CHAPTER 7 MODAL ANALYSIS

Transcription

1 CHAPER 7 MODAL ANALYSIS 7. Itroducto I Chater we systeatcally defed the equatos of oto for a ult dof (dof) syste ad trasfored to the s doa usg the Lalace trasfor. Chater 3 dscussed frequecy resoses ad udaed ode shaes. Chater 5 dscussed the state sace for of equatos of oto wth arbtrary dag. It also covered the subject of colex odes. Heavly daed structures or structures wth exlct dag eleets, such as dashots, result colex odes ad requre state sace soluto techques usg the orgal couled equatos of oto. Lghtly daed structures are tycally aalyed wth the oral ode ethod, whch s the subject of ths chater. he ablty to th about vbratg systes ters of odal roertes s a very owerful techque that serves oe well both erforg aalyss ad uderstadg test data. he ey to oral ode aalyss s to develo tools whch allow oe to recostruct the overall resose of the syste as a suerosto of the resoses of the dfferet odes of the syste. I aalyss, the odal ethod allows oe to relace the -couled dfferetal equatos wth -ucouled equatos, where each ucouled equato reresets the oto of the syste for that ode of vbrato. If atural frequeces ad ode shaes are avalable for the syste, the t s easy to vsuale the oto of the syste each ode, whch s the frst ste beg able to uderstad how to odfy the syste to chage ts characterstcs. Suarg the odal aalyss ethod of aalyg lear echacal systes ad the beefts derved: ) Solve the udaed egevalue roble, whch detfes the resoat frequeces ad ode shaes (egevalues ad egevectors), useful theselves for uderstadg basc otos of the syste. ) Use the egevectors to ucoule or dagoale the orgal set of couled equatos, allowg the soluto of -ucouled sdof robles stead of solvg a set of -couled equatos.

2 3) Calculate the cotrbuto of each ode to the overall resose. hs also allows oe to reduce the se of the roble by elatg odes that caot be excted ad/or odes that have o oututs at the desred dof s. Also, hgh frequecy odes that have lttle cotrbuto to the syste at lower frequeces ca be elated or aroxately accouted for, further reducg the se of the syste to be aalyed. 4) Wrte the syste atrx, A, by secto. Asseble the ut ad outut atrces, B ad C, usg arorate egevector ters. Frequecy doa ad forced traset resose robles ca be solved at ths ot. If colete egevectors are avalable, tal codto traset robles ca also be solved. For lghtly daed systes, roortoal dag ca be added, whle stll allowg the equatos to be ucouled. 7. Egevalue Proble 7.. Equatos of Moto We wll start by wrtg the udaed hoogeeous (uforced) equatos of oto for the odel Fgure 7.. he we wll defe ad solve the egevalue roble. F F 3 F 3 3 Fgure 7.: Udaed tdof odel. Fro (.5) wth ad c c 0: && + 0 (7.)

3 0 0 && && && (7.) 7.. Prcal (Noral) Mode Defto Sce the syste s coservatve (t has o dag), oral odes of vbrato wll exst. Havg oral odes eas that at certa frequeces all ots the syste wll vbrate at the sae frequecy ad hase,.e., all ots the syste wll reach ther u ad axu dslaceets at the sae ot te. Havg oral odes ca be exressed as (Weaver 990): Where: j t ( ) ω +φ s ω t +φ I(e ) (7.3) vector of dslaceets for all dof s at the th frequecy the th egevector, the ode shae for the th resoat frequecy ω the th egevalue, th resoat frequecy φ a arbtrary tal hase agle For our tdof syste, for the th frequecy, the equato would aear as: s ω t+φ ( ) 3 3, (7.4) where the dces the ter rereset the th dof ad the th ode of the odal atrx Egevalues / Characterstc Equato Sce the equato of oto && + 0 (7.5)

