Walter D. Pilkey. Dmitry V. Balandin. Nikolai N. Bolotnik. Sergey V. Purtsezov. Introduction. Nomenclature

Transcription

1 Waler D. Pilkey e al Waler D. Pilkey wdp@virgiia.edu Uiversiy of Virgiia Dep. of Mechaical ad Aerospace Eg Charloesville, VA. USA Diry V. Baladi baladi@pk.u.rue.ru izhy ovgorod Sae Uiversiy Dep. of Cop. Maheaics ad Cybereics 6395 izhy ovgorod. Russia ikolai. Boloik boloik@ipe.ru Isiue for Probles i Mechaics of he Russia Acadey of Scieces 956 Moscow. Russia Exreal Disurbace Aalysis for Dyaical Syses wih Ucerai Ipu The philosophy of he exreal disurbace aalysis for dyaical syses wih ucerai ipus is described. This aalysis ivolves solvig opial corol probles i which he ie hisories of he ipus (exeral disurbaces) are regarded as he corol fucios ad a respose easure of he syse serves as he perforace idex. The perforace idex should be axiized or iiized over he disurbaces wihi a prescribed class. Ofe his class is specified by lower ad upper bouds (a corridor) bewee which he values of he disurbace us lie. The axiizaio ad iiizaio probles are referred o as he wors disurbace ad bes disurbace probles, respecively. The soluios of hese probles provide he exree values bewee which he respose easure lies for ay disurbace fro he specified class. The exreal disurbace aalysis is ipora, i paricular, whe desigig sadards for esig devices for he proecio of fragile objecs fro ipac loads. This approach is illusraed for a sigledegree-of-freedo syse ha ca be regarded as a siplified odel of he equipe for sled ess of auoobile resrai syses. Keywords: Ucerai ipu, wors disurbace, bes disurbace, ipac isolaio, sled ess Sergey V. Pursezov pusv@virgiia.edu Uiversiy of Virgiia Dep. of Mechaical ad Aerospace Eg Charloesville, VA. USA Iroducio There are uerous srucural dyaics probles where he loadig is o fully defied. Typically, such probles are reaed as sochasic probles usig he heory of probabiliy (Hass, 5; Iyeger, 97). This paper discusses a aleraive deeriisic approach i which he available iforaio o he loadig is uilized o calculae bouds o possible resposes of he srucure. The lower boud would be he leas axiu respose ha ca occur. The correspodig loadig is referred o as he bes disurbace. The upper boud is he greaes possible peak respose ad he correspodig loadig is he wors disurbace (Koivuiei, 966; O Hara, 973). The respose of he srucure o ay oher loadig fallig wihi he prescribed loadig bouds will lie bewee he exree values of he respose. Oe exaple occurs if a explosive eviroe is characerized i ers of a oal ipulse. This is sufficie iforaio o deerie wha us happe o soe equipe, e.g., iliary equipe, ad he axiu respose of his equipe. Oher exaple occurs i he esig of safey devices for crashig vehicles. For isace, ie-dow syses for wheelchairs are esed o high-speed sleds for which he crash pulse us adhere o sadards ha are defied i ers of a evelope i which he crash pulse us lie (Kag ad Pilkey, 998). A eve ore coo exaple occurs i he sled esig of child seas, where he sled pulse us lie wihi prescribed ie-varyig bouds (Cradall e al., 996). The proposed schee provides upper ad lower bouds for criical resposes. These resposes ca be used o help gauge wheher he prescribed pulse corridor is oo igh or oo broad Preseed a XI DIAME Ieraioal Syposiu o Dyaic Probles of Mechaics, February 8h - March 4h, 5, Ouro Preo. MG. Brazil. Paper