SIMPLE SHOCKS HAVE SIMILAR SHOCK SPECTRA WHEN PLOTTED AS PVSS ON 4CP

Transcription

1 SIMPLE SHOCKS HAVE SIMILAR SHOCK SPECTRA WHEN PLOTTED AS PVSS ON 4CP By Howar A. Gaberson, Ph.D., P.E. Consulan 234 Corsicana Dr.; Oxnar, CA 93036; (805) ABSTRACT The pseuo velociy shock specrum ploe on four coorinae paper (PVSS on 4CP) emphasizes shock moion severiy. This paper applies his fac o he heoreical half sine an oher simple shocks. By aing he rop acceleraion an comparing simple rop able shock machine shocks wih he same velociy change uring he impac, I show all have essenially he same PVSS. Bu when he calculaions are applie o shaker generae simple shock moions, hey urn ou no similar an hey seriously lack low frequency conen. I also consier how simple shocks can be relae o acual explosive ransiens. INTRODUCTION Previous work has shown he raiional four coorinae pseuo velociy shock specrum (PVSS on 4CP) is he bes forma for violen machine ransien founaion moion analysis. [1] The pseuo velociy shock specrum ploe on four coorinae paper emphasizes shock moion severiy. This paper applies his fac o he heoreical half sine an oher simple shocks. By aing he rop acceleraion an comparing simple rop able shock machine shocks wih he same velociy change uring he impac (which means same rop heigh), I show all have essenially he same PVSS. The rick urns ou o be inclusion of pre an pos shock moion in he calculaion an comparing shocks wih he same impac velociy change. Bu when he calculaions are applie o shaker generae simple shock moions, hey urn ou no similar; hey seriously lack low frequency conen. I also show how he simple shocks can be relae o acual explosive ransiens. Focusing on he velociy change uring impac helps. I'll say here are five simple shocks, an hey are he half sine, haversine, iniial an final peak saw ooh, an rapezoial shocks. You will also fin analyses of riangular shocks wih various asymmeries, an square shocks. Many pioneers of our business wen on an on calculaing posiive an negaive, iniial an resiual acceleraion shock specra of he simple shocks, an he calle hem shock response specra (SRS's). The misake hey mae was o consier only he SRS, an o normalize all of he specra o peak acceleraion. Typical references inclue Minlin[2], Jacobsen an Ayre[3], Ayre[4], Rubin[5], all he famous of shock analysis auhors. Their analyses no longer nee o be suie. Lalanne [6] i a sensible careful analyses of he kinemaics, as he erms i, of he simple shocks, bu never rie o calculae PVSSs of hem; a he ime he wroe his book, he was no emphasizing PVSS heory. One auhor pair, Gerel an Hollan [7], go i correc bu everyboy ignore hem. Anyhow, o ry o se maers sraigh, his paper shows you why he simple shocks are all basically he same, an equally goo es shocks. This paper is an expansion of a paper I wroe in [8] To make he poin of simple shock PVSS similariy, I have o calculae he PVSS on 4CP (pseuo velociy shock specrum ploe on four coorinae paper) for all he simple shocks. The way I show hem similar is o show he PVSSs separaely an hen superpose all of heir specra on a single composie plo. Below I will escribe he characerisics of each shock an he compromises I ha o make o esablish he ruh of he ile of his paper. I have assume no reboun in he following simple shock analyses. Expers have ol me ha his is generally no he case. However, o keep hings uncomplicae I will procee wih he no reboun analyses. Reboun affecs he low frequency asympoe an his is carefully iscusse in an appenix available from me. HALF SINE SHOCK Half sine shocks are usually specifie by a peak acceleraion an a uraion, such as an 11 millisecon 30g half sine. To evelop a formula, I'll say he shock has peak acceleraion, x, an uraion. The frequency associae wih he half sine uraion has a perio of wice his uraion, an since he frequency is one over he perio, he frequency associae wih he half sine is given by Eq (1a). 1