4 ad the for of the oto ( t ) s ω +φ (7.6) are ow, ca be dfferetated twce ad substtuted to the equato of oto: ( t ) && ω s ω +φ (7.7) ( ) ( ) ω s ω t + φ + s ω t +φ 0 (7.8) Cacelg the se ters: ω + (7.9) 0 ω (7.0) Equato (7.0) s the egevalue roble ostadard for, where the stadard for s (Strag 998): A λ (7.) he soluto of the sultaeous equatos whch ae u the stadard for egevalue roble s a vector such that whe s ultled by A, the roduct s a scalar ultle of tself. he ostadard roble s ostadard because the ass atrx falls o the rght-had sde. he for of the atrx resets o roble for had calculatos, but for couter calculatos t s best trasfored to stadard for. Rewrtg the ostadard for egevalue roble as a hoogeeous equato: ( ) ω (7.) 0 A trval soluto, 0, exsts but s of o cosequece. he oly ossblty for a otrval soluto s f the deterat of the coeffcet atrx s ero (Strag 998). Exadg the atrx etres:

5 ω 0 (7.3) Perforg the atrx subtracto: ω 0 ω 0 (7.4) 0 ω Settg the deterat of the coeffcet atrx equal to ero: ω 0 ω 0 0 ω (7.5) he deterat results a olyoal ω, the characterstc equato, where the roots of the olyoal are the egevalues, oles, or resoat frequeces of the syste. ω + ω ω ( ) 3 4 ω ω + ω (7.6a,b) wo of the roots are at the org: Solvg for ω as a quadratc (7.6b) above: ω ω 0 (7.7) ( ) 4 ± ± 3

6 6, 3, (7.8) ω ± ω ± 3 3 (7.9) For each of the three egevalue ars, there exsts a egevector, whch gves the ode shae of the vbrato at that frequecy Egevectors o obta the egevectors of the syste, ay oe of the degrees of freedo, say, s selected as a referece. he, all but oe of the equatos of oto s wrtte wth that value o the rght-had sde: ( ) ω (7.0) 0 ( ω ) 0 ( ω) ( ) ω (7.) Exadg the frst ad secod equatos, drog the subscrts ad : ( ) ( ) ω 0 + ω 0 3 (7.a,b) Rewrtg wth the rato fro (7.a): ter o the rght-had sde ad solvg for the ( / ) ( ) ω (7.3)

7 Solvg for the ( 3 / ) rato fro (7.b): ω (7.4) ( ) ω (7.5) 3 ω ( ) 3 ω 3 ω ( ) (7.6) (7.7) 3 ( ω)( ω) (7.8) ω 3ω + (7.9) 4 3 We ow have the geeral equatos for the egevector values. If a value s chose for, say.0, the the two ratos above ca be solved for corresodg values of ad 3 for each of the three egevalues. Sce at each egevalue there are (+) uows ( ω, ) for a syste wth equatos of oto, the egevectors are oly ow as ratos of dslaceets, ot as absolute agtudes. For the frst ode of our tdof syste the uows are ω,, ad 3 ad we have oly three equatos of oto. Substtutg values for the three egevalues to the geeral egevector rato equatos above, assug, : For ode, ω 0 (7.30)

8 (7.3) 3 ( )( ) (7.3) (7.33) 3 Arbtrarly assgg : (7.34) Rgd-Body Mode, 0 rad/sec Fgure 7.: Mode shae lot for rgd body ode, where all asses ove together wth o stress the coectg srgs. For ode, ω 0 (7.35) 0 (7.36) 3 ( )( 0) (7.37)

9 (7.38) 3 0 (7.39) - Secod Mode, Mddle Mass Statoary, rad/sec Fgure 7.3: Mode shae lot for secod ode, ddle ass statoary ad the two ed asses ove out of hase wth each other wth equal altude. For ode 3, ω (7.40) (7.4) ( )( ) (7.4) (7.43) 3 3 (7.44)

