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two-dimensional finite element method
Wave Motion 51 (2014) 1193–1208
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Wave Motion journal homepage: www.elsevier.com/locate/wavemoti
Wave propagation modeling of fluid-filled pipes using hybrid analytical/two-dimensional finite element method Je-Heon Han a , Yong-Joe Kim a,∗ , Mansour Karkoub b a
Acoustics and Signal Processing Laboratory, Department of Mechanical Engineering, Texas A&M University, 3123 TAMU, College Station, TX 77843-3123, USA b
Department of Mechanical Engineering, Texas A&M University at Qatar, Texas A&M Engineering Building, P.O. Box 23874, Doha, Qatar
highlights • The first hybrid approach combining the HAFEM and ATF methods is proposed. • Long, fluid-filled, composite pipes with multiple cross-sectional shapes are modeled. • Experimental and numerical data are used to validate the proposed hybrid method.
article
info
Article history: Received 8 November 2013 Received in revised form 7 July 2014 Accepted 16 July 2014 Available online 23 July 2014 Keywords: Acoustic transfer function (ATF) Fluid-filled, composite pipes Hybrid Analytical/Finite Element Method (HAFEM) Structural wave propagation Fluid–structure interactions
abstract In this paper, a Hybrid Analytical/Two-Dimensional Finite Element Method (2-D HAFEM) is proposed to analyze wave propagation characteristics of fluid-filled, composite pipes. In the proposed method, a fluid-filled pipe with a constant cross-section is modeled by using a 2-D finite element approximation in the cross-sectional area while an analytical wave solution is assumed in the axial direction. Thus, it makes possible to use a small number of finite elements even for high frequency analyses in a computationally efficient manner. Both solid and fluid elements as well as solid–fluid interface boundary conditions are developed to model the cross-section of the fluid-filled pipe. In addition, an acoustical transfer function (ATF) approach based on the 2-D HAFEM formulation is suggested to analyze a pipe system assembled with multiple pipe sections with different cross-sections. An ATF matrix relating two sets of acoustic wave variables at the ends of each individual pipe section with a constant cross-section is first calculated and the total ATF matrix for the multisectional pipe system is then obtained by multiplying all individual ATF matrices. Therefore, the HAFEM-based ATF approach requires significantly low computational resources, in particular, when there are many pipe sections with a same cross-sectional shape since a single 2-D HAFEM model is needed for these pipe sections. For the validation of the proposed method, experimental and full 3-D FE modeling results are compared to the results obtained by using the HAFEM-based ATF procedure. © 2014 Elsevier B.V. All rights reserved.
1. Introduction For the purpose of nondestructively evaluating the structural health of a long pipe system, e.g., used to transport a fluid such as petroleum or natural gas, it is important to scan a long pipe section with a single measurement. Thus, it has gained
∗
Corresponding author. Tel.: +1 979 845 9779; fax: +1 979 845 3081. E-mail addresses: [email protected] (J.-H. Han), [email protected] (Y.-J. Kim), [email protected] (M. Karkoub).
http://dx.doi.org/10.1016/j.wavemoti.2014.07.006 0165-2125/© 2014 Elsevier B.V. All rights reserved.
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significant interest to identify the characteristics of guided waves that can propagate long distances with small spatial decay rates, making the nondestructive evaluation (NDE) of long pipe systems feasible. In this article, dispersion curves of fluid-filled pipes with constant cross-sections are identified by using a Hybrid Analytical/Two-Dimensional Finite Element Method (2-D HAFEM) to understand the structural wave propagation characteristics of the pipes, e.g., in terms of frequency-dependent phase speeds and wavenumbers. Although the HAFEM models are computationally efficient, requiring a small number of finite element (FE) meshes to analyze the wave propagation characteristics of fluid-filled, multi-layered composite pipes even at high frequencies (e.g., ultrasonic frequencies), the cross-sectional shapes should not be changed in the axial direction. In order to address this limitation, an acoustical transfer function (ATF) approach based on the 2-D HAFEM procedure is proposed for analyzing a long, fluid-filled, composite pipe with multiple cross-sections, while previous waveguide FE methods [1–11] cannot be applicable to these multi-cross-sectional pipe systems. When considering the only 2-D HAFEM separated from the ATM approach, it is similar to the waveguide FE methods in that all of these methods are based on 2-D FE approximations in the cross-sectional directions. However, the formulation of the 2-D HAFEM includes a general solution in the axial direction, allowing the application of any boundary conditions along the axial direction. Regarding previous analytical approaches, Gazis [12] obtained the dispersion curves of an isotropic, hollow, circular pipe and Fuller et al. [13] studied wave propagation characteristics in fluid-filled, thin circular shells. Long et al. [14] investigated the dispersion curves of fluid-filled pipes surrounded by soil using an analytical