Half Sine Shock Tests to Assure Machinery Survival in Explosive Environments

Transcription

1 Half Sine Shock Tess o Assure Machinery Survival in Explosive Environens By Howar A. Gaberson, Ph.D., Chairan MFPT Diagnosics an Signal Analysis Coiee 34 Corsicana Drive Oxnar, CA hagaberson@a.ne ABSTRACT Previous work has shown ha raiional four coorinae pseuo velociy shock specru is he bes fora for violen achine ransien founaion oion analysis. I ephasizes he oion severiy. This paper applies hese resuls o he heoreical half sine an oher siple es pulses. I shows he siple pulses srongly relae o acual explosive ransiens. The rick urne ou o be inclusion of pre an pos pulse oion in he calculaion. The calculaions applie o shaker shock pulse oions are also iscusse. INTRODUCTION Previous work has shown ha raiional four coorinae pseuo velociy shock specru is he bes fora for violen ransien founaion oion analysis if one is rying o inicae severiy [1,, 3]. I ephasizes he oion severiy. This paper applies hese resuls o he heoreical half sine an oher siple es pulses, an in so oing, esablishes hree unappreciae characerisics of he siple shock pulses. Inclusion of he rop an any reboun in he pseuo velociy shock specru ploe on four coorinae paper (PVSS on 4CP) shows he siple pulses siilar, as observe by Gerel an Hollan [4]. Shock specra for explosive environens are shown siilar o hose of siple pulse ess prouce on a rop able excep for irecionaliy. The rick urne ou o be inclusion of pre an pos pulse oion in he shock specru calculaion. Shock specru calculaions applie o shaker generae shock pulse oions show ha siple pulse ess conuce on a shaker wih liie isplaceen o no ach he low frequency severiy of rop able shocks. A ifficuly wih his paper is as follows. You can confir wha I have one unless you have a shock specru progra an he abiliy o plo he resuls on four coorinae paper. In ha sense his is a "rus e" paper. Goo shock specru progras are no coon. I will give you y MATLAB progra, [5, 6], an assure you ha I have been using i for 0 years or ore; I see no bugs in i. Only recenly have I prograe goo 4CP, bu ha correcness is easily verifie. PSEUDO VELOCITY AND FOUR COORDINATE PAPER. The shock specru is a plo of an analysis of a shock oion (i.e., ransien oions ue o explosions, earhquakes, package rops, railroa car buping, vehicle collisions, ec.) ha calculaes he iu response of any ifferen frequency ape single egree of freeo syses (SDOF) expose o he oion. The plo is a graph of iu response versus frequency. Pseuo velociy is specifically he iu relaive isplaceen ies frequency in raians. I is surprising an no rivial, bu his is he iporan quaniy o plo for shock specra when you wan o inicae severiy or capaciy o cause aage. The bes way o plo hese is on