10 - hrd Mode,.73 rad/sec Fgure 7.4: Mode shae lot for thrd ode, wth two ed asses ovg hase wth each other ad out of hase wth the ddle ass, whch s ovg wth twce the altude of the ed asses Iterretg Egevectors For the frst ode, f all the asses start wth ether ero or the sae tal velocty ad wth tal dslaceets of soe scalar ultle of [ ], where s the trasose, the syste wll ether rea at rest or wll cotue ovg at that velocty wth o relatve oto betwee the asses. For the secod ad thrd odes, f the syste s released wth ero tal veloctes but wth tal dslaceets of soe scalar ultle of that egevector, the the syste wll vbrate oly that ode wth all the asses reachg ther u ad axu ots at the sae ot te. Ay other cobato of tal dslaceets wll result a oto whch s a cobato of the three egevectors Modal Matrx Now that the three egevectors have bee defed, the odal atrx wll be troduced. he odal atrx s a (x) atrx wth colus corresodg to the syste egevectors, startg wth the frst ode the frst colu ad so o:

11 ode: 3 3 DOF 3 DOF (7.45) DOF 3 3 For our tdof roble: 0 (7.46) 7.3 Ucoulg the Equatos of Moto At ths ot the syste s well defed ters of atural frequeces ad odes of vbrato. If ay further forato such as traset or frequecy resose s desred, solvg for t would be laborous because the syste equatos are stll couled. For traset resose, the equatos would have to be solved sultaeously usg a uercal tegrato schee uless the roble were sle eough to allow a closed for soluto. o calculate the daed frequecy resose, a colex equato solvg route would have to be used to vert the colex coeffcet atrx at each frequecy. I order to facltate solvg for the traset or frequecy resoses, t s useful to trasfor the -couled secod order dfferetal equatos to - ucouled secod order dfferetal equatos by trasforg fro the hyscal coordate syste to a rcal coordate syste. I lear algebra ters, the trasforato fro hyscal to rcal coordates s ow as a chage of bass. here are ay otos for chage of bass, but we wll show that whe egevectors are used for the trasforato the rcal coordate syste has a hyscal eag; each of the ucouled sdof systes reresets the oto of a secfc ode of vbrato. he -ucouled equatos the rcal coordate syste ca the be solved for the resoses the rcal coordate syste usg well-ow solutos for sgle degree of freedo systes. he -resoses the rcal coordate syste ca the be trasfored bac to the hyscal coordate syste to rovde the actual

12 resose hyscal coordates. hs rocedure s show scheatcally Fgure 7.5. PHYSICAL COORDINAES Couled Equatos of Moto Ital Codtos Forcg Fuctos rasfor PRINCIPAL COORDINAES Ucouled Equatos of Moto Ital Codtos Forcg Fuctos Soluto Bac-rasfor PHYSICAL COORDINAES Soluto Fgure 7.5: Roada for Modal Soluto he rocedure above s aalogous to usg Lalace trasfors for solvg dfferetal equatos, where the dfferetal equato s trasfored to a algebrac equato, solved algebracally, ad bac trasfored to get the soluto of the orgal roble. We ow eed a eas of dagoalg the ass ad stffess atrces, whch wll yeld a set of ucouled equatos. he codto to guaratee dagoalato s the exstece of -learly deedet egevectors, whch s always the case f the ass ad stffess atrces are both syetrc or f there are -dfferet (oreeated) egevalues (Strag 998). Gog bac to the orgal hoogeeous equato of oto: && + 0 (7.47)

13 Havg oral odes eas that at frequecy : s ( ω t +φ ) (7.48) Dfferetatg twce to get accelerato: && ω s ( ω t +φ ) (7.49) Substtutg bac to the equato of oto: { } { } ω s( ω t + φ ) + s( ω t +φ ) 0 (7.50) Cacelg se ters: ω + (7.5) 0 Rearragg ad wrtg the above equato for both the th ad j th odes: ω (7.5) ω (7.53) j j j ad j are the th ad j th egevectors, the th ad j th colus of the odal atrx. Preultlyg (7.5) by the trasose of, : j j ω (7.54) j j ag the trasose of (7.53), where the trasose of a roduct s the roduct of the dvdual trasoses tae reverse order,.e., [ ] AB B A : j ωj j, (7.55) sce ad are syetrcal,, ad : ω (7.56) j j j