acceped: Jue, 5. Techical Ediors: J.R.F. Arruda ad D.A. Rade. o provide effecive sadards. Thus, he procedure here develops a echology o sudy he sesiiviy of dyaic resposes o icopleely prescribed es codiios. We choose o use he sled esig of child seas as a exaple of usig exreal disurbace aalysis o illusrae he priciples of his sesiiviy echology. oeclaure A, B, C, D = coordiaes of he verices of he rapezoid, defied by Eq. (8), (ie, acceleraio), (s, /s ) a = heigh of he rapezoid, defied by Eq. (7), acceleraio, /s c = dapig coefficie of he resrai syse, dapig coefficie, kg/s F = force applied o he sled, force, f = force applied o he duy by he resrai syse, force, g( ) = ipulse respose of he syse, s h = ie bewee saples, ie, s i = saple uber, diesioless j = desig variable uber, diesioless J = peak agiude of he absolue acceleraio of he duy, acceleraio, /s K( ) = fucio, defied by Eq. (), /s k = siffess coefficie of he resrai syse, siffess coefficie, kg/s L = uber of saples, diesioless M = ass of he sled, ass, kg = ass of he duy, ass, kg = uber of desig variables, diesioless p = paraeer defied by Eq. (), diesioless 5 / Vol. XXVIII, o. 4, Ocober-Deceber 6 ABCM

2 Exreal Disurbace Aalysis for Dyaical Syses wih Ucerai Ipu R = sesiiviy of he peak force rasied o he duy relaive o he variaio i he crash pulse, diesioless R d = prescribed value of he sesiiviy of he peak force rasied o he duy relaive o he variaio i he crash pulse, diesioless T = duraio of he crash pulse, ie, s = ie, s u = absolue acceleraio of he duy wih ius sig, acceleraio, /s W = class of he desig variables w, acceleraio, /s w = absolue acceleraio of he sled wih ius sig reaed as a desig variable, acceleraio, /s w ( ) = lower boud of he desig variable w, acceleraio, /s w ( ) = upper boud of he desig variable w, acceleraio, /s x = displacee of he duy relaive o he sled, displacee, z = displacee of he sled relaive o he fixed frae, displacee, Greek Sybols v = velociy chage durig he crash, velociy, /s v = lower boud of he velociy chage durig he crash, velociy, /s v = upper boud of he velociy chage durig he crash, velociy, /s δ ( ) = Dirac dela fucio, diesioless ς = fracio of criical dapig, diesioless λ = siilariy facor, diesioless λ = siilariy facor equal o.4, diesioless µ = reduced ass of he syse, defied by Eq. (), ass, kg τ = variable of iegraio, ie, s τ = paraeers of he rapezoid, defied by Eq. (7), ie, s 4 ψ ( ) = fucio, represeig a rapezoid ad defied by Eq. (7), acceleraio, /s ω = agular udaped aural frequecy of he syse, agular frequecy, rad/s ω D = agular daped aural frequecy of he syse, agular frequecy, rad/s Subscrips i relaive o saple uber j relaive o desig variable uber relaive o he opial soluio of he bes disurbace proble relaive o he lower boud of he desig variable w relaive o he upper boud of he desig variable w Superscrips relaive o he opial soluio of he wors disurbace proble relaive o he upper boud of he desig variable w Basic Coceps of he Bes ad Wors Disurbace Aalyses; Saee of he Opial Corol Probles A sled es is ofe eployed o ivesigae proecive characerisics of safey devices. For exaple, auoobile resrai syses ca be sudied wih sled ess. I paricular, cosider a sled es for a auoobile resrai ha resrics he oio of a occupa o preve a viole coac wih copoes iside he car durig a crash. Typically, he es sled equipe cosiss of a