Transient Vibration of the single degree of freedom systems.

Transcription

1 Trasiet Vibratio of the sigle degree of freedo systes. 1. -INTRODUCTION. Trasiet vibratio is defied as a teporarily sustaied vibratio of a echaical syste. It ay cosist of forced or free vibratios, or both (1). Trasiet loadig, also ow as ipact, or echaical shoc, is a operiodic ecitatio, which is characterized by a sudde ad severe applicatio. I real life, echaical shoc is very coo. Soe eaples of shoc could be a forgig haer, a autoobile passig trough a road bup, the free drop of a ite fro a height, etc. I aalysis of systes ivolvig echaical shoc, ost of ties is ecessary to idealize the forcig fuctio (displaceet, velocity, acceleratio or force) of such syste, as a step or pulse fuctio. Several shapes of step ad pulse fuctios are discussed here, as well the respose of the syste, regardig tie history, ad frequecy respose. I first istace, SDOF syste without dapig are aalyzed, the, the effect of dapig will be cosidered DEFINITIONS To avoid cofusio, it is ecessary to defie soe useful teriology, which will be used durig this discussio. The spectru of a give forcig fuctio is a plot of a respose quatity, chose to represet oe aspect of the effect of the force fuctio i a sigle degree of freedo oscillator, agaist the ratio of the characteristic period or frequecy of the forcig fuctio to the atural period or frequecy of the oscillator (2). Regardig shoc pheoea, it is ore coveiet to use the period (duratio) of the ipulse, ad the atural period of the syste ivolved, rather tha frequecies. This approach is coveiet because i trasiet loadig the ecitatio is relatively short i duratio, or has the ature of a sigle pulse. This lead to use a period spectru (2). The aiu absolute displaceet, velocity, or acceleratio of the syste occurrig at ay tie as a result of the forcig fuctio, is called the aia respose, deoted by ν. (2) The aiu displaceet of the syste durig the residual vibratio era, called the residual aplitude, ad easured with respect to the fial positio of equilbriu, deoted by ν r (2).

2 3. - SINGLE DEGREE OF FREEDOM SYSTEMS Udaped, liear syste. Lets cosider a SDOF syste, liear, ad with o dapig. The iput is a fuctio of tie, ad ay be a force fuctio actig o the ass, or a displaceet of the base or foudatio. Soeties is ore coveiet to epress it as groud acceleratio. (Fig. 1) F(t) 2 2 X 1(t) a b c Fig. 1.- Sigle sprig-ass syste, subjected to ipulsive ecitatios. (a) Force F(t), (b) Groud displaceet, (c) Groud acceleratio. The differetial equatio goverig the otio of the systes show i figure 1, are: = + F() t or F() t + = (1) = [ u() t ] or + = u() t (2) δ + = δ or δ u () t + δ = (3) Where is the absolute displaceet of the ass relative to a fied referece,

3 ad δ is the displaceet relative to a ovig groud. The relatioship betwee these displaceets ad the groud displaceet is = u + δ. I this report, the otio of a syste will be described usig a ore geeral for of the equatios (1, 2, 3). The equatio is: v+ v = ξ (4) Where v is the respose of the syste, ad ξ the ecitatio, both fuctios of tie. However, soeties it is ecessary to epress ecitatio ad respose usig ore specific otatios. The alterate fors of ecitatio ad respose are give i table 1(1). Force Ecitatio ξ(t) F( t) Respose v(t) Absolute displaceet Groud displaceet u ( t) Absolute displaceet Groud acceleratio u ω 2 Relative displaceet δ Groud acceleratio u Absolute acceleratio Groud velocity u Absolute velocity th derivative of d u groud displaceet dt th derivative of absolute displaceet d dt Table 1. - Alterate fors of ecitatio ad respose of equatio 4.

4 Steplie ad pulselie ecitatio. The ipulsive ecitatios cause vibratioal resposes i elastic systes, ad the aiu values of these resposes ay be less tha, equal to, or greater tha the correspodig static resposes (3). I geeral, the respose depeds upo the syste properties, ad the ature of the load. For sigle degree of freedo systes, the characteristic that deteries the respose is the atural period (or atural frequecy). I additio, the shape ad duratio of the ipulse plays a iportat role i the respose. Shoc pheoea ca be odelated usig ideal step ad pulse fuctios, which represet very well the behavior of real trasiet iputs. The followig aalysis is based o the assuptio that the syste is iitially at rest. Step type ecitatio fuctios. The ost fudaetal trasiet ecitatio is the for of the step fuctio. To be fully realistic, these fuctios ust describe the traslatio of the syste through a fiite distace, i a fiite tie, with fiite acceleratio ad deceleratio (4). May fuctios rise to their aiu ξ c i a fiite tie t, called rise tie. Cosider the three et fuctios, the ecitatio fuctios ad the epressios for aia respose are give by (1): a) Costat slope frot. ξ t ξc [ 0 t τ ] = τ ξc [ τ t] (5) ν ξ c T πτ = 1+ si πτ T (6) a) Versed sie frot frot. ξ ξc π t 1-cos 0 t = 2 τ ξc [ τ ] [ τ t] (7)

