Modeling a vibrating string terminated against a bridge with - PowerPoint PPT Presentation

Modeling a vibrating string terminated against a bridge with arbitrary geometry Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa V¨ alim¨ aki Institute of Cybernetics at Tallinn University of Technology, Centre for Nonlinear Studies (CENS), Tallinn, Estonia & Department of Signal Processing and Acoustics, Aalto University, Espoo, Finland August 3, 2013 Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa V¨ SMAC SMC 2013 alim¨ aki (CENS) August 3, 2013 1 / 18

Motivation In numerous musical instruments the collision of a vibrating string with rigid spatial obstacles, such as frets or a bridge is present. Biwa Shamisen Sitar Jawari (bridge) Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa V¨ SMAC SMC 2013 alim¨ aki (CENS) August 3, 2013 2 / 18

Motivation Medieval and Renaissance bray harp and bray pins Audio example of bray harp timbre (15 s) Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa V¨ SMAC SMC 2013 alim¨ aki (CENS) August 3, 2013 3 / 18

Motivation Capo bar (Capo d’astro) of the piano cast iron frame Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa V¨ SMAC SMC 2013 alim¨ aki (CENS) August 3, 2013 4 / 18

String description � ∂ 2 u ∂t 2 = c 2 ∂ 2 u T ∂x 2 , c = (1) µ u (0 , t ) = u ( L, t ) = 0 (2) Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa V¨ SMAC SMC 2013 alim¨ aki (CENS) August 3, 2013 5 / 18

String description � ∂ 2 u ∂t 2 = c 2 ∂ 2 u T ∂x 2 , c = (1) µ u (0 , t ) = u ( L, t ) = 0 (2) Solution to Eq. (1) is famous d’Alembert’s solution: u ( x, t ) = 1 2 [ u r ( x − ct ) + u l ( x + ct )] (3) Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa V¨ SMAC SMC 2013 alim¨ aki (CENS) August 3, 2013 5 / 18

Geometric termination condition (TC) TC is an absolutely rigid unilateral constraint of the string’s transverse deflection. Support profile geometry is described by an arbitrary function U ( x ) . Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa V¨ SMAC SMC 2013 alim¨ aki (CENS) August 3, 2013 6 / 18

Geometric termination condition (TC) TC is an absolutely rigid unilateral constraint of the string’s transverse deflection. Support profile geometry is described by an arbitrary function U ( x ) . Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa V¨ SMAC SMC 2013 alim¨ aki (CENS) August 3, 2013 6 / 18

Geometric termination condition (TC) TC is an absolutely rigid unilateral constraint of the string’s transverse deflection. Support profile geometry is described by an arbitrary function U ( x ) . Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa V¨ SMAC SMC 2013 alim¨ aki (CENS) August 3, 2013 6 / 18

Bridge-string interaction model Since the termination is rigid, it must hold u ( x ∗ , t ) � U ( x ∗ ) . (4) Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa V¨ SMAC SMC 2013 alim¨ aki (CENS) August 3, 2013 7 / 18

Bridge-string interaction model Since the termination is rigid, it must hold u ( x ∗ , t ) � U ( x ∗ ) . (4) In order to satisfy condition (4) for u ( x ∗ , t ) > U ( x ∗ ) a reflected traveling wave is introduced � t − x ∗ � � t + x ∗ � u r = U ( x ∗ ) − u l , (5) c c Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa V¨ SMAC SMC 2013 alim¨ aki (CENS) August 3, 2013 7 / 18

Bridge-string interaction model Since the termination is rigid, it must hold u ( x ∗ , t ) � U ( x ∗ ) . (4) In order to satisfy condition (4) for u ( x ∗ , t ) > U ( x ∗ ) a reflected traveling wave is introduced � t − x ∗ � � t + x ∗ � u r = U ( x ∗ ) − u l , (5) c c here the waves u l and u r correspond to any waves that have reflected from the terminator earlier. Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa V¨ SMAC SMC 2013 alim¨ aki (CENS) August 3, 2013 7 / 18

Single wave reflection from the terminator u r ( t − x ∗ /c ) = U ( x ∗ ) − u l ( t + x ∗ /c ) u ( x ∗ , t ) = U ( x ∗ ) = u r ( t − x ∗ /c ) + u l ( t + x ∗ /c ) (6) Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa V¨ SMAC SMC 2013 alim¨ aki (CENS) August 3, 2013 8 / 18

Single wave reflection from the terminator u r ( t − x ∗ /c ) = U ( x ∗ ) − u l ( t + x ∗ /c ) u ( x ∗ , t ) = U ( x ∗ ) = u r ( t − x ∗ /c ) + u l ( t + x ∗ /c ) (6) Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa V¨ SMAC SMC 2013 alim¨ aki (CENS) August 3, 2013 8 / 18

Single wave reflection from the terminator u r ( t − x ∗ /c ) = U ( x ∗ ) − u l ( t + x ∗ /c ) u ( x ∗ , t ) = U ( x ∗ ) = u r ( t − x ∗ /c ) + u l ( t + x ∗ /c ) (6) Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa V¨ SMAC SMC 2013 alim¨ aki (CENS) August 3, 2013 8 / 18

