Egeerg Vbrato. Itroducto he study of the moto of physcal systems resultg from the appled forces s referred to as dyamcs. Oe type of dyamcs of physcal systems s vbrato, whch the system oscllates about certa equlbrum postos. hs moto s redered possble by the ablty of materals the system to store potetal eergy by ther elastc propertes. Most physcal systems are cotuous tme ad ther parameters are dstrbuted. I may cases the dstrbuted parameters ca be replaced by dscrete oes by sutable lumpg of the cotuous system. hs lumped parameter systems are descrbed by ordary dfferetal equatos, whch are easer to solve tha the partal dfferetal equatos descrbg cotuous systems. he umber of degrees of freedom specfyg the umber of depedet coordates ecessary to defe the system ca be establshed. Oscllatory systems ca be classfed to two groups accordg to ther behavor: lear ad o-lear systems. For a lear system, the prcple of superposto apples, ad the depedet varables the dfferetal equatos descrbg the system appear to the frst power oly, ad also wthout ther cross products. Although oly lear systems are dealt wth aalytcally, some kowledge of o-lear systems s desrable, sce all systems ted to become o-lear wth creasg ampltudes of oscllato. A physcal system geerally exhbts two classes of vbrato: free ad forced. Free vbrato takes place whe a system oscllates uder the acto of forces heret the system tself, ad whe the exteral forces are abset. It s descrbed by the soluto of dfferetal equatos wth ther rght had sdes set to zero. he system whe gve a tal dsturbace wll vbrate at oe or more of ts atural frequeces, whch are propertes of the dyamcal system determed by ts mass ad stffess dstrbuto. he resultg moto wll be the sum of the prcpal modes some proporto, ad wll cotue the absece of dampg. hus the mathematcal study of free vbrato yelds formato about the dyamc propertes of the system, relevat for evaluatg the respose of the system uder forced vbrato. Forced vbrato takes place whe a system oscllates uder the acto of exteral forces. Whe the exctato force s oscllatory, the system s forced to vbrate at the exctato frequecy. If the frequecy of exctato cocdes wth oe of the atural frequeces, resoace s ecoutered, a pheomeo whch the ampltude bulds up to dagerously hgh levels, lmted oly by the dampg.
All physcal systems are subject to oe or other type of dampg, sce eergy s dsspated through frcto ad other resstaces. hese resstaces appear varous forms, after whch they are amed: vscous, hysteretc, Coulomb, ad aerodyamc. he propertes of the dampg mechasms dffer from each other, ad ot all of them are equally ameable to mathematcal formulato. Fortuately, small amouts of dampg have very lttle fluece o the atural frequeces, whch are therefore ormally calculated assumg o dampg.
. SINGLE DEGREE OF FREEDOM SYSEM. Vscous Dampg A complex structure ca be cosdered as a umber of masses, tercoected by sprgs ad dampg elemets to facltate the aalytcal soluto of the dyamc behavour of the structure. Sce the dampg forces a real structure caot be estmated wth the same accuracy as the elastc ad erta forces, a rgorous mathematcal smulato of the dampg effects s futle. Nevertheless, to accout for the dsspatve forces the structure, assumptos of the form of dampg have to be made, that gve as good as possble a estmate of the dampg forces practce. Furthermore, the form has to be coducve to easy mathematcal mapulato, specfcally adaptable to lear equatos of moto, mplyg that the dampg forces are harmoc whe the exctato s harmoc. wo such sutable forms of dampg are vscous ad hysteretc. he respose of a sgle degree of freedom system to vscous dampg wll be descrbed ths secto ad to hysteretc dampg the ext secto. x(t) k m F(t) c Fg.. Sgle degree of freedom system. Fg. shows a sgle degree of freedom system, where a mass-less dashpot of dampg coeffcet c ad a sprg of stffess k are mouted betwee the mass m ad the fxed wall. he dashpot exerts a dampg force stataeous velocty ad s postve the postve drecto of moto for forced harmoc exctato may be wrtte as where x s the dsplacemet x s the velocty x s the accelerato F s the exctato force 3 cx whch s proportoal to the x. he equato of j t mx cx kx Fe ()
j s ad s the exctato frequecy. Dvdg equato () by m oe obta F x x x e k jt () where k udamped atural frequecy, m c c dmesoless dampg rato, m cc cc s the crtcal dampg. Cosder the soluto of the form j t x Xe where X s the ampltude of the steady state vbrato, t ca be show that X F / k F / k j / j / (3) hus jt Fe x Xe / j / k he dsplacemet x s proportoal to the appled force, the proportoalty factor beg H / j / whch s kow as the complex frequecy respose fucto (FRF). Equato (4) llustrates that the dsplacemet s a complex quatty wth real ad magary parts. hus jt (4) (5) 4
x j / / / / / / Fe k jt (6) hs shows that the dsplacemet has oe real compoet Re x / / / Fe k jt (7) whch s -phase wth the appled force ad aother compoet Im x / / / Fe k jt (8) whch has a phase lag of 90 behd the appled force. he ampltude of the total dsplacemet s gve by Re Fe k / / X Im X he total dsplacemet lags behd the force vector by a agle gve by x ta Im / Re x,.e. jt (9) / ta / (0) he steady state soluto of Eq.() ca therefore also be wrtte the form x where s gve by Eq. (0). / / Fe k j( t) () he quatty the square brackets of Eq. () s the absolute value of the complex frequecy respose, H, ad t s called the magfcato factor, a 5
