Shock n on Shakers George Fox Lang, ssociae Edior Elecrodynamic and hydraulic shakers have become a commonly used plaform for shock esing. his aricle reviews curren conroller echnology o safely and reliably compensae for he mechanical limiaions of boh ypes of shakers in reproducing desired shock exciaions. Conrolled hydraulic and elecrodynamic shakers have become he preferred es plaforms for modes shock ess. While drop-es (and oher) faciliies remain necessary for he simulaion of exreme shock pulses, he conrolled shaker has proven very cos effecive for more rouine produc qualificaion and seismic evaluaion work. Modern DSP shaker conrollers now do an ousanding job of reproducing desired ransien pulses safely, reliably and repeaably. heir use saves he enormous ime of ieraively designing mechanical drop-arges o provide a required shock profile. One now merely keys in or selecs he desired acceleraion-versus-ime shock profile or is shock response specrum (SRS) and runs he es. However, a shaker presens some physical barriers o shock esing. hese devices have a limied range of displacemen sroke and exhibi velociy limis (valve or amplifier induced) ha canno be exceeded wihou loss of conrol. he shaker conroller compensaes for hese shorcomings by employing a process ermed compensaion, he focus of his aricle. While his compensaion process is no wihou flaws, i opens he range of esing ha can be performed on a single laboraory sysem enormously. he Problem Shock es profiles are ypically described by an acceleraionversus-ime hisory of a few milliseconds duraion. Cerain shapes have become he Classical Shock library. hese basic pulse forms sem from prior drop-esing wherein he es objec sars in free-fall and collides wih a arge whose elasic and crush properies deermine he es acceleraion profile. he resuling pulse shapes are almos invariably unipolar as displacemen and velociy were no consideraions in heir developmen. When hese same profiles are run on a shaker, velociy and displacemen are he primary concern. Consider he classic half-sine acceleraion shock pulse shown in Figure. Since he acceleraion is solely in he posiive direcion, he velociy a he pulse s conclusion is posiive and he es objec coninues o move a his velociy even hough he acceleraion has reurned o zero. he es objec has displaced during he shock pulse and will coninue o displace a consan velociy unil arresed by some barrier. Wihou smarer inervenion, he sroke limis of a shaker could provide his barrier wih likely expensive and dangerous resuls. shaker can only operae over a limied range of displacemen, is sroke. Wihin his range i can only operae in a conrolled fashion if is velociy limi is no exceeded. vailable amplifier-volage limis he maximum conrolled velociy of an elecrodynamic shaker, while he flow gain of he conrol valve dicaes he velociy limi of a hydraulic shaker. Hence, a shock-es can only be run if he resuling displacemens fall wihin he shaker s sroke range wih peak velociy wihin sysem limis. s a maer of pracicaliy, he es mus sar from condiions of zero acceleraion, velociy and displacemen and reurn o his sae a he conclusion. his can be accomplished by he use of pulse compensaion, which is auomaically inroduced by he conroller. Mos conrollers also perform a feasibiliy analysis before aemping o conduc he es and will sop impracical applicaion wihou risking he shaker sysem in any way. Classical Pulses and heir Properies You will noe ha he moional responses of Figure were presened in normalized form. ime was divided by he pulse duraion, acceleraion was divided by he peak acceleraion, velociy was divided by and displacemen by. hese normalizaions are indusry-sandards, allowing an acceleraion pulse and is inegrals o be easily over-ploed wihin a reasonable graphic range, wihou concern for he peak value of he pulse or is