SHOCK AND VIBRATION RESPONSE SPECTRA COURSE Unit 21 Base Excitation Shock: Classical Pulse

Transcription

1 SHOCK AND VIBRAION RESPONSE SPECRA COURSE Ui 1 Base Exciaio Shock: Classical Pulse By om Irvie omirvie@aol.com Iroucio Cosier a srucure subjece o a base exciaio shock pulse. Base exciaio is also referre o as suppor moio. he pulse is ypically efie i erms of acceleraio, alhough i coul be efie i erms of isplaceme of velociy. Examples of base exciaio are 1. A auomobile ravelig ow a washboar roa.. A builig subjece o a earhquake or seismic moio. 3. A crysal oscillaor moue o a circui boar subjece o shock. For simpliciy, he auomobile, builig, or crysal oscillaor ca be moele as a sigleegree-of-freeom sysem. he respose of he objec o a give base exciaio ca he be calculae give he aural frequecy a he ampig raio. Noe ha he mass value is o explicily require. ypically, he peak respose acceleraio is he esire oupu parameer. he aural frequecy of he objec, however, may be ukow. hus, he respose calculaio ca be performe for a series of aural frequecies. he peak acceleraio respose ca he be ploe agais he aural frequecy. he resulig fucio is a shock respose specrum. he shock respose specrum is useful for evaluaig he amage poeial of he shock pulse. he purpose of his Ui is o prese he shock respose specrum for he case of a classical base ipu pulse, such as a half-sie pulse. Derivaio of Equaios Cosier he sigle-egree-of-freeom sysem subjece o base exciaio show i Figure 1. he free-boy iagram is show i Figure. 1

2 m &&x k c &y Figure 1. Sigle-egree-of-freeom Sysem he variables are m = mass, c = viscous ampig coefficie, k = siffess, x = absolue isplaceme of he mass, y = base ipu isplaceme. he ouble-o oaio iicaes acceleraio m &&x k(y-x) c(& y x&) Figure. Free-boy Diagram Defie a relaive isplaceme as follows. Le = x y & = x& y& && = x&& y&& x&& = && y&& he followig equaio of moio for he relaive isplaceme was erive i Ui 8. && ξ & = && y (1) Cosier he half-sie pulse give by equaio ().

3 3 > π =,, A si y&& () () he equaio of moio becomes, A si π = ξ & && (3) Le π = (4) ( ), Asi = ξ & && (5) Equaio (5) may be solve via Laplace rasforms, as show i Referece 1. Afer may seps, he absolue acceleraio urig he pulse is ( ) [ ] ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) { } ( ) ( ) { } ( ) ( ) for, A si si 1 cos exp A cos si A si 1 () cos () () exp x() ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ = & & && (6)

4 he absolue acceleraio afer he pulse is x && () = exp exp ( ξ ( ){[ () ξ() & ] cos( ( )} [ ] si ( ) ( ( ) ξ ξ () ( 1 ξ ) & ( ) ( ), for > (7) Example Cosier he example i able 1. he base pulse is show i Figure 3. he calculaios were mae usig equaios (6) a (7), as implemee i he ahsie.exe program. able 1. Shock Respose Specra, Q=1, Acceleraio Base Ipu =.1 sec, 1 G, Half-sie Pulse Naural Frequecy (H) Peak Posiive Acceleraio (G) Peak Negaive Acceleraio (G) Figure

5 Noe ha oly he peak posiive a egaive values are reaie for each ime hisory respose. he peak values are fou via a simple search meho raher ha a calculus meho. Furhermore, oe ha he peak respose ca occur eiher urig or afer he half-sie pulse. he overall shock respose specrum is show i Figure 7. I is cosruce by ploig he peak posiive a egaive acceleraio ampliues versus aural frequecy i (H). he are oher ypes of shock respose specra which coul be ploe. For example, he absolue value acceleraio respose specrum coul be ploe, isea of he iiviual posiive a egaive specra. Furhermore, he peak isplaceme or peak velociy coul be ploe versus he aural frequecy. I aiio, his process coul be repeae for oher classical pulses, such as rapeoial, sawooh, a recagular pulses. BASE INPU:.1 sec, 1 G HALF-SINE PULSE 1 ACCEL (G) IME (SEC) Figure 3. 5

6 SDOF RESPONSE: FN = 1 H, Q=1 BASE INPU:.1 sec, 1 G HALF-SINE PULSE 15 1 ACCEL (G) IME (SEC) Figure SDOF RESPONSE: FN = 8 H, Q=1 BASE INPU:.1 sec, 1 G HALF-SINE PULSE ACCEL (G) IME (SEC) Figure 5. 6

7 SDOF RESPONSE: FN = 1 H, Q=1 BASE INPU:.1 sec, 1 G HALF-SINE PULSE 15 1 ACCEL (G) IME (SEC) Figure 6. SHOCK RESPONSE SPECRUM Q=1 BASE INPU:.1 msec, 1 G HALF-SINE PULSE 1 Negaive Posiive PEAK ACCEL (G) NAURAL FREQUENCY (H) Figure 7. 7

8 Noe ha he posiive shock respose specrum i Figure 7 coverges o he peak ipu ampliue a higher aural frequecies, say above H. Compoe Shock esig Some shock ess are specifie i erms of a base ipu half-sie pulse. he es may be performe o a rop ower or o a shaker able. here is a sigifica ifferece bewee he posiive a egaive shock respose specra of a half-sie pulse, however, as show i Figure 5. hus, hese specificaios ypically require ha he es be performe i each irecio of each axis. esig o shaker able preses a furher challege. he e isplaceme a e velociy of he shaker able mus each be ero. hus, pre a pos pulses mus be use o achieve hese requiremes. Refereces 1.. Irvie, Respose of a Sigle-egree-of-freeom Sysem Subjece o a Classical Base Exciaio, Vibraioaa.com Publicaios, Homework 1. Repea he example for a half-sie base ipu of.8 secos, 1 G. Assume a ampig value of Q = 1. Use program ahsie.exe for boh he ime hisory respose a shock respose specra calculaios.. Wha is he aural frequecy which has he highes absolue value respose for problem 1? 3. A compoe has alreay bee ese o a 5 G,.11 sec base ipu half-sie pulse. A ew requireme is ha he compoe mus wihsa a 1 G,. sec half-sie pulse. Compare he specificaios i erms of heir correspoig shock respose specra. Wha coclusios ca be mae abou he ee for reesig? 4. Rea uorial srs_ir.pf. 8