dmesoless rato betwee the ampltude of dsplacemet X ad the statc dsplacemet F/k. 0 0. 0. 0.3 0.5 (a) o 80 0 o 50 0. 0. o 0 0.3 0.5 o 90 o 60 o 30 o 0 (b) Fg. (a) Magfcato factor H as a fucto of the dmesoless frequecy rato / for varous dampg rato, ad (b) phase lag of dsplacemet behd force as a fucto of. 6
Fg. (a) shows the absolute value of the complex frequecy respose fucto H as a fucto of the dmesoless frequecy rato / for varous dampg rato. It ca be see that creasg dampg rato teds to dmsh the ampltudes ad to shft the peaks to the left of the vertcal through /. he peaks occur at frequeces gve by () where the peak value of H s gve by H (3) For lght dampg ( < 5%) the curves are early symmetrc about the vertcal through /. he peak value of H occurs the mmedate vcty of / ad s gve by H Q (4) where Q s kow as the qualty factor. Fg. (b) shows the phase agle agast / for varous plotted from Eq. (0). It should be oted that all curves pass through the pot 90 ( / ), / ;.e., o matter what the dampg s, the phase agle betwee force ad dsplacemet at the udamped atural frequecy / s 90. Moreover, the phase agle teds to zero for / 0 ad to 80 for /. Cosder the system wth =0% for example, Fg.3.(a) defes the pots P ad P where the 7
o Xk/F (db) 0 j 5 0 3dB Peak P P d 5 0-5 -0 0 0. 0.4 0.6 0.8..4.6.8 / (a) (b) Fg.3 (a) Magfcato factor H as a fucto of the dmesoless frequecy rato / for 0%, ad (b) complex s-plae of the poles. ampltude of H reduces to Q / of ts peak value are called the half power pots. If the ordate s plotted o a logarthmc scale, P ad P are pots where the ampltude of H reduces by 3 db ad are thus called the -3 db pots. he dfferece the frequeces of pots P ad P s called the 3 db badwdth of the system, ad for lght dampg t ca be show that (5) where s the 3 db badwdth s the frequecy at pot P s the frequecy at pot P From Eqs. (4) ad (5) where s called the Loss Factor. c (6) c Q c o exame the varato of the -phase Re(x) ad quadrature Im(x) compoets of dsplacemet, Eqs. (7) ad (8) are plotted as a fucto of Fgs. 4(a) ad 4(b) respectvely. he curves of the real compoet of dsplacemet Fg. 4(a) have a 8
zero value at / depedet of dampg rato, ad exhbts a peak ad a otch at frequeces (7) As the dampg decreases, the peak ad the otch crease value ad become closer together. I the lmt whe 0, the curve has a asymptote at /. he frequeces ad are ofte used to determe the dampg of the system from the equato / / (8) he curves of the magary compoet of dsplacemet have a otch close vcty of / ad they are sharper tha those of H Fg. (a) for correspodg values of. 0.0 0. 0.3 (a) 9
0.3 0. 0.0 (b) Fg.4.(a) Real compoet of dsplacemet as a fucto of the dmesoless frequecy rato for varous values of, ad (b) magary compoet of dsplacemet agast for varous values of. 0
. Hysteretc (structural) Dampg Aother type of dampg whch permts settg up of lear dampg equato, ad whch may ofte gve a closer approxmato to the dampg process practce, s the hysteretc dampg, sometmes called structural dampg. A large varety of materals, whe subjected to cyclc stress (for stras below the elastc lmt), exhbt a stress-stra relatoshp whch s characterzed by a hysteress loop. he eergy dsspated per cycle due to teral frcto the materal s proportoal to the area wth the hysteress loop, ad hece the ame hysteretc dampg. It has bee foud that the teral frcto s depedet of the rate of stra (depedet of frequecy) ad over a sgfcat frequecy rage s proportoal to the dsplacemet. hus the dampg force s proportoal to the elastc force but, sce eergy s dsspated, t must be phase wth the velocty ( quadrature wth dsplacemet). hus for smple harmoc moto the dampg force s gve by x j kx k (9) where s called the structural dampg factor. he equato of moto for a sgle degree of freedom system wth structural dampg ca thus be wrtte k mx x kx Fe j t (0) where s called the complex stffess. he steady state soluto of Eq. () s gve by j t mx k j Fe () jt Fe x Xe / k j correspodg to Eq. (4) for vscous dampg. he real ad the magary compoets of the dsplacemet ca be obtaed: x / j / / jt Fe k jt () (3)
hus Re x / / Fe k jt (4) ad Im x / Fe k jt (5) he total dsplacemet s gve by x / Fe k jt (6) whch lags behd the force vector by a agle gve by ta / (7) 0 0. 0.5 (a)
o 80 o 50 o 0 0 0. 0.5 o 90 o 60 o 30 o 0 (b) Fg.5 (a) Magfcato factor as a fucto of / for varous values of the structural dampg factor, ad (b) phase lag of dsplacemet behd force as a fucto of / for varous values of. he term the square brackets of expresso (6) (magfcato factor) ad are plotted agast / for varous values of Fgs.5(a) ad (b) respectvely. he curves of Fgs.5(a) ad (b) ca be see to be smlar to those of Fgs.(a) ad (b) respectvely for vscous dampg; however, there are some mor dffereces. For hysteretc dampg t ca be see from Fg.5(a) that the maxmum respose occurs exactly at / depedet of dampg. At very low values of / the respose for hysteretc dampg depeds o ad the phase agle teds to ta whereas t s zero for vscous dampg. 3