duraion or he sysem of measuremen unis employed. he erminal normalized acceleraion, velociy and displacemen values of such a plo are ermed k, k V and k D, respecively. hese coefficiens permi a simple and indusry-acceped means of describing he gross moional characerisics of a shock-pulse. Graphical able presens hese coefficiens (and ohers) for a variey of Classical Shock waveforms, including hose employed in compensaion. he moional changes induced by a pulse of able may be convered o a physical uni basis in accordance wih: cceleraion change = k Velociy change = k V Displacemen change = kd Noe ha all pulses in his able sar and end a zero acceleraion. For his reason, each has a k coefficien of zero. Pulses wih non-zero k will be discussed subsequenly. able also includes wo less common indicaors, k SE and k ME. hese will be discussed in conex of a specral represenaion of he pulses. hese indices may be convered o physical unis in accordance wih: Signal Energy = kse (4) Mechanical Energy = kme M where M is he mass of he es objec Our focus now reurns o uiliy of he hree moional coefficiens k, k V and k D. ime =, he displacemen, velociy and acceleraion of he objec under es are y, y and y, respecively. pplying a pulse of ampliude and duraion changes all of hese parameers. he conclusion of he pulse, hese sae-variables relae o he iniial condiions in accordance wih: Ï È Ï y Ì = k y (6) D Ì y kv y y Î k y In our iniial consideraions, we will deal wih pulses characerized by k = and we can simplify he sae relaionships o: Ï È Ï Ì = y kd Ì y y Î kv y Eq. 7 will provide he basis o undersand shock-pulse compensaion. ouncing ack asic Compensaion Figure illusraes applicaion of a half-sine pulse o a es () () (3) (5) (7) SOUND ND VIRION/SEPEMER 3
.8.6.4...4.6.8 Figure. Shock esing on an elecrodynamic shaker. Mars Exploraion Rover # undergoing verical axis landing loads pulse esing a JPL on an LDS Model 994 shaker. Phoo couresy of he Je Propulsion Laboraory, California Insiue of echnology. objec iniially a equilibrium. he conclusion of he pulse, he specimen is displaced posiively and moving wih a posiive velociy. his can be modeled by Eq. 7 and he coefficiens of able as: Ï Ï È Ï Ì = Ì y =. 383 Ì. 383 y. 6366 Î. 6366 Clearly, we need o do somehing o arres he moion if his es is o be conduced on a shaker. he obvious soluion is o apply a pulse of opposie sign. o comply wih he es specificaions, his added pulse should be low in ampliude. For sake of example, le s resric his pulse o be no more han percen of he ampliude of he es pulse. For a given peak ampliude, he mos influenial pulse we can use is a recangle. his can be concluded by examining he k V and k D facors in able. Classical shock pulses and various non-dimensional properies. recangle k =. k V =. k D =.5 k SE =. k ME =.5 erminal-pk sawooh k =. k V =.5 k D =.667 k SE =.3333 k ME =.5 riangle half-sine sine k =. k V =.5 k D =.5 k SE =.3333 k ME =.5 k =. k V =.6366 k D =.383 k SE =.5 k ME =.6 k =. k V =. k D =.59 k SE =.5 k ME =. Fandrich iniial-peak sawooh k =. k V =.5 k D =.3333 k SE =.3333 k ME =.5 rapezoid haversine damped sine k =. k V =.7797 k D =.3898 k SE =.6855 k ME =.5 k =..5 < k V <..667 < k D <.5.3333 < k SE <..5 < k ME <.5 k =. k V =.5 k D =.5 k SE =.375 k ME =.5 k =.. < k V < k D. < k D <.59. < k SE <.5. < k ME <.7976 (8) Figure. Normalized responses o a half-sine acceleraion pulse. lack acceleraion/, blue velociy/, red displacemen/..4..8.6.4...5.5.5 3 3.5 4 4.5 Figure 3. Using a negaive pulse o drive he velociy o zero leaves a posiive displacemen of he shaker armaure. able ; hose for a recangle are he larges. We again apply Eq. 7, now using he recangle k V and k D facors in he marix and he resuls of Eq. 8 as he inpu saevecor. he resuls are: È Ï -5 Ì =. 383. Î - 6. 366 (9) Ï + Ì. 383 +. 6366 -. 5. 