5 ν ξ c 1 πτ = 1+ cos 4 1 T 2 2 ( τ T ) (7) a) Cycloidal frot. ξ ξc 2πt 2πt -si 0 t = 2 τ τ ξc [ τ t] [ τ ] (8) ν 1 πτ = 1+ si ξ πτ τ T c 2 2 ( 1 T ) (9) Where T is the atural period of the syste ivolved. The plots for the ecitatio fuctio, ad the tie respose curves are superiposed i fig. 2. a b c Figure 3.- Tie respose curves ad ecitatio fuctios for (a) costat slope frot, (b)versed sie frot ad (c)cycloidal frot. Syetrical pulses. A pulselie ecitatio is a ore cople fuctio, beig equivalet to the superpositio of two or ore successive step fuctios. Cosider three sigle syetrical pulses; rectagular, half sie ad versed sie. The ecitatio fuctios ad tie respose equatios are give by the followig equatios (1). Residual respose factors are set i bracets.

6 a) Rectagular ξ = ξp ν = ξp ω b) Half cycle sie ( 1 cos( t) ) ξ = 0 πt τ ν = ξp 2si siω t T 2 [ 0 t τ ] [ τ t] (10) (11) πt ξ() t = ξp si τ ξ p πt T ν = si siω 2 2 t 1 T 4τ τ 2τ c) Versed sie ξ = 0 ( T τ) cos( πτ T) τ ν = ξ p siω 2 2 t ( T 4τ ) 1 2 ξ p 2π t ξ = 1 cos 2 τ 2 2 ξ p 2 τ τ 2πt ν = 1 + cos cos τ T T T τ ξ = 0 si ( πτ T ) τ ν = ξp siω 2 2 t 1 τ T 2 ( ω t) [ 0 t τ ] [ τ t] [ 0 t τ ] [ τ t] (12) (13) (14) (15) Tie respose curves ad ecitatio fuctios are show i fig. 5, for differet values of the ratio τ T.

7 Ratio τ T Rectagular Half sie Versed sie 1/4 1/2 1 3/2 2 5/2 Figure 5.- Tie respose curves for several syetrical pulses; rectagular, half sie, ad versed sie, for differet values of τ T

8 3.2. Shoc spectru. Ay echaical syste is capable of trasiet vibratios i the for of otios of oe part with respect to aother, ad ivolvig resoaces at a uber of frequecies (5). The priary effect of shoc is to ecite the equipet at its resoace frequecies. This vibratio dies away depedig of the aout of dapig preset i the syste. As a result, daage or alfuctio ay tae place. The pea acceleratio ad pea relative displaceet of the syste are particularly iportat. Spectru was defied as a plot of a respose quatity (displaceet, velocity or acceleratio), agaist the ratio of the period or frequecy of the forcig fuctio, to the atural period or frequecy of the syste. I this report, we will refer to shoc spectra, also regarded as respose spectra. Shoc spectra are siply the pea acceleratio or displaceet produced by the shoc i the ass as a fuctio of the resoat frequecy (4). Shoc spectru ca be eperietally easured, coputed fro wavefors, or deteried theoretically. Such diagras are of iterest i desig, because they provide the possibility of predictig the aiu dyaic stress (3), ad the potetial daage i the syste. I fact, the shoc spectru gives a full ad realistic easure of the daagig potetial of a shoc disturbace. To select a daage criterio (acceleratio or displaceet), the duratio of the pulse ad the atural period of the syste are of great iportace. If T τ, the syste behavior becoes stiff, ad the otio of the ass closely follows the otio of the support. The acceleratio becoes the quatity of cocer. Otherwise, whe T τ, the eleet becoes soft, ad the ass reais substatially at rest util otio of the support has bee ceased, is the aiu displaceet the value that deteries potetial daage. Moreover, if the trasiet disturbace is either fast or slow eough to fit ito oe of the previous cases, o siple daage criterio ca be foud (4) Shoc spectra for particular pulse ad step shapes. Further isight ito the sigificace of shoc spectru ca be obtaied by studyig the spectra correspodig to a uber of siply shapes (5). Cosider the fuctios discussed o sectio ad Fig. 6 shows the spectra for aia respose resultig fro the step fuctios discussed o sectio 3.2.1, plotted as a fuctio of the ratio betwee rise tie ad atural period of the syste. Maia respose occurs after the ecitatio as reached its costat aiu, ad is related to the residual aplitude by(1): ν = νr + ξc (16)