.3 Complex Frequecy Soluto k x(t) m F(t) c It has bee show that the steady state soluto to Eq.() for forced vbrato of j t such a system was gve by x Xe. I the case of free damped vbrato, for whch the equato of moto s gve by mx cx kx 0 (3.) the above soluto ca be exteded to a more geeral type of the form x where s j s kow as the complex frequecy. st Xe (3.) Before solvg Eq.(3.) t s mportat to exame the varable s. Sce s s complex t s best llustrated the complex frequecy plae, where the real axs represets s the decay rate (amout of dampg), ad the magary axs represets j, the frequecy. Every pot ths plae defes a partcular form of oscllato. For postve the magtude creases expoetally wth tme, ad for egatve t decreases expoetally wth tme, Values of s whch are complex, thus lyg j t aywhere the plae, gvg solutos of the form x Xe descrbe oscllatory moto creasg or decreasg expoetally wth tme depedg o the sg of. he atcpated soluto (3.) s serted (3.), leadg to the algebrac equato s m c s 0 (3.3) k k whch s kow as the characterstc equato of the system. I the absece of dampg the atural frequecy of the system s gve by k/ m whch may be substtuted Eq. (3.3) to yeld 4
s c s 0 km (3.4) or where s s 0 c c c km s the dampg rato. c Sce Eq. (3.5) s a quadratc, the two roots of the equato are gve by s, (3.5) (3.6) ad the values of s obtaed, thus defe two forms of oscllato the complex plae. Obvously the ature of the roots s ad s depeds o the value of.he effect of the varato of o the roots ca be llustrated the complex plae (s-plae) the form of a dagram represetg the locus of roots plotted as a fucto of the parameter. From Eq. (3.6) t ca be see, that for 0 the roots are gve by s j, whch le o the magary axs, correspodg to udamped oscllatos, at the atural frequecy of the system. For 0 or c km whch s kow as the uderdamped case, the roots are gve by s, j (3.7) hus s ad s are always complex cojugate pars, located symmetrcally wth respect to the real axs. As the values of s / are complex, the magtude s (3.8) dcatg that the locus of the roots s a crcle of radus, cetered at the org. Furthermore, whe the complex cojugate pars of the roots are assocated wth each other, they ca be terpreted as a real oscllato, for example 5
jt jt t e e e cost (3.9) As approaches uty, c km the two roots approach the pot o the real axs, whch s kow as the crtcally damped case. For or c km kow as the overdamped case, the roots are gve by s (3.0), whch le o the egatve real axs. As, s 0ad s. o obta the tme respose of free vbrato for gve tal codtos, Eq.(3.6) s substtuted Eq.(3.) to gve st x t X e X e s t (3.) For the overdamped case, the soluto s gve by t x t X e X e t t t.e. xt Xe X e e (3.) whch descrbes aperodc moto decayg expoetally wth tme. X ad X are determed from the tal dsplacemet ad velocty whch tur gover the shape of the decayg curve. For the crtcally damped case, Eq. (3.5) has a double root s s ad the soluto s gve by x t X tx e t t (3.3) he respose aga represets aperodc moto decayg expoetally wth tme. However, a crtcally damped system approaches the equlbrum posto the fastest for gve tal codtos. For the uderdamped case 0 the soluto s gve by j t j t xt Xe X e e t 6
x t X e X e e.e. jd t jd t t (3.4) where d s called the frequecy of damped free vbrato. Eq.(3.4) represets expoetally decayg oscllatory moto wth costat frequecy d. 7
3. MULI DEGREE OF FREEDOM SYSEMS 3. FREE VIBRAION he umber of degrees of freedom chose dctates the umber of dfferetal equatos ecessary to characterze the system. As these equatos are ormally coupled to each other, they must be decoupled before ther soluto s attempted. o do ths, the orthogoal propertes of the Prcpal Modes are exploted, eablg the orgal dfferetal equatos to be rewrtte terms of the prcpal coordates. However, calculatg the respose uder forced vbrato, ot oly s t ecessary to make assumptos about the type of dampg, but also the dstrbuto of dampg--proportoal or o-proportoal. hs s because the latter type of dstrbuto, geerally ecoutered complex structures, s cosderably more dffcult to resolve mathematcally, gvg what s called complex modes, cotrast to real modes obtaed wth proportoal dampg. F F k k k 3 c c c3 x x 3.. Natural Frequecy ad Vbrato Mode he equatos of moto of the system show Fg.9 are m x c c x cx k k x k x F m x c x c c x k x k k x F 3 3 (3.) whch ca be wrtte matrx form as m 0 x c c c x k k k x F 0 m x c c c 3 x k k k 3 x F (3.) o determe the atural frequeces ad atural mode shapes of the system, the udamped free vbrato of the system s frst cosdered. hus the equatos reduce to 8
