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Transducer 2 transducer 3 transducer 1 hyperbola 1 2

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Transducer 2 Transducer 3 Transducer 1 Hyperbola 1-2 Hyperbola 1-3 Source Z 2
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monitoring one half of the vessel. Mathematical calculations for the location of an acoustic emission source in a three-dimensional space become simpler with this arrangement. As shown in Fig. 12, four transducers are located at ( r 0 , [± π ÷ 2], 0) and ( r 0 , 0, ± h ) in cylindrical coordinates. When values of i = 1 and 2, the distance from an acoustic emission source at an arbitrary position ( r , θ , z ) to each transducer is: (18) where: (19) and: (20) For i = 3 and 4, the distance from an acoustic emission source at an arbitrary position ( r , θ , z ) to each transducer is: (21) where a 3 2 = a 4 2 and a 3 2 = r 0 2 + r 2 – 2 r 0 r cos θ . When j = 1, 2, 3 and 4 and the subscripts i and j indicate transducer numbers, then: (22) The basic relations in Eqs. 18 to 22 lead to a quadratic equation for z in terms of r and θ for each pair of transducers. The solution of the quadratic equation for transducer 1 and transducer 2 is: (23) For transducer 1 and transducer 3: (24) Equations 25 to 27 define the terms used in Eq. 24: (25) (26) (27) To make Eq. 24 valid for transducer 1 and transducer 4, h and Δ t 13 in Eqs. 25 to 27 should be replaced by – h and Δ t 14 , respectively. The purpose for the development of Eqs. 23 and 24 is to simplify computer programming. Both equations can be used to calculate values of z for each pair of transducers from given values of r and θ . Therefore, all possible solutions for z can C r r v t v t r r r r v t h v t h r r h = ( ) + + ( ) + + ( ) + + ( ) + 4 4 4 2 2 2 0 2 2 2 4 13 4 2 13 2 0 2 2 0 2 13 2 2 2 13 2 2 0 4 cos sin cos sin sin cos θ θ θ θ θ θ Δ Δ Δ Δ B hr r hA = ( ) 4 2 0 cos sin θ θ A h v t = ( ) 4 2 2 13 2 Δ z B B AC A = ± 2 4 2 z r r v t v t r r = + 4 1 4 0 2 2 2 2 12 2 2 12 2 0 2 2 0 5 sin . θ Δ Δ d d v t i j ij + = Δ d h z a i i = ± ( ) + 2 2 a r r r r 2 2 0 2 0 2 = + + 2 sin θ a r r r r 1 2 0 2 0 2 = + + 2 sin θ d z a i i = + 2 2 129 Acoustic Emission Source Location F IGURE 12. Source location in three dimensions for thick walled cylinder. Transducer 3 ( r o , 0, h ) Transducer 2 ( r o , [– π ÷ 2], 0) Transducer 1 ( r o , [ π ÷ 2], 0) Transducer 4 ( r oa , 0, – h ) h z d 3 a 3 d 2 d 1 z a 2 a 1 r a 3 θ Legend a 1 , a 2 , a 3 = horizontal distance (meter) from source to transducers 1, 2 and 3, respectively d 1 , d 2 , d 3 = direct distance (meter) from source to transducers 1, 2 and 3, respectively h = vertical length or height (meter) of cylinder in z direction r = cylindrical coordinate for radius (meter) z = cylindrical coordinate for vertical distance (meter) θ = cylindrical coordinate for angular distance (radian)
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be obtained from two loops in a computer program. The radius r varies from the inner radius to the outer radius and the angle θ varies from (– π ÷ 2) to ( π ÷ 2). Each loop only needs a limited number of steps to achieve sufficient spatial resolution.
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