2 f 1 (1a) 2 The formula for our half sine shock is given by Eq (1). x x sin 2 f ( 1) The velociy change uring he half sine pulse is very imporan. Assume zero iniial velociy an inegrae over he half cycle o ge he final velociy, which is he velociy change. 1 x x sin 2 cos 2 cos 1 2 f 2 0 f 0 f x x f x f (2a) I'll express he resul in erms of uraion using Eq (1a) in Eq (2). 2x x (2) This resul is imporan. I relaes hree imporan pulse properies: velociy change, peak acceleraion, an uraion. Two of hese, x, x, can be rea off he pseuo velociy shock specrum. The shock specrum of his pulse by iself is wha is usually compue. A package is on he able an somehing elivers a half sine shock upwar in a zero graviy siuaion an he package an able coninue going off ino space a consan velociy. This is unrealisic, an leaves he low frequency asympoe or isplacemen limi ou. Ye we fin his in every book on shock ha presens he half sine shock specrum. I wan you o see why his is incomplee. I'm going o plo many shock specra in wha follows an I'm going o selec fairly severe parameers so you can ge familiar wih severe shock specra. I will use a velociy change of 100 ips an peak acceleraion level of 200 g. Le's plo he above shock along wih is wo inegrals, an hen calculae he PVSS on 4CP for his unrealisic half sine shock. Using Eq (2) we fin he uraion o be ms. Figure 1 shows he acceleraion shock an is wo inegrals. The PVSS for i unampe an wih amping of 5%, is given in Figure 2. Figure 1. Acceleraion, velociy, an isplacemen 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. 2

3 Figure 2. The PVSS of a 200 g 100 ips half sine acceleraion shock. In he lower righ corner he specra are asympoic o 200 g. From abou 100 Hz own o 0.1 Hz he unampe specrum shows a consan velociy of 100 ips. In Figure 2, he consan velociy of 100 ips own o 0.1 Hz probably i no happen. The PVSS ploe on 4CP inicaes ha he shock woul cause he 0.1 Hz SDOF a eflecion of 160 inches or 13.3 f. Assuming i was one in rop able shock machine; we woul selom have 13 f available, an if so a 13 f rop woul cause a much greaer velociy change han 100 ips..this wha you are liable o see, an I on' like. The velociy change of 100 ips shows up as i shoul an he acceleraion level is asympoic o 200 g's, as expece. The plo oesn' show how low a frequency coul be excie o 100 ips. I's an excellen plo of an analysis of a unreal pulse. Now we consier a realisic half sine shock. We rop i hrough a isance, such ha a shock programmer elivering our half sine jus brings i o res. The velociy begins an ens a zero, hus he shock acceleraion will have a zero mean. To calculae he rop heigh, consier a 1 g rop for a ime rop so ha he area (g imes rop ) equals he impac velociy change, hus Eq (3a) will give rop. g rop x x, or rop (3a) g 1 2 For a freely falling boy saring from res, he velociy will be g, an he isplacemen will be h g. 2 Subsiuing he rop value from Eq (3a), we ge Eq (3b) for he necessary rop heigh. The ime hisory plo an he inegrals are shown in Figure 3. 2 x h 2g (3b) 3

4 Figure 3 A 200 g, 100 ips half sine shock precee by a inch rop, an is wo inegrals. The op subplo has an expane x-axis o show he acual pulse shape. This is a shock ha coul occur. Is unampe an 10% ampe PVSS on 4CP are shown in Figure 4. Noice he low frequency limi of he 5% ampe shock specrum is abou 2 Hz, an he high frequency limi is 200 Hz. This is he frequency range for which he shock is severe. Noice also, as he specrum roops ownwar o he righ heaing for he acceleraion asympoe, ha for a shor range of frequencies, Hz, ha he acceleraion nearly oubles over he 200 g asympoe level. This occurs o some exen on all five simple shock PVSSs. I will refer o his as oubling in he roop zone, an will bring i up four more imes. SIMPLE SHOCK PVSS CHARACTERISTICS Figure 4 illusraes imporan conceps ha I emphasize when eaching shock aa analysis. The PVSS on 4CP of a simple shock has a shape like a flaene hill. The op of he hill, he plaeau, is he impac velociy change for an unampe PVSS, somewha less in a ampe PVSS analysis. The hill slopes own an o he righ wih an asympoe equal o he peak acceleraion of he shock. The high frequency asympoe is he imum shock acceleraion. The hill slopes own an o he lef wih an asympoe equal o he imum shock isplacemen. The low frequency asympoe is he imum isplacemen. You will see his on all simple shocks. I carefully explain his in [1]. Simple shocks begin an en wih zero velociy, an have a imum isplacemen. If hey in' en wih zero velociy he isplacemen woul increase wihou boun. Tha is why we will see ha all simple shocks have similar PVSS of 4CP when scale o he same impac velociy change. Tha's wha I'm going o show you, an is he main poin of he paper. 4