mx kx 0 (3.3) where m m 0 0 m ad k k k k k k k 3 Assumg harmoc moto { } { } j x v e t wth Eq. (3.3) becomes mx kx 0 or m k v ( ) 0 (3.4) Premultplyg Eq. (4.4) by m ad rearragg we obta m k I v 0 (3.5) where m k s called a dyamc matrx ad m m I s a ut matrx. Eq. (3.5) s a set of smultaeous algebrac equatos v. From the theory of equatos t s kow, that for a o-trval soluto {v} = 0, the determat of the coeffcets of Eq. (3.5) must be zero. hus m k I 0 (3.6) whch s kow as the characterstc equato of the system. Eq. (3.6) whe expaded ca be rewrtte as (3.7) a a... a 0 whch s a polyomal for a -degree-of-freedom system. he roots of the characterstc equato are called egevalues ad the udamped atural frequeces of the system are determed from the relatoshp (3.8) 9
By substtutg to matrx equato (3.5) we obta the correspodg atural (or Prcpal) mode shape v whch s also called a egevector. he mode shape represet a deformato patter of the structure for the correspodg atural frequecy. As Eq. (3.5) are homogeeous, there s ot a uque soluto for the{ v} s. hus the atural mode shape s defed by the rato of the ampltudes of moto at the varous pots o the structure whe excted at ts atural frequecy. he actual ampltude o the other had depeds o the tal codtos ad the posto ad magtude of the exctg forces. Cosder a umercal example for the system show Fg.9 where Substtutg Eq. (3.) we get m 5 kg; m 0 kg; k k N / m; k 4 N / m 3 5 0 x 4 x 0 0 0 x 6 x 0 (3.9) hus Eq. (3.5) becomes 5 0 4 0v 0 0 0 6 0 v 0 4 / 5 / 5 v 0 / 5 3/ 5 v 0 (3.0) For a o-trval soluto the determat of the above equato must equal zero, thus the characterstc equato 7 0 5 5 he roots of the above equato are, / 5 ad hus / 5 ad ad the two atural frequeces are gve by 0
/ 5 ad Substtuto of ad Eq. (3.0) wll gve the two atural mode shapes. hus the mode shape for the atural frequecy s v v v where v s arbtrary. Smlarly the mode shape for the atural frequecy s v For = rad/s the masses move - phase 5 v v / For a arbtrary deflecto of v v the two mode shapes would be v ad v / for / 5 for For = rad/s the masses move out -of - phase -- Fg. 0. Frequecy ad mode shapes for the two-degree-of-freedom system. he system ca thus vbrate freely wth smple harmoc moto whe started the correct way at oe of two possble frequeces as show Fg.0. Note that the masses move ether phase or 80 out of phase wth each other. Sce the masses reach ther maxmum dsplacemets smultaeously, the odal pots are clearly defed.
3.. Orthogoal Propertes of Egevectors It was show the prevous secto that soluto of Eq. (3.4) yelds egevalues ad correspodg egevectors. hus a partcular elgevalue ad the egevector v wll satsfy Eq. (3.4);.e, kv mv (3.) Premultply Eq. (3.) by the traspose of aother mode shape v j,.e., v j kv v j mv, where the superscrpt deotes a traspose matrx. We ow wrte the equato for the th mode,.e., th j mode ad premultply by the traspose of the v kv j j v mv j (3.) Ask adm are symmetrc matrces v j kv v kv j ad v j mv v mv j herefore subtractg Eq. (3.3) from Eq. (3.) we obta 0 (3.3) (3.4) j v mv j (3.5) (mplyg two dfferet atural frequeces) If j ad from Eq. (3.3) t ca be see that 0 v m v (3.6) 0 j v k v (3.7) j
Equatos (3.6) ad (3.7) defe the orthogoalty propertes of the mode shapes wth respect to the system mass ad stffess matrces respectvely. 3..3 Geeralzed Mass ad Geeralzed Stffess It ca be see that f j Eq. (3.5) the the two modes are ot ecessarly orthogoal ad Eq. (3.6) s equal to some scalar costat other tha zero, e.g. m ad from Eq. (3.3) t follows that m ad,,3... v m v m (3.8) v k v m m k,,3... (3.9) k are called the geeralzed mass ad geeralzed stffess respectvely. he umercal values of the mode shapes calculated above wll be used to determe the geeralzed mass ad geeralzed stffess. he mode shapes were foud to be v for / 5 ad v / for Substtutg v Eq. (3.8) we obta the geeralzed mass m for the frst mode: 5 m.smlarly substtutg v Eq. (3.8) we obta m 5/. hus the geeralzed masses m ad m for the frst ad secod modes are 5 ad 5/ respectvely. he geeralzed stffesses k ad k for the frst ad secod modes are k m ad 6 k m. /5 3..4 Normal Mode If oe of the elemets of the egevector v s assged a certa value, the rest of the elemets are also fxed because the rato betwee ay two elemets s costat. hus the egevector becomes uque a absolute sese. hs process of adjustg the elemets of the atural modes to make ther ampltude uque s called ormalzato, ad the resultg scaled atural modes are called orthoormal modes, or ormal modes. here are several ways to ormalze the mode shapes, ad the 3
commo practce egeerg s that the mode shapes ca be ormalzed such that the geeralzed mass or modal mass m Eq. (3.8) s set to uty. hs method has the advatage that Eq. (3.9) yelds drectly the egevalues frequeces. ad thus the atural he ormalzato wll be llustrated by the umercal example of the system show Fg.9. he mode shapes were show to be v v v for / 5 ad v v v / for herefore substtuto of v Eq. (3.8) yelds v 5 0v m v 0 0 v Ad the ormalzed mode shape for / 5 s v v /5 v /5 (3.0) Smlarly for v we get ormalzed mode shape for s v 5 0 v v / 0 0 v / m,ad the v v /5 v / / 5 / (3.) It ca ow be see that these ormalzed mode shapes could also have bee obtaed by dvdg the atural modes by the square root of ther respectve geeralzed masses calculated the prevous secto,.e., for ormalzato of the frst atural mode 4
