CEE49b Chapter - Free Vbrato of Mult-Degree-of-Freedom Systems - II We ca obta a approxmate soluto to the fudametal atural frequecy through a approxmate formula developed usg eergy prcples by Lord Raylegh As wth sgle-degree-of-freedom systems, MDOF systems ca also use ths approxmato: ω Q u m u g g Qu mu where u the statc deflecto uder the dead load of the structure Q, actg the drecto of moto, ad g the accelerato due to gravty hus, the frst mode s approxmated shape by the statc deflecto uder dead load For a buldg, ths ca be appled to each of the X ad Y drectos to obta the estmates of the fudametal sway modes Fg a) Deflecto for Raylegh s Formula Appled to Buldgs Lewse for a brdge, by applyg the dead load each of the vertcal ad horzotal drectos, the fudametal lft ad drag modes ca be obtaed he torsoal mode ca also be approxmated by applyg the dead load at the approprate radus of gyrato ad determg the resultg rotato agle Page -
CEE49b Fg b) Deflecto for Raylegh s Formula Appled to Brdges Eve whe performg a detaled dyamc aalyss usg computer software le SAP, ANSYS or ALGOR, a chec usg Raylegh s method s advsable Ofte, for most prelmary desgs, a detaled dyamc aalyss s ot requred ad a frstorder aalyss usg Raylegh s method s all that s requred Geeralzed Coordates Geeralzed coordates are a meas of smplfcato of a mult-degree-of-freedom system to a seres of equvalet sgle-degree-of-freedom systems Exteral Load Dstrbuto P ( z, t) (N/m) a cos ω t y( z, t) a cosω tφ ( z) m(z) Page -
CEE49b Cotuous Structures ca be dealzed mode-by-mode Geeralzed Parameters : EQUIVALEN SDF SYSEM K a cos ω t M P ( t) D Geeralzed Mass: M H m( z) φ ( z) dz Geeralzed Stffess: K D ω ω M D ot geeralzed ω ot geeralzed Geeralzed Force: H P ( t) P( z, t) φ ( z) dz he respose of the actual structure mode s the same as that of ts equvalet SDF system mode whe defed by ts Geeralzed Propertes Stffess, Mass ad Force Page -3
CEE49b Orthogoalty of Modes Orthogoalty of modes s a very mportat relatoshp betwee ay two modes of free vbrato It meas that each mode s truly depedet of aother Mode Mode Recall that we foud that the atural frequeces be determed algebracally: ([ ] [ m] ){ u }, where { u } ω ad correspodg modes ca u( ) u( ) ω the Egevectors or Mode Shapes u ( ) Egevalues or Natural Frequeces Wrtg ths equato for two modes ad, (for example the st ad 3 rd mode): [ m ]{ u } [ ]{ u } ω (-) [ m ]{ u } [ ]{ u } ω (-) Now, traspose equato (-), ad postmultply by { u } ( ω [ m ]{ u }) { u } [ ]{ u } ( ) { u } he, because of the Reversal Law, {[ ][ b] } [ b] [ a] { u } [ m] { u } { u } [ ] { u } ( ) a, the ths s also equal to: ω (-3) Page -4
CEE49b Matrces [ m ] ad [ ] are symmetrc ad so [ m] [ m] ad [ ] [ ] } premultply equato (-) by { u : If we the { u } [ m]{ u } { u } [ ]{ u } ω (-4) We otce ow that the rght had sdes of equatos (-3) ad (-4) are equal ad therefore subtractg equato (-4) from (-3) yelds: ( ω ω ){ u } [ m]{ u } Sce ω ω, the { u } [ m]{ u } for (-5) hs s the Orthogoalty Codto for mode shapes { u } { } ad cludg the mass u matrx he examg equato (-4) usg the orthogoalty codto that results from equato (-5), we see that: { } [ ]{ u } u for hs s the secod Orthogoalty Codto cludg the stffess matrx Equato (-5) whe expaded, s of the form: m { u u u } m u u m u Ad f oe carres out the multplcato, the orthogoalty codto volvg mass s obtaed the form: m u u for (-6) Multplyg Equato (-6) by the atural frequecy ω ad realze that ω mu s the erta force assocated wth mode ad hece ω muu s force x dsplacemet or wor he, equato (-6) suggests that the total wor doe by erta forces of oe mode o dsplacemets of ay other mode vashes Page -5
CEE49b I further cosderatos we wll deote the modal dsplacemets by φ ad all modes lsted as colums of a square matrx [ φ ], φ φ φ3 φ φ φ φ3 φ [ φ ] φ φ φ3 φ mode: st d 3 rd th he modes are Orthogoal or Idepedet We ca exame some stadard trgoometrc fuctos ad ther tegrals for a aalogy to the Orthogoalty Codto cos x cosxdx s x sxdx Itegrals volvg products of harmoc fuctos s x cosxdx he trgoometrc dettes of sums are: a) cos( x + y) cos x cos y s x s y b) cos( x y) cos x cos y + s x s y Addg a) ad b) we obta: ( cos( x y) + cos( x )) cos x cos y + y, so for x y, ( cos( ) x + cos( x) cos x cosx + ) ad for (aalogous to dfferet modes): π π cos x cosxdx s( ) s( ) x + + x ( ) + or, orthogoal ad for (aalogous to the same mode): π π cos xdx (+ cos xdx or, fte Page -6
CEE49b Geeralzato of Orthogoalty Codtos It was foud that betwee two dfferet modes, where Orthogoalty Codto s: ad ω ω that the { } [ m]{ φ } or m φ φ φ (-7) { φ } [ m ]{ φ } m φ M > > Now for, m φ, because m ad φ hus:, the geeralzed mass of the th mode he Geeralzed Mass, recall, s the equvalet mass of mode f treated as a sgle-degree-of-freedom system More geerally, for all modes: [ φ] [ m][ φ] [ M ], the dagoal matrx of Geeralzed Masses (-8) hs ca be verfed by re-wrtg [ φ ] terms of parttoed matrces ad treatg the sub-matrces that are created by ths parttog as elemets f they are coformable, as follows: [ φ] [ m][ φ] symbolzes the followg trple matrx product: { φ} { φ } st mode d mode [ m] [{ φ} { φ} { φ} ] th mode { φ} x x x ad s coformable, e: x { φ} [ m] { φ } [ m] { φ } [ m] [{ φ } { φ } { φ }] M M M x x Page -7
CEE49b he secod Orthogoalty Codto volves the stffess matrx: { } [ ]{ φ }, whe φ, or φ φ (-9) Where { } [ ]{ φ } K, whe φ, or φ K s the Geeralzed Stffess (a x matrx or a scalar quatty) Usg Equato (-9), a relato ca be derved volvg all modes, wrtte as colums [ φ ]: [ φ ] [ ][ φ] [ K ] [ ω ][ M ] (-) ad f we loo at equato (-4), the [ K ] s the Geeralzed Stffess Matrx; ω s the th atural frequecy o prove equato (-), [ φ ] ad [ ] accordg to the modes ad the the matrces multpled: [ ] [ ][ φ] { φ} { φ } φ { φ} { φ} [ ] { φ } [ ] { φ } [ ] [ ] [{ φ } { φ } { φ }] x x x φ ca be parttoed K K M K [{ φ } { φ } { φ }] [ K ] [ ω ][ ] x x Wth respect to equato (-3), wrtte for : K { φ } [ ]{ φ } ω { φ } [ m]{ φ } ω M Page -8