14 Postultlyg (7.56) by ω (7.57) j j j Now, subtractg (7.57) fro (7.54): ω j j j j j ( ω ) 0 ( ω ω ) j j (7.58) Whe / j, the ter ( ω ω j ) caot be equal to ero, eag that the ter j ust be equal to ero. j 0 (7.59) Loog at the ses of the atrces ultled: j x x x (7.60) Equato (7.59) ca be rewrtte: ( x) x( x) x( x) ( x) scalar (7.6) 0, (7.6) j j where j s a off-dagoal ter the ass atrx of the rcal coordate syste. he two egevectors j ad are sad to be orthogoal wth resect to, where orthogoalty s defed as the roerty that causes all the off-dagoal ters the rcal ass atrx to be ero. j, ω ω 0. hus the roduct ca be set equal to ay arbtrary costat,, a dagoal ter the rcal ass atrx. Returg to (7.6), for ( j )

15 (7.63) hs s where varous oralato techques for egevectors coe to lay, dscussed the ext secto. he stffess atrx,, s oraled the sae aer. I ractce, stead of dagoalg the ass ad stffess atrces ter by ter by re- ad ostultlyg by dvdual egevectors, the etre odal atrx s used to dagoale oe oerato usg two atrx ultlcatos: (7.64) (7.65) 7.4 Noralg Egevectors Because egevectors are oly ow as ratos of dslaceets, ot as absolute agtudes, we ca choose how to orale the. U to ow, whe calculatg egevectors we have arbtrarly set the altude of the frst dof to. We wll ow dscuss two of the ost cooly used egevector oralato techques. Dfferet oralg techques result dfferet fors of the resultg ucouled dfferetal equatos Noralg wth Resect to Uty Oe ethod s to orale wth resect to uty, ag the largest eleet each egevector equal to uty by dvdg each colu by ts largest value. We ow add the otato, where the refers to a oraled odal atrx (7.66) 0.5 Usg the uty oraled odal atrx to trasfor the ass atrx two atrx ultlcatos:

16 (7.67) (7.68) Slarly trasforg the stffess atrx: (7.69) (7.70) Note that the orgal flled stffess atrx s ow dagoal. Also ote that f the dagoal eleets of the stffess atrx (7.70) are dvded by the corresodg dagoal eleets of the ass atrx (7.69), the three ters are the squares of the resectve egevalues Noralg wth Resect to Mass Aother ethod s to orale wth resect to ass usg the equato:, (7.7).0 ag each dagoal ass ter equal.0. hs s the ethod used by default ANSYS. Oce aga, ote that odal atrx subscrt sgfes the oraled th egevector. Each oraled egevector s defed as follows: q (7.7)

17 Where q s defed as: q j j j (7.73) For a dagoal ass atrx, q ca be slfed sce all the ero: j ters are q (7.74) hus, by oeratg o by,the ass atrx should be trasfored to the detty atrx. Startg wth ad the q values fro above: 0 (7.75) () () () q () ( ) ( ) q () ( ) () q (7.76a,b,c) he odal atrx oraled wth resect to ass becoes: (7.77) Usg to trasfor the ass atrx:

18 (7.78) (7.79) (7.80) he orgal ass atrx has bee trasfored to the detty atrx. Slarly trasforg the stffess atrx:

19 (7.8) (7.8) (7.83)

20 (7.84) Note that the oraled stffess atrx s ow dagoal ad that the dagoal ters are the squares of the corresodg three egevalues. he oraled stffess atrx s also ow as the sectral atrx (Weaver 990). Because oralg wth resect to ass results a detty rcal ass atrx ad squares of the egevalues o the dagoal the rcal stffess atrx, we wll use oly ths oralato the future. Sce we ow the for of the rcal atrces whe oralg wth resect to ass, o ultlyg of odal atrces s actually requred: the hoogeeous rcal equatos of oto ca be wrtte by secto owg oly the egevalues. 7.5 Revewg Equatos of Moto Prcal Coordates Mass Noralato 7.5. Equatos of Moto Physcal Coordate Syste 0 0 && && && [ ] (7.85) Egevalues: ω 0 (7.86) ω ± ω ± 3 3 (7.87a,b) Egevectors, oraled wth resect to ass:

21 3 6 0 (7.88) Equatos of Moto Prcal Coordate Syste && 0 0 && && [ ] (7.89) Exadg Matrx Equatos of Moto Both Coordate Systes Physcal Coordates && + 0 && && Prcal Coordates && 0 && && hese equatos are couled ad have to be solved sultaeously. hese hoogeeous equatos are ucouled ad ca be solved deedetly. able 7.: Suary of equatos of oto hyscal ad rcal coordates.