ad for the secod mode v /5 5 /5 m v /5 / 5/ / /5 / m whch are the same as calculated Eq. (3.0) ad (3.) respectvely. 3. Forced Vbrato he equatos of moto of the two-degree-of-freedom system show Fg. 9 wthout dampg ca be wrtte as mx mx kk x kx k x F k k x F 3 (3.) or matrx form as m 0 x k k k x F 0 m x k k k3 x F (3.3) I solvg the above equatos for the respose {x} for a partcular set of exctg forces, the major obstacle ecoutered s the couplg betwee the equatos;.e., both coordates x ad x occur each of the Eq. (3.). I Eq. (3.3) the couplg s see by the fact that whle the stffess matrx s symmetrc, t s ot dagoal (.e, the off-dagoal terms are o-zero). hs type of couplg s called elastc couplg or statc couplg (o-dagoal stffess matrx) ad occurs for a lumped mass system, f the coordates chose are at each mass pot. If the equatos of moto had bee wrtte terms of the extesos of each sprg, the stffess matrx would have bee dagoal but ot the mass matrx. hs kd of couplg s termed ertal couplg or dyamc couplg (o-dagoal mass matrx). It s thus see that the way whch the equatos are coupled depeds o the choce of coordates. If the system of equatos could be ucoupled, so that we obtaed dagoal mass ad stffess matrces, the each equato would be smlar to that of a sgle degree of freedom system, ad could be solved depedet of each other. Ideed, the process of dervg the system respose by trasformg the equatos of moto to a depedet set of equatos s kow as modal aalyss. 5
hus the coordate trasformato we are seekg, s oe that decouples the system ertally ad elastcally smultaeously, ad therefore yelds us dagoal mass ad stffess matrces. It s here that the orthogoal propertes of the mode shapes dscussed above come to use. It was show by Eq. (3.8), that f the mass or the stffess matrx s post ad pre-multpled by a mode shape ad ts traspose respectvely, the result s some scalar costat. hus wth the use of a matrx v whose colums are the mode shape vectors, we already have at our dsposal the ecessary coordate trasformato. he x coordates are trasformed to by the equato x v p (3.4) where v v v v v v v v v v... (3.5) v s referred to as the modal matrx ad p s called prcpal coordates, ormal coordates or modal coordates. Eq. (3.3) ca be wrtte as ad substtutg Eq. (3.4) to (3.6) yelds mx kx F (3.6) m p kvp F (3.7) Pre-multplyg Eq. (3.6) by the traspose of the modal matrx,.e, v, we obta v mvp v kvp v F (3.8) I Eq. (3.8) the mass matrx was post ad pre-multpled by oe mode shape ad ts traspose, gvg a scalar quatty, whle Eq. (3.8) the mass matrx s post ad premultpled by all the mode shapes ad ther traspose. hus the product s a matrx 6
M whose dagoal elemets are some costats whle all the off-dagoal terms are zero,.e. v m v m (3.9) Smlarly m k (3.30) where m ad k are dagoal matrces. Hece Eq. (3.8) ca be wrtte as Eq. (3.3) represets -equatos of the form where v s the m p k p v F (3.3) m p k p v F f (3.3) th colum of the modal matrx,.e., the th mode shape. m ad k ca be recogsed as the th modal mass (geeralzed mass) ad th modal stffess (geeralzed stffess) respectvely. Eq. (3.3) s the equato of moto for sgle degree of freedom systems show Fg.. m f k Fg.. Sgle degree of freedom system defed by Eq. (3.3). Sce k m, Eq. (3.3) ca be wrtte as 7
v F f p p (3.33) m v m v Oce the soluto (tme resposes) of Eq. (3.3) for all p s obtaed, the soluto terms of the orgal coordates x ca be obtaed by trasformg back,.e. substtutg for Eq. (3.4) x v p. It should be oted that whe the modal matrx v of Eq. (3.5) s made up of colums of the ormalzed mode shapes (such that m ). Eq. (3.33) would be smplfed to p p f v F (3.34) hus the modal mass would be uty ad the modal stffess would be the square of the atural frequecy of the th mode. Let us cosder our umercal example of the system of Fg.9 wth forcg fuctos F ad F. hus Eq. (4.9) becomes 5 0 x 4 x F 0 0 x 6 x F (3.35) he atural frequeces ad atural mode shapes were / 5 v v / hus the modal matrx v usg atural mode shapes v / he x coordates are trasformed by the equato 8
x v p (3.36).e. x p x / p (3.37) Substtutg Eq. (4.4) to Eq. (4.40) ad pre-multplyg by v gves 5 0 p 4 p F 0 0 p 6 p F v v v v v he products v mv ad v k v are calculated to be v 5 0 5 0 mv / 0 0 / 0 5/ 4 6 0 / 6 / 0 5/ v kv Substtutg these products to the equato above we obta 5 0 p 6 0 p F 0 5/ p 0 5/ p / F hus the equatos of moto are 5p 6 p F F p p F F 5/ 5/ / (3.38) he orgal set of equatos (3.38) are show to be ucoupled; other words the two degree of freedom system s broke dow to two sgle degree of freedom systems show Fg.. 9
F F 5 5 0 4 x x 5 F F 5/ F F / 6 5/ Fg.. he udamped two-degree-of-freedom system show Fg. 9, s trasformed to two sgle degree of freedom systems. As metoed above, the modal matrx ca also be made up of colums of the ormalzed mode shapes (such that m ). Usg the ormalzed mode shapes from Eqs. (3.0) ad (3.), the ormalzed modal matrx v s gve by v /5 /5 /5 /5 / hus the products v mv ad v k v are gve by /5 /5 5 0 /5 /5 0 v m v /5 /5 0/ 0 /5 /50 / /5 /5 4 /5 /5 / 5 0 v k v /5 /5 / 6 /5 /50 / It ca thus be see that, by usg the ormalzed modal matrx for coordate trasformato, the mass matrx becomes a ut matrx ad the stffess matrx s dagoalzed wth the dagoal terms equal to the egevalues (the square of the atural frequeces). 30