approach that was implemented in a computer program referred to as DISPERSE. However, for pipes with complicated cross-sectional configurations such as multiple composite layers with non-circular cross-sections, it is almost impossible to obtain analytical dispersion relations and associated mode shapes. In order to analyze these complex pipes, it is required to apply numerical methods such as finite element methods (FEMs). However, these approaches are not always applicable to relatively high frequency analyses that require a large number of FE meshes, resulting in high computational costs [15–17]. In order to reduce computational costs and perform high frequency analyses with a small number of FE meshes, hybrid approaches combining analytical and numerical procedures have been developed. For example, a hybrid finite strip element formulation was presented by Cheung in 1976 [1]. Nelson and Dong presented a hybrid analytical/1-D FE model to obtain the displacements and dispersion relations in orthotropic composite plates with 2-D nodal displacements [2,3]. Taweel et al. [4] investigated the wave propagation characteristics of rectangular bars and a circular cylinder by modeling them using a hybrid analytical/2-D FE approach. In order to study tire vibration and its radiated noise, Kim and Bolton [5,6] modeled a tire using 2-D shell elements in tire cross-sectional directions and an analytical solution in the circumferential direction. Hayashi et al. [7] obtained the dispersion curves of a railroad by using a hybrid method referred to as the Semi Analytical Finite Element Method (SAFEM). Shorter [8] improved the previous Nelson and Dong’s approach by considering 3-D nodal displacements and calculated the dispersion relations of infinite-size composite plates. Kim and Han [9,10] studied the acoustic characteristics of honeycomb sandwich panels by developing the Hybrid Analytical/One-Dimensional Finite Element Method (1-D HAFEM) and considering the boundary conditions of finite-sized panels. In order to consider damping effects in a structure, a damping loss factor was considered by using complex Young’s modulus for analyzing dynamic responses of plates [11] and acoustic and structural wave propagation characteristics of the honeycomb sandwich panels [9,10]. When a pipe system is used to transport a fluid, the dispersion curves obtained from a ‘‘hollow’’ pipe model can mislead NDE results of the pipe system. Therefore, the dispersion curves including fluid loading effects are essential for obtaining the accurate NDE results. The aforementioned hybrid approaches may be used to analyze the fluid loading effects by modifying their formulations for analyzing viscoelastic materials to fluid materials. In Ref. [18], Nilsson et al. succeeded to obtain the dispersion curves of fluid-filled ducts for the first time by using platestrip elements for shell structures and fluid-strip elements in the triangular coordinates for fluid with linear shape functions. However, the coarse approximation used in this fluid element limits the frequency range of the dispersion curves up to 1 kHz, which may make it not suitable for a NDE application. In this paper, the existing 2-D HAFEM procedure with the solid elements in Ref. [19] is extended by developing fluid elements and solid–fluid boundary conditions, resulting in the highfrequency dispersion curves of fluid-filled pipes. In the proposed procedure, quadratic interpolation functions are used to improve the computational efficiency and accuracy. Similar to the hybrid solid elements in Ref. [19], the hybrid fluid element uses a 2-D finite element approximation in the cross-section, while an analytical wave solution is used in the axial direction. In order to consider a pipe system assembled with multiple pipe sections with different cross-sections, an ATF matrix is derived from the HAFEM formation. Although an ‘‘analytical’’ ATF matrix was formulated previously from the thick shell vibration equations in Ref. [20] to model pipe systems, this analytical approach cannot be used to consider fluid loading effects and multi-layered composite pipes with complex cross-sectional shapes. In the proposed ATF procedure, each pipe section with a constant cross-section is modeled by using an ATF matrix relating two sets of acoustic variables at the two ends of this pipe section. The total ATF matrix is then calculated by multiplying all of the individual ATF matrices. After the matrix multiplication, thus, only two sets of acoustic variables at the ends of the multi-cross-sectional pipe system are related by the total ATF matrix as the final form of the model equation. Although there are existing investigations on the dispersion curves and vibration responses of solid elastic pipes using one of hybrid modeling approaches or acoustical/vibrational transfer function approaches, this paper contributes to the modeling of long, fluid-filled, composite pipes with different cross-sectional shapes by combining both the HAFEM and ATF approaches, creating a novel, efficient approach for the modeling of the pipes.
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Fig. 1. Coordinate transformation of quadratic element.
Fig. 2. Illustration of 2-D HAFEM model for analyzing hollow, cylindrical pipe: (a) finite element approximation on cross-section and (b) analytical solution in axial direction.