highspeed sled wih a ock-up of he car ierior, a occupa sea, a resrai syse, e.g., sea bel, ad a duy as he occupa. The car ock-up wih ierior ad seabels are rigidly aached o he sled. The sled, duy, ad he sea bels are equipped wih sesors ha easure acceleraios, displacees, srais, forces, ad oher characerisics of he respose of he syse. The sled is acceleraed o a prescribed velociy ad he is subjec o a deceleraio pulse siulaig he crash ipac. The various easured resposes ca be used o for ijury crieria ha ca be copared wih esablished olerace levels. To fix ideas, we cosider a siplified odel of he esig equipe i which he sled ad he duy are regarded as rigid bodies (Fig. ). The sled oves alog a horizoal sraigh lie relaive o a fixed (ierial) referece frae ad he duy oves relaive o he sled alog he sae lie. Le z deoe he displacee (coordiae) of he sled relaive o he fixed frae, x he displacee (coordiae) of he duy relaive o he sled, M he ass of he sled, he ass of he duy, F he force applied o he sled durig he crash siulaio, ad f he force applied o he duy by he resrai syse. The ass of he sea bels is egleced as copared wih he asses of he sled ad he duy. The force f is assued o be a fucio of he paraeers of he relaive oio of he duy, he relaive coordiae x ad he relaive velociy xɺ, i.e., f = f ( x, xɺ ). This fucio akes io accou he elasic ad dissipaive properies of he seabels. The oio of his echaical syse is govered by he siulaeous equaios Mz ɺɺ = F f, ( ɺɺ x ɺɺ z) = f, () which represe ewo s secod law for he sled ad he duy, respecively. The duy is aced upo oly by he resrai syse force f, whereas he sled is aced upo by he ipac force F ad he force f as he sea bel ieracs wih he sled. Solve he syse of Eq. () for xɺɺ ad zɺɺ o obai M µ ɺɺ x = f ( x, xɺ ) F, Mz ɺɺ = F f ( x, xɺ ), µ =. () M M The quaiy µ is called he reduced ass of he syse. I he syse of Eq. (), he equaio goverig he oio of he duy relaive o he sled is idepede of he equaio of oio of he sled relaive o he fixed frae. If oe is ieresed oly i he relaive oio of he duy, ad he ipac force is a kow fucio of ie, i.e., F = F( ), oly he firs relaio of () is used. I he heory of ipac isolaio, he ipac disurbace specified as he force pulse acig o he base (he sled) is referred o as he dyaic disurbace. z M x c k Figure. Sigle-degree-of-freedo odel. J. of he Braz. Soc. of Mech. Sci. & Eg. Copyrigh 6 by ABCM Ocober-Deceber 6, Vol. XXVIII, o. 4 / 53

3 Waler D. Pilkey e al I a uber of cases, i is reasoable o assue ha he ipac disurbace is characerized by he acceleraio of he base, raher ha by he force applied o i. For isace, for he sled ess, he ipac deceleraio of he sled is easured by acceleroeers. The he relaive oio of he body o be proeced (he duy) is govered by he secod relaio of Eq. () where ɺɺ z( ) is reaed as a kow fucio of ie. The ipac disurbace specified as he base acceleraio pulse is referred o as he kieaic disurbace. For boh ypes of disurbaces, he equaio of he relaive oio of he body o be proeced ca be represeed i a uified for as ɺɺ x u( x, xɺ ) = w( ), (3) where u( x, xɺ ) = f ( x, x ɺ )/ µ, w( ) = F( ) / M for he dyaic disurbace ad u( x, xɺ ) = f ( x, xɺ )/, w( ) = ɺɺ z( ) for he kieaic disurbace. I he heory of ipac isolaio, he fucio u( x, xɺ ) is frequely called he characerisic of he ipac isolaor. As a rule, Eq. (3) is subjeced o zero iiial codiios x() =, xɺ () =. (4) These codiios iply ha he objec o be proeced does o ove relaive o he base uil he ipac pulse has bee applied o he base. To copare he disurbaces i ers of he respose of he objec o be proeced, oe should have a uerical easure of his respose. We selec J ( w) = ax u( x( ), xɺ ( )) (5) where x( ) sads for he soluio of he differeial equaio (3) subjec o he iiial codiios of Eq. (4) for a give w = w( ). The quaiy J characerizes he peak agiude of he force rasied o he duy by he resrai syse. (Acually, he peak force is equal o µ J or J for he case of he dyaic or kieaic disurbace, respecively.) The easure J ( w ) is a fucioal of he disurbace, depedig o he eire ie hisory of w. Cosider ow he saee of he wors ad bes disurbace probles for he syse of Eqs. (3) ad (4) wih he respose easure of Eq. (5). Le he disurbaces belog o a prescribed class W, i.e., w W. oe ha W is a se of fucios raher ha a se of values of hese fucios. I ers of he sled es, he class W ca characerize a se of disurbaces ha are represeaive of he es, akig io accou he ieviable iaccuracy i he reproducio of he desired deceleraio pulse. Usually, he class of represeaive disurbaces ivolves a corridor or evelope (Fig. ) i which hese disurbaces us lie, i.e., W = { w( ) : w ( ) w( ) w ( ), [, T ]}, (6) where w ( ) ad w ( ) are prescribed fucios ha defie he lower ad upper bouds of he corridor ad T is a fixed ie. I addiio, he velociy of he sled should chage by a cerai value durig he crash pulse ie T. I he case of he kieaic disurbace, his codiio has he for T v = w( τ ) dτ, (7) where v is a specified posiive quaiy. Usually, he quaiy v is prescribed wih a allowace for a error i he easuree of he velociy of he sled, v v v, (8) where he ierval [ v, v] characerizes he uceraiy i he easuree of he velociy decrease. Sice v is defied as he iegral of w, he iequaliies of Eq. (8) ipose addiioal cosrais o he deceleraio pulse ie hisory. Therefore, he oal class o which he allowed deceleraio pulses us belog is defied by T W = { w( ) : w ( ) w( ) w ( ), v w( τ ) dτ v}. (9) w - () Tie w () Figure. Corridor i which a ipu pulse w() us lie. The wors disurbace proble is ha of he deeriaio of he upper boud of he fucioal J ( w ) for he disurbaces of he class W. Proble (Wors Disurbace Proble). For he syse govered by Eqs. (3) ad (4), fid he disurbace fucio w ( ) belogig o he class W such ha J w = () ( ) ax w W J ( w). The bes disurbace proble is ha of he deeriaio of he lower boud of he fucioal J ( w ) for he disurbaces of he class W. Proble (Bes Disurbace Proble). For he syse govered by Eqs. (3) ad (4), fid he disurbace fucio w( ) belogig o he class W such ha J ( w ) = i J ( w). () w W Fro he aheaical poi of view Probles ad are opial corol probles wih a axiu ype fucioal, sice he perforace idex i hese probles is he axiu of he absolue value of he force rasied o he duy over he ipac respose ie. The wors ad bes disurbace aalyses eable oe o ivesigae he sesiiviy of he resuls of he sled es o he variaio of he ipac pulse wihi he corridor prescribed by he sadards for such ess. If he differece bewee he upper ad lower bouds is large, he es ay have a udesirably low value of he peak force rasied o he duy due o he occurrece of he disurbace close o he bes oe. Such a es ca be decepive whe assessig he qualiy of he resrai syse ad, hece, is illdesiged. I his case, he sadards for he defiig allowable ipac pulses should be revised. Soeies, he sesiiviy of he peak force rasied o he duy relaive o he variaio i he ipac pulse is characerized by he raio 54 / Vol. XXVIII, o. 4, Ocober-Deceber 6 ABCM