I geeral v mv ad v kv where 0 0... 0 0 0... 0 0 0... Oce the tme resposes for p ad p have bee determed from Eq. (3.38), they ca be substtuted Eq. (3.37) to gve the tme respose terms of the orgal coordates x, thus x t p t p t x t p t p t (3.39) Eq. (3.39) fact llustrates a very mportat prcple vbrato, amely that ay possble free moto ca be wrtte as the sum of the moto each prcpal mode some proporto ad relatve phase. I geeral for a -degree-of-freedom system x v v v x v v v p cos t p cos t... p cos t x v v v (3.40) If the two-degree-of-freedom system dscussed above s gve arbtrary startg codtos, the resultg moto would be the sum of the two prcpal modes some proporto ad would look as show Fg.3. 3
Fg. 3. Respose of the two-degree-of-freedom system whe gve arbtrary startg codtos. 3.3 Proportoal Dampg he assumpto that systems have o dampg s oly hypothetcal, sce all structures have teral dampg. As there are several types of dampg, vscous, hysteretc, coulomb, aerodyamc etc., t s geerally dffcult to ascerta whch type of dampg s represeted a partcular structure. I fact a structure may have dampg characterstcs resultg from a combato of all the types. I may cases, however, the dampg s small ad certa smplfyg assumptos ca be made. 3.3. Vscous Dampg he equatos of moto of the two degree of freedom system wth dampg, show Fg. 9, are gve by Eq. (3.) m 0 x c c c x k k k x F 0 m x c c c 3 x k k k 3 x F (3.4) I short form they ca be wrtte as 3
mx cx kx F (3.4) wo assumptos are take for grated before attemptg soluto of these equatos. Frstly, that the type of dampg s vscous, ad secodly that the dstrbuto of dampg s proportoal. By proportoal dampg t s mpled that the dampg matrx [c] s proportoal to the stffess matrx or the mass matrx, or to some lear combato of these two matrces. Mathematcally t meas that ether c m or c k or c m k (3.43) where ad are costats. Because of the assumpto of proportoal dampg, the coordate trasformato usg the modal matrx for the free udamped case whch dagoalzes the mass ad stffess matrces, wll also dagoalze the dampg matrx. hus the coupled equatos of moto for a proportoally damped system ca also be ucoupled to sgle degree of freedom systems as show the followg. Substtutg the coordate trasformato of Eq. (3.4) to Eq. (3.4) we obta mvp cvp kvp F (3.44) Pre-multplyg Eq. (3.44) by the traspose of the modal matrx,.e, v we obta v mvp v cvp v kvp v F (3.45) It was show before Eq. (3.9) ad (3.30) that because of the orthogoal propertes of the mode shapes the mass ad stffess matrces are dagoalzed,.e. v m v m ad v k v k Because of proportoal dampg.e. c m k we have v cv v m kv 33
.e. v mv v kv v c v m m c where c s a dagoal matrx. hus substtuto to Eq. (3.45) gves m p c p k p v F (3.46) Eq. (3.46) represets a ucoupled set of equatos for damped sgle degree of freedom systems. he th equato s m p c p k p v F f (3.47) whch represets the equato of moto of a sgle degree of freedom system. m f k c Fg. 4. Sgle degree of freedom system defed by Eq. (3.47) Sce k m, from Eq. (3.9), Eq. (3.47) ca be wrtte as v F f p p p (3.48) m m Where c (3.49) km he soluto of a damped sgle degree of freedom system, as descrbed by Eq. (3.48) has bee dscussed prevously. Oce the soluto of Eq. (3.48) s obtaed for all p, 34
the soluto terms of the orgal coordates x ca be deduced by trasformg back,.e. substtutg for p. Eq. (3.4). It should be oted that f the dampg matrx s proportoal to the stffess matrx,.e, c k the from Eq. (3.48) we see that k km whch meas that the hgher frequecy modes wll have hgher dampg ratos. 3.3. Hysteretc Dampg Hysteretc or structural dampg was dscussed uder sgle degree of freedom systems. It was show, that ths case the dampg force s proportoal to the elastc force, but as eergy s dsspated, the force s phase wth the velocty. hus for smple harmoc moto the dampg force s gve by j kx. For a mult-degree of freedom system, the equatos of moto wth hysteretc dampg ca be wrtte as mx j kx kx F (3.50) Chagg to Prcpal Coordates as show the secto above leads to hus each equato s of the form m p j k p v F (3.5) m p j k p v F p j p.e. If j t 35 v F (3.5) m F F e the j t Substtutg Eq. (3.5) we obta p p e v F F p j p (3.53) m m
the soluto of whch has bee dscussed prevously. 3.4 No-proportoal Dampg 3.4. State-Space Method Whe the dampg matrx s ot proportoal to the mass or the stffess matrx, ether the modal matrx or the weghted modal matrx wll dagoalze the dampg matrx. I ths geeral case of dampg, the coupled equatos of moto have to be solved smultaeously, or they eed to be ucoupled usg the state-space method. By ths method the set of secod order dfferetal equatos are coverted to a equvalet set of frst order dfferetal equatos, by assgg ew varables (referred to as state varables) to each of the orgal varables ad ther dervatves. o llustrate the procedure, the equatos of moto for the two degree of freedom system show Fg.9 are wrtte as m 0 x 0 m x m 0 x 0 m x 0 0 m 0 x c c c x k k k x F 0 m x c c c x k k k x F or parttoed matrx form as (3.54) 0 0 m 0 x m 0 0 0 x 0 0 0 0 m x 0 m 0 0 x 0 m 0 c c c x 0 0 k k k x F 0 m c c c x 0 0 k k k x F Substtutg x x z z x z z 3 x z z 4 x x z 3 z 4 we get 36