22 7.6 rasforg Ital Codtos ad Forces Now that we ow how to costruct the hoogeeous ucouled equatos of oto for the syste, we eed to ow how to trasfor tal codtos ad forces to the rcal coordate syste. We ca the solve for traset ad forced resoses the rcal coordate syste usg the ucouled equatos. Startg wth the orgal o-hoogeeous equatos of oto hyscal coordates: Preultlyg by && + F (7.90), the trasose of the odal atrx: && + F (7.9) Isertg the detfy atrx, I : Rewrtg ad regroug ters: && { + { F (7.9) I I && { + {{ F {, (7.93) 3 && F where ad were show to dagoale the ass ad stffess atrces the revous secto. Defg ters: (x) dagoal rcal ass atrx (x) dagoal rcal stffess atrx && && accelerato vector rcal coordates dslaceet vector rcal coordates

23 F F force vector rcal coordates I the revous secto, the deftos for acceleratos ad dslaceets hyscal ad rcal coordates were show to be: && && (7.94) he sae relatoshs hold for tal codtos of dslaceet ad velocty: & o o & o o (7.95) I (7.95), o ad & o are vectors of tal dslaceets ad veloctes, resectvely, the rcal coordate syste, ad o ad & o are vectors of tal dslaceets ad veloctes, resectvely, the hyscal coordate syste. ag the verse of the odal atrx to covert tal codtos requres that the odal atrx be square, wth as ay egevectors as uber of degrees of freedo. We wll see future chaters that there are staces where ot all egevectors are avalable. I oe case, we ay choose to oly calculate egevalues ad egevectors u to a certa frequecy order to save calculato te or because the roble oly requres owledge of resose a certa frequecy rage. I aother case, we ay buld a reduced odel where oly the ost sgfcat odes are retaed. Fortuately, a large ajorty of real lfe robles volve ero tal codtos. 7.7 Suarg Equatos of Moto Both Coordate Systes he two sets of equatos, hyscal ad rcal coordates, are show able 7.:

24 Physcal Coordates Prcal Coordates && + F && + F 3 && + F IC's:,,,&,&,& 3 3 && F && + F 3 && F3 IC's:,,,&,&,& 3 3 able 7.: Suary of equatos of oto hyscal ad rcal coordates. he varables hyscal coordates are the ostos ad veloctes of the asses. he varables rcal coordates are the dslaceets ad veloctes of each ode of vbrato. he equatos rcal coordates ca be easly solved, sce the equatos are ucouled, yeldg the dslaceets. We ow eed to bac trasfor the results the rcal coordate syste to the hyscal coordate syste to get the fal aswer. 7.8 Bac-rasforg fro Prcal to Physcal Coordates We showed revously that the relatosh betwee hyscal ad rcal coordates s: (7.96) Preultlyg by : ( ) 3 I (7.97) (7.98) hus, the dslaceet vector hyscal coordates s obtaed by reultlyg the vector of dslaceets rcal coordates by the oraled odal atrx.

25 Slarly for velocty: & & (7.99) 7.9 Reducg the Model Se Whe Oly Selected Degrees of Freedo are Requred So far we have hted at the fact that oly ortos of the egevector atrx are eeded f selected dof s have forces aled ad other (or the sae) dof s are eeded for outut. hs secto wll show how the reducto dof s occurs. hs reducto s oe of the ey stes to be used later the boo whe we cover how to reduce the se of odels derved fro large fte eleet sulatos. Revewg the stes the odal soluto, startg wth the equatos of oto ad tal codtos hyscal coordates: && + F && + F 3 && + F (7.00) Ital Codtos:,,,&,&,& Solve for egevalues: ω, ω, ω 3 Solve for egevectors, orale wth resect to ass ad for the odal atrx fro colus of egevectors: 3 3 (7.0) rasfor forces fro hyscal to rcal coordates: F F (7.0) Wrte the equatos of oto rcal coordates:

