RESPONSE OF DUFFING OSCILLATOR UNDER NARROW-BAND RANDOM EXCITATION

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1 Th rd Intrnational Confrnc on Comutational Mchanics and Virtual Enginring COMEC 9 9 OCTOBER 9, Brasov, Romania RESPONSE O DUING OSCILLATOR UNDER NARROW-BAND RANDOM EXCITATION Ptr STAN, Mtallurgical High School, Slatina, Romania, -mail: tr_stan_marian@yahoo.com Abstract: Nonlinar, dynamic systms subjct to random xcitations ar frquntly mt in nginring ractic. Th sourc of randomnss can vary from surfac randomnss in vhicl motion and nvironmntal changs, such as arthquaks or wind xciting high ris buildings or wav motions at sa xciting ofshor structurs or shis, to lctric or acoustic nois xciting mchanical structurs. Th rsarch goals ar, firstly, th comutation of stochastic, nonlinar rsons charactristics (with accuracy and fficincy as imortant critria) and, scondly, th invstigation and thorough undrstanding of stochastic, nonlinar rsons hnomna. Th dsir to comut rsons charactristics, such as th owr sctral dnsity of th rsons of ths systms, lads to th dvlomnt of mthods that can b usd to aroximat this rsons. Th xcitations, that will b studid, ar stationary, Gaussian rocsss. 1. INTRODUCTION W rsnt a mthod for stimating th owr sctral dnsity of th stationary rsons of oscillator with a nonlinar rstoring forc undr xtrnal stochastic wid-band xcitation. An quivalnt linar systm is drivd, from which th owr sctral dnsity is dducd. Th mthod of th stochastic quivalnt linarization is basd on th ida that a nonlinar systm may b rlacd by a linar systm by minimizing th man squar rror of th two systms. This mthod has sn th broadst alication bcaus of thir ability to accuratly catur th rsons statistics ovr a wid rang of rsons lvls whil maintaining rlativly light comutational burdn. SOLUTION O EQUATION O MOTION Considr th quation of motion... m( t) + c( t) + k ( t) + αk ( t) ( t). (1) Th rducd quation is... ( t) + ξ ( t) + ( t) + α ( t) f ( t). () As a nxt lt us considr xcitation dscribd by subsqunt corrlation function[9] D λτ R ( τ ) cos βτ, () whr aramtrs D>, λ>, β. Powr sctral dnsity function [1] of xcitation w obtain from th rlation: 1 ( ) R ( τ ) dτ π () By substitution of th () in th () and intgration w obtain or Dλ + β ( ) π ( i) + λ( i) + β Dλ ( ) π + β. (6) + + λ ( λ β ) (5) 75

2 ig. 1. Th owr sctral dnsity π of xcitation for D 5 N, λ 1 s, β s ig.. Th owr sctral dnsity π of xcitation for D 5 N, λ 1 s, β, 5 s. 1 1,5,5,5 1,5 1,5,,8 1, 1,6,,8,,6,,8 5, 5,6 6 ig.. Th owr sctral dnsity π of xcitation for D 5 N, λ s, β, 5 s

3 7,8 7, 6,6 5, 6,8,,6, 1,8 1,,6,,81,1,6,,8,,6,,85,5,6 6 6,6,8 ig. Th owr sctral dnsity π of xcitation for D 5 N, λ 1 s, β 6, 5 s. 1 1 Powr sctral dnsity function of outut w can obtain from th rlation ( ) / m + ξ S ( ) ( ) So w obtain. (7) Dλ ( + β ) S ( ). (8) πm { + α σ + ξ } ( β ) + λ Th dislacmnt varianc [] of th singl-dgr of frdom systm undr Gaussian whit nois xcitation can b xrssd as, σ R () S ( ) d. (9) Substitution of th (8) in th (9) and obtain Dλ ( + β ) σ d mπ. (1) { + α σ + ξ } ( β ) + λ Intgration [,] obtain + d π ( bo h1+ h1 h h ) d b h1 hh boh1 d h 1 +, (11) ( i) λ( i) d ( i) b ( i) b ( ) whr h1 b1+ λ, h b+ λb1+ d, h λb+ db1. (1) In this cas 1 ξ + λ + λξ + β λ + ξ β h ( ), h, h ( ) ασ 1 b (1+ ), b ξ. A σ, (1) B whr A Dλσ [1 α ( ξ + λ) 6 αλ ] + Dλ{ ( ξ + λ) λξ ( ξ + λ) + ( ξ + λ)( β ) λ ξ ( β )} (1) (15) 77

