese terms cotrol the fluece of & u& g o the modal respose. You may otce that (f both modes are ormalzed to.0 at roof level for example) L L > ( sce ad are of the same sg, whle ad are of opposte sg). herefore, the frst mode s more lkely to play a more promet role overall respose (frequecy cotet of the put groud moto also affects ths ssue). 5
Dampg Now, you ca add ay modal dampg you wsh (whch s aother bg plus, sce you cotrol the dampg each mode dvdually). If you choose ξ 0.0 or 0.05, the equatos become: & q + ξ q& + q L & u g,,, NDOF OK, go ahead ad solve for q (t) the above ucoupled equatos (usg a SDOF-type program), ad the fal soluto s obtaed from: u Φq u & Φq& u & Φq& t u & u&& + & u& g 6
ult-degree-of-freedom (DOF) Respose Spectrum Procedure. Oce you have geeralzed coordates ad ucoupled equatos, use respose spectrum to get maxmum values of respose (r ) max for each mode separately. Calculate expected max respose ( ) usg r rmax ( r ) max root sum square formula where,, N degrees of freedom of terest (maybe frst 4 modes at most) ad r s ay quatty of terest such as u max or SD (ote that summg the maxma from each mode drectly s ofte too coservatve ad s therefore ot popular; because the maxma occur at dfferet tme stats durg the earthquake exctato phase) See Chopra Dyamcs of Structures for mproved formulae to estmate r max. 7
Respose Spectrum odal Resposes ax relatve dsplacemet u or u j (j th floor, th mode) u j L Sd j (S d s S d evaluated at frequecy or perod ) Estmate of maxmum floor dsplacemet u j u j ( umber of modes of terest) axmum Equvalet statc force f or f j (jth floor, th mode) f j L S a m j j 8
herefore, modal base shear V 0 ad momet 0 N V 0 f j j base # of floors 0 N where d j Dstace from floor j to base Estmate of maxmum base shear ad momet j f j d j V 0 V 0 0 0 9
Dampg atrx for DOF System of Equatos (Ch., Chopra, Dyamcs of Structure) Classcal dampg matrx ) Raylegh dampg c a o m ad c a k he stffess proportoal dampg matrx appeals to tuto because t geerates a dampg based o story deformatos, but mass proportoal s eeded as wll be show. 0
I ay modal equato, we have && q + C q& + K q 0 where, K ad C herefore, a o ca be specfed to obta ay desred ξ for ay gve mode such that ξ ξ a 0 or a or 0 ξ ξ a o (e.g. at π radas, ξ.05) fd a 0
hs form of mass proportoal dampg results the tred show below. ξ ξ a o c a m o 3 4
Smlarly we ca fd a such that ξ a ξ or a or ξ a 3
Such stffess proportoal dampg dsplays the tred show below wth dampg creasg learly wth frequecy (ot so good physcally). ξ c ak ξ a 3 4
Physcally, we ofte observe ( frst approxmato) a early equal value of dampg for the frst few modes of structural respose (e.g., frst 5 modes), ad we wat to model that. herefore, we use: c a m + a k 0 ad defe dampg ratos ξ ad ξ j two modes ad get the dampg show by the combed curve, whch s somewhat uform wth the rage of terest (say Hz to 7 Hz or π to 4π radas). 5
ξ Combed Rage of Iterest Stffess early uform dampg ass j for example 4 th mode for example 6
herefore, (sce) or ad for ay two modes ad j, we get 7 0 a a ξ + K a a ξ o + 0 j j ξ ξ a a solve to get a 0 ad a ad costruct your dampg matrx (whch s a lear combato of mass ad stffess matrces)
Notes ) For a choce of ξ ξ ξ sgle value the two modes, we get, a 0 ξ j + j a ξ + j ) Classcal dampg ad s attractve because of combato of mass ad stffess, allowg the odampg free-vbrato mode shapes to u-couple the matrx equato of moto. 8
) Caughey dampg he above procedure was geeralzed by Caughey to allow for more cotrol over dampg the specfed modes of terest (.e. to be able to specfy ξ for more tha modes ad j) I ths geeralzato, you ca stay wth the scope of classcal dampg by usg c m N 0 a [ ] m k to fd a coeffcets to match ξ modal dampg ratos (see Ch., Chopra, Dyamcs of Structures) 9
Dsadvatages:. c ca become a full matrx stead of beg a baded matrx (f m ad k are baded) as wth c a 0 m + a k. You must check to esure that you do t ed up wth a egatve ξ some mode where you have ot specfcally specfed dampg (because dampg varato wth frequecy mght dsplay sharp oscllatos). I summary, c a 0 m + a k s the usual choce at preset despte the lmtatos dscussed above. 30