5 Figure 4. PVSS of 200 g, 100 ips half sine, precee by a 1 g rop. Noice now we see he 13" asympoe a he low frequencies. The 100 ips velociy change an he 200 g peak asympoe clearly. Noice however, as he specrum roops ownwar o he righ heaing for he acceleraion asympoe, ha for a shor range of frequencies, Hz, ha he acceleraion nearly oubles over he 200 g asympoe level. This occurs on all five shock PVSS's. I call his as oubling in he roop zone. TRAPEZOIDAL SHOCK Abrup rise imes, while ineresing o he acaemic communiy, on' occur in shocks. Trying o eal wih hese pracically requires iscussion an I'll eal wih ha here. As I hope you realize, he PVSS on 4CP shows he poenial velociy or sress he shock can evelop in es iems. Thus our ool for evaluaing sligh changes in simple shock shapes is he effec on he PVSS. A m T r T f T Figure 5. This is a rapezoial shock of uraion T, an imum acceleraion ampliue, A m, expresse in g. I has a rise ime, T r, an a fall ime of T f. Nex o i is a copy of he rawing from Mil S 810 F. [9] Figure 5 shows a rapezoial shock of uraion T, an imum acceleraion ampliue, A m, expresse in g. I has a rise ime, T r, an a fall ime of T f. These symbols an efiniions are aken from MIL-STD 810F, [9] Figure , on page which is reprouce above. The area of his pulse is equal o he velociy change i causes. Les a up hese wo riangular areas an he recangular cener in Eq (4a) an ake hree lines o show he simplificaions. Here I'll ake g 0 o mean he numerical value in/sec 2, an A m o mean he acceleraion in g. 5

6 V A gt A g T T T A gt A g T T T (4a) m 0 r m 0 r f 2 m 0 f m 0 2 r 2 f For he ieal case we are calculaing we will ake he rise an fall imes equal an given by imes he pulse uraion. This subsiuion of,, imes he shock uraion, T makes he resul iy as shown in Eq (4b). 1 1 m m 0 V A g T T T (1 ) A g T (4b) Using beer symbols we have he linear ramp rapezoial equaion relaing velociy change, peak acceleraion, an uraion given in Eq (5a). x 1 x (5a) The acceleraion equaions for he hree ime inervals are given in Eqs (5 b, c,.) x x x x x x,, 0, 1, 1, (5 b,c,) I wroe a linear ramp rapezoi shock MATLAB scrip, halfrapz.m, o calculae he ime hisory precee by 200 zeros, he 1 g rop o aain he 100 ips, he ramp up, he fla porion, an he ramp own followe by 200 zeros. (I wroe programs o o his for all 5 simple shocks. I'll be happy o sen hem o you.) This ime hisory for 0.1 acceleraion an is wo inegrals are given in Figure 6 Noice ha he op subplo has an expane ime scale o show he shock shape. Figure 6. This is a ime hisory for a 10% linear ramp rapezoial shock. The rop is inches. The pulse uraion urne ou o be 1.44 mili-secons. 6

7 To illusrae "oubling in he roop zone", I wan o firs show he unampe PVSS of a 5% linear rampe rapezoi in Figure 7a. This is wha I mean by he ifficuly wih abrup rise imes. The 100 ips fla porion is severe from abou Hz. Noice, however, ha he acceleraion is mosly a 400 g from abou 400 Hz o abou 4000 Hz, insea of he 200 g asympoe I wan you o expec. The specrum is beginning is ive o 200 a jus abou 4000 Hz. I'll show you ha his occurs on all he calculae abrup rise ime simple shocks I've one. I'm sure his is ue o he impulsive rise ime of he rapezoial acceleraion. Figure 7a. Unampe shock specrum for 200g, 100 ips velociy change rapezoial shock wih a 5% linear ramp. Noice ha he acceleraion asympoe is a 400 g mos of he way. Figure 7. The 0 an 5% ampe shock specra for he 10% linear ramp rapzoial shock of Figure 6. 7

8 The ampe an unampe shock specra for Figure 6, he 10% linear ramp rapezoial shock are given in Figure 7. The 100 ips fla porion is severe from abou Hz. Noice ha he acceleraion asympoe is mosly a 400 g from abou 400 Hz o 2000 Hz, insea of he 200 we are expecing. This rapezoial shock ha a 10% linear rise an fall, an now a 5000 Hz i is almos own o 200 g's. Even wih change o 10%, hings improve bu we are sill no own o 200 g's a 5000 Hz. Now I reo he rapezoial shock wih a half cosine ramp up; again le he ramp uraion be. Figure 8 shows he porion of he cosine I use. Figure 8. Cosine ramp geomery. The frequency of his cosine ramp has o be such ha uring he rise ime inerval, 0 o, he argumen of he cosine increases by. Thus he frequency of he ramp has o given by Eq f r, or f r (6) 2 The acceleraion uring he ramp inerval is given by Eq (7). The argumen of he cosine has o look like 2f; using f r from Eq (6), we evelop Eq (7). 1 x x 1 cos, for 0 2 (7) The area (or he velociy change) uring he cosine ramp is foun by inegraing Eq (7) in Eq (8a). 1 x x x x c cos sin 0 (8a) Evaluaing he limis yiels he simple answer shown in Eq (8b) x ; or x x 2 2 x 2 c c (8b) Thus he oal velociy change is (wih a cosine ramp on he beginning an he en of he rapezoi) is evaluae in Eq (8c). 8