For the validation of the proposed method, the dispersion curves and frequency response functions (FRFs) obtained by using the proposed method are compared to experimental or full 3-D FE analysis results for both hollow and fluid-filled pipes. 2. Hybrid analytical/two-dimensional finite element method for modeling fluid-filled pipes 2.1. Governing equation As shown in Fig. 1, a coordinate transformation from the x–y to ξ –η coordinates in the domain of ξ = −1 to 1 and η = −1 to 1 is defined to simplify a spatial integral in the x–y domain [21]. Fig. 2 illustrates a 2-D HAFEM model of a pipe or joint. Here, an analytical solution is assumed in the axial direction (i.e., z-direction in Fig. 2), while FE approximations are applied to the cross-section of the pipe or joint. Then, the vibration displacements of the HAFEM element in the ξ –η coordinates shown in Fig. 1 are approximated by multiplying the interpolation functions, N and the analytical nodal displacement vector, u: i.e.,
u(ξ , η, z , t ) v(ξ , η, z , t ) w(ξ , η, z , t )
ψ(ξ , η, z , t ) =
= N(ξ , η)u(z , t ) =
N1 0 0
0 N1 0
0 0 N1
···
Nn 0 0
0 Nn 0
u 1 v1 0 w1 . 0 .. , Nn un vn wn
(1)
where n is the number of nodes in an element. Then, a strain vector can be expressed by using the strain–displacement relation as
∂u e= ∂x
∂v ∂y
∂w ∂z
∂v ∂u + ∂x ∂y
∂w ∂v + ∂y ∂z
∂w ∂u + ∂x ∂z
T
.
(2)
For a solid element, the stress vector is related to the strain vector as s = Ce.
(3)
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The matrix C for the case of orthotropic materials can be expressed as
C
C12 C22 C23 0 0 0
11
C12 C13 C= 0 0 0
C13 C23 C33 0 0 0
0 0 0 C44 0 0
0 0 0 0 C55 0
0 0 0 . 0 0 C66
(4)
For a fluid element, the constitutive equation for a Newtonian fluid [22] is
σij = −pδij + µ
∂ u˙ i ∂ u˙ j + ∂ xi ∂ xj
+ λδij e˙ kk .
(5)
By approximating the volume changing as dV /V ∼ = exx + eyy + ezz and using Eq. (5), the stress vector for the fluid is related to the strain vector as s = C1 e + C2 e˙ .
(6)
That is,
σ
κ σyy κ σzz κ = τxy 0 τyz 0 τzx 0 xx
κ κ κ
κ κ κ
0 0 0
0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
exx 0 2µ + λ 0 eyy λ 0 ezz + λ 0 γxy 0 0 γyz 0 0 γzx 0
λ 2µ + λ λ
λ λ 2µ + λ
0 0 0
0 0 0
0 0 0
µ 0 0
0 0 0 0
µ 0
e˙ xx 0 0 e˙ yy 0 e˙ zz , 0 γ˙xy γ˙yz 0
µ
(7)
γ˙zx
where κ is the bulk modulus (p = −κ dV /V ), µ is the dynamic viscosity coefficient, and λ is the bulk viscosity coefficient. By applying the Variational Principle in Eq. (8), the 2-D HAFEM equation of motion (EOM) for the single element can be obtained as in Eq. (13).
δ(U − T − W ) = 0,
(8)
where the potential energy U, the kinetic energy T , and the work W are represented, respectively, as Usolid = Ufluid =
T =
Kzz
2 1
t
x
y
2
eH Cedxdydzdt ,
2
t
x
t
x
y
1
2
t
(9)
z
y
1
W =
1
(eH C1 e + eH C2 e˙ )dxdydzdt ,
(10)
z
ρ z
∂ uH H ∂ u N N dxdydzdt , ∂t ∂t
uH f dz dt .
(11)
(12)
z
∂ 2u ∂u ∂ 2u + K + K u + M = fi + fe , ( x + y ) z x + y ∂ z2 ∂z ∂t2
(13)
where K is the element stiffness matrix, M is the element mass matrix, u is the nodal displacement vector, and fi and fe represent the internal and external force vectors, respectively. The detailed information on the stiffness and mass matrices for the solid element is presented in Ref. [19]. The stiffness and mass matrices of the fluid element can be expressed as Kzz (ω) = −
1
ξ =−1
(BHz C1 Bz − iωBHz C2 Bz )|J|dξ dη,
(14)
1
η=−1
(BHξ +η C1 Bz − BHz C1 Bξ +η − iωBHξ +η C2 Bz + iωBHz C2 Bξ +η )|J|dξ dη,
(15)
1
η=−1
(BHξ +η C1 Bξ +η − iωBHξ +η C2 Bξ +η )|J|dξ dη,
(16)
1
M= ξ =−1
ξ =−1 1
η=−1
1
1
ξ =−1
K(x+y)z (ω) = Kx+y (ω) =
1
η=−1
ρ NH N|J|dξ dη,
(17)
J.-H. Han et al. / Wave Motion 51 (2014) 1193–1208
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where Bξ +η = B(ξ +η),1 B(ξ +η),2 B(ξ +η),3 B(ξ +η),4 B(ξ +η),5 B(ξ +η),6 B(ξ +η),7 B(ξ +η),8 B(ξ +η),9 ,
