150
Y. Liu et al.
Reusser et al. [7, 8] found that the Lamb wave has different transmission energy
at different frequencies when passing through the stiffeners, and it has a tendency
to oscillate. By comparing the signals through the 2 aspect ratio stiffeners, this
phenomenon is more pronounced in stiffened panels with a large aspect ratio. When
the S0 mode Lamb wave frequency is near the resonant frequency of the stiffened
panel, the signal attenuation is severe. However, their research is mainly on the influence of a single stiffener on the Lamb wave signal, and the height of the stiffener is
not high. Santhanam et al. [9] studied the relationship between the incident angle,
frequency and reflection coefficient when the Lamb wave encounters the end of the
panel. In addition, some scholars studied the modal transition when the Lamb wave
passes through the notches [10].
However, the above scholars’ research has the following inadequacies: Less
research has been done on the variation law of Lamb waves after passing through
multiple stiffeners; The height of the stiffeners studied is not too high; There are fewer
studies on the S0 mode Lamb wave with fast wave velocity and less interference.
Based on previous studies, this paper studies the propagation law of Lamb waves
after different number of stiffeners based on finite element simulation. The height of
the stiffeners is up to 22 mm. The variation of Lamb wave velocity, attenuation and
energy is analyzed in time domain and frequency domain. In the structural health state
detection and non-destructive testing of the stiffened panel structure, the proposed
method can provide reference for the selection of the Lamb wave frequency band.
14.2 Math
According to the motion mode of particles relative to the panel, Lamb waves
can be divided into symmetrical mode (S) and antisymmetrical mode (A) Lamb
waves. With the change of frequency, each form of Lamb waves can be further
divided into multiple modes with different phase velocities and group velocities. This
phenomenon is called dispersion characteristics. The dispersion curve of symmetrical
and antisymmetrical mode are represented by (14.1) and (14.2)
tan(qh)/q + 4k
2 p tan( ph)/
q
2
− k
2
2 = 0
(14.1)
q tan(qh) + (q
2
− k
2
)
2 tan( ph)/(4k
2 p) = 0
(14.2)
where h = d/2, p
2
= ω
2
/c
2
l − k
2 , q
2
= ω
2
/c
2
t − k
2 , k = ω/c p , c l =
√
E(1 − μ)/[ρ(1 + μ)(1 − 2μ)], c t =
√
E/[2ρ(1 + μ)]. d is the panel thickness.
k, ω, c p are the wave number, the angular frequency and the phase velocity of Lamb
wave, respectively. c l is the longitudinal wave velocity in the material, and c t is the
transverse wave velocity. The parameters of the 5A06 magnesium-aluminum alloy
are shown in Table 14.1.
Y. Liu et al.
Reusser et al. [7, 8] found that the Lamb wave has different transmission energy
at different frequencies when passing through the stiffeners, and it has a tendency
to oscillate. By comparing the signals through the 2 aspect ratio stiffeners, this
phenomenon is more pronounced in stiffened panels with a large aspect ratio. When
the S0 mode Lamb wave frequency is near the resonant frequency of the stiffened
panel, the signal attenuation is severe. However, their research is mainly on the influence of a single stiffener on the Lamb wave signal, and the height of the stiffener is
not high. Santhanam et al. [9] studied the relationship between the incident angle,
frequency and reflection coefficient when the Lamb wave encounters the end of the
panel. In addition, some scholars studied the modal transition when the Lamb wave
passes through the notches [10].
However, the above scholars’ research has the following inadequacies: Less
research has been done on the variation law of Lamb waves after passing through
multiple stiffeners; The height of the stiffeners studied is not too high; There are fewer
studies on the S0 mode Lamb wave with fast wave velocity and less interference.
Based on previous studies, this paper studies the propagation law of Lamb waves
after different number of stiffeners based on finite element simulation. The height of
the stiffeners is up to 22 mm. The variation of Lamb wave velocity, attenuation and
energy is analyzed in time domain and frequency domain. In the structural health state
detection and non-destructive testing of the stiffened panel structure, the proposed
method can provide reference for the selection of the Lamb wave frequency band.
14.2 Math
According to the motion mode of particles relative to the panel, Lamb waves
can be divided into symmetrical mode (S) and antisymmetrical mode (A) Lamb
waves. With the change of frequency, each form of Lamb waves can be further
divided into multiple modes with different phase velocities and group velocities. This
phenomenon is called dispersion characteristics. The dispersion curve of symmetrical
and antisymmetrical mode are represented by (14.1) and (14.2)
tan(qh)/q + 4k
2 p tan( ph)/
q
2
− k
2
2 = 0
(14.1)
q tan(qh) + (q
2
− k
2
)
2 tan( ph)/(4k
2 p) = 0
(14.2)
where h = d/2, p
2
= ω
2
/c
2
l − k
2 , q
2
= ω
2
/c
2
t − k
2 , k = ω/c p , c l =
√
E(1 − μ)/[ρ(1 + μ)(1 − 2μ)], c t =
√
E/[2ρ(1 + μ)]. d is the panel thickness.
k, ω, c p are the wave number, the angular frequency and the phase velocity of Lamb
wave, respectively. c l is the longitudinal wave velocity in the material, and c t is the
transverse wave velocity. The parameters of the 5A06 magnesium-aluminum alloy
are shown in Table 14.1.