6 Vol. No.6 005 ENGNEENG MEHANS Dec. 005 000-4750(005)06-0046-06 * (. 00084. 000) O47.4, P5. A THEE-DMENSONAL VSO-ELAST ATFAL BOUNDAES N TME DOMAN FO WAVE MOTON POBLEMS * LU Jng-bo, WANG Zhen-yu, DU Xu-l, DU Y-xn (. Deartment of vl Engneerng, Tsnghua Unversty, Bejng 00084, hna;. Bejng Unversty of Technology, Bejng 000, hna) Abstract: Three-dmensonal vsco-elastc artfcal boundares n tme doman are develoed by means of theory of elastc wave moton n ths aer. Three-dmensonal vsco-elastc artfcal boundary equatons along normal and tangent drectons are derved based on three-dmensonal wave moton equatons. Numercal smulaton technques of vsco-elastc artfcal boundares n tme doman are studed. Numercal examles of classc wave moton roblems demonstrate that hgh recson s acheved by use of three-dmensonal vsco-elastc boundares, and that the boundares can be used n analyss of three-dmensonal wave moton roblems easly. ey words: wave moton; vsco-elastc artfcal boundary; three-dmensonal meda; tme doman; nfnte doman [~] () [~5] 00--0 004-05-4 97 (00B4706) (5007808) (8000) *(956) (E-mal: lujb@tsnghua.edu.cn) (976) (96) ; (979).
47 [6,7] [8] Deeks [9] [8,0]. (P ) φ = c ( φ) () φ c P () φ (, = f ( cp + g( + cp () f( ) g( ) φ u = φ = = f ( c f ( c () u σ = ( λ + µ ) + λ (4) λ µ () = f ( c f ( c + (5) f ( c u = f ( c f ( c (6) (5) (6)(4) f( ) λ + µ σ = f ( c f ( c + (7) f ( c σ u c c u& = = f ( c + f ( c t (8) u c c u& = = f ( c f ( c (9) σ λ + µ = c f ( c + c f ( c c f ( c (0) () (7) (0) 4G ρ σ + & σ = u + u& + u&& () c c 4G () G = µ ρ () λ + µ = ρc () () λ µ ρ () 4G σ = u () () []. --
48 u + u& u& ) = σ (4) ( M M u& M + ( u& M u& ) = 0 (5) (4) (5) u u M σ M Fg. Physcal system mosed on vertcal boundary (4) u & M = ( u + u& σ ) (6) u & M = ( u& + u&& σ& ) (7) (6) (7)(5) M σ M M u σ + = u + + (8) t () (8) = 4G = ρc M = ρ (9) (9). (S ) [] u(, = f ( cs + g( + cs (0) c s (S ) u u M u(, = f ( cs () () u γ (, = = f ( c f ( c s + s () τ (, = Gγ = G f ( c + f ( c s s () (, cs u& = = f ( cs (4) () () (4) τ G (, = u(, cs u(, ρ & (5) (5) G = µ ρ (5) G = ρc s. (5) - = G = ρcs (6) (6) (9) (6) M ( ) M M () ( ) (Vscous-Srng Boundary)
49 X Y Z = = 4G = = ρc G = = ρc = = = s = A, A A, A, (7) A =4 A A Y A A4 X Z Y Fg. Three dmensonal vscous-srng boundary 4 4. Lamb Lamb δ δ δ ( τ ) = 6[ G4( τ ) 4G4 ( τ ) + 6G4( τ ) 4 (8) 4G4( τ ) + G4( τ )] 4 t G 4 ( τ ) = τ H ( τ ), τ = T X T H (τ ) Heavsde y ( Lamb Fg. Lamb s roblem δ Lamb [,4] (D vscous-srng boundary)(vscous boundary)(fxed boundary) (mass-vscoussrng boundary) x b = x b = 0. 5 z b =.0 x = y = z = 0. s =4 ρ =.0 ν =0.5 A(r=0.) B(r=0.4) 4 5 A B z x
50 /m 0.045 0.040 0.05 0.00 0.05 0.00 0.05 0.00 0.005 0.000-0.005 0.0 0. 0.4 0.6 0.8.0..4.6.8.0 /s 4 A(r=0.) Fg.4 Dslacement tme-hstory of observaton ont A (r=0.) 0.00 0.05 0.00 /m 0.05 0.00 0.005 0.000-0.005 0.0 0. 0.4 0.6 0.8.0..4.6.8.0. /s 5 B(r=0.4) Fg.5 Dslacement tme-hstory of observaton ont B (r=0.4) 4. 6 y o δ( z 6 Fg.6 nner source roblem x b = xb = zb = 0. 5 x x=y=z=0. Lamb 7 z 7 Lamb 0.5 0.5 =0.707 /m 0.00 0.05 0.00 0.05 0.00 0.005 0.000 0.0 0. 0.4 0.6 0.8.0..4.6.8.0 /s (=0.5) (=0.707) 7 δ z Fg.7 Dslacement tme-hstory n z drecton of observaton ont under δ-functon n nner source roblem 5
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