6366 - Clearly, o arres moion, he erminal velociy mus equae o zero. ha is, he added negaive recangle mus be of such duraion ha he area above is curve equals ha of he area under he iniial half-sine es pulse. ha is: ẏ () + =. 6366 - = fi =. 6366 However, we also wan he shaker o come o res a is original mid-sroke posiion. If we fail in his ques, he conroller is obligaed o hold he shaker a an offse posiion by applying a DC command... forever! he displacemen a he end of he pulse, where velociy is zero, is given by: y + =. 383 +. 6366 -. 5 = () Ê ˆ. 383 +. 6366 Á his is clearly a non-zero resul. For a % compensaion DYNMIC ESING REFERENCE ISSUE 3
Figure 4. Using a longer negaive pulse o drive he displacemen back o zero leaves a negaive velociy a ha poin..4..8.6.4. -. -.4 -.6 -.8.4..8.6.4...4.6 3 4 5 6 7 8 3 4 5 6 7 8 9 Figure 5. pplying a negaive % ampliude recangle pulse 5.763 imes as long as he half-sine, followed by a posiive % pulse.58 imes as long, reurns he shaker o mid-sroke a zero velociy and acceleraion in 9.343 half-sine duraions. pulse ( =.) he res posiion of he shaker is: y + =. 334. his resul is illusraed in Figure 3. Simply increasing he duraion of he compensaing pulse will no correc he problem. he shaker could be driven o zero displacemen, bu would arrive here wih considerable negaive velociy as illusraed by Figure 4. he soluion o his conundrum is no inuiively obvious. he answer is o apply he negaive pulse for longer han necessary o bring he velociy o zero, hen apply a posiive pulse o drive he velociy back o zero. his soluion is illusraed in Figure 5. Noe he erminal sae has zero acceleraion, velociy and displacemen. he duraion of he compensaing pulses canno be chosen capriciously. gain apply Eq. 7, his ime wih a marix represening a posiive recangular pulse acing upon he inpu saevecor provided by Eq. 9 o achieve he final pos-es sae. Specifically: È Ï C + 5. CC Ì = C. 383 +.. 6366-5 Î CC. 6366 - Ï + + C È. 5CC +. 383 +. 6366 Ì () Î -. 5 +. 6366C -C C + - C. 6366 Figure 6. Preceding he half-sine by a posiive recangular pulse of % ampliude and.58 imes duraion, followed by a negaive % ampliude pulse 5.763 as long, leaves he shaker a mid-sroke wih zero velociy and acceleraion in 9.343 duraions. We require he erminal velociy and displacemen o be zero. ha is: For simpliciy, we have used compensaion pulses of equal ampliude and opposie signs. ha is, = C =.. Imposing his simplificaion on Eqs. 3 and 4 resuls in he simulaneous soluions: s inuiion would sugges, hese same soluion imes can be used for a pre-pulse compensaion as illusraed in Figure 6. When we reverse he sequence of pulses in ime, he saevecors are differen. he erminal sae-vecor for he es of Figure 6 is given by Eq. 7. gain we demand he erminal velociy and displacemen o be zero. ha is: and.4..8.6.4...4.6.8 3 4 5 6 7 8 9 ẏ = C +. 6366 - = C y =. 5CC +. 383 +. 6366 -. 5 +. 6366C -C = = 5. 736 C =. 58 (3) (4) (5) (6) Ï È È Ì = y. 383-5. y Î. 6366 Î - È Ï C 5. CC Ì = C Î CC (7) Ï + + C Ì. 383 -. 5 +. 5C C + CC - + CC. 6366 - + CC ẏ = C +. 6366 - = (8) (9) While he velociy of Eq. 8 is idenical o ha of he pospulse compensaion consrain of Eq. 3, he displacemen of Eq. 9 differs from he corresponding consrain, 4. Noneheless, he soluions previously saed in 5 and 6 saisfy Eqs. 8 and 9. C y =. 5CC +. 383 - - 5. + C C + CC = 4 SOUND ND VIRION/SEPEMER 3