2 four coorinae paper (4CP) (also calle riparie paper). Four coorinae paper is a logarihic graph paper ha has four ses of lines relaing frequency, isplaceen, velociy, an acceleraion of sinusoial oions. The shock specru algorih fins he peak relaive isplaceen for a base excie SDOF. Tha's he iu energy sore in he elasic eber uring he ransien even. A ha insan of ie he velociy was zero, hus here was no aping force, so he force in he spring a ha insan ies he ass was he acceleraion a ha insan oo. If ha iu elasic energy sore in he spring uring he shock were convere o velociy (kineic energy) in he nex or previous quarer cycle, ha velociy woul be (equaing he kineic an elasic energies): kz v k, or v z (1) Since k/ ω, we fin: v ωz (1a) Tha v is he pseuo velociy. We calculae z for each frequency, an we plo ωz on four coorinae paper. The reasons why PV is so goo are a lile clouy an ifficul. The ypical engineering srucures explanaion is given in Reference [7]. I nees ore hough, bu I'll give you y bes now. Peak oal velociy in elasic srucure is proporional o peak sress, an no acceleraion as os sill o no know [, 8, 9, 10, 11, 1]. Say we have a seel chair, oune on a shaker se o shake i horizonally an fin is firs oe. The op of he back of he chair will have he highes oal velociy (easure he peak acceleraion an ivie i by he frequency in raians). The peak sress is probably where one of he chair legs is wele o he sea. Tha peak sress in psi will be a consan beween 1 an 5 ies 146 ies ha velociy in ips. Tha is a fac [, 9], an i is also rue for secon an hir an all of he oes if we know for each, he peak velociy locaion an he peak sress locaion. Pseuo velociy is he bes quaniy o preic he poenial o generae oal velociies in srucure. Relaive velociy oes no inicae oal velociy a low or high frequencies; i aches PV only in cener severe secion of he shock specru. The asypoic behavior of he PVSS is suarize as follows [, 3]. When he PVSS is ploe on 4CP, he isplaceen is exac. One can expec o see wo asypoic values: a he high frequency en of he shock specru he curve shoul approach he peak pulse acceleraion, an a he low frequency en of he specru i shoul approach he peak shock eflecion. For inereiae values of he frequency, he peak pseuo velociy is ofen alos consan. Four coorinae paper (4CP) (riparie paper) is iporan o he PVSS presenaion. For a sinusoial oion, he isplaceen, velociy, an acceleraion are: z z sin ω, z& ωz cosω, & z ω z sin ω (a) By equaing he iu values we have: & && && & ω z z, ω z z, z ωz (b) The four quaniies: z, z&, && z, anω or πf, are relae by Eq. b. Knowing any wo, you can calculae he oher wo. By aking logs he lines of consan z, z&, & z, versus ω, or πf, are all sraigh lines. Tha is why we can copue he plo of Figure 1.

3 THE SIMPLE PULSES HAVE SIMILAR SHOCK SPECTRA WHEN PLOTTED AS PVSS ON 4CP To ake he poin of siple pulse siilariy, I have o calculae he PVSS on 4CP for all he siple shocks an here is no space o presen iniviual plos for each shock. The obvious way o show he siilar is o superpose all of heir specra on a single coposie plo an ha is one in Figures an 3. Below I will escribe he characerisics of each shock an he coproises I ha o ake o prove his poin. a. Half Sine Shock Half sine shocks are usually specifie by a peak acceleraion an a uraion, such as an 11 illisecon 30g half sine. The shock has peak acceleraion, & x&, an uraion ; here is a frequency associae wih his uraion since i is half of one perio of a sine wave wih frequency, f 1/. 1 & x & x sin πf, where f (1) The velociy change uring he half sine pulse is very iporan. Assue zero iniial velociy an inegrae over he half cycle o ge he final velociy, which is he velociy change. && x x && x& () πf π This resul is iporan. I relaes hree iporan pulse properies: velociy change, peak acceleraion, an uraion. Two of hese, x&, && x, are apparen on he pseuo velociy shock specru. The shock specru of his pulse by iself is wha is usually copue. A package is siing on he able an soeboy bels he able upwar wih a rubber alle in a zero g siuaion, an he package coninues going off ino space a consan velociy. I is unrealisic, an leaves he low frequency asypoe (conen or lii) ou. Ye we fin his in every book on shock ha presens he half sine shock specru. I a going o plo any shock specra in his paper an I a going o selec fairly severe paraeers so you can ge failiar wih severe shock specra. I will use a velociy change of 100 ips an peak g level of 00gs. Le's plo he above shock along wih is wo inegrals, an hen calculae he shock specru for his nue half sine shock. Using Eq. () we fin he uraion o be.034 s. Figure 4 shows he acceleraion shock an is wo inegrals. This is wha I avise you o learn o expec. The velociy change of 100 ips shows up as i shoul an he acceleraion level is assypoic o 00gs, as expece. The fac ha he pulse we analyze inicaes consan velociy of he shock able for ever, eans ha an SDOF syse wih a naural frequency of 0.1 Hz woul have a peak eflecion of abou 140 inches, or abou 1 f. I is an excellen plo of an analysis of a ub pulse. Now we consier a realisic half sine shock. We rop i hrough a isance, such ha a shock prograer elivering our half sine, jus brings i o res. The velociy begins an ens a zero, hus he shock will have a zero ean. A 1g fall hrough for a ie rop so ha he area (g x rop ) equals he velociy change, an since v gh, he rop heigh is also foun fro Eq. (3). The ie hisory plo an he inegrals are shown in Figure 5. g rop x& x& x, & or rop, an h (3) g g This is a shock ha coul occur Is shock specra are shown as he re curves on Figures an 3. Noice now we see he 13-inch assypoe a he low frequencies. We also see he 100 ips velociy change an he 00g peak assypoe clearly.