4 6 7 B 18mσ λξα + 6 σ mα { λα[ + λξ + ( β )]( ξ + λ) + + ( ξ + λ) [ + λξ + ( β )][ λ+ ξ( β )] + α[ + ξλ ( β ) + ( ξ + λ) ( β )] + λα } + 1 mσ { αξ ( + λ)[ λ+ 5 + ξ( β )][ + λξ + ( β )] ξ α( β ) ξ λα( 6 + β ) λ α αξ ( + λ) ( β )} + { λα[ + λξ + ( β )]( ξ + λ) + + ( ξ + λ) [ + λξ + ( β )][ λ+ ξ( β )] + α[ } + ξλ ( β ) + ( ξ + λ) ( β )]} + m( ξ + λ)[ λ+ ξ( β )][ + + λξ + ( β )] ξ m( β ) ξ mλ( β ) λ m m( ξ + λ) ( β ). Using th notation 7 { 5 6 l 18mλξα (17) n 6 αm{ λα[ + λξ + ( β )]( ξ + λ) + ( ξ + λ) [ + λξ + ( + β )][ λ+ ξ( β )] + α[ ξλ ( β ) + ( ξ + λ) ( β )] + λα } { α ξ λ λ ξ λ β λξ λ β ξ α λ β r 1 m { ( + )[ + ( + )][ + + ( + )] ( + ) 5 6 ξ λα( β ) λ α α( ξ + λ) ( β )} + { λα[ + λξ + ( + β )]( ξ + λ) + ( ξ + λ) [ + λξ + ( β )][ λ + ξ( β )] + α[ + ξλ ( β ) + ( ξ + λ) ( β )]}} { s m ( ξ + λ)[ λ + ξ( β )][ + λξ + ( β )] ξ ( β ) 5 6 ξ λ( β ) λ ( ξ + λ) ( β ) Dλ[1 α ( ξ + λ) 6 αλ ] } q Dλ{ ( ξ + λ) 8 λξ ( ξ + λ) ( ξ + λ)( β ) + λ + ξ ( β )} (1) obtain th quation 8 6 lσ + nσ + rσ + sσ + q. () W can always find a way to dcomos th nonlinar rstoring forc to on linar comonnt lus a nonlinar comonnt h( ) ( + G( ) α), () whr α is th nonlinar factor to control th ty and dgr of nonlinarity in th systm. Th ida of linarization is rlacing th quation by a linar systm:... ( t) + ξ ( t) + ( t) f ( t), () whr ξ ξ. (5) is th daming ratio of quivalnt linarizd systm and (16) (18) (19) () is th natural frquncy of th quivalnt linarizd systm. To find an xrssion for, it is ncssary to minimiz th xctd valu of th diffrnc btwn quations () and () in a last squar sns. Now th diffrnc is th diffrnc btwn th nonlinar stiffnss and linar stiffnss trms, which is h( ( t)) ( t). (6) Th valu of can b obtaind by minimizing th xctation, of th squar rror: de{ }. (7) d Substituting th quation (6) into quation (7) rforming th ncssary diffrntiation, th xrssion of can b obtaind as: E{ G( )} (1 + α ) (1+ ασ )., (8) σ 78

5 whr σ is th standard dviation of ( t). This quation shows how th nonlinar comonnt of th stiffnss lmnt affcts th valu of.. NUMERICAL RESULTS: Considr in this xaml, N Ns m 1 kg, k 6, c, m. m m α (9) Lt us st th subsqunt valus of xcitation aramtrs 1 1 D 5 N, λ 1 s, β s. () Obtain: or σ 1+ 16σ, (1) σ + 781,91 1 σ + 85, 5 1 σ σ + 781,91 1 σ + 85,5 1 σ + 16σ 1, () σ, 5m. () Substituting th quation () into quation (8), obtain 1 (1 ασ ) 7, 6 + s. () In litratur, vry littl attntion has bn aid to th frquncy domain charactristics of nonlinar, dynamic systms xcitd by stochastic rocsss. It will b shown that this information can b of grat valu for th undrstanding of th systm's stochastic bhaviour. In th figurs 1,,, and 5, th owr sctral dnsity of th xcitation, [ N s], is lottd for th diffrnts aramtrs D, λ, β. igur 6 dscribs th harmonic ak with th sam aramtr valus. 7,5 7 6,5 6 5,5 5,5,5,5 1,5,5 1,,8 1, 1,6,,8,,6,,8 5, 5,6 6 6, π ig. 5 Th owr sctral dnsity [ N s] of xcitation 1 1 N Ns D 5 N, λ 1 s, β s, m 1 kg, k 6, c, m. m m α 79

6 ,5,,5,,5,,15,1,5,,,6,8 1 1, 1, 1,6 1,8,,,6,8 ig.6. Th owr sctral dnsity π N Ns S [ m s] of rsons. m 1 kg, k 6, c, m. m m α. CONCLUSION Th statistical linarization tchniqu can also tackl a wid varity of roblms and also rovids aroximat information on th frquncy domain charactristics of th stochastic rsons. In this tchniqu, a linar modl, which otimally ts th original, nonlinar systm (in som statistical sns), is constructd. Du to th fact that rsons statistics of such a modl can, in gnral, b valuatd analytically, statistical linarization is comutationally vry ffcint. Howvr, it only rovids accurat aroximation of th rsons statistics for wakly nonlinar systms. In this chatr, it is shown that th statistical linarization tchniqu structurally undrstimats th varianc of th rsons of th ic-wis linar systm (vn for a modrat nonlinarity). This is dangrous whn ths stimats ar usd in failur critria for ractical systms. Th caus for this undrstimation of th varianc can b found by comaring accurat, simulatd frquncy domain charactristics with thos dtrmind using th linar modl.. REERENCES: [1] Pandra, N., Parlac, S., Mchanical vibrations, Pitsti Univrsity,. [] Muntanu, M., Introduction to dinamics oscilation of a rigid body and of a rigid bodis sistms, Clusium, Cluj Naoca, 1997 [] Zhao, L., Chn, Q., Equivalnt linarization for nonlinar random vibration, Probabilistic Enginring Mchanics,9(199). [] Butscu, V., Stan, P., Alid mathmatics, Sigma, Bucharst,. 7