9 x 0.5 x x 0.5 x x x 1 (8c) This is acually he same as Eq (5a) which can be seen by rearranging o obain Eq (8). The cosine ramp has he same inegral as he linear ramp. x ; or x 1 x 1 x (8) The acceleraion equaions for he cosine ramp up, an he fla cener region are given in Eqs (8 e). I in' ry o figure ou he relaion for he ramp own because Malab has a flip lef righ funcion, an I jus flippe he ramp up an appene i o ha fla porion. 1 x x 1 cos, 0, 2 x x, 1 (8 e) Again I wroe a cosine ramp rapezoi shock MATLAB scrip, halfrapcos.m, o calculae he ime hisory precee by 200 zeros, he 1 g rop o aain he 100 ips, he ramp up an he fla porion. The ramp own is obaine by flipping he ramp up lef righ an appening o he fla porion, followe by he 200 zeros. Now o jusify how much cosine ramp is reasonable, I graphically place a 30% ramp rapezoi shock wihin he confines of he IEC Specificaion, accoring o Figure 3, on p 40. [10] I i his in Malab an show he resul in Figure 9. Figure 9, The figure shows how 30% cosine rampe rapezoi fis in he limis of he IEC shock specificaion. [10] Now suying his picure, heir nominal rapezoi has a linear ramp wih phi = 0.1; he blue an green limis allow one o increase by 0.4, an hen use a cosine wih a = 0.3. I seems like raher han ge all confuse by he nominal an a permissible uraion, he I jus use his as proof ha i is accepable in some circumsances o use a = 0.3, an see wha his yiels in he specrum. Les look a he ime hisory of a 30% rapezoi shock in Figure 10. This ime hisory for phi = 0.3 an is wo inegrals are given in Figure 10. Noice ha he op subplo has an expane ime scale o show he shock shape. 9

10 Figure 10. Time hisory an inegrals for a 30% cosine ramp rapezoial shock. The ime uraion for his pulse came ou o be 1.85 mili-secons. Now his is sufficien rise ime o be able o see he 200 g asympoe of he PVSS shown below in Figure 11. Figure 11. PVSS for a 30 % cosine ramp rapezoial shock. Noice ha finally i reaches is 200 g asympoe a 2500 Hz. Noice ha finally i reaches is 200 g asympoe a 2500 Hz. Noice ha his 30% ramp smoohes i enough o permi i o ive own o 200 g a abou 2500 Hz. The five percen ampe specrum is now as expece. I have gone over his o show ha all he simple shocks have he expece simple shock asympoes, The abrup rise imes ouble he peak g level over a range before he peak g asympoe appears. HAVERSINE SHOCK The haversine shock (also calle a verse sine) is a raise cosine shape. I is shown in Figure 12. The haversine shock is anoher classical simple shock. I happens o be a cosine rampe rapezoi shock wih = 50%, so I can 10

11 use my Malab scrip halfrapcos.m wih = 50%, o generae i. The equaion relaing peak acceleraion, velociy change, an uraion is obaine from Eq (8) wih 0.5, an is shown in Eq (8f). 2x x x x 2 (8f) x Figure 12. Drawing of Haversine, or verse sine shock. Figure 13. shows a ime hisory of he 100 ips, 200 g haversine shock we will use for comparison. Finally he PVSS of his haversine shock is shown in Figure 14. I is wha I expec from he more genle iniial rise han he half sine. Figure 13. Time hisory an inegrals of haversine shock. Top subplo shows expane ime scale o see shock shape eail. 11

12 Figure 14. PVSS of haversine shock. I comes back o is 200 g asympoe a jus over 1000 Hz. Comparing Figure 14, he haversine PVSS, an Figure 4 he half sine PVSS, he haversine PVSS has a lile less oubling in he roop zone, an comes o he 200 g asympoe slighly sooner han he half sine. TERMINAL PEAK SAWTOOTH The saw ooh shocks are also common shocks, erminal peak being he mos common. Papers were wrien claiming his was a very goo es shock for various reasons; hey are unimporan P T Figure 15a. Terminal peak saw ooh rawing wih symbols from Mil-S-810-F, Meho 516.5, SHOCK, an Figure Figure 15b. This is he exac figure of he erminal peak shock reprouce from Mil S 810F [9] The velociy change associae wih he ieal shock is is area, one half of he heigh imes he uraion as expresse in Eq (9). (g 0 = in/sec 2 ) V (9) 0.5Pg0T I'll wrie his relaion in erms I like beer in Eq (9a). 12