m 0 c c c z 0 0 k k k z F 0 m c c c z 0 0 k k k z F 0 0 m 0 z 3 m 0 0 0 z3 0 0 0 0 m z 4 0 m 0 0 z 4 0 whch ca be abbrevated to (3.55) Az Bz Q (3.56) It ca be see that whle the secod order equatos have bee reduced to frst order equatos, the umber of equatos have bee doubled. he soluto of above equatos for free vbrato reveals that damped atural modes do exst, however, they are ot detcal to the udamped atural modes. For the udamped modes, varous parts of the structure move ether phase or 80 out-of-phase wth each other. For the o-proportoally damped structures, there are phase dffereces betwee the varous parts of the structure, whch result complex mode shapes. hs dfferece s mafested by the fact that for udamped atural modes all pots o the structure pass through ther equlbrum postos smultaeously, whch s ot the case for the complex modes. hus the udamped atural modes have well-defed odal pots or les ad appear as a stadg wave, whle for complex modes the odal les are ot statoary. 3.4. Forced Normal Modes of Damped Systems For a -degree-of-freedom system wth vscous dampg, the equatos of moto for steady state susodal exctato ca be wrtte ts geeral form as s m x c x k x F t (3.57) where the system erta, dampg ad stffess matrces [m], [c] ad [k] respectvely, ad they are assumed to be real symmetrc ad postve defte. If the dampg s hysteretc, the secod term would be gve by 37 / dx, where [d] s the hysteretc dampg matrx. I the geeral case dampg would be o-proportoal ad thus the dampg matrx caot be dagoalzed usg the ormal mode trasformato. For a
arbtrary set of forces F ad exctato frequecy the soluto of Eq. (3.57) s rather complcated. Although the resposes at each coordate x are harmoc wth the exctato frequecy, they are ot all phase wth each other or wth the exctato force. If, however, a system wth -degree-of-freedom s excted by umber of forces whch are ether 0 or 80 out-of-phase (ofte called moophase or coheretly phased forces), the for a partcular rato of forces, the respose at each of the coordates wll be phase wth each other ad lag behd the force by a commo agle (called the characterstc phase lag). hus we have to determe the codtos whch wll produce a soluto of the form x X x X x3 X 3s t v s t x X (3.58) For ay gve exctato frequecy, there exst solutos of the type gve by Eq. (3.58), where each of the modes v s assocated wth a defte phase ad a correspodg dstrbuto of forces whch s requred for ts exctato. he respose uder these codtos s called the forced ormal modes of the damped system, sce every pot of the system moves phase ad passes through ts equlbrum posto smultaeously wth respect to the other pots. Substtutg Eq. (3.58) Eq. (3.57) gves s t k m v cos t c v F st (3.59) Expadg the s t ad the cos t ad cos t terms we obta terms ad separatg the s t k mv s cv F cos (3.60) 38
k mv cv s cos 0 (3.6) hese equatos cota three ukows F, v ad sce s gve. If cos 0, Eq. (3.6) may be dvded by cos to gve k m cv ta 0 (3.6) Eq. (3.6) has a o-trval soluto f the determat k m c ta 0 (3.63) It s evdet that for a gve there are values of ta,, correspodg to the egevalues ad for each ta there s a correspodg egevector v satsfyg the equato k m cv ta 0 (3.64) If Eq. (3.64) s premultpled by the traspose v ad rearraged, we obta ta v cv v k mv (3.65) From Eq. (3.65) t ca be see that each of the roots ta s a cotuous fucto of. For low values of, ta s small,.e, s a small agle. As creases ad approaches the udamped atural frequecy, oe of the roots (whch ca be amed ) approaches the value /. As s creased above the deomator of Eq. (3.65) gets egatve ad Whe teds to,,3,,3, s equal to / gets larger tha /. teds to. I a smlar maer the remag roots ca be plotted as a fucto of frequecy, where at the th udamped atural frequecy. hus 39
k s that root whch has the value / at the udamped atural frequecy k. Havg examed the varato of egevalues ta as a fucto of frequecy, the mode shapes ca ow be vestgated. It ca be see from Eqs. (3.63) ad (3.64) that at ay oe frequecy the mode shapes deped oly o the shape of the dampg matrx ad ot o ts testy. If every elemet the matrx [c] s multpled by a costat factor, the Eq. (3.63) shows that the roots ta wll all be creased by the same rato. hus Eq. (3.64), whch determes the mode shapes, wll be multpled throughout by the same factor ad the mode shape v wll be uchaged. Eq. (3.64) ca be re-wrtte as c k m v ta 0 (3.66) Whe s equal to oe of the udamped atural frequeces, say, the oe of the roots s 90 as show above. hus Eq. (3.66) whch determes the mode shape for ths root becomes k mv 0 (3.67) It ca thus be see, that whe the frequecy s equal to oe of the udamped atural frequeces, the mode shape for oe of the roots (whch s equal to /) s detcal to the Prcpal or Normal mode shape. Atteto ca ow be pad to the force rato that s requred to excte ay oe mode v for the correspodg root ta at ay oe frequecy. he force rato requred ca be calculated from Eq. (3.60) amely k mv s cv cos (3.68) I the specal case whe oe of the udamped atural frequeces, oe of the roots 90 ad Eq. (3.68) reduces to cv (3.69) 40