4 Exreal Disurbace Aalysis for Dyaical Syses wih Ucerai Ipu R = J w J w () ( ) / ( ). Resrai Syse wih Liear Characerisics; uerical Soluio of he Opial Corol Probles Le he resrai syse have liear elasic ad dapig properies, i.e., f ( x, xɺ ) = cxɺ kx, (3) where c ad k are he dapig ad siffess coefficies, respecively. For a ipac disurbace of he kieaic ype, Eq. (3) becoes ɺɺ x ςω xɺ ω x = w, (4) ( ) where ω is he aural frequecy of he syse ad ς is he dapig raio, / / = ( / ), ς = /(( ) ). (5) ω k c k The crierio J of Eq. (5) (he peak agiude of he occupa s absolue acceleraio) becoes J w ςω xɺ ω x. (6) ( ) = ax ( ) ( ) Eve for he liear syse of Eq. (4), he soluio of he bes ad wors disurbace probles (Probles ad ) requires uerical ehods. To his ed, he coiuous-ie forulaio of hese probles is replaced by a discree-ie approxiaio. The ie axis is discreized wih sep size h. O he ie iervals ( i ) h < ih, i =,,..., he fucio w( ) is assued o be cosa: wi, if ( i ) h < T w( ) =, ( i ) h < ih. (7), if ( i ) h T The cosa paraeers wi, i =,...,, play he role of desig variables whe solvig he opiizaio probles. The uber of hese desig variables is defied as follows: T / h, if { T / h} = =, (8) [ T / h], if { T / h} where he square brackes ad he braces deoe he ieger ad fracioal pars, respecively, of he expressio eclosed. The soluio of Eq. (4) subjec o he iiial codiios of Eq. (4) has he covoluio for where x( ) = g( τ ) w( τ ) dτ, xɺ ( ) = gɺ ( τ ) w( τ ) dτ, (9) ωd exp( ζω)si ωd, if ς <, / g( ) = exp( ςω), if ς =, ωd = ω ς. () ωd exp( ζω)sih ωd, if ς >, The fucio g( ) is referred o as he ipulse respose fucio (fudaeal soluio) for Eq. (4). I saisfies he differeial equaio of (4) wih w( ) = δ ( ), where δ ( ) is he Dirac dela fucio, subjec o he iiial codiios x () = ad xɺ ( ) =. This fucio ca also be defied as he soluio of Eq. (4) wih zero righ-had side ( w( ) ), subjec o he iiial codiios x () = ad x ɺ () =. Subsiue he relaios of Eq. () io Eq. (6) o obai ax ( τ ) ( τ ) τ, ( ) ςω ( ) ω ( ). J = K w d K = gɺ g () The discree-ie approxiaio of he respose easure of Eq. () has he for jh ( j ) h i{ i, } J ( w) = ax p w, i [: L] ij j j = p = K( ih τ ) dτ, j i, w = [ w,..., w ], ij () where L is he uber of discree iervals o he ie axis o which he respose o he ipac pulse is o be calculaed. To obai reliable resuls, i is ecessary o cosider he respose o he ie ierval subsaially exceedig ha of he crash pulse. Sice he crash pulse has a raher shor duraio, he respose easure ca aai is axiu afer he disurbace has ceased o ac. For his reaso, he uber of ie isas a which he respose is easured should exceed he uber of he discreizaio pois i he crash pulse ierval [, T ] ad, hece, L >. The crierio of Eq. () is a fucio of variables w = [ w,..., w ]. The cosrais of Eq. (9) are discreized as follows: w ( ih) w w ( ih), i [: ]; v h w v. (3) i j j= I he discree-ie forulaio, he wors-disurbace proble is reduced o he axiizaio of he fucio of Eq. () ad he bes-disurbace proble o he iiizaio of his fucio, subjec o he cosrais of Eq. (3). These cosrais are liear relaive o he desig variables w,..., w, ad he fucio o be axiized or iiized is a axiu of he absolue values of