of guiding ultrasonic waves to be localized on the weld. As a prelude to the analysis of the weld problem and in view of its complexity, a simplified model for the ...
VIBRATIONAL MODES IN EQUILATERAL TRIANGULAR SOLID BARS Omar Asfar1, and Bruno Morvan2 1 Department
of Electrical Engineering, Jordan University of Science and Technology, Box 3030, Irbid 22110-Jordan 2 Laboratoire Ondes et Milieux Complexes, LOMC, Universite du Havre, Le Havre, France
Motivation This work is related to the non-destructive testing of seam-welded pipes. Since welds run helically around pipes of large diameter, it is necessary to find a method of guiding ultrasonic waves to be localized on the weld. As a prelude to the analysis of the weld problem and in view of its complexity, a simplified model for the Vweld that is amenable to mathematical analysis will be adopted in the form of an equilateral triangular waveguide. The shear and compressional acoustic modes of the triangular guide are studied both theoretically and experimentally.
Helical pipe weld
Waveguide Model for Vibrational Modes Symmetric solutions of the problem of wave propagation in equilateral triangular waveguides are obtained by using a combination of three plane waves such that each side of the triangle is excited by two waves. The plane wave configuration is either a delta (∆), with waves traveling parallel to the sides of the triangle, or a wye (Y) with waves traveling perpendicular to the sides. The Y configuration corresponds to the field of the longitudinal wave (Dirichlet problem), while the ∆ configuration corresponds to the fields of an SV wave (Neumann problem).
Case of SV and Longitudinal Acoustic Waves In view of symmetry of the equilateral triangle, the vector potentials of the SV waves on side AB are representative of the fields on the other two sides and are given by
D and Y configuration of plane waves C y
=
Longitudinal Waves SV Waves
z x
V - weld
=−
−
( −√ −
[(
)
( +√
)
)]
−
[(
−
)] .
aN: unit normal to plane of incidence of the ray
=√
For the longitudinal wave, we can write the sum of the two scalar potential waves traveling towards side AB in the form
B
A
=
[
√
−
+
[−
√
+
[ (
−
)],
Normal displacement for the combined SV and Longitudinal waves = −√ =
√
√
+
( +
) (
( −
−
).
Normal displacement of first combined Longitudinal + SV mode at 97.9 kHz in Aluminum triangular bar (a=3cm, vl=6337 m/s, vs=3130 m/s).
).
The value of ω is obtained from the condition that the rays of both waves on surface AB lie in the same plane of incidence, and hence their projections onto the z-x plane fall on the same line. Relative Amplitudes of SV and Longitudinal Waves : =
(
+ )
+
Eigen frequency of SV + longitudinal Mode ⎧
−
=
+
√
⎫
⎨ ⎩
a
⎬ ⎭
−
Experimental and FEM simulated normal displacements Emitter transducer
Laser velocimeter Polytec OFV505
20 cm
y
z x
Experimental setup
25
Normal velocity is measured at 930 points on the free end face of waveguide using laser velocimeter. FFT in time is then performed to calculate the normal displacements at each frequency.
20
y(mm)
Triangular waveguide
15
10
5
0
0
5
10
15 x(mm)
20
25
Normal stress of longitudinal mode excitation from experiment f=85kHz
FEM simulation with COMSOL software at f= 98.9kHz
Summary & future perspectives The experiments performed on an equilateral triangular bar made of aluminum confirmed the theory for the fundamental acoustic mode around f=97kHz. Difference in the measured frequency may be attributed to the fact that experimental sample had smooth corners resulting in an effective side length a larger than the design value. Future work is concerned with ultrasonic wave propagation on a periodically corrugated triangular waveguide that will emulate an actual weld and to develop the analytical and experimental techniques that will be used in non-destructive testing of welded pipes.
