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PONAŠANJE BRODA NA TALASIMA____________________________5.1. Spektri i statističke vrednosti ljuljanjabrodaIz izraza za prenosnufunkciju ljuljanjamoguće je dobitiZPZ( ωn)=Aon( ω )n( ω )2 2Zo( ωn) 2 An( ωn)2PZ ( ωn) PZ ( ωn) ST( ωn)2⋅Δω= 2⋅Δω= ⋅ppri čemu je iskorišćenop1ST( ωn) =2n2A nn( ω)ΔωDefinisaćemo, analogno spektru talasa, spektarljuljanja broda (spektar valjanja, poniranja,posrtanja...)21 Z ( )oωS ( ) =nBωn2 ΔωSledi formula2S ( ω ) = P ( ω ) ⋅S( ω )B n Z n T nppUkolikoΔω→ dωpdiskretne vrednostiS ( ω ), P ( ω ), S ( ω )T n Z n B npostaju kontinualne funkcije prividnefrekvencije ω pS ( ω ), P ( ω ), S ( ω )tako da važiT p Z p B p2S ( ω ) = P ( ω ) ⋅S( ω)B p Z p T pTo je ključna relacija proračuna ljuljanjabroda na neregularnim talasima...Omogućava da se, iz zadatog spektraneregularnih talasa i poznate prenosnefunkcije ljuljanja broda,odredi spektar ljuljanja broda naneregularnim talasima.pPREDAVANJA 2009. 149
PONAŠANJE BRODA NA TALASIMA____________________________Ukoliko, kod rotacionih kretanja, koristimoprenosnu funkcijuZo( n) Zo( n) PZ( n)Pω ω ωZ( ωn)= = =α ( ω ) k ( ω ) A ( ω ) k( ω)n n n n n n n nsledi alternativna formula222SB ( ωp) = PZ( ωp) ⋅ k ST( ωp) = PZ( ωp) ⋅S α( ωp)gde je Sα ( ω p) tzv. spektar nagiba talasa...Ukoliko se vrednosti spektra ljuljanja nanesunad ogdovarajuće frekvencije ω n sa korakomΔω p ...dobiće se stepenasta linija...Površina pod svakim od ovako dobijenihihstepenika iznosi1 2SB ( ωn) ⋅ Δωp = Zo( ωn)2a ukupna površina ispodstepenaste linije je∞∞S∑B n p∑o nn= 1 n=11 2( ω ) ⋅ Δω = Z ( ω )2Dokazano je ranije da za neregularnetalase važi relacija∞( T)1 2σ = ∑oAn( ωn)2n=1To je tzv. Persevalova teorema... kojavaži i za ljuljanja brodaPREDAVANJA 2009. 150
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