his simple exercise has highlighed several imporan observable facs:. Compensaion is no an opion; i is absoluely required o run an unipolar shock pulse on an elecrodynamic or hydraulic shaker.. Compensaion pulses may be added eiher before or afer he es-pulse o force he es o boh sar and end a zero acceleraion, velociy and displacemen. 3. In general, wo pulses of opposie sign mus be added o bring boh erminal velociy and displacemen back o zero. 4. he compensaion pulses may be of much lower ampliude han he desired es pulse (and hey need no be of equal ampliude o one anoher). hey do no need o have he same shape as he es pulse. 5. he lower he ampliude of he compensaion pulses, he longer he oal conrolled-pulse duraion will become. (his can pose a conroller problem.) 6. Maximum sroke and velociy occur ouside he desired es pulse, wihin he compensaion inerval. 7. When (exclusive) pre- or pos-pulse compensaion is used, only half of he available shaker sroke will be uilized...4.8.6.4. 4 3 3 4 5 Figure 7. n example of combined pre- and pos-pulse compensaion using four equal-ampliude rapezoidal pulses. Noe symmeric sroke and velociy peaks. Firs, Las or oh? Should I compensae before or afer my specified es pulse? Is here any meri in doing boh? hese are wo very percepive quesions, each suffering es-specific answers. Shock ess are run for a variey of reasons. Some profi from pre-es compensaion, some from pos-es compensaion. If he shock is large wih respec o he shaker s capabiliy, hese specific advanages may need o be se aside. combinaion of pre- and pos-pulse compensaion may be required, simply o allow he shaker o generae he pulse wihin is sroke and velociy limis. Consider qualificaion of an air-bag deploymen sensor. he shock-es is likely run o deermine he g-level a which he sensor swich closed, causing bag deonaion. In his siuaion, a clear picure of evens during he es-pulse rise is required. Pos-pulse compensaion is he righ answer here, providing an unconaminaed rising inpu from he desired zero-g iniial condiion. Now consider esing a compuer disk drive, obliged o operae in a hosile ravel environmen. he shock-es is likely o include monioring read/wrie funcioning hrough and afer he simulaed bump. In his insance, pre-pulse compensaion is he righ answer so ha afershock effecs raise no quesion of coninued proper funcion. Oher ess are more pedanic hey are merely run o demonsrae he es objec can survive he even and his deerminaion is no made during he shock-es. Here he use of pre or pos-pulse compensaion is a moo poin. However, es labs are always called upon o simulae increasingly hosile environmens wihou commensurae upgrade of he available es faciliies. Evenually, you will be faced wih running a pulse oo aggressive for he half-sroke of your shaker, or one ha requires peak velociy beyond is conrol range. his is where combined pre- and pos-pulse compensaion can save he day. Figure 7 illusraes combined pre- and pos-pulse compensaion of a half-sine using equal ampliude rapezoidal compensaion pulses of % ampliude. Compare his wih Figures 5 and 6 ha illusrae he same es-pulse and noe:. he peak displacemen values are cenered abou he shaker s mid sroke, doubling he available displacemen range.. he oal peak-o-peak sroke used is significanly less han ha of a pre- or pos-pulse (only) compensaed es. 3. he peak velociies are significanly lower for he pre- and pos-compensaed es. 4. he oal conrolled-pulse imes are all abou equal. firs blush, his all seems oo good o be rue. However, i is he naural resul of using shorer compensaion pulses. In fac, he symmeric soluion presened is far from opimum. Modern conrollers can combine a myriad of pulse shapes o provide opimizaion for sroke, velociy and he energy imposed on he es objec. y using wo preceding and wo following pulses (or following pulses of asymmeric naure), he conroller can posiion he shaker armaure o a negaive displacemen prior o he espulse wih a negaive velociy equal o abou half of he velociy change induced by he desired es waveform. I will selec hese pulses so ha he sroke used is cenered in he shaker s range. hus, a modern conroller allows he shaker sysem o deliver he mos aggressive es-pulse possible wihin is physically limied sroke and velociy capabiliies. Why ll of hese Pulse Shapes? he half-sine, haversine, riangle, rapazoid, erminal-peak