13 ` 2x x x x 2 (9a) One minor poin is ha his can' be exac because when you igially raw i, here has o be a iny aiional area of he lile riangle from he peak own o he firs zero sample. I'll ge ou of his clumsy complicaion by aing eiher a sraigh line ramp or a cosine ramp from x own o zero. Noice ha he iniial peak sawooh will also have he same velociy change. The equaion for he ieal acceleraion of a erminal peak sawooh shock from 0 o x, is shown in Eq (10). xab,@ 0, x0; x x, a x x x (10) For he iniial peak sawooh we ge he resul in Eq (10a). x xab,@ 0, x x; b x0 a x, a x x 1 (10a) Now le's consruc a half cosine fall off from he peak since he insananeous rop o zero is pracically unaainable. We will have he fall off occur in a ime inerval. Here means he oal pulse uraion incluing he linear ramp up from 0 o acceleraion an he cosine fall off back own o zero. We wan o inser a half cosine of uraion, ; his cosine will have a frequency, f c efine by Eq (11). The argumen of a half of a perio of cosine is. 2f f c c 1 2, or (11) The acceleraion uring his cosine fall off is given by Eq (12) x x 1cos2 x 1 cos, for (12) Inegraing his o velociy over he cosine uraion gives Eq (12a). 13

14 1 1 x x sin x cos 0 1 xcos x 2 (12a) This velociy change (over he inerval ), which gives Eq (13). ) mus be ae o he velociy change of Eq (9a) (over he inerval x x x x 1, or x x 2 (13) Eq (13) gives he velociy change for a erminal peak sawooh shock wih a cosine fall off of Analysis of Figure 16, shows ha a = 0.1 easily fis he olerance levels of Mil S 810. [9] Figure 16. Scale rawing from Figure of Mil STD 810F. [9] The black cosine fall off ramp is a 10% ramp. Noice ha I can slie he re curve wih he black cosine fall off o he lef much more, cerainly enough o allow for a 20% fall off cosine ramp which we will nee for he analysis of he iniial peak shock. I'll o he analysis of erminal peak an iniial peak shocks wih his = 0.1 cosine ramp o save rouble. Now using Eq (13) o efine he for our 100 ips velociy change, 200 g shock, an Eqs (10) an (12) for he shock shape, we ge he ime hisory, an is wo inegrals shown in Figure 17. Again, I wroe a Malab program, halfermpkcos.m, ha generaes he acceleraion ime hisories shown in Figures 17 an 19. The PVSS for he 10% cosine ramp is shown in Figure

15 Figure 17. The secon subplo shows a 200 g, 100 ips erminal peak sawooh shock wih a 10% cosine fall off o zero, precee by a inch rop. The rop an shock are in he secon subplo, an is wo inegrals are in he hir an fourh subplos. Figure 18. This is he PVSS of he erminal peak saw ooh shock wih he 10% cosine fall off. I has a surprisingly small roop zone oubling region because of he genle linear rise. I is very nicely behave. I his our asympoes of 13 inches, 100 ips, an 200 g's. I on' hink we woul noice any ifference if we ha use a linear fall of insea of he cosine ramp fall off o zero. 15

16 In Figure 18 he 100 ips velociy change plaeau an he 200 g peak asympoe show clearly. Noice he roop zone, (as he specrum roops ownwar o he righ heaing for he acceleraion asympoe,) covers a shorer range of frequencies, Hz, an ha he acceleraion oesn' come close o oubling. For he erminal peak sawooh shock we observe less han 1.5 imes he peak g level in he roop zone. INITIAL PEAK SAWTOOTH Trouble evelops hough when we ry he iniial peak saw ooh. Figure 19 shows he iniial peak ime hisory wih a 10% cosine ramp up o he iniial peak. Figure 19. This is a 10% cosine ramp up iniial peak saw ooh shock. The cosine is ifficul o see bu we'll procee. We're going o have rouble wih he shock specrum because of he seep iniial rise. The PVSS for his 10% cosine ramp up iniial peak sawooh shock is shown in Figure 20. As he specrum roops ownwar o he righ heaing for he acceleraion asympoe, i oesn' quie ge here even a 5000 Hz, an he acceleraion oes ouble. A 5000 Hz i is efiniely heaing for he 200 g line. If I increase o 0.2, we make i, as is illusrae below. The ime hisory wih he 20% cosine ramp is shown Figure 21. Figure 20. Iniial peak sawooh PVSS for he shock of Figure 19. We see he, 13 inch rop, 100 ips velociy change, bu he 200 g imum acceleraion asympoe is no reache. The roop zone rouble evelops wih he iniial peak because of seep iniial rise. 16