liear fucios of hese variables. I his case, he cosraied iiizaio (axiizaio) proble ca be reduced o ha of he liear prograig, which subsaially faciliaes he soluio, sice here are rapidly covergig reliable liear prograig algorihs available i os opiizaio sofware. For a ore deailed descripio of he soluio of opiizaio probles siilar o Probles ad o he basis of he liear prograig, see Baladi, Boloik, ad Pilkey (). uerical Exaple Assue he dapig ad siffess coefficies of he resrai syses are: c s k 5 =, =, = 3 kg, (4) J. of he Braz. Soc. of Mech. Sci. & Eg. Copyrigh 6 by ABCM Ocober-Deceber 6, Vol. XXVIII, o. 4 / 55

5 Waler D. Pilkey e al which are reasoably realisic for resrai syses. The ass of he occupa of 3kg correspods o he 6-year-old child duy. Figures 3 o 5 deosrae he resuls of he soluio of Probles ad for he sled deceleraio pulse corridor show i Fig. 3. The corridor correspods o he sadard of HTSA (979) ha specifies es codiios for child resrais. The uceraiy ierval of Eq. (8) for he sled velociy chage is defied as 45.k/h v 48.3 k/h (8 ph v 3 ph), (5) Deceleraio (g) 3 Bes Pulse Wors Pulse Figure 3 shows he ie hisories for he bes ad wors deceleraio pulses. The shaded area i Fig. 3 represes he corridor i which deceleraio pulses us lie. If here were o cosrais such as of Eq. (5) o he velociy chage, boh he bes ad wors disurbace pulses would have swiched bewee he lower ad upper bouds of he corridor. This follows sice he soluio of a liear prograig proble always lies o he boudary of he doai cosraiig he desig variables. Fro he ie hisories show i Fig. 3 i is appare ha boh he bes disurbace ad wors disurbace pulses lie wihi he walls of he corridor o soe ie iervals. Therefore, he velociy chage assues oe of he boudary values prescribed by Eq. (5), specifically, v = 45. k/h for he bes disurbace ad v = 48.3 k/h for he wors disurbace. Figure 4 depics he ie hisory of he force applied o he duy by he sea bel i he case of he wors (solid curve) ad bes (dashed curve) disurbaces. I is appare ha i boh cases he peak agiude of he force is aaied before he ed of he disurbace. This is because he duraio of he pulse is close o he udaped vibraio period of he syse. I accordace wih figs. 3, he duraio of he deceleraio pulse lies bewee.75 ad.9s, while he vibraio period for he paraeers of Eq. (4) is.96s. The ie hisories of he displacee of he duy for he wors ad bes disurbaces are show i Fig. 5. I fac, he curves of Fig. 5 repea hose of Fig. 4 scaled by he siffess coefficie 5 ( / ). This is due o he fac ha he dapig does o have uch effec durig he ie ierval uder cosideraio. The dapig raio ς of Eq. (5) calculaed for he syse wih he paraeers of Eq. (4) equals -.66 s ad, accordigly, he characerisic ie of he dapig equals 5.s, which subsaially exceeds he udaped vibraio period. The sesiiviy raio of Eq. () for he forces rasied o he duy is R =. /7.6 =.34. (6) This reasoably low value would probably be accepable i pracice. Force (k) Displacee () Tie (s) 5 Figure 3. Bes ad wors disurbaces... Bes Wors.5..5 Tie (s) Figure 4. Force applied o he duy by he sea bel Bes Wors.5..5 Tie (s) Figure 5. Displacee of he duy relaive o he sled. Raioal Desig of Sled Tes Sadards I he previous secio, we have illusraed he applicaio of he exreal disurbace aalysis o check he sesiiviy of he respose easure (he force rasied o he occupa) o he variaio of he sled deceleraio pulse wihi a prescribed corridor. This cocep suggess a echique for he raioal desig of such a corridor so ha he sesiiviy raio does o exceed a allowed value. Cosider a exaple. Le he walls of he corridor have a shape of a rapezoid ABCD show i Fig. 6. A geeric rapezoid ca be represeed aalyically by he piecewise liear fucio 56 / Vol. XXVIII, o. 4, Ocober-Deceber 6 ABCM