4 b. Trapezoial Shock We nex consier a rapazoial pulse of uraion, an iu acceleraion apliue, A, expresse in gs. I has a rise ie, T r, an a fall ie of T f. These efiniions are aken fro Reference [13], Figure , on page For he ieal case we are calculaing we will ake he rise an fall ies o be φ ies he pulse uraion,. The uraion inclues he raps up an own. The area of his pulse is equal o he velociy change i causes. The linear rap rapazoial equaion relaing velociy change, peak acceleraion, an uraion is given in Eq. (4). x& (4) ( 1 φ) && x I use MATLAB o calculae he ie hisory preceee by 00 zeros, he 1g rop o aain he 100 ips, he rap up, he fla porion, an he rap own followe by 00 zeros. Tha ie hisory for φ 0.1 an is wo inegrals are given in Figure 6. The ape an unape shock specra are given in Figure 7. The 100 ips fla porion is severe fro abou 1.5 o 00 Hz. Noice ha he high frequency acceleraion sars as asypoic o abou 400gs, an hen sars heaing for he 00g a 5,000 Hz. This is ue o he ipulsive rise ie of he rapazoial acceleraion. To prove his poin I re-i he rapezoial shock wih a half cosine rap wih a rap uraion of φ. The oal velociy change is (wih a cosine rap on he beginning an he en of he rapezoi) he sae as given in Eq. (4). To figure ou how uch of a cosine rap woull be reasonable wihin he confines of he IEC Specificaion, [14] accoring o Figure 3., on p40, I ploe cosine raps insie he olerances given in heir Figure 3. This easily perie use of a a φ 0.3. The resuling ie hisory is shown in Figure 8. This is sufficien rise ie aelioraion o be able o see he 00g asypoe of he shock specra. The specru is shown as he green curve on Figures 1 an 4. Noice ha his soohs i enough o peri i o ive own o 00gs a abou 500 Hz. The 5% ape specru is also as expece. All one can say is ha an abrup rise oubles he peak g level over a range before he peak g asypoe appears. c. Saw Tooh Shocks In he case of hese riangular shocks, he velociy change uring he pulse will be is area, one half of he heigh ies he uraion which yiels: & x (5) && x This can no be exac because when you igially raw i, here has o be an aiional area of he lile riangle fro he peak own o he firs zero saple. We will ge ou of his clusy coplicaion by aing a cosine rap as was one wih he rapezoial shock. Noice ha he iniial peak sawooh will have he sae velociy change. In hese pulses we assue ha shock prograers on he shock achines bring he able o zero velociy so ha he velociy change is equal o he velociy fro Eq. (5). The cosine fall off will occur in a ie inerval φ. Here eans he oal pulse uraion incluing he linear rap up fro 0 o acceleraion an he cosine fall off back own o zero. Carrying ou he inegraion yiels he sae velociy change equaion, Eq. (5), inepenen of φ. A linear rap has he sae area as a cosine rap. Graphically checking he olerance levels of MIL STD 810F, Figure [13], shows ha a φ 0.1 cosine is easily accepable. I will o he analysis of erinal peak an iniial peaks shocks wih his φ 0.1, cosine rap o save rouble. Now using Eq. (5) o efine he for our 100 ips velociy change, 00g shock, we ge he ie hisory, an i is wo inegrals shown in Figure 9. This also requires a 1.95-inch rop. The shock specru for wo aping values, 0 an 5%, are given as he blue curves of Figures an 3. The 13-inch rop an he 100 ips severe region are righ where hey shoul be.

5 The curves coe own o he 00g asypoe a he lowes frequency of any of he siple pulses. This is because of he graual rise in acceleraion. Trouble evelops wih he iniial peak saw ooh an is abrup rise. Le s look a he iniial peak ie hisory wih a 10% cosine rap up o he iniial peak shown in Figure 10. The 10% cosine is ifficul o see bu i is here. We have rouble wih he shock specru, shown in Figure 11, because of he seep iniial rise. Noice an iniial oubling of he acceleraion assypoe in os of he high frequency region. A 5,000 Hz i is efinielly heaing for he 00g line, bu oes no ge here. If I increase φ o 0., we will solve he proble. The ie hisory wih he 0% cosine rap is illusrae in Figure 1. Coparing his o he erinal peak olerance figure in Reference [13], i is clear ha a 0% rise is wihin he specificaion. The resuling shock specra are shown as he black curves in Figure an 3. This is ore graual rise is aequae o peri he 00 Hz asypoe o appear.. Coposie plos an he siilariies of he shock specra Now we have coplee he exainaion of he PVSS of he siple pulses: he half sine, he rapezoi, he iniial peak an he erinal peak saw ooh. We observe ha boh ape an unape, he PVSS's are siilar. The reason ha we can see ha hey are siilar is ha I scale he o have equal velociy change an peak acceleraion. The unape an 5% ape coposie plos are shown in Figures 11 an 1. This is very ineresing an iporan. Afer a huge aoun of copuing an ploing, he shock specra reveal he siilariy. Only in he high frequency region, where he velociy levels are becoing less severe o hey iverge. Gerel's [4] off he cuff coen ha ha all he siple pulses are siilar is confire if no proven. The erinal peak saw ooh coes own o he 00g asypoe faser han he oher hree because i has a ore graual rise. The velociy change uring he pulse was he siilar hing abou he. They all require he sae rop heigh so hey all have he sae low frequency asypoe. All of he siple pulses evelope on a rop able shock achine by a prograer ha resuls in zero velociy when he pulse is over will have a velociy change of he square roo of gh. They will all have he sae rop heigh or iu isplaceen, hence he sae low frequency asypoe. Since hey all have he sae velociy change, hey all have he sae cenral region. Since I ajuse he pulses o have he sae peak acceleraion, hey all us have he sae high frequency asypoe. The only way heir shock specra can iffer are a he wo corners, an his can be seen in Figures an 3. I ha rouble geing he acceleraion asypoes o appear in he rapezoi an he iniial peak saw ooh. These have abrup rise ies ha cause an iniial oubling of he peak acceleraions. I ha o ecrease he rise ie abrupness by using a half cosine rap rise of 30% in he rapezoi an 0% in he iniial peak wave for. Two oher iporan conclusions have o be esablishe. The half sine, an hence he oher siple pulses are relae an in a sense siilar o he explosive shocks, he rop is very iporan. Shock siulaion on shaker where he rop is no aainable weakens if no ruins he frequency range over which he shock is severe. SHOCK SPECTRA FROM EXPLOSIVE EVENTS ARE SIMILAR TO SIMPLE PULSE TESTS One exaple will be given o explain he siilariy. Figure 13 is an acceleraion ie hisory of an explosive even an is wo inegrals. The ean has been reove fro he acceleraion ie hisory o assure ha he velociy ens a zero. The fac he exree values are all inia akes no ifference. Figure 14 gives is shock specra for hree apings. Noice on he lef all hree curves are asypoic o jus uner 9 inches, ha he cener severe region is hovering near 300 ips, alhough he oal velociy change on Figure 13 is close o 400 ips. In he iniial par of he shock here is 300 ips ore or less rapi change. A he highes frequencies he ape curves are a leas heaing for