17 Figure 21. Iniial peak sawooh shock wih a 20% cosine ramp o he peak. Examining Figure 21, an comparing i o Figure 16, i is clear ha a 20% rise is wih in he specificaion [9]. The resuling PVSSs are shown in Figure 22. We ge back own o 200 g a abou 2500 Hz. This compromise of he iniial peak sawooh woul fi well in he re an green limis of Figure 16, an is sufficien o permi he 200 Hz asympoe o appear, an complees he analysis of he five simple shocks. They all are similar. Figure 22. PVSS of he iniial peak sawooh wih a 20% cosine ramp o he peak. SIMPLE SHOCK COMPOSITE PVSS PLOTS Now we have complee an examinaion of he PVSS of he simple pulses: he half sine, he rapezoi, he haversine, an he iniial peak an he erminal peak saw ooh. We observe ha boh ampe an unampe, he PVSS's are similar. The reason ha we can see ha hey are similar is ha I scale hem o have equal velociy change an peak acceleraion. Figure 23 is he unampe composie plo. Nex we o he 5% ampe PVSS composies in Figure 24, which suppors he same conclusion. 17

18 Figure 23. This is an unampe composie PVSS plo of he five simple shocks. Noice ha he shocks only iffer in he roop zone. Figure 24. This is a 5 % ampe composie PVSS plo of four of he simple shocks. Again noice ha he shocks only iffer in he roop region. This is ineresing an imporan. Afer a huge amoun of calculaing an ploing he PVSSs of he simple shocks show he similariy. ONLY THE PVSS ANALYSIS WITH THE TABLE DROP ACCELERATION INCLUDED AND SCALED TO VELOCITY CHANGE WITH THE SAME PEAK ACCELERATION SHOWS THE SIMPLE SHACKS SIMILAR. THIS FACT HAS ESCAPED ALMOST ALL SHOCK EXPERTS. I HAVE TO EMPHASIZE THIS BECAUSE IT IS GENERALLY NOT KNOWN. Only in he high frequency region, where he velociy(severiy) levels are becoming less severe o hey iverge. Gerel's [5] off he cuff commen ha ha all he simple pulses are similar is confirme if no proven. The erminal peak saw ooh comes own o he 200 g 18

19 asympoe faser han he oher four because i has a more graual rise. The velociy change uring he pulse was he similar hing abou hem. They all require he same rop heigh so hey all have he same low frequency asympoe. All of he simple pulses evelope on a rop able shock machine by a programmer ha resuls in zero velociy when he pulse is over will have a velociy change of he square roo of 2 g h. 2gh They will all have he same rop heigh or imum isplacemen, hence he same low frequency asympoe. Since hey all have he same velociy change, hey all have he same plaeau region. Since I ajuse he pulses o have he same peak acceleraion, hey all mus have he same high frequency asympoe. The only way heir shock specra can iffer, are a he wo corners an his can be seen in Figures 23 an 24. I ha rouble geing he acceleraion asympoes o appear in he rapezoi an he iniial peak saw ooh. These have abrup rise imes ha cause a oubling of he peak acceleraions in he roop zone. I ha o ecrease he rise ime abrupness by using a half cosine ramp rise of 30% in he rapezoi an 20% in he iniial peak wave form. I'll summarize he formulas for peak acceleraion, velociy change, an uraion for convenience an comparison. Half Sine Shock 2x x (2) Cosine ramp rapezoi shock (rise ime: ) x x (8) 1 Haversine shock x x (8f) 2 Cosine rampe saw-ooh shock x x (13) 2 I wan o convince you of wo oher imporan ieas. Simple shocks are frequenly simulae on a shaker, bu he rop canno be simulae, an he rop is very imporan. This ruins he frequency range over which he shock is severe. Also, he half sine, an he oher simple shocks have a PVSS ha can be hough of as similar o ha of he explosive shocks, an can be use o simulae hem. BUT SHAKER GENERATED SHOCKS ARE NOT SIMILAR! Shaker generae shocks have a severely limie low frequency asympoe an can have almos a negligible plaeau. Half sine shock ess can be conuce on an elecrically riven shaker bu shakers have a limie isplacemen capabiliy. The PVSS on 4CP of a shaker generae shock will reflec his wih a grealy reuce low frequency capabiliy. Le s consier a half sine wih recangular pre an pos pulses. The magniue of he pre an pos pulses is se a imes he imum acceleraion. See Figure 25. The area of he pre an pos pulses mus equal he half sine velociy change we evelope in Eq (2) which is shown in Eqs (14 a an b). This yiels he pre an pos pulse uraion, p in Eq. (14b). Again I wroe a shor Malab program, halfsin4.m ha generaes he recangular pre an pos pulse halfsine shaker shock acceleraion ime hisory. The ime hisory an is inegrals are shown in Figure