6 Exreal Disurbace Aalysis for Dyaical Syses wih Ucerai Ipu, (, τ) [, ), a( τ)/ τ, [ τ, τ τ ) [, ), ψ ( ) = a, [ τ τ, τ τ 3) [, ), a( τ τ 4) /( τ 3 τ 4), [ τ τ 3, τ τ 4) [, ),, τ τ 4. (7) The coordiaes of he verices of his rapezoid o he ieacceleraio plae are give by, (, τ ) [, ), a( τ ), [, τ τ λτ ) [, ), τ w ( ) = λa, [ τ λτ, τ λτ 3) [, ), a, τ λτ 4, ( τ λτ 4), [ 3, τ λτ τ λτ 4 ) [, ), ( τ 3 τ 4) (3) A = ( τ,), B = ( τ τ, a), C = ( τ τ, a), D = ( τ τ,). 3 4 (8) I ers of he geoery of he rapezoid, τ 4 ad τ 3 τ are he leghs of he lower ad upper bases, respecively, a is he heigh, / / ( τ a ) ad (( τ τ ) a ) are he leghs of he laeral 4 3 sides AB ad CD, respecively. The paraeer τ characerizes he shif of he rapezoid alog he ie axis. The walls (he upper ad lower bouds) of he corridor for sled ess show i Fig. 3 have he shape of rapezoids wih for he lower boud ad τ = 4s, τ = 9s, τ = 4s, 3 τ = 7s, a=9g 4 τ = s, τ = s, τ = 53s, 3 τ = 9s, a=5g 4 (9) (3) for he upper boud. Wih a high degree of accuracy, he rapezoid of he upper boud is siilar o he rapezoid of he lower boud wih he siilariy facor close o.3. Deceleraio a A B C D τ τ τ τ τ 3 τ τ 4 Tie Figure 6. A rapezoidal boud for he force applied o he duy by he sea bel. We will cofie ourselves o he case where he fucios w ( ) ad w ( ) of Eq. (6) are represeed by siilar rapezoids. Le he lower boud be fixed ad le w ( ) = ψ ( ), where ψ ( ) is defied by Eq. (7). Le he upper boud be described by he fucio where λ ad τ are he paraeers of he upper boud, he siilariy facor ad he ie coordiae of he lef-had boo verex (poi A ) of he respecive rapezoid. For he curve w ( ) of Eq. (3) o lie above he curve w ( ) = ψ ( ) of Eq. (7) i is ecessary ha λ, τ ( λ) τ τ τ. (3) 4 I is appare ha for λ =, he lower ad upper bouds of he corridor ach oe aoher. By varyig he paraeers λ ad τ wihi he doai of Eq. (3) oe ca assure ha he raio R of Eq. () does o exceed a prescribed value. The variaio ca be orgaized i various ways. For exaple, oe ca cosrai he paraeer τ o lie i he iddle of he ierval allowed for his paraeer by Eq. (3), i.e., τ = τ ( λ) τ /. (33) 4 I his case, he variaio is perfored wih respec o oly oe paraeer, λ. As λ icreases, he corridor becoes wider ad, herefore, he bes-disurbace respose easure decreases while he wors-disurbace respose easure icreases. Accordigly, he sesiiviy raio R is a oooically icreasig fucio of λ. The search for he axiu λ ha assures ha he quaiy R does o exceed he prescribed value R is reduced o he soluio of he equaio R( λ ) = R d. This equaio ca be solved by various ehods, for exaple, by he ierval bisecio ehod. For each rial λ, oe should solve he bes disurbace ad wors disurbace probles o calculae R( λ ). Figure 7 preses he curve R( λ ), calculaed for he corridor defied by Eqs. (7), (9), (3), ad (33), copleed by he velociy chage ierval of Eq. (5). The curve becoes ore fla as λ icreases. oe ha his curve begis wih λ = λ.4, raher ha wih λ =, i which case he upper ad lower bouds of he