PSI - Issue 42

Sebastian Henschel et al. / Procedia Structural Integrity 42 (2022) 110–117

114

S. Henschel and L. Kru¨ger / Procedia Structural Integrity 00 (2019) 000–000

5

bars apart (IB)

bars apart (TB)

bars together

S

proposed method

N

c

c

U I , U R , U T , u 1,OE, u 2,OE

S ,

v

U T,max

v S , U I,max

ε I,max , ε T,max

M

K 1 , K 2

K ε,I

K ε,T

K σ

C

rearrange bars

rearrange bars

R

calibration during actual test possible

perform actual test

T

Fig. 3. Sequence of calibration steps for conventional calibration (left) and proposed new method (right). S: name of step, N: necessary value, M: measured values, C: calibration constants, R: rearrangement step, T: actual test.

6

6

v 0 = 8.7 m/s PS: none

v 0 = 7.0 m/s PS: Cu ∅ 5x1.0

1.5

1.5

TB

Displacement u OE / mm 2 4

Displacement u OE / mm

4

1.0

1.0

IB

u 2,OE

IB

2

TB

0.5

0.5

u 1,OE

Bar signal U / V

Bar signal U / V

-200 0 200 400 0

0.0

-200 0 200 400 600 0

0.0

(a)

(b)

Time / µs

Time / µs

Fig. 4. Measured incident, reflected and transmitted pulses and displacements at positions 1 and 2 (without specimen). (a) Without pulse shaper (PS), (b) With Cu pulse shaper, diameter 5 mm, thickness 1 mm.

This equation is applied for both incident (index 1) and transmitted bars (index 2) by taking Eqs. 7 and 8 into account:

u 1 , SG ( t ) = K 1 · c t 0 u 2 , SG ( t ) = K 2 · c t 0

! = u 1 , OE ( t )

( U I ( τ − t I ) − U R ( τ + t I )) d τ

(10)

! = u 2 , OE ( t )

U T ( τ + t T )d τ

(11)

The incident pulse does not necessarily have a rectangular shape. In this general case, the calibration factors K 1 and K 2 can not be directly evaluated. Hence, these calculations are performed by regression analyses. Both equations have the principle form K · x ( t ) = y ( t ). By using the method of least squares, the coe ffi cients K 1 and K 2 are found. The steps needed for the di ff erent types of calibrations are summarized in Fig. 3. The name of each step is given in the blue boxes followed by the required wave speed of the bars (yellow). As will be shown later, the knowledge of c is not necessary in the new proposed method. The values that must be measured are in the green boxes. The results (red), i.e. the calibration constants, are followed by a possible rearrangement of the bars (orange). From Fig. 3, it can be summarized that the dynamic calibration according to the new method does not need a rearrangement of the bars and, moreover, can be performed during an actual test.

4. Results and discussion

Fig. 4 shows the measured incident, reflected and transmitted pulses as well as the measured displacements at the positions 1 and 2 for tests without specimen. It is observed that incident, reflected and transmitted pulses are a ff ected by the pulse shaper. A nearly rectangular incident pulse was observed in Fig. 4a. Since there was no specimen between