26 && F && +ω F && +ω F IC's:,,,&,&,& (7.03a,b,c,d) Solve the equatos rcal coordates ether te or frequecy doa ad the bac trasfor to hyscal coordates: & & (7.04) Note that the two crtcal trasforatos (assug ero tal codtos) volve reultlyg by the trasose of the odal atrx ( F F ) (7.0) or the odal atrx ( ) (7.04). Let us frst exae the force trasforato by exadg the equatos: F F (7.05) 3 F 3 F 3 3 F F F F F F + F + 3F3 F F 3F F + 3F + 33F3 (7.06) Note that the ultlers of F the frst colu are the eleets of the frst row of the odal atrx, the ultlers of F the secod colu are the eleets of the secod row of the odal atrx ad the ultlers of F 3 the thrd colu are the eleets of the thrd row of the odal atrx. Suose that force s to be aled at oly ass, F, the oly the frst row of the odal atrx s requred to trasfor the force hyscal coordates to the force rcal coordates.

27 Now let us exae the dslaceet trasforato by exadg the equatos: (7.07) (7.08) Note that the coeffcets of the rcal dslaceets the frst row above are the eleets of the frst row of the odal atrx. Slarly, coeffcets of the secod ad thrd rows are the eleets of the secod ad thrd rows of the odal atrx. Suose that the oly hyscal dslaceet we are terested s that of ass,, the oly the secod row of the odal atrx s requred to trasfor the three dslaceets,, 3 rcal coordates to. hs leads to the followg cocluso about reducg the se of the odel: Oly the rows of the odal atrx that corresod to degrees of freedo to whch forces are aled ad/or for whch dslaceets are desred are requred to colete the odel. For ths tdof odel, reducg the se of the roble s ot requred; however, we wll see later that a realstc fte eleet odel, wth hudreds of thousads of degrees of freedo, resets a etrely dfferet roble. Havg the ablty to reduce the roble se s crtcal order to use the detaled results of a colcated fte eleet odel to rovde accurate results a lower order MALAB odel. 7.0 Dag Systes wth Prcal Modes 7.0. Overvew Dag colex bult-u echacal systes s ossble to redct wth the reset state of the art. We wll dscuss ths secto the codtos whch detere f a dag atrx ca be dagoaled, ad the crtero to eable the daed equatos to be dagoaled. I geeral, a arbtrary dag atrx caot be dagoaled by the udaed egevectors, as the

28 ass ad stffess atrces ca. hs leads to usg what s called roortoal dag ost fte eleet sulatos. If a echacal syste s desged wth a secfc vscous dag eleet, for exale a dashot, that doates the sall aout of heret structural dag reset, the that eleet ca be added to the syste as a vscous daer. he resultg syste s lear, but robably does ot exhbt oral odes as dscussed Secto 7... I geeral ths leads to the ablty to dagoale ad ucoule the equatos of oto, requrg a state sace soluto of the orgal, couled equatos of oto. Vscoelastc dag treatets (dag elastoers) have bee used for years ds drves, ost tycally as costraed layer daers o the th sheet etal susesos whch suort the read/wrte head. he effect of ths vscoelastc dag ca be aroxated at a secfc teerature ad frequecy as roortoal dag by usg the odal stra eergy techque assocato wth a fte eleet structural odel (Johso 98). Igorg secfc vscous, coulob, ad vscoelastc dag eleets, dag tycal structures arses fro hysteress losses the aterals as they are straed, soe cases fro vscous losses due to structure/flud teracto but ore ortatly fro relatve oto at the terfaces ad boudares where dfferet arts are attached or grouded. Uless a secfc dag eleet s used a structural desg, ost structures have dag whch vares fro ode to ode ad wll be the rage of 0.05% to % of crtcal dag. he odes ths chater are all real or oral odes as defed earler. Oce aga, havg oral odes eas that at certa frequeces all ots the syste wll vbrate at the sae frequecy ad hase,.e., all ots the syste wll reach ther u ad axu dslaceets at the sae ot te. Chater 5 dscussed colex odes, odes whch all ots the syste do ot reach ther u ad axu dslaceets at the sae ot te Codtos Necessary for Exstece of Prcal Modes Daed Syste Wth a coservatve (o dag) syste, oral odes of vbrato wll exst. I order to have oral odes a daed syste, the ode shaes ust be the sae as for the udaed case, ad the varous arts of the syste ust ass through ther u ad axu ostos at the sae stat te, exressed as:

29 cos ( t ) th ω +φ for the ode (7.09) A suffcet codto for the exstece of daed oral odes s that the dag atrx be a lear cobato of the ass ad stffess atrces. We ow that ad are dagoaled by oeratg o the wth the odal atrx. Whe c s a lear cobato of ad, the the dag atrx c s also ucouled (dagoaled) by the sae re- ad ostultlcato oeratos by the odal atrx as wth the ad atrces (Weaver 990, Crag 98). he daed equatos of oto the becoe: where the dag atrx s a lear cobato of && + c& + F, (7.0) ad : c a+ b (7.) c c, (7.) ad where s the oraled (wth resect to ass) odal atrx. Wrtg out the colete equato: && + c& + F (7.3) { + + {{ {{ { && c & F (7.4) 3 I && c & F Loog at the c to c coverso where c a+ b : c a+ b (7.5) c a + b ai+ b, (7.6) where s a dagoal atrx whose eleets are the squares of the egevalues.

30 he equato for the th ode s: Rewrtg, defg ( ) && + a+ bω & +ω F (7.7) c, the ( a b ) + ω ter, usg otato: c a+ bω ζ ω (7.8) Where ζ s the ercetage of crtcal dag for the th ode, defed as: c c c ζ c ω cr (7.9) he: a+ bω ζ ω (7.0) Rewrtg the equato rcal coordates: && + ζ ω & +ω F (7.) hs tye of dag s ow as roortoal dag, where the dag for each ode (they ca all be dfferet) s roortoal to the crtcal dag for that ode. Sce the dag s also roortoal to velocty, t s of a vscous ature. If the sae dag value s used for all odes, t wll be referred to as ufor dag. Dag whch the dag value for each ode ca be set dvdually wll be referred to as o-ufor dag Dfferet yes of Dag Sle Proortoal Dag Vscous dag each ode s tae to be a arbtrary ercetage, ζ, of crtcal dag:

31 && + ζω & +ω F + ζ & + && F (7.) hs s aalogous to the falar otato used for a sgle degree of freedo syste: && + c& + F c F && + & + (7.3) Defe crtcal dag c cr ad defe the ter ultlyg velocty to be: c ζω c c cr c c (7.4) Rewrtg: F && + ζω & +ω (7.5) Proortoal to Stffess Matrx Relatve Dag Recogg that the hgher odes of vbrato da out qucly, relatve dag yelds dag roorto to frequeces oral odes, bascally lettg the a ter for ζ go to ero: a+ bω bω ζ ω a 0 7.6) If a value of ζ, for the frst ode, s assued, a value ca be defed for b :

32 b ζ, (7.7) ω ad the value for ay other ode s: ω ζ ζ (7.8) ω Proortoal to Mass Matrx Absolute Dag Absolute dag s based o ag b equal to ero, whch case the ercetage of crtcal dag s versely roortoal to the atural frequecy of each ode. hs wll gve decreasg dag for odes as ther frequeces crease. a+ bω a ζ ω ω b 0 (7.9) If a value of ζ, for the frst ode, s assued, a value ca be defed for a : a ζ ω, (7.30) ad the value for ay other ode s: ω ζ ζ (7.3) ω