(18)
Bz = Bz ,1 Bz ,2 Bz ,3 Bz ,4 Bz ,5 Bz ,6 Bz ,7 Bz ,8 Bz ,9 ,
(19)
∂ Ni
B(ξ +η),i
(J−1 )11 +
∂ Ni −1 (J )12 ∂η
∂ξ 0 0 ∂ N ∂ Ni −1 i − 1 = (J )21 + (J )22 ∂ξ ∂η 0
0
0
∂ Ni −1 ∂ N i −1 (J )22 + (J )21 ∂η ∂ξ
0
0
0 0 0
B z ,i =
∂x ∂ξ J= ∂x ∂η
0 0 0
Ni 0 0
0 0 0
0 Ni 0
0 0 Ni
0
∂ Ni −1 ∂ N i −1 (J )11 + (J )12 ∂ξ ∂η 0 0
0 0
∂ Ni −1 (J )21 + ∂ξ ∂ Ni −1 (J )11 + ∂ξ
, ∂ Ni −1 (J )22 ∂η ∂ Ni −1 (J )12 ∂η
(20)
T ,
T ∂ N (ξ , η) ∂y x ∂ξ ∂ξ = ∂ y ∂ NT (ξ , η) x ∂η ∂η
(21)
∂ NT (ξ , η) y ∂ξ . T ∂ N (ξ , η) y ∂η
(22)
For a multi-element system, a global HAFEM EOM is obtained by assembling the local stiffness matrices, the local mass matrices, and the force vectors in the global coordinate. At the interface of solid and fluid elements, the shear forces of the solid element are balanced with the viscous shear forces of the fluid element. Likewise, the normal forces on the solid element have the same amplitudes but opposite directions to those of the neighbored fluid element per the Newton’s Third Law. In the same way, the internal forces at the interface of same type elements are canceled out during this global assembly process. For convenience, the same notation is used for the global matrices after dropping the underlines in Eq. (13): i.e., Kzz
∂u ∂ 2u ∂ 2u + K(x+y)z + Kx+y u + M 2 = f. 2 ∂z ∂z ∂t
(23)
In order to identify the wave propagation characteristics of a pipe modeled by using the proposed HAFEM procedure, wavenumbers are calculated by substituting a wave solution into Eq. (23) at each frequency. Here, it is assumed that there is no reflection wave in the z-direction for identifying the wave propagation characteristics of a HAFEM modeled pipe. Then, the wave solution with the only positive z-direction propagating wave component can be represented as u = u0 expi(kz −ωt ) .
(24)
By substituting the assumed solution into Eq. (23) for the free vibration condition, an eigenvalue problem can be derived as
(−Kzz k2 + ikK(x+y)z + Kx+y − ω2 M)u0 = 0.
(25)
For a non-trivial solution, the determinant of the coefficient matrix in Eq. (25) should be zero and the characteristic equation can be then obtained as det(−Kzz k2 + ikK(x+y)z + Kx+y − ω2 M) = 0.
(26)
2.2. Dispersion relation in low audible frequencies From the characteristic equation in Eq. (26), the wavenumbers associated with longitudinal, torsional, and flexural waves can be obtained at each frequency of interest: i.e., at a single frequency, two longitudinal wavenumbers, kL+ and kL− , two torsional wavenumbers, kT + and kT − , and four flexural wavenumbers, kFR+ , kFR− , kFI + , and kFI − can be obtained by solving Eq. (26). The wave types associated with the calculated wavenumbers can be determined by comparing their real part amplitudes. That is, at a frequency, the real parts of kL+ and kL− are smaller than those of kT + and kT − since the wave speed of a longitudinal wave is faster than that of a torsional wave. Similarly, in low frequencies, the real parts of kFR+ , kFR− , kFI + , and kFI − are larger than those of kT + and kT − . When there are N nodes in a 2-D HAFEM model, 3N wavenumbers are obtained from Eq. (26). Except the wavenumbers associated with the longitudinal, torsional, and flexural waves at low frequencies, other wavenumbers can be ignored due to their large imaginary parts. As shown in Fig. 3, the dispersion relations for the
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Fig. 3. Dispersion relations of longitudinal and torsional waves for empty and water-filled pipes in Table 1. Table 1 Material properties of hollow pipe. Young’s modulus (GPa) Density (kg/m3 ) Thickness (mm) Outer diameter (mm) Poisson’s ratio Structural damping coefficient
201 7400 3.55 42.2 0.285 0.0026
Table 2 Material properties of water filled inside of pipe in Table 1.