1.0 1.0 1.0 Displacement u ( x,t ) Displacement u ( x,t ) Displacement u ( x,t ) 0.5 0.5 0.5 0.0 0.0 0.0 0.5 0.5 0.5 Time 0.70 Time 0.80 Time 1.00 1.0 1.0 1.0 0.0 0.0 0.0 0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 0.6 0.8 0.8 0.8 1.0 1.0 1.0 Distance along the string x Distance along the string x Distance along the string x Single wave reflection from the terminator u r ( t − x ∗ /c ) = U ( x ∗ ) − u l ( t + x ∗ /c ) u ( x, t ) = u r ( t − x/c ) + u l ( t + x/c ) (7) 1.0 Displacement u ( x,t ) 0.5 0.0 0.5 Time 0.40 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Distance along the string x Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa V¨ SMAC SMC 2013 alim¨ aki (CENS) August 3, 2013 9 / 18

1.0 1.0 Displacement u ( x,t ) Displacement u ( x,t ) 0.5 0.5 0.0 0.0 0.5 0.5 Time 0.80 Time 1.00 1.0 1.0 0.0 0.0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1.0 1.0 Distance along the string x Distance along the string x Single wave reflection from the terminator u r ( t − x ∗ /c ) = U ( x ∗ ) − u l ( t + x ∗ /c ) u ( x, t ) = u r ( t − x/c ) + u l ( t + x/c ) (7) 1.0 1.0 Displacement u ( x,t ) Displacement u ( x,t ) 0.5 0.5 0.0 0.0 0.5 0.5 Time 0.40 Time 0.70 1.0 1.0 0.0 0.0 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1.0 1.0 Distance along the string x Distance along the string x Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa V¨ SMAC SMC 2013 alim¨ aki (CENS) August 3, 2013 9 / 18

1.0 Displacement u ( x,t ) 0.5 0.0 0.5 Time 1.00 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Distance along the string x Single wave reflection from the terminator u r ( t − x ∗ /c ) = U ( x ∗ ) − u l ( t + x ∗ /c ) u ( x, t ) = u r ( t − x/c ) + u l ( t + x/c ) (7) 1.0 1.0 1.0 Displacement u ( x,t ) Displacement u ( x,t ) Displacement u ( x,t ) 0.5 0.5 0.5 0.0 0.0 0.0 0.5 0.5 0.5 Time 0.70 Time 0.80 Time 0.40 1.0 1.0 1.0 0.0 0.0 0.0 0.2 0.2 0.2 0.4 0.4 0.4 0.6 0.6 0.6 0.8 0.8 0.8 1.0 1.0 1.0 Distance along the string x Distance along the string x Distance along the string x Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa V¨ SMAC SMC 2013 alim¨ aki (CENS) August 3, 2013 9 / 18

Single wave reflection from the terminator u r ( t − x ∗ /c ) = U ( x ∗ ) − u l ( t + x ∗ /c ) u ( x, t ) = u r ( t − x/c ) + u l ( t + x/c ) (7) 1.0 1.0 1.0 1.0 Displacement u ( x,t ) Displacement u ( x,t ) Displacement u ( x,t ) Displacement u ( x,t ) 0.5 0.5 0.5 0.5 0.0 0.0 0.0 0.0 0.5 0.5 0.5 0.5 Time 0.40 Time 0.80 Time 0.70 Time 1.00 1.0 1.0 1.0 1.0 0.0 0.0 0.0 0.0 0.2 0.2 0.2 0.2 0.4 0.4 0.4 0.4 0.6 0.6 0.6 0.6 0.8 0.8 0.8 0.8 1.0 1.0 1.0 1.0 Distance along the string x Distance along the string x Distance along the string x Distance along the string x Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa V¨ SMAC SMC 2013 alim¨ aki (CENS) August 3, 2013 9 / 18

Single wave reflection from the terminator 1.0 0.5 Amplitude [1] 0.0 0.5 Time 0.00 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Distance along the string [1] Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa V¨ SMAC SMC 2013 alim¨ aki (CENS) August 3, 2013 10 / 18

Model application: Biwa String length L = 0 . 8 m String plucking point l = 3 / 4 L = 0 . 6 m Linear mass density of the string µ = 0 . 375 g/m String tension T = 38 . 4 N Velocity of the traveling waves c = 320 m/s Fundamental frequency f 0 = 200 Hz Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa V¨ SMAC SMC 2013 alim¨ aki (CENS) August 3, 2013 11 / 18

Bridge profiles studied Profile shapes 1.0 case 1 0.5 Case 1: Linear 0.0 bridge with 0 5 10 20 x c 1.0 sharp edge U ( x ) (mm) case 2 0.5 Case 2: Linear 0.0 bridge with 0 5 15 20 x b 1.0 curved parabolic case 3 0.5 edge 0.0 Case 3: Bridge 0 x a 5 15 20 x b Distance along the srtring x (mm) with minor defect Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa V¨ SMAC SMC 2013 alim¨ aki (CENS) August 3, 2013 12 / 18

Result: Time series u ( l, t ) 6 case 1 Displacement u ( l,t ) (mm) 3 0 3 6 6 case 2 3 0 3 6 6 case 3 3 0 3 6 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Time t (s) Nonperiodic and almost periodic vibration regimes. Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa V¨ SMAC SMC 2013 alim¨ aki (CENS) August 3, 2013 13 / 18