6 abou 900gs. I woul have o calculae uch higher frequencies o see he asypoe on he unape specru. (To o his, one has o inerpolae he ie hisory using MATLABs Inerp funcion o increase he sapling rae) The poin here, is ha a siple pulse wih a peak g level of 918gs, an a velociy change of 303 ips, woul have a shock specru aching he severe an high frequency regions. To aain he velociy change wih only a 9-inch rop woul require a bungee or spring assis, bu i coul be one, if a g level of 13.8gs coul be obaine. This woul give he 9-inch isplaceen low frequency asypoe. In his sense I ake he saeen ha he siple pulse ess are siilar o explosive shock specra. SHAKER GENERATED HALF SINE SHOCK AND REBOUND REDUCE LOW FREQUENCY CONTENT a. The Half Sine Shock wih a Reboun In he case of he half sine shock, here is ofen a coefficien of resiuion o eal wih because i can be fore by ipac wih a rubber like pa. In exaining pas aa, I have foun values fro 0.3 o 0.5. In his case he require rop heigh is reuce because he velociy change is he su of he falling velociy an he reboun velociy. The velociy change uring he half sine shock is sill given by Eq. (), bu now he falling an reboun velociy are given in Eq. (6). x& v v r f ev + v x& v (1 + e) f f r x& v f 1+ e The rop an reboun heighs, assuing he able is caugh when he velociy goes o zero are given by Eq. (7). (6) v v f r gh gh f r (7) The ie hisory for our 00g, 100 ips half sine shock ipacing wih a coefficien of resiuion of 0.33, as an exaple is given in Figure 15. This reuces he rop heigh fro abou 13 inches o 7.3 inches. The unape an 5% ape shock specra are given in Figures 16 an 17 as he black curves. The re curves on Figures 18 an 19 are for a half sine shock fore by a prograer wih no reboun. The isplaceen asypoe is now a 7.3 inches, an he low frequency severe velociy range is ecrease or he lowes severe velociy increase fro abou 1.5 Hz o abou 3 Hz. b. Shaker Generae Half Sine Shocks Half sine shock ess are also conuce on an elecrically riven shaker an hese have a liie isplaceen capabiliy. Therefore, he shock specra of shaker generae shock will reflec his wih a reuce low frequency capabiliy of shaker generae shock. Shaker generae shocks will have inaequae low frequency severiy. Lang [15] consiers a hos of pre an pos pulses ha allow he shaker araure o sar fro is cener posiion, perfor he half sine shock, an reurn he araure o is cener posiion. Le s consier a half sine wih recangular pre an pos pulses wih a agniue of θ ies he iu acceleraion of he half sine o zero he ean. The area of he pre an pos pulses us equal he velociy change fro Eq. (), which yiels he pre an pos pulse uraion, p in Eq. (8). The ie hisory an is inegrals are given in Figure 18.