sawooh and iniial-peak sawooh are all elemens of he Classical Shock waveform library. (I should noe here ha he recangle, riangle and sawooh waveforms are all specific sub-ses of he rapazoid, deermined by he rise-ime and fall-ime duraions.) Each of hese pulse-shapes has a place in he hisory of shock-esing and each is he likely focus of your nex specified es, being someone s noion of he proper simulaion of an environmenal even your produc is likely o suffer. In conras, he recangle, Fandrich, sine and damped-sine pulses are all ypically employed as compensaion pulses. Each has desirable characerisics in he eyes of he es feasibiliy designer. R.. Fandrich may have been he prooype es feasibiliy designer. In 98 he sudied he problem of performing a MIL-SD-8C half-sine pulse of 3 g peak and msec duraion on an elecrodynamic shaker wih a in. PP sroke. His soluion is embedded in many conrollers and spawned a hos of independenly invened, improved soluions o he general problem. One of Fandrich s concerns was he addiional damage poenially induced by he compensaing pulses. While he admired he posiioning efficiency of he recangular pulse, he feared is rich specral conen (caused by he abrup rising and falling edges) migh induce unwarraned sress in he es objec. His soluion was o approximae he square pulse wih only firs and hird harmonic componens. (Noe ha his soluion is no simply he runcaion of Fourier Coefficiens for a recangular pulse.) He chose o approximae a recangular pulse of ampliude and duraion by: Ê ˆ Ê ˆ ẏ ( ) =. 55sin. sin Áp P + 3 Á3p for < < and elsewhere (a) I believe ha Mr. Fandrich relied heavily on numerical inegraion in his landmark work. I respecfully submi ha a closed-form soluion provides a slighly more precise saemen, which is: È Ê ˆ Ê ˆ ẏ () =. 4834sin sin Áp P + Á3p Î 5 (b) DYNMIC ESING REFERENCE ISSUE 5
.6 recangle Fandrich.5 accel vel disp ESD / (Hz )......4.3.... 3 4 5 6 7 8 9 f Figure 8. Comparison of he normalized energy specral densiy of a recangle and Fandrich pulse of he same peak ampliude and duraion. Indeed, he Fandrich pulse has less frequency conen han a recangular pulse, alhough he difference may be less profound han he equaion suggess. Figure 8 compares he Energy Specral Densiy specra of a Fandrich pulse and recangular pulse of he same peak ampliude and duraion. Clearly, he Fandrich pulse aenuaes more rapidly as frequency increases. lso clear is he fac ha his pulse (like he recangle) conains disribued energy across a wide band, no merely conen a wo frequencies. he recangle specrum exhibis a zero a every frequency muliple of /. he firs lobe is 3.4 d below he DC value and subsequen lobes roll-off a 6 d/ocave. he Fandrich pulse has a wider primary lobe; wih he firs zero a.5/. Subsequen zeros occur a.5/, 3.5/ and so on. he firs sidelobe ampliude is 4.8 d less han he primary lobe and subsequen side-lobes decay a d/ocave. Mr. Fandrich also proposed he use of a damped-sine as a pos-pulse compensaion. He chose his single asymmeric pulse as an alernaive o wo pos-es compensaion pulses of opposie sign. he inen here was o have a single pulse wih k D /k V raio selecable by specifying he damping facor. You will noe from able ha he (undamped) sine is unique in exhibiing a k V of zero. s damping is applied, his rises rapidly relaive o k D. I is ineresing o noe ha Fandrich prescribed an unusual damped-sine equaion of he form: p Ê ˆ ẏ () = sin, () Áp for < < elsewhere He describes evaluaing his for posiive exponen p and hen applying he waveform in ime-reversed sequence. Unforunaely, evaluaing closed-form soluions for k V and k D of his waveform is difficul; an infinie series of sub-inegrals resuls. able presens a more convenional damped-sine (in vibraion parlance) descripion in accordance wih: -pd Ê ˆ ẏ () = e sin, () Áp for < < elsewhere Figure 9 illusraes