20 Figure 25. Shaker shock simplifie acceleraion geomery. The negaive area of he pre an pos fla regions mus equal he area of he half sine. 2 x p m p 2 x m (14 a,b) Figure 26. Time hisory an inegrals for a recangular pre an pos pulse half sine shaker shock Noice in Figure 26, ha he peak isplacemen is only abou 0.2 inch, an his will limi he low frequency severe porion of is PVSS plaeau. The 5% ampe shock specrum is shown in Figure 27. The shock now is only severe from abou 70 o 200 Hz. Also he shaker armaure moion is all in one irecion. Cenering he shaker armaure so i is reurn o zero, furher reuces he isplacemen o abou 0.11 inch an affecs he plaeau a lile ifferenly. This is very ba. A greay eal of equipmen has is firs moe frequency below 70 Hz an ha means ha for his equipmen he firs moe woul no be exciie by he shock es. 20

21 Figure 27 5% ampe PVSS of he shaker shock version of ou 100 ipsa 200 g half sine. Noice is almos non exisaan plaeau. I has a severiy of near 90 ips from 70 o 200 Hz. Comparing shaker shocks wih he rop able shocks in Figure 24, i is easy o see he reuce frequency range of he shaker shock plaeau. However, here is no way his can be noice unless he shock specrum is ploe as a PVSS on 4CP, an compare o a rop able half sine wih he rop inclue. Shaker simulae half sines can be inaequae for machinery an equipmen wih lower moal frequencies. This is incluing he shocks synhesizing a shock specrum wih a collecion of oscillaory moions. The beauy of shaker shock is ha he irecion of he shock or is polariy can be reverse wihou urning he equipmen over. Tha convenience can no make up for he loss of low frequency plaeau. Figure 28. The acceleraion ime hisory an is wo inegrals for an example explosive shock. SHOCK SPECTRA FROM EXPLOSIVE EVENTS ARE SIMILAR TO SIMPLE PULSE TESTS One example will be given o explain he similariy. Figure 28 is an acceleraion ime hisory of an explosive even an is wo inegrals. The mean has been remove from he acceleraion ime hisory o assure ha he velociy ens a zero. The fac he exreme values are all minima makes no ifference. Figure 29 gives is shock specra for 3 amping values. 21

22 Figure 29. Shock specra of an example explosive shock es. Noice on he lef all hree curves are asympoic o jus uner 9 inches, ha he cener severe region of he mile 5% ampe curve is hovering jus above 200 ips, an ha a he highes frequencies he ampe curves a leas are heaing for abou 900 g's. I woul have o calculae o higher frequencies o see he high frequency acceleraion asympoe on he unampe specrum. To use a half sine o approximae he 5% ampe curve, I rie a 300 ips, 1500 g half sine wih a coefficien of resiuion of 1, o reuce he rop heigh, which is sill oo high. This is Figure 30. 5% Dampe PVSS of HW4 in re an 300 ips, 1500 g half sine in black. I coul increase g in my hafsingv program o reuce he rop heigh an he low frequency asympoe. The roop zone oubling of he 1500 g halfsine oes a nice job of almos enveloping HW4. shown in Figure 30. The poin here is ha a simple pulse wih a peak g level 1500 g, an a velociy change of 300 ips, has a shock specrum maching he severe an high frequency regions of HW4. I woul provie a es of equal severiy. To aain he velociy change wih a only 9 inch rop woul require an arificial high g level in he 22