7 θ&& x p p πθ && x πf x && π (8) Fro he shaker owner's poin of view, his is no goo. The shaker araure oion is all in one irecion, however he peak isplaceen is only abou 0. inch, an his liis he low frequency severe porion of is shock specru. The unape an 5% ape shock specra are shown as he green curves on Figures 16 an 17. The shock now is only severe fro abou 60 o 00 Hz. In Figure 7, of Reference [15], Lang shows a recanguar pre-pre posiive pulse of abou 1/3 he uraion of he pre pulse, o cener shaker. Le s say he pre-pre, he pre, he pos, an he pos pos pulses are recangular an have agniues of θ ies he iu acceleraion of he half sine. For syery ake he uraion of he pulses p, an p /3. The pre an pos pulses us again accoplish he sae velociy change as he half sine an p is given by Eq. (9). The ie hisory an is wo inegrals are given in Figure 19. θ&& x p p θ&& x 3 3 πθ p x && π (9) The one-hir esiae was no correc since I i no achieve equal posiive an negaive isplaceen. I akes he poin ha we can reuce he shaker excursion; see Reference [15] for he exac resul. Since he isplaceen is furher reuce, he shock specru us show a low frequency asypoe of 0.11-inch which can be seen in he blue curves on Figures 16 an 17. Coparing shaker shocks wih he rop able shocks, one noes a reuce high velociy severe region. Shaker siulae half sines woul be inaequae for achinery an equipen wih lower oal frequencies. This is excluing he shocks synhesizing a shock specru wih a collecion of oscillaory oions. The beauy of shaker shock is ha he irecion of he shock on is polariy, can be reverse. CONCLUSIONS The PVSS on 4CP (pseuo velociy shock specru ploe on four coorinae paper) ephasizes a velociy change secion of he specru, enione by Robers [16] as severe secion. The IEC Specificaion [14] calls he siple pulse velociy change he severiy of he pulse, which I copleely agree wih. This view of he specru was he basis of equipen insallaion esign wihou acually calling i he velociy change region. [1, 3] The shock specra of he siple pulses are siilar an have siilar aage poenial. Reference [4] is he only reference I have foun ha acually akes he saeen: (page 6)"...he i of he shock specru curve of all siple pulses have he sae general shape." They are righ. They ake i so casually ha one igh assue i was coon knowlege. Mos auhors [17, 18] ephasize he uniporan high frequency ifferences of he unape specra because hey plo acceleraion shock specra. The low frequency conen inaequacy of shaker generae siple pulse shock ess for equipen fragiliy esing akes clear he nee o inclue he rop heigh in half-sine shock calculaions. The general siilariy of he PVSS on 4CP of he siple shock pulse ess an explosive ess argues for heir usefulness in general fragiliy esing.

8 REFERENCES 1. Eubanks, R.A. an Juskie, B.R., Shock Harening of Equipen, Shock an Vibraion Bullein 3, Par III, 1963, pp "Reasons for Presening Shock Specra wih Velociy as he Orinae", by H.A. Gaberson, an R. H. Chalers; Proceeings of he 66h Shock an Vibraion Syposiu, Vol , pp Gaberson, H.A., an Eubanks, Ph.D., S.E.,R.A.,"Siplifie Shock Design for Equipen Insallaion," NCEL Technical Noe, N-16, March 198. (ADA AD114331) 4. Gerel, Mike, an Hollan, R.,"A Suy of Selece Shock Analysis Mehos", A Repor Allie Research Associaes, Inc, Concor, MA, one uner conrac for U.S. Ary, Qualiy Assurance Direcorae, Frankfor Arsenal, Philaelphia, PA April (ADAD81480) 5. Malab for Winows, Version 5.3, 1999, High Perforance Nueric Copuaion an Visualizaion Sofware, The MahWorks, Inc., 4 Prie Park Way, Naick, MA. 6. Gaberson, H.A., Shock Specru Calculaion fro Acceleraion Tie Hisories, Technical Noe N-1590, Civil Engineering Laboraory, Por Huenee, CA, (ADA09716) 7. Hall, W. J., "Vibraion of Srucures Inuce by Groun Moion", Chaper 4 of "Harris Shock an Vibraion Hanbook", 5h E. by Harris, C. M. an Piersol, A.G., 00. pp Hun, F. V. Sress an Srain Liis on he Aainable Velociy in Mechanical Vibraion; J. Acousical Soc. A. Vol 3, no. 9, Sep 1960, pp Gaberson, H.A. an Chalers, R.H., Moal Velociy as a Crierion of Shock Severiy, Shock an Vibraion Bullein 40, Par, Dec 1969, pp Ungar, E.E., Maxiu Sresses in Beas an Plaes Vibraing a Resonance, Trans ASME, J. Engrg. In., 196, v 3, n1, pp , Feb. 11. Cranall, S.H., Relaion beween Srain an Velociy in Resonan Vibraion, J. Acous. Soc. Aer., Vol. 34, No. 1, Dec 1960, pp Lyon, R.H., Saisical Energy Analysis of Dynaical Syses, MIT Press, Cabrige, MA, 1975, p MIL-STD-810F: Meho Shock; IEC Shock Specificaion, 68--7; Basic environenal esing proceures; Par : Tess - Tes Ea an Guiance: Shock. Thir Eiion Lang, George Fox, "Shock'n on Shakers", Soun an Vibraion, v37, n9, Sepeber Robers, W.H., Explosive Shock, Shock an Vibraion, Bullein 40, Par, Dec 1969, pp Minlin, R.D., Dynaics of Package Cushioning, Bell Syse Technical Journal, Vol. 4, Jul-Oc 1945, pp Ayre., R.S., "Transien Response o Sep an Pulse Funcions", Chaper 8 of "Harris' Shock an Vibraion Hanbook", 5h E. by Harris, C. M. an Piersol, A.G., 00.