Water (at 15 °C)
Density (kg/m3 ) Bulk viscosity (Pa s) Dynamic viscosity (Pa s) Bulk modulus (GPa)
998.2 3.1 × 10−3 1.155 × 10−3 2.2
longitudinal and torsional waves can be obtained for the given material properties of a hollow pipe in Table 1. The HAFEM model of this pipe is built by using 5 nodes in the r-direction and 16 nodes in the θ -direction. This model is validated by comparing its dispersion relations to the analytical ones obtained by using ω = k(E /ρ)1/2 for the longitudinal waves and ω = k(G/ρ)1/2 for the torsional waves where G and ρ are the shear modulus and density, respectively. As shown in Fig. 3, the dispersion curves obtained from the HAFEM model match well with the analytical ones for the empty pipe. When the pipe is filled with water (see Table 2 for the water properties), the wavenumber slightly increases as the frequency increases for the longitudinal and torsional waves as shown in Fig. 3. In Fig. 4, the dispersion relations for the flexural waves are plotted. The real and imaginary wavenumbers are plotted in Figs. 4(a) and 4(b), respectively. The imaginary wavenumbers are associated with exponentially-decaying evanescent waves. Thus, the two real and two imaginary wavenumbers are required to describe the flexural waves properly at a single frequency. As the frequency increases, the discrepancy between the analytical and HAFEM results increases since the analytical solution obtained from the Euler–Bernoulli beam theory (i.e., k = (ω2 ρ A/E /I )1/4 ) is valid only for thin beams in low frequencies where the wavelength are much larger than the cross-sectional dimensions. Therefore, the dispersion curves obtained from both the analytical and HAFEM analyses agree with each other at only low frequencies. In addition to the dispersion relations obtained by using the Euler–Bernoulli beam theory, the dispersion curves obtained by using the thick shell theory are compared with the HAFEM dispersion results in Fig. 4 (see Ref. [20] for the thick shell theory and Refs. [23,24] for the Timoshenko beam theory). The HAFEM dispersion curves for the empty pipe match well with those obtained from the thick shell theory that includes the coupling effects between the propagating waves and the effects of shear deformation and rotational inertia as in the Timoshenko beam theory in the entire frequency range of Fig. 4. In addition, the dispersion relations of the water-filled pipe are presented in Fig. 4 (see Table 2 for the water properties). When the pipe is filled with water, the flexural wavenumber increases much more than the previous longitudinal and torsional wavenumbers at each frequency as shown in Fig. 4 and, therefore, the associated phase speed decreases. 2.3. Dispersion relations in ultrasonic frequencies In ultrasonic frequencies, an extensively large number of Lamb wave modes can be excited in a pipe. Thus, the HAFEM dispersion relation in Eq. (26) has many corresponding k solutions at a single ultrasonic frequency, making it extremely
J.-H. Han et al. / Wave Motion 51 (2014) 1193–1208
a
1199
b
Fig. 4. Dispersion relations of flexural wave for empty pipe and water-filled pipe in Tables 1 and 2: (a) real parts of wavenumbers and (b) imaginary parts of wavenumbers.
Fig. 5. Dispersion curves of axisymmetric longitudinal wave modes for empty and water-filled pipe in Ref. [14] (see also Table 3): (a) empty case and (b) water-filled case.
Table 3 Material properties of pipe in Fig. 5 and Ref. [14]. Young’s modulus (GPa) Density (kg/m3 ) Thickness (mm) Outer radius (mm) Poisson’s ratio
113.6 7100 16 143 0.28
difficult to study all of the individual Lamb wave modes. Therefore, in order to validate the HAFEM procedure in ultrasound frequency ranges, the dispersion curves for only axisymmetric longitudinal wave modes obtained from a HAFEM model are compared with the results presented in Ref. [14]. The material properties and geometry information of the water-filled pipe in Ref. [14] are presented in Table 3. The HAFEM model of this pipe is built by using 9 nodes in the r-direction for the pipe structure, 11 nodes in the r-direction for water, and 36 nodes for both the pipe and water in the θ -direction. Thus, the total of 684 nodes is used for this model. The dispersion curves of both the empty and water-filled pipe cases in an ultrasonic frequency range up to 25 kHz are presented in Fig. 5. It is shown that the dispersion curves obtained from the proposed HAFEM agree well with the analytical dispersion curves presented in Ref. [14]. In addition to the dispersion curves for the pipe in Ref. [14] (see also Table 3), the dispersion curves of the axisymmetric longitudinal wave modes for the water-filled pipe in Tables 1 and 2 are estimated from the HAFEM model up to 200 kHz in Fig. 6. When the pipe is filled with water, as shown in Fig. 6(b), the second mode of the hollow pipe is separated to several modes and an additional mode referred to as α mode3 can be observed. In low frequencies (e.g., below 30 kHz), the phase
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Fig. 6. Dispersion curves of axisymmetric longitudinal wave modes obtained by using HAFEM procedure for empty and water-filled pipes in Tables 1 and 2: (a) empty case and (b) water-filled case.