9 Maia respose for step fuctios ν /ξ c Costat Slope Versed Sie Cycloidal Ratio τ/t Figure 5.- Maia respose spectru for three step fuctios; costat slope, versed sie, ad cycloidal. Fro the aalysis of these graphs we ca observe that the etree values of the ratio of aiu respose to step height ν ξ c are 1 ad 2. Whe the ratio of step rise tie to atural period τ T teds to 0, ν ξ c teds to its aiu of 2, ad whe τ T approaches ifiity, the step is o loger a dyaic ecitatio, i cosequece the iertia forces of the syste teds to zero, ad ν ξc approaches the lower value of 1(1). It is clear that for certai values of τ T, the ratio ν ξ c is equal to 1. The lowest value of τ T for which ν ξ c =1, are: costat slope, 1.0; versed sie, 1.5; cycloidal, 2.0 (1). Respose spectra for syetrical pulses (rectagular, half-sie ad versed sie) are show i fig. 6. A further aalysis of these plots ad the tie respose curves (Fig. 5) reveals that for values of τ T less tha ¼ (this is the case of short duratio pulses), the shape of the pulse is of less iportace i deteriig the aiu value of the respose, but, i cotrast, if τ T is larger tha ½, the shape ay be of great sigificace. The aiu value of the residual respose aplitude for the shapes discussed is ofte a good approiatio to aiu of aia respose, ad they occur at values of τ T ot greatly differet fro each other (1). The residual respose aplitude geerally has zero values for certai fiite values of τ T (1).

10 2.5 Respose spectru for rectagular pulse Sigle degree of freedo, udaped 2.0 Ratio ν/ξ Residual respose Maia respose Ratio τ/t (a) Respose spectru for half sie pulse Sigle degree of freedo, udaped 2.0 Ratio ν /ξ Maia respose Residual respose Ratio τ/t (b) Respose spectru for versed sie pulse SIgle degree of freedo udaped 2.0 Ratio ν/ξ Residual respose Maia respose Ratio τ/t (b) Figure 6.- Shoc respose spectru for aia ad residual aplitude for three syetrical pulses; (a)rectagular, (b)half sie. (c)versed sie.

11 Liear syste with viscous dapig. Regardig steady forced vibratio, eve if the syste has little values of dapig, it has great iportace i liitig the syste respose ear resoace. However, if the ecitatio is a pulse or step fuctio, the effect of dapig ay be of less iportace, uless the syste is highly daped (1). Nevertheless, as a result of the itroductio of dapig i the syste, aia respose decreases, as well the residual respose. Although, less wor had bee doe i daped systes uder shoc ecitatios. For daped systes, the equatio of otio uder geeral otatio, is: c v+ v+ v= ξ t 2 or () 1 2ζ v+ v+ v= ξ t ω ω Figure 6 shows a respose spectru for a sigle degree of freedo syste with viscous dapig, subjected to a half sie pulse ecitatio (1). () Figure 6. Maia respose spectru for a sigle degree of freedo syste with viscous dapig uder subjected to a half sie pulse ecitatio, for various values of dapig ratio(1). 3. CONCLUSIONS AND REMARKS. The ost iportat quatities i shoc easureets are the aia respose ad the residual respose, both used to deterie the severity of the shoc. A powerful tool i the aalysis of shoc otios is the shoc respose spectru, which gives iforatio about the relatioship betwee aia respose, ad the duratio of the shoc ad the atural period of the syste.

12 Fro the shoc respose spectru ad tie resposes for step fuctios, it is clear that the aia respose occurs always after the ecitatio has reached its aiu. O the other had, for pulse fuctios, the respose depeds upo the ratio betwee the duratio of the pulse ad the atural period of the syste. Or, i other words, if the pulse is of short duratio, the shape of the pulse has ot iportace (for τ T less tha 1/2), ad the aia occurs after the ecitatio has ceased. I cotrast, whe the pulse is log, the shape plays a iportat role, ad aia respose ca occur durig the pulse, ad after it has ceased. For soe values of τ T, residual respose its zero. The iclusio of viscous dapig i the syste causes a decrease i the aia respose. Most of the wor has bee carried out cosiderig oly viscous dapig. Further research ca be doe i ivestigatig the behavior of the syste whe the dapig is structural. 3. REFERENCES. 1. Harris ad Creede: Shoc ad vibratio hadboo, Ayre, R. S.: Egieerig vibratios, Tiosheo S, Youg D H, Weaver W.: Vibratio probles i egieerig, Sowdo J C.: Vibratio ad shoc i daped echaical systes, Morrow: Shoc ad vibratio egieerig, 1963