33 7.0.4 Defg Dag Matrx Whe Proortoal Dag s Assued Fgure 7.6: wo degree of freedo for dag exale. F F c c c 3 A terestg questo to as s what the eleets of the dag atrx should be the two degree of freedo (dof) roble show Fgure 7.6 order to be able to dagoale the equatos of oto. We wll use the egevectors fro the udaed case to orale the dag atrx. he we wll solve for the secfc values of the dvdual daers whch wll allow the dagoalato. We wll see how o-tutve the values of c,c adc 3 are order to be able to dagoale. (See Crag [98] for a geeral exresso to calculate the hyscal dag atrx whe gve roortoal dag values, the orgal ass atrx, the dagoaled ass atrx ad the egevalues ad egevectors.) Solvg for Dag Values Startg wth the udaed egevalues ad egevectors: 0 c + c c 0 c c c + c (7.3) Solve for the dagoaled dag atrx, assug roortoal dag, ad owg that the dagoaled stffess atrx eleets are squares of the egevalues:

34 ω 0 c c ζ ζ 0 ω (7.33) Preultlyg by ( ) ad ostultlyg by ( ) : ( ) c ( ) ζ( ) ( ) 443 I 443 I (7.34) ( ) ( ) c ζ (7.35) Solvg for the verses above, otg that for ths dof syste, the erforg the oeratos o :, ad he verse of a x atrx ca be foud by:. Iterchagg the two dagoal eleets.. Chagg the sgs of the two off-dagoal eleets. 3. Dvdg by the deterat of the orgal atrx. d b a b c a c d a b c d able 7.: Iverse of x atrx. ( ) ( ) (7.36)

35 (7.37) (7.38) c ζ ζ (7.39) Now we ca solve for the secfc values for the three daers: ( ) c ζ 3 ( ) c ζ 3 (7.40) ζ ( ).73

36 3 ( ) c + c c + c ζ + 3 ( ) c c ζ + 3 c 3 ( ) ( ) ζ (7.4) ζ ( ) ζ Suarg: c c3 ζ (7.4) ( ) c ζ.73 (7.43) Note that the values for the three daers are ot at all tutve ad would have bee very dffcult f ossble to guess to be able to costruct a dagoalable dag atrx. If defg the dagoalable dag atrx for ths x roble s dffcult, age tryg to defe t for a real lfe fte eleet roble wth thousads of degrees of freedo. Also, t s hghly robable that the bac-calculated dag values hyscal coordates would atch the actual dag the structure Checg Raylegh For of Dag Matrx We have ow defed the values of the c,c adc 3, daers whch allow dagoalg the equatos of oto. Aother terestg questo s whether the Raylegh for has bee satsfed: Is c a lear cobato of ad? + 3 3? 0 c ζ a + b (7.44) + We have two uows, a ad b, ad essetally two equatos, sce the two dagoal eleets are the sae ad the two off dagoal eleets are the sae. Frst, let us loo at the two off dagoal ters, equatg ters o the two sdes above:

37 ( ) ( ) ( ) ζ 3 a 0 + b ( ) 3 b ζ ζ 3 ( ) (7.45) Now, equatg the dagoal ters: ( ) ζ + 3 a + b a + ζ ( 3 ) (7.46) ( ) a + ζ 3 ( ) ( ) a ζ + 3 ζ 3 ζ (7.47) ξ 3 3 a ζ 3 3 (7.48) Checg the two values for a ad b by substtutg bac to (7.46).

38 ζ ζ ( ) ( ) +ζ 3 ( )( ) ζ + 0 ( 3 3 ) ζ (7.49) So c s a lear cobato of ad ad the Raylegh crtero holds. Probles Note: All the robles refer to the two dof syste show Fgure P.. P7. Set, ad solve for the egevalues ad egevectors of the udaed syste. Norale the egevectors to uty, wrte out the odal atrx ad had lot the ode shaes P7. Norale the egevectors P7. wth resect to ass ad dagoale the ass ad stffess atrces. Idetfy the ters the oraled ass ad stffess atrces. Wrte the hoogeeous equatos of oto hyscal ad rcal coordates. P7.3 Covert the followg ste forcg fucto ad tal codtos hyscal coordates to rcal coordates: a) F, F 3 0,&,,& b) P7.4 Usg the results of P7. ad P7.3, wrte the colete equatos of oto hyscal ad rcal coordates assug roortoal dag.