9 Figure 1. The PVSS os a 00-g, 100 ips half sine aceleraion shock. In he lower righ corner he specra are asypoic o 00gs. Fro a lile uner 00 Hz own o 0.1 Hz he specra show a consan velociy of 100 ips. Tha proably can no be. Figure. Coposie unape shock specru plos of he half sine, he rapezoi, he iniial peak an he erinal peak saw ooh shocks. Figure 3. Coposie 5% ape shock specru plos of he half sine, he rapezoi, he iniial peak an he erinal peak saw ooh shocks. Figure 4. Acceleraion, velociy, an isplaceen of a half sine acceleraion shock precee an followe by zeros. The shock has a velociy change of 100 ips; an coninues a his velociy forever. This is an unrealizable shock; bu he shock specru for i unape an wih aping of 5 %, is given in Figure. Figure 5. A 00-g, 100 ips half sine shock precee by a 1.95-inch rop, an is wo inegrals. Figure 6. Ten percen linear rap rapazoial shock precee by a 1.95-inch rop.. The pulse uraion urne ou o be 1.44 ili-secons.

10 Figure 7. The 0 an 5 % ape shock specra for he 10 percen linear rap rapezoial shock of Figure 4. Figure 8. Trapezoial shock wih 30% cosine rap precee by a 1.95-inch rop. The ie uraion for his pulse cae ou o be 1.85 ili-secons. Figure 9. The ie hisory of a 00-g, 100 ips, erinal peak saw ooh shock wih a 10% cosine rop back o zero, an is iegrals o velociy an isplaceen. Drop heigh is 1.95 inches, an pulse uraion is.59 s. Figure 10. The ie hisory of a 00g, 100 ips iniial peak saw ooh shock wih a 10% cosine rap up o ha iu acceleraion, an is iegrals o velociy an isplaceen. Drop heigh is 1.95 inches, an pulse uraion is.59 s. Figure 11. The shock specru of a 00-g, 100 ips iniial peak saw ooh shock wih a 10% cosine rap up o he iu acceleraion. Drop heigh is 1.95 inches, an pulse uraion is.59 s. Figure 1. The ie hisory of a 00-g, 100 ips iniial peak saw ooh shock wih a 0% cosine rap up o he iu acceleraion, an is iegrals o velociy an isplaceen. Drop heigh is 1.95 inches, an pulse uraion is.59 s.

11 Figure 13. The acceleraion ie hisory an is wo inegrals for an exaple explosive shock. Figure 14. Shock specra of an exaple explosive shock es. Figure 15. The acceleraion ie hisory an is wo inegrals for a 00g, 100 ips half sine shock rebouning wih a coefficien of resiuion of The crop an reboun heighs are 7.3 inches an 0.8 inch, respecively. Figure 16. Shock specru coparison of shaker an rop able 00g, 100 ips, half sine shocks for he unape case: Figure 17. Shock specru coparison of shaker an rop able 00g, 100 ips, half sine shocks for he 5% ape case. Figure 18. Acceleraion ie hisory wih 10% recangular pre an pos pulses o accoplish he 00g, 100 ips half sine shock on a shaker.

12 Figure 19. Acceleraion ie hisory wih wo 10% recangular pre an pos pulses o accoplish a 00g, 100 ips half sine shock on a shaker, an ore nearly have equal posiive an negaive shaker isplaceen.