ypical normalized responses of he convenional damped-sine of able. his plo presens responses wih a damping facor of d =.375. k V and k D may be uned by selecing d in accordance wih: -pd kv = - e p + d (3) and ( ) - d kv kd = p + d ( ) (4) Figure illusraes k D, k V and heir raio as a funcion of seleced damping facor d....4.6.8 Figure 9. Normalized acceleraion, velociy and displacemen responses of a damped-sine for damping value, d =.35.... k V k D k D /k V... damping coefficien, d Figure. Range of damped-sine k D and k V coefficien variaion provided by selecion of damping facor, d. Some More Recen Conribuions In he years since he landmark Fandrich paper, conroller designers have been busy. In addiion o he dauning ask of poring shaker conrol from dedicaed racks o friendly personal compuer screens, some designers have reurned o he basic physics of he problem. One area of recen exploiaion is he use of pulses wih nonzero k coefficiens. For example, consider he half-sine pulse previously discussed. llow his pulse o overshoo by evaluaing i for more han a half-cycle excursion as shown in Figure. his allows he es-pulse iself o be par of he pos compensaion. s he sine funcion passes hrough zero, is slope is nearly consan. Exending he pulse by % provides a very linear decrease in acceleraion o.3 of he pulse magniude. his is in he neighborhood of he pos-pulse level ha mus be applied o mee MIL-SD-8 requiremens. One ges o ha level wih no furher curve inflecions, minimizing specral conen due o compensaion. Since he pulse no longer sars and ends a he same acceleraion, a non-zero k equal o he erminal normalized acceleraion resuls. he pulse sequence is obligaed o conain anoher asymmeric pulse wih equal and opposie k o reurn he shaker o zero acceleraion a he es end. his can be as simple as a ramp beween he erminal value and zero, perhaps wih a sine (or oher waveform) added o i. lernaively, a parial sine cycle migh be seleced as he (single) complemenary pos-es pulse. s shown in Figure, he variaion of k wih overshoo is highly linear up o 3% of he es-pulse ampliude, owing o he near-linear slope of a sine passing hrough zero. When pulses wih non-zero k are included in he es sequence, he general sae-vecor of Eq. 6 applies in lieu of he simplified saemens of Eq. 7. I is refreshing o see ha he refinemen of shock-es conrol sill includes reurn o he basic physics of he problem. 6 SOUND ND VIRION/SEPEMER 3
Normalized cceleraion..9.8.7.6.5.4.3.. -. -. -.3 -.4...3.4.5.6.7.8.9. Figure. Half-sine pulse wih % period overshoo ends a.3 acceleraion level..7.6.5.4.3.....3.4...3.4.5.6.7.8.9. / spec Figure. Variaion of k, k V and k D coefficiens wih overshoo of halfsine period. We all appreciae he advancemens PCs offer, bu only when he underlying soluion is echnically asue! n r-sae udi I have alluded o coninued developmen effor in his field and feel obliged o provide some hard evidence of i. o his end, I posed a challenge o conroller manufacurers and hey responded wih a vengeance. My challenge was o provide heir 3 soluion o he problem ha plagued Fandrich in 98 perform a 3 g by msec half-sine es on a shaker of in. (PP) sroke capaciy while respecing all olerance limis of MIL-SD-8C (yes, I know we re up o F!). Each manufacurer was asked o provide his soluion as a g- versus-ime Excel file. hese have been collecively ploed in Figure 3. I hink we learn from he similariies and from he differences. I is clear ha he general form of alernaing posiive and negaive compensaion pulses before and afer he espulse is common o all soluions. I is also clear ha differen designers have aken differen roues o solving his problem, and solving i well. Presening Pulse Specra Properly Specrum (FF) analysis is applied o all kinds of signals. he appropriae ampliude scaling is differen for periodic, random and ransien signals. Measuring