23 program, which coul be one. In pracice I hink we coul obain he 300 velociy change 9 inches wih bungee cor. This woul give he 9 inch isplacemen low frequency asympoe. In his sense I make he saemen ha he simple shock ess are similar o explosive shock specra. Since he PVSS of he shock shows is capaciy o inuce moal velociies in equipmen es iems, he es has he same severiy as he explosive shock.. The simple shock woul have o be applie in boh he posiive an negaive irecions because he simple shocks are he mos highly irecional or polarize. CONCLUSIONS The very imporan final poin I'll make is ha all he simple shocks are similar when ploe on he same page an scale o have he same velociy change an peak acceleraion. Unampe an ampe composie plos of he five shocks are given in Figures 23 an 24 o show how similar hey are. All simple shocks have ha same PVSS on 4CP; an his means ha here is no sense in using ifferen simple shock shapes. The half sine is fine. If you are making a high frequency high acceleraion shock, an he half sine eerioraes ino a haversine, ha's fine oo. I repea wha I emphasize on page 18: ONLY THE PVSS ANALYSIS WITH THE TABLE DROP ACCELERATION INCLUDED AND SCALED TO VELOCITY CHANGE WITH THE SAME PEAK ACCELERATION SHOWS THE SIMPLE SHACKS SIMILAR. THIS FACT HAS ESCAPED ALMOST ALL SHOCK EXPERTS. I HAVE TO EMPHASIZE THIS BECAUSE IT IS GENERALLY NOT KNOWN. Generaing a simple shock on a shaker sacrifices a huge amoun of low frequency conen, even hough ha is exremely convenien (convenien because you can changes he irecion wihou urning he objec over). You won' have any low frequency conen an will likely allow meeing shock specificaion ha couln' be me on a rop able shock machine. Compare Figures 27 an 24 o see he shocking ifference. The PVSS on 4CP of all zero mean shocks are shape like a hill ans have a high PV plaeau. The simple shock PVSS on 4CP is very simple; is 5% ampe plaeau is a 93% of he impac velociy change, is high frequency asympoe slopes own an o he righ on a consan acceleraion line a he peak shock acceleraion, an he plaeau low frequency asympoe slopes own an o he lef on a consan isplacemen line a he peak shock isplacemen. The plaeau ienifies he severe frequency range of he shock. To es for an explosive shock PVSS shape like a hill, an envelope by y, y, y use a simple shock wih a 93% velociy change ha covers y, an a peak acceleraion equal o y, an a imum isplacemen equal o y. There is a roop zone, righ afer he inersecion of he plaeau an he imum acceleraion asympoe. Abrup rise ime shocks can ouble he acceleraion asympoe for a range of frequencies. A wors case is shown in Figure 7, where he oubling begins a abou 70% of he plaeau level, an oubling coninues unil o 10% of he plaeau level. I consier his a minor poin because i occurs well below he severe PV levels. This roop zone oubling region significance is wha was exaggerae by he auhors of he normalize simple shocks analyses in [2, 3, 4, 5]. The paper is a summary of resuls of many Malab calculaions. I wroe shor programs o generae all he simple shocks, a ploing program ha inegraes an plos he shock acceleraion, velociy an isplacemen. Oher slighly longer programs were use o calculae an plo he PVSSs an craw he 4CP. I will make hese available o any who are inerese. REFERENCES 1. Gaberson, H. A, "Pseuo Velociy Shock Specrum Rules an Conceps", Proceeing of he Annual Meeing of he Mechanical Failure Prevenion Technology Sociey [ April 19, Also publishe as Gaberson, H. A,, "Pseuo Velociy Shock Specrum Rules for Analysis of Mechanical Shock"; IMAC XXV, Orlano, FL; Sociey of Experimenal Mechanics; Behel, CT, Feb 2007; p Minlin, R.D., Dynamics of Package Cushioning, Bell Sysem Technical Journal, vol 24, Jul-Oc 1945, pp Jacobsen, L. S., an Ayre, R.S., "Engineering Vibraions", McGraw- Book Co

24 4. 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., Rubin, S. Conceps in Shock Daa Analysis, "Harris' Shock an Vibraion Hanbook", 5h E. by Harris, C. M. an Piersol, A.G., pp Lelanne, Chrisian, "Mechanical Shock", Volume II in "Mechanical Vibraion an Shock", Hermes Science Publicaions, Paris 1999; English Eiion, Hermes Penon L Taylor an Francis Books Inc., 29 Wes 35h Sree, New York, NY, 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. Army, Qualiy Assurance Direcorae, Frankfor Arsenal, Philaelphia, PA April 1967; AD# AD Gaberson, H.A. "Half Sine Shock Tess o Assure Machinery Survival in Explosive Environmens". IMAC XXII, Dearborn, MI; Sociey of Experimenal Mechanics, Jan 29, MIL-STD-810F: Meho Shock; Jan [Polariy: para 2.3.3, p ][Terminal peak sawooh shock. p , an Figure IEC Shock Specificaion, ; Basic environmenal esing proceures; Par 2: Tess - Tes Ea an guiance: Shock. Thir Eiion