Fig. 7. Experimental setup for measuring ultrasonic wave speeds in hollow pipe.
speed of this α mode converges to the non-dispersive leak noise propagation velocity [14] defined as
V = cL
1+
κ 2a
−1/2
Ed
,
(27)
where κ is the bulk modulus of water, a is the inner radius of the pipe, d is the thickness of the pipe, and cL is the wave speed in water. In order to validate the proposed HAFEM modeling procedure experimentally in this ultrasonic frequency range, eight piezoelectric transducers (APC-851 manufactured by APC International, Ltd.) are attached on the pipe in the circumferential direction to generate longitudinal wave modes as shown in Fig. 7. The material properties and geometric information of this pipe are the same as the previous simulation case in Table 1 and the size of each transducer is 7 × 7 mm. A National Instruments (NI) system equipped with a PXIe-5122 ultrasonic data acquisition (DAQ) module, a PXI-5421 signal generator, and an in-house LabView code is used to generate a burst sinusoidal excitation signal. This signal is fed to drive the eight transducers. The excited waves are then propagating and measured by using another piezoelectric transducer placed 40 cm apart from the eight excitation transducers. A Brüel & Kjær Type 2693 Nexus conditioning amplifier is used to amplify the measured ultrasonic wave signals before the signals are fed to the NI DAQ system. The measured ultrasonic wave signals are recorded for 0.05 s at the sampling frequency of 20 MHz. In order to compensate a time-lag caused by the signal conditioner, the excitation signal is also fed directly to the signal conditioner and measured with the NI DAQ system as described in Fig. 7. In order to determine the wave speeds of the longitudinal modes, the Hilbert transformation is applied to the excitation and measured signals as shown in Fig. 8(a) and (b) to extract the envelopes of the signals. Then, the peak envelope locations are used to calculate group velocities. The group velocity can be also obtained by using a finite difference approximation from predicted dispersion relations [25] as f ≃
f1 + f2 2
,
vg =
∂ω ω2 − ω1 ≃ , ∂k k2 − k1
(28)
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Fig. 8. Measured wave signals and group speeds: (a) excitation signal, (b) measured signal, and (c) comparison of experimental and HAFEM-predicted group speeds.
where the subscripts represent two adjacent data points. The HAFEM-predicted group velocities in Fig. 8(c) are obtained by applying Eq. (28) with ∆f = f2 − f1 = 5 kHz to the HAFEM results in Fig. 6(a). Fig. 8(c) shows that the measured group velocities match well with the HAFEM-predicted ones. 3. HAFEM-based acoustic transfer function approach for modeling multi-cross-sectional pipe systems Although the HAFEM modeling approach is useful to understand the wave propagation characteristics of fluid-filled, multi-layered composite pipes, the cross-sectional shapes of the pipes should not change in the axial direction. In order to consider a pipe system assembled with multiple pipe sections with different cross-sections, it is proposed that an acoustic transfer function (ATF) matrix is derived from the HAFEM formation. In particular, the wavenumbers in the analytical ATF matrix calculated in Ref. [20] are obtained from the 2-D HAFEM based characteristic equation in Eq. (26) of a fluid-filled, multi-layered composite pipe. By solving the characteristics equation in Eq. (26) at each frequency, the wavenumbers for the longitudinal, torsional, and flexural waves (i.e., kL+ , kL− , kT + , kT − , kFR+ , kFR− , kFI + , and kFI − ) are obtained for the fluid-filled pipe. Then, the ATF matrix between two axial locations in a constant cross-sectional pipe is derived by using the uncoupled three wave equations as in Ref. [20]. The acoustic variables for deriving the ATF matrix are described in Fig. 9. By using the procedure described in Ref. [20], the 8 × 8 ATF matrix, T can then be obtained as
− →
TL
− → 0 − →
0
− → − →
U z =L = 0
TT
0
− →
0
1 TF 1 TF 2 T− F1
0
Uz =0 = TUz =0 ,
(29)
where U = uz F u θ T u y θ M V
TL =
1 kL+ − kL−
(30)
kL− EAikL+ eikL+ z − kL+ EAikL− eikL− z
1 kT + − kT −
,
−kL− eikL+ L + kL+ eikL− L
TT =
T
−kT − eikT + L + kT + eikT − L
1
ikFR+ TF 1 = −EIFR k2FR+ iEIFR k3FR+
kT − JikT + eikL+ z − kT + JikT − eikT − z 1 ikFR−
−EIFR k2FR− iEIFR k3FR−
1 ikFI + −EIFR k2FI + iEIFR k3FI +
i
(eikL+ z − eikL− z )
,
(31)
(eikT + z − eikT − z ) , J kT + eikT + z + kT − eikT − z
(32)
EA kL+ eikL+ z + kL− eikL− z i
1 ikFI − , −EIFR k2FI − iEIFR k3FI −
(33)
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Fig. 9. Acoustic variables: (a) Longitudinal and torsional waves and (b) flexural waves. Table 4 Material properties and diameters of drill pipe system for experimental setup in Ref. [19]. Pipe Young’s modulus (Pa) Density (kg/m3 ) Structural damping coefficient Outer diameter (mm) Inner diameter (mm)
eikFR+ z 0 = 0 0
TF 2
θ=
0 eikFR− z 0 0
0 0 eikFI + z 0
0 0 0 e
ikFI − z
,
104.78 50.8
(34)
(35)
∂ 2 uy , ∂ z2
V = −EIFR
73.03 54.65
∂ uy , ∂z
M = EIFR
Joint 2.08 × 1011 7856 0.0044