specral ampliude in gs (or vols or any oher ransduced linear uni) is proper for a periodic signal or a mixed signal dominaed by essenially saionary ones. he preferred scaling for his ype of measuremen is roo-mean-square (rms) ampliude. When a coninuous random signal is measured, we require power specral densiy (PSD) ampliude scaling in g /Hz o provide an insrumen-independen ampliude descripion. k k V k D Reference, g 35 3 5 5 5 5 Specral Dynamics Labworks Vibraion Research Dacron/LDS Daa Physics 8 6 4 4 6 8 ime, msec Figure 3. Curren commercial soluions o he 3 g by msec halfsine problem. Dashed lines upper and lower bounds per MIL-SD- 8C. When a ransien pulse is analyzed, he correc specral ampliude uni is energy specral densiy (EDS), expressed in g sec/hz. ny oher scaling provides an ampliude answer dependen on he specifics of he hardware making he measuremen, never a desirable hing. he area under a PSD curve is he coninuous random signal s mean-square value (overall-rms ), ofen ermed he signal power. he area under an acceleraion ESD curve is he signal energy defined by: SE = Ú y d (5) he acceleraion signal energy hus has unis of g sec (or relaed acceleraion and ime unis). his provides he g sec porion of he g sec/hz unis of an ESD. he per Hz dimensional componen is he same as ha of a PSD, he noise bandwidh of he specrum analyzer making he measuremen. able liss he (oal) signal energy of each pulse ype discussed in his aricle as a normalized k SE coefficien. Muliplying k SE by he square of he pulse ampliude and he duraion of he pulse ( ) provides he signal energy in physical unis. ll specral plos presened herein are normalized. he frequency axis is muliplied by pulse duraion, so ha he horizonal axis is dimensionless. Energy specral densiy is presened verically. his is divided by so ha he verical axis has dimension Hz. Figure 8 presens a comparison of he ESDs of he recangular pulse and is band-limied Fandrich approximaion. We will now examine he ESD specra of he various Classical Shock exciaion pulses. In Figure 4, he half-sine pulse exhibis zeros a.5/,.5/, 3.5/ and so on. he firs side-lobe is 3 d below he maximum value and subsequen lobe peaks fall off a d/ocave. he haversine (i.e. he Hanning window shape) has a broader primary lobe and more rapidly decaying side-lobes. Is firs zero occurs a / and subsequen zeros occur a ineger muliples of /. he firs side-lobe peak is 3. d below he main lobe. Higher side-lobes decay a 8 d/ocave. Figure 5 presens hree races, he riangle pulse, he iniial-peak (IP) sawooh and he erminal-peak (P) sawooh. s inuiion migh sugges, he specral magniudes of he IP and P sawooh waveforms are idenical. (he corresponding phase specra are reflecions of one anoher as hese wo complex specra are a conjugae pair.) he riangle wave has wide lobes, wih zeros occurring a ineger muliples of /. he firs side-lobe is 6.5 d below he maximum and subsequen lobes aenuae a d/ocave. In conras, he sawooh waveforms exhibi no zeros. he specrum follows ha of he riangle s primary lobe o a frequency of / and hen rolls off smoohly a a rae of 6 d/ocave. Figure 6 presens he damped-sine a hree differen damping levels (d =.,. and.375.) he undamped sine (d = ) exhibis a unique primary lobe ha sars wih a zero a DC. DYNMIC ESING REFERENCE ISSUE 7
