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An example is shown fig 3 for the predicted out of

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An example is shown (Fig. 3) for the predicted out-of-plane displacement for a cracklike acoustic emission source (set of double sources) connected to the same surface as the sensor. For this example, the source is 0.9 m (3 ft) from the transducer. Inverse Problem Equation 34 may be rewritten in terms of the coefficients Q of distinct bessel functions: (37) (38) (39) (40) The response v ( t ) is then given by: (41) Thus, there are four distinct amplitudes needed for each k and b. All four are needed for an exact calculation of waveforms. They are functions of the parameters M •• that represent source orientation and strength; they are functions also of the depth of the source represented by the coordinate z and of the wavenumber and branch number of the mode of interest. However, for an estimate of the relative importance of each mode at asymptotic distances rk >> 1, it suffices to consider only a single effective amplitude Q eff for each type of strain nucleus source and each source depth and each mode. At asymptotic distances, J 0 ( kr ) is about – J 0 ( kr ); J 0 is π /2 radians (90 deg) out of phase with J 0 and J 0 ( kr )·( kr ) –1 is negligible — hence the definition of a single effective amplitude: (42) This single quantity represents the contribution of a given mode { k,b }. It is useful to plot it versus ω for each branch b , for each of several parameters M •• relating source orientation to strength and for source coordinate z — thereby illustrating the relative contributions of different sources to different modes. If source character is to be ascertained through examination of the waveform, it must necessarily be by means of this quantity Q eff ; all information about the source is transmitted through Q eff . This observation forms the basis of the inverse procedure below. Q k k k Q k Q k Q k b eff ( ) = ( ) × ( ) [ ] + ( ) ( ) [ ] ω 2 2 2 1 3 2 v k dk Q k J kr Q k J kr Q k J kr Q k J kr kr k t k b b b = ( ) ( ) [ + ( ) ( ) + ( ) ′′ ( ) + ( ) ( ) ( ) [ ] ( ) – cos 1 1 1 0 2 0 3 0 4 0 2 π ω ω Q k k M U z W h 4 2 ( ) ( ) ( ) = θθ ρ M Q k k M U z W h rr 3 2 ( ) ( ) ( ) = ρ M Q k M kW z U z W h rz z 2 2 ( ) ( ) + ∂ ( ) [ ] × ( ) = ρ M Q k M z W h zz z 1 2 ( ) ( ) [ ] ( ) = ρ M 2 2 2 1 R R R R J kr r dr J k k 0 ( ) = ( ) 117 Modeling of Acoustic Emission in Plates F IGURE 3. Out-of-plane displacement prediction for a cracklike source located 0.9 m (3 ft) from the transducer and connected to the same surface, in Unified Numbering System G43400 alloy steel. 0.3 0.2 0.1 0 –0.1 –0.2 160 200 240 280 320 360 400 440 480 520 Displacement (arbitrary unit) Time (μs)
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The case of a step vertical surface load (Eq. 35) is also of interest: (43) where Q 1 stepload ( k ) is: (44) where sign = –1 if the branch is an antisymmetric branch and if the source and receiver are on opposite sides of the plate; sign = +1 otherwise. Thus, Q 2 = Q 3 = Q 4 = 0 is identified for the step force source. The effective Q is given again by Eq. 42: (45) The inverse problem thus reduces to estimating, from measured waveforms, the effective modal amplitudes Q eff at each { k,b }. A waveform in the time domain at given time t may be understood as a superposition of the many modes { k,b } that propagate at the same group velocity. 2
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  • Fall '19
  • Acoustic Emission

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