crash pulse corridor coicide. This is because of he cosrai of Eq. (5) o he velociy chage. The lower boud for he velociy chage is 45.k/h, whereas he iegral of he lower boud of he pulse corridor (he area of he rapezoid represeig he respecive wall of he corridor) is 34.9k/h. Therefore, he pulse of he lower boud is uable o decelerae o a coplee sop eve for he leas crash velociy allowed by he ierval of Eq. (5). Moreover, o pulse of he corridor is able o provide a velociy chage of 45.k/h while he area of he upper boud rapezoid is less ha his value. This is he case for λ < λ. For λ = λ, he area of he upper boud rapezoid is equal o 45.k/h. I his case, he oly pulse of he corridor able o decelerae he sled crashig a his velociy o a coplee sop is ha of he upper boud ad, d J. of he Braz. Soc. of Mech. Sci. & Eg. Copyrigh 6 by ABCM Ocober-Deceber 6, Vol. XXVIII, o. 4 / 57

7 Waler D. Pilkey e al accordigly, he wors ad bes disurbaces coicide, i.e., R( λ ) =. R λ Figure 7. Sesiiviy of he peak force rasied o he duy relaive o he variaio i he ipac pulse R versus he siilariy facor λ. Coclusios The exreal disurbace aalysis eables oe o evaluae he sesiiviy of he respose of a dyaical syse o he variaio i he exeral disurbaces wihi a prescribed uceraiy class. To ha ed oe should solve wo opial corol probles, he bes disurbace proble o calculae he lower boud ad he wors disurbace proble o calculae he upper boud of a respose easure. The raio of hese bouds ca serve as he sesiiviy idex. The closer his raio is o uiy, he lower he sesiiviy. The exreal disurbace aalysis is, i paricular, ipora for he validaio of sadards for ipac ess of fragile objecs ad he equipe for he proecio of such objecs. Such sadards cooly prescribe a corridor for ipac pulses o be reproduced o he es faciliy. The es daa are cosidered o be reliable if he spread i he resposes o he ipu pulses wihi he corridor is reasoably sall. This spread is characerized by he sesiiviy raio resulig fro he exreal disurbace aalysis. I addiio, he exreal disurbace aalysis suggess a echique for raioal desig of a corridor for loadig pulses so ha he sesiiviy raio does o exceed a prescribed value. Ackowledges This research was parly suppored by SF (gra BES- 3337), ATO (gra CBP.R.RCLG 988), ad he Russia Foudaio for Basic Research (gras ad 4- -). Refereces Baladi, D.V., Boloik,.., ad Pilkey, W.D.,, Opial Proecio fro Ipac, Shock ad Vibraio, Taylor ad Fracis Publishers, Philadelphia, PA, 437 p. Cradall, J.R., Pilkey, W.D., Kag, W., ad Bass, C.R., 996, Sesiiviy of Occupa Subjec o Prescribed Corridors, Joural of Shock ad Vibraio, Vol. 3, o. 6, pp Hass, M., 5, Applied Fuzzy Ariheic - A Iroducio wih Egieerig Applicaios, Spriger, Berli, 56 p. Iyeger, R.., 97, Wors Ipus ad a Boud o he Highes Peak Saisics of a Class of oliear Syses, Joural of Soud ad Vibraio, Vol. 5, o., pp Kag, W., ad Pilkey, W.D., 998, Crash Siulaios of Wheelchair Occupa Syses i Traspor, Joural of Rehabiliaio Research ad Develope, Vol. 35, o., pp Koivuiei, A.J., 966, Paraeer Opiizaio i Syses Subjec o Wors Bouded Disurbace, IEEE Trasacios o Auoaic Corol, Vol., o. 3, pp aioal Highway Traffic Safey Adiisraio (HTSA), 979 (Ocober, ed.), FMVSS 3 Child Resrai Syses, Vol. 49 CFR O Hara, G.J., 973, Maxiizaio ad Miiizaio of Dyaic Load Facors, Shock ad Vibraio Bullei, Vol. 43, Par, pp / Vol. XXVIII, o. 4, Ocober-Deceber 6 ABCM