. half-sine haversine. d = d =. d =.375 ESD / (Hz )... ESD / (Hz )...... 3 4 5 6 7 8 9 f. 3 4 5 6 7 8 9 f Figure 4. Comparison of he half-sine and haversine pulse specra. ESD / (Hz )....... IP saw riangle P saw 3 4 5 6 7 8 9 f Figure 5. ESD specra of sawooh and riangle waveforms. he upper-side of he primary lobe is bounded by a zero a frequency /. Subsequen zeros occur every / hereafer. he firs side-lobe is 8.3 d lower han he primary, and following side-lobes diminish a he rae of d/ocave. s damping is added, he peaks and zeros become less definie and he specrum evenually converges o a d/ocave line (above frequency = /) for high damping. s a maer of pracicaliy, use of damping in excess of d =.375 is unlikely. his value, k D and k V converge (see Figure ) and heir raio remains uniy while boh diminish hereafer. Many conrollers provide for FF specrum analysis (as well as SRS analysis) of shock-es measuremens. However, I am unaware of any curren conroller ha properly scales he ampliude of such ransien measuremens o ESD forma as a maer of course. his is an unforunae omission as es labs coninue o accumulae libraries of sysem-dependen g-versus- Hz documenaion. Wha causes damage? he Holy Grail of package design is a means of predicing if a componen or sysem will fail when exposed o a given shock pulse, before applying he pulse. I know of no one who claims o have his answer. In a landmark work, Gaberson, e al addressed he problem of wha o measure during an observable shock o indicae he damage poenial of he even. he proposed modificaion o he Shock Response Specrum (SRS) algorihm has ye o be incorporaed in any commercial conroller or analyzer. Dr. Howard Gaberson has promised o apply hese mehods o invesigae he effecs of compensaion pulses on damage poenial, hopefully in a fuure issue of Sound and Vibraion. he previously menioned signal energy (k SE ) coefficiens of able and he corresponding energy specral densiy disribuions jus discussed are probable inpus o he damage predicion process. However, signal energy is no synonymous wih he mechanical energy inpu o a srucure under es. he remaining non-dimensional coefficien k ME in able aemps o esimae his. Figure 6. ESD specra of damped-sine a various damping values d. he shaker applies a ransien force equal o he es-mass acceleraion produc F = My o he device under es. he inegral of his force wih respec o he resuling displacemen is he work done by he shaker on he es aricle, which mus evenually equae o all resuling energy dissipaed by he srucure s plasic behavior. While some of he dissipaion occurs wihou damage, he overall energy impared o he es iem seems a very likely index of damage poenial. k ME is derived from: M kme = ME = Work = Fdx F dx = Ú d = Fyd = M yyd d Ú Ú Ú (6) Looking a he k ME values in able suggess ha he Fandrich pulse is no less damaging han he recangle i replaces. I furher implies ha eiher of hese pulses is four imes as likely o produce a failure as a haversine, riangle or sawooh of he same peak value and duraion. I also denoes a half-sine as being 6% as aggressive as a haversine of like proporion. Does his simple one-number saisic provide a harbinger of poenial failure? I suspec no, reflecing on he observaion ha a full-sine pulse has a k ME equal o zero. However, k ME in consor wih oher pulse (and srucural) properies may provide a few more lumens as we char he dark cave of ransien-induced mechanical failure. Closure his aricle has reviewed shock-esing on shakers from several aspecs. I has provided some lessons in basic physics, in signal processing and in he hisory of our indusry. I has also given many manufacurers dedicaed o our indusry an opporuniy o demonsrae heir commied suppor of our work hrough heir own research and developmen. I hank hem for heir candor and rus in conribuing o his aricle. References. R. Fandrich, Opimizing Pre- and Pos-Pulses for Shaker Shock esing, he Shock and Vibraion ullein, Number 5, Par, May 98.. H. Gaberson, D. Pal, and R. Chapler, Classificaion of Violen Environmens ha Cause Equipmen Failure, Sound and Vibraion, Volume 3, Number 5, May. 3. W. E. Frain, Shock Waveform esing on an Elecrodynamic Vibraor, he Shock and Vibraion ullein, Sepember 977. 4. R. Lax, New Mehod for Designing MIL-SD Shock ess, es Engineering & Managemen, June/July. he auhor can be conaced a george@foxlang.com. 8 SOUND ND VIRION/SEPEMER 3