o llustrate the cocepts dscussed above cosder the same umercal example of the two degree of freedom system of Fg. 9, but wth the values of dampg added as show Fg. 6. F st F st 4N/m N/m N/m 4c Ns/m c Ns/m 7c Ns/m x x Fg. 6. wo degree of freedom system wth o-proportoal dampg. hus the equatos of moto accordg to Eq. (3.) become 5 0 x 5c c x 4 x F st 0 0 x c 8c x 6 x F For a o-trval soluto the determat of Eq. (3.63) must be equal to zero,.e. 4 5 0 5 6 0 0 8 ta c 0.e, c c ta 4 5 5 ta c ta ta 6 0 8c 0.e, ta 4 5 5c ta 6 0 8c ta c 0 whch reduces to 0 70 50 4 ta 9 45 cta 39 c 0 (3.70) he udamped atural frequeces of the system are gve by lettg c = 0,.e., by the equato 4 0 70 50 0 4
hey are foud to be / 5 0.63 rad/s ad rad/s, whch obvously should be the same as those calculated uder secto 3.. Dvdg Eq. (3.70) by ta ad substtutg c we get ta 4 39 9 45 0 70 50 0 he roots of ths equato are gve by 9 45 9 45 390 70 50 4 c (3.7) ta 39 Whe s equal to oe of the udamped atural frequeces or the equato reduces to hus whe we get 9 45 9 45 c ta 39 9 45 9 45 c ta 39 so that 90 for the egatve sg ad 39c ta 9 45 for the postve sg. Smlarly whe we get 9 45 9 45 c ta 39 so that 90 for the egatve sg ad 4
39c ta 9 45 for the postve sg. he varato of ad ca be plotted as a fucto of frequecy usg Eq. (3.7). he curves are show Fg.7 for three values of dampg c, correspodg roughly to lght, medum ad heavy dampg. he shape of the curves are see to be smlar to those of Fg.(b). Fg. 7. Varato of the roots ad as a fucto of frequecy for three values of dampg. he mode shapes are obtaed by substtutg umercal values for the matrces [m], [c] ad [k] Eq. (3.64),.e, 4 5 0 5 v 0 ta c 6 0 0 8 v 0.e. c ta 4 5 5c ta c v 0 ta ta 6 0 8cv 0 (=, ) Expadg the frst equato we get cv ta 4 5 5 ta c v 0 43
hus v ta c c/ ta v ta 4 5 5c 4 5 5 c/ ta Substtutg for c / ta from Eq. (3.7) we obta 4 v 49 45 60 75 v 4 30 5 60 75 (3.7) v v v v v v v v Fg.8. he characterstc phase lag modes as a fucto of frequecy he characterstc phase lag modes are plotted as a fucto of frequecy usg Eq. (3.7) Fg.8. he postve sg ow correspods to = for the frst mode ad the egatve sg correspods to = for the secod mode. Note, the specal case whe s equal to the frst udamped atural frequecy / 5 for = we obta v v v whch s the same as the frst prcpal mode shape v. 44
At / 5 for = we obta v 56 3 Smlarly, whe s equal to the secod udamped atural frequecy for = we obta At for = we obta v 0 v whch s the same as the secod prcpal mode shape v. he force rato requred to excte ay oe mode v for the correspodg root ta at ay oe frequecy ca be calculated from Eq. (3.68),.e., cos 4 5 0 v 5 v F s 6 0 0 c v 8 v F (3.73) he force rato for each root ca be plotted as a fucto of frequecy by substtutg the value of for each ad the correspodg mode shape. Fg.9 shows the force ratos requred as a fucto of frequecy for the two s. 45
Fg. 9. Force ratos requred to excte the two modes as a fucto of frequecy. It s, however, terestg to calculate the force ratos at the udamped atural frequeces. As show prevously, at ay oe udamped atural frequecy, oe of the roots ta,.e, oe of the 90, ad for that root the mode shape obtaed s the Prcpal Mode Shape. hus Eq.(3.73) reduces to 5 v F c 8 v F whch gves the force rato requred to excte the prcpal mode shape at the correspodg udamped atural frequecy. Substtutg the frst udamped atural frequecy / 5 ad v/ v we obta 5 F / 5c 8 F.e, F / F 4 / 7 46
Substtutg the secod udamped atural frequecy ad v/ v / we obta 5 F c 8 F.e, F / F /0 Before cocludg ths secto t s mportat to recaptulate the followg pots: () For each frequecy of exctato there are as may characterstc phase agles as there are umber of degrees of freedom, correspodg to certa sets of forces. () For each characterstc phase lag there s a correspodg mode shape whch vares wth frequecy. At the udamped atural frequeces oe of the mode shapes s detcal to the correspodg udamped Prcpal mode. (3) he mode shapes deped o the shape of the dampg matrx ad ot o the testy of dampg. (4) I each mode the resposes at the coordates are all phase, but lag behd the exctato force by a agle. At the udamped atural frequecy, 0 90 for oe of the modes whch s the prcpal mode. (5) Orthogoal propertes of phase lag modes also exst. he orthogoal propertes of the prcpal modes of vbrato were demostrated prevously secto 3..3. o derve aalogous propertes of forced modes. Eq.(3.6) ca be wrtte for the th egevalue ad egevector ad premultpled by the traspose of the j th egevector. he procedure s repeated wth ad terchaged gvg the followg two equatos, v k mv v cv ta 0 j j (3.74) v k mv v cv ta j 0 j j (3.75) Sce [m], [c] ad [k] are symmetrc matrces Eq.(3.75) ca be trasposed to obta 47
v k mv v cv ta j 0 j j (3.76) Subtractg Eq.(3.76) from Eq.(3.74) we obta the orthogoal propertes as v k mv j 0 (3.77) Ad 0 v c v (3.78) j provded ta ta. j Combg Eqs.(3.77) ad (3.78) wth Eq.(3.60) we obta the thrd relato v F 0 v F j j (3.79) 48