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• W1909283576 abstract "A crack in a thin plate reflects ultrasonic waves; therefore, it is reasonable to determine the location of the crack by measuring the reflected waves. The problem of locating the crack can be reformulated in purely geometric terms. Previously, time-consuming iterative numerical methods were used to solve the resulting geometric problem. In this paper, we show that explicit (and fast to compute) formulas can be used instead. Formulation of the engineering problem. One of the most common problems in aging aircraft structures is the presence of cracks. These cracks are often not visible because they are hidden inside the structure or covered with paint. It is therefore necessary to use techniques of non-destructive testing (NDT) such as ultrasonic Lamb waves. Lamb waves in thin plates are very convenient in detecting cracks in large-scale structures because these waves can propagate long distances and thus, can help us explore large portions of the plate; see, e.g., (Viktorov 1967). In a faultless plate, a Lamb wave can travel long distances without dispersion or reflection. Defects reflect and scatter these waves; as a result, the very presence of a reflected wave indicates a defect. It is reasonable to determine the location of the crack by measuring the reflected waves. Reduction to a geometric problem. To locate the crack, we generate a wave pulse that is sent, via a transmitter T, to the plate. This pulse propagates through the plate and reaches a sensor S. In a faultless plate, the only signal we receive at S is a signal that goes directly from T to S; this signal is received at a time t1 = t0 + d0/v, where t0 is the moment of time when the original signal was sent, d0 is the distance between T and S, and v is the (known) velocity with which the Lamb waves propagate. In a plate with defects, in addition to this direct signal, we also observe the signal reflected from a defect; this reflected signal arrives at S at a moment t2 = t0 + d/v, where d is the length of the path TFS = TF + FS from T to S via a reflecting point F on the fault. Since we measure t2 and we know the values t0 and v, we can therefore determine the distance d as v · (t2 − t0). If we move the sensor a little bit, to a new location S′ at a small distance s from the old one, then the reflection point shifts a little bit to a new point F′, and the path length changes from d to a new value d′. On a large scale, a crack is usually reasonably smooth. Therefore, between the two close points F and F′, the shape of a crack can be approximated by a straight line segment. Thus, we arrive at the following geometric problem (see Fig. 1): • We know the location of three points T, S, and S′ on the plane. • We know that there is a segment FF′ of a straight line ` on the same plane. • We know the length d of the two-line-segment path that starts at T, gets reflected by ` at a point F ∈ `, and ends at S. • We also know the length d′ of the two-line-segment path that starts at T, gets reflected by ` at a point F′ ∈ `, and ends at S′. • Our objective is to locate the points F and F′. How this problem was solved before. For the (unknown) reflection point F, we know the sum TF + FS of the distances from two known points: T and S. It is a known geometrical fact that for any given two points T and S, the set of all points F with a given sum TF + FS is an ellipse. Due to Snell’s law describing wave reflection, the angle between the incoming wave and the crack must be the same as between the crack and the outcoming wave." @default.
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• W1909283576 date "2003-01-01" @default.
• W1909283576 modified "2023-09-27" @default.
• W1909283576 title "Detecting Cracks in Thin Plates by Using Lamb Wave Scanning: Geometric Approach" @default.
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