∂ 3 uy . ∂ z3
(36)
(37)
For a fluid-filled pipe system with multiple pipe sections and joints, the total ATF matrix can be determined by multiplying all individual ATF matrices, each is obtained from a single pipe or joint. Therefore, once a measurement is made at one end of the combined system, the acoustic wave variables at the other end can be estimated from the total ATF matrix without making another measurement at this end. Furthermore, by controlling the length of a pipe virtually, the acoustic wave variables at arbitrary axial locations can be also estimated once acoustic data are given at one end. 4. Experiments and finite element analyses for validation In order to validate the HAFEM-based ATF matrix approach, the existing experimental and full 3-D FE analysis results for the two empty pipes connected with one joint in Ref. [19] are reused in this article. Table 4 shows the dimensions and material properties of the pipe system. The damping coefficient in Table 4 is obtained by applying the half power method to the experimental data [26] and the complex Young’s modulus [20] is used to consider the structural damping. In addition, an experiment with a water-filled pipe is conducted as described below. An experimental setup with the water-filled pipe is shown in Fig. 10. The 1.83 m pipe is hanged by using two steel cables at z = 0.3 m and 1.53 m. Table 1 shows the material properties of the pipe. The density is determined by measuring its weight and dimensions. The Young’s modulus is decided by fitting predicted and measured natural frequencies for a longitudinal excitation case with the empty pipe and the damping coefficient is obtained by applying the half power method to the experimental data [26]. Similar to the experimental setup described in Ref. [20], longitudinal and flexural waves are generated by exciting the left pipe end with a Brüel & Kjær (B&K) Type 8206 impact hammer. For the purpose of generating the flexural waves, a vertical force is applied to the pipe end by using the impact hammer. In order to generate the longitudinal waves, a circular cover is
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Fig. 10. Experimental setup for fluid-filled single pipe.
a
b
c
d
Fig. 11. Experimental and HAFEM-predicted FRF results for ‘‘longitudinal’’ excitation (L = 9.74 m): (a) z = 0, (b) z = 0.219L, (c) z = 0.5L and (d) z = L.
glued to the left end of the pipe and the impact hammer is then used to excite the center of the glued circular cover. On the other side of the water-filled pipe, a thin plastic wrap is installed to hold the water inside the pipe. The acceleration data is measured by installing a PCB Piezotronics triaxial accelerometer (Model: 356A24) on the pipe for 2 s at a sampling frequency of 6400 Hz. A B&K PULSE system (Model: 3560-B-130) is connected to the accelerometer for acquiring three-directional acceleration data for the axial and transversal excitation cases at the three measurement points at z = 0.15 m, 0.9 m, and 1.68 m as shown in Fig. 10. Ten measurements are repeatedly made at each location and then linearly averaged. 5. Results and discussion 5.1. Existing experiment results with two hollow pipes connected by joint The HAFEM model is built with 160 nodes (i.e., 5 nodes in r-direction × 16 nodes in θ -direction × 2 pipe and joint sections). For the ‘‘longitudinal’’ excitation cases in Fig. 11, the FRFs estimated by using the proposed HAFEM-based ATF matrix method agree well with the experimental FRF results except for some anti-resonance locations at approximately 140, 160, and 380 Hz in Fig. 11(a) and (b). Due to the low signal-to-noise-ratio (SNR) and the high sensitivity of accelerometer position error on the FRF results at these anti-resonance locations, the measured amplitudes at these locations are expected to
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a
b
c
d
Fig. 12. Full 3-D ANSYS FE analysis and HAFEM-predicted FRF results for ‘‘torsional’’ excitation (L = 9.74 m): (a) z = 0, (b) z = 0.219L, (c) z = 0.5L, and (d) z = L.
a
b
c
d
Fig. 13. Experimental, HAFEM-predicted, and analytical transfer matrix results at for ‘‘flexural’’ excitation (L = 9.74 m): (a) z = 0, (b) z = 0.219L, (c) z = 0.5L, and (d) z = L.
be imprecise. For the FRFs at the accelerometer locations of z = 0 m, 0.219L, and L, there are the large longitudinal peaks approximately at 280 Hz while this peak becomes small at the joint (i.e., z = 0.5L) as shown in Fig. 11. For the ‘‘torsional’’ excitation case in Fig. 12, the FRFs obtained from the proposed HAFEM-based method are also well in line with the full 3-D ANSYS FE analysis results. For the ‘‘flexural’’ excitation case in Fig. 13, there is the discrepancy between the predicted and measured results at low frequencies (e.g., below 100 Hz). Since the pipe system is hanged by using the two steel cables as described in Ref. [19],
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a
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b
Fig. 14. Experimental FRF results at z = 0.15 m: (a) longitudinal excitation and (b) flexural excitation.
it cannot be assumed as the free–free boundary condition, in particular, in the low frequencies, while the predicted FRFs are based on the free–free boundary condition. However, at the high frequencies above 100 Hz, the discrepancy becomes insignificant and the predicted FRF results agree well with the experimental results. As shown in Fig. 4, the wavenumbers obtained from both the HAFEM model and the analytical solution (i.e., k = (ω2 ρ A/E /I )1/4 ) are slightly different. In order to investigate these differences in detail, the analytical FRF results obtained from an analytical transfer matrix method [20] based on the analytical solution are also plotted in Fig. 13. At high frequencies above 250 Hz, the HAFEM-predicted results are matched better to the experimental results than the analytical FRF results since the HAFEM-predicted wavenumbers are more accurate than the analytical solution based on the Euler–Bernoulli beam theory for thin pipes. As shown in Fig. 4(a), at high frequencies, the Euler–Bernoulli beam theory results in lower wavenumbers (i.e., higher stiffness) than the other two methods. Therefore, the resonance frequencies estimated by using this analytical approach are higher than the HAFEMpredicted ones in Fig. 13. Since the flexural wave equation used for the derivation of the ATF matrix is based on the pure bending deformation in low frequencies, the proposed ATF approach becomes inaccurate in high frequencies. In the near future, this limitation will be addressed by obtaining an ATF matrix from a general governing equation. Regardless of this limitation, the proposed modeling approach can be a power tool for the computationally efficient modeling of an extremely long, fluid-filled pipe with multiple cross-sectional shapes like a drill string system filled with mud, while any existing pure FE or hybrid FE approaches cannot be used to model this long, fluid-filled pipe. A pure FE approach requires an extensively large number of FE meshes to model this pipe, making it infeasible in terms of computational costs, while a hybrid FE approach cannot be used to model the fluid-filled, multi-cross-sectional pipe system. 5.2. Experiment results with water-filled single pipe In order to see the fluid loading effects, the measured FRFs for both the empty and water-filled pipe cases are compared in Fig. 14. The resonance frequencies, in particular, for the flexural wave excitation case decrease when the pipe is filled with water as shown in Fig. 14(b), although this frequency shift is insignificant for the longitudinal excitation case in Fig. 14(a). The FRFs of the water-filled pipe are estimated by using the proposed HAFEM-based ATF approach and compared with the experimental results in Figs. 15 and 16. Although the resonance frequencies are shifted significantly, in particular, for the flexural excitation case as shown in Fig. 14 (b), it is shown that in Figs. 15 and 16, the HAFEM can be used to estimate the FRFs of the water filled pipe precisely. Unlike the previous results for the flexural excitation case in Section 5.1, the FRF results match well even at low resonance frequencies. The mass of the multi-sectional pipe system in Section 5.1 is approximately 150 kg, while the mass of this single pipe is only 5.84 kg. Therefore, the reaction forces at the cable hanging positions in the multi-sectional pipe system cannot be ignored. Thus, the multi-sectional pipe system cannot be assumed to have the free–free boundary conditions which may cause the discrepancy in the low frequencies below 100 Hz. On the other hand, since the reaction forces in the single pipe system are much smaller than the multi-sectional pipe case, the assumption of the free–free boundary condition is valid even at the lowest resonance frequency. In addition, in order to see the fluid loading effects in a multi-sectional pipe, a 10 mm-thick joint with the same inner diameter is inserted to the center of the water-filled pipe. The length of the joint is 0.4 m and the total length of the pipe system is unchanged as 1.83 m. At x = 0.9 m, the FRFs of the water-filled pipe with the joint are estimated by applying the HAFEM and compared with the no joint case. As shown Fig. 17(a), for a longitudinal excitation case, the resonance frequencies are unchanged while peak levels become changed due to the wave interactions at the joint. For the flexural excitation case in Fig. 17(b), the resonance frequencies are shifted to the higher frequencies by inserting the joint in the
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a
b
c
Fig. 15. Experimental and HAFEM-predicted FRF results of the water-filled pipe for ‘‘longitudinal’’ excitation: (a) z = 0.15 m, (b) z = 0.9 m, and (c) z = 1.68 m.
middle of the pipe since the bending stiffness of the water-filled, multi-sectional pipe system becomes higher than that of the uniform cross-sectional pipe due to the thick joint in the middle. 6. Conclusion In this article, the wave propagation characteristics of pipes are investigated by developing the HAFEM procedure. The proposed HAFEM approach can be used to build computationally-efficient models by combining the finite element approximation and the analytical wave solution. For the validation of the proposed HAFEM procedure, the analytical solutions for the simple pipes are compared to the HAFEM-predicted results. The developed HAFEM procedure can be applied to model multi-layered composite pipes, identifying the wave speeds and cross-sectional mode shapes. The wave speeds predicted by using the HAFEM procedure match well with the analytical ones for the empty pipe case. In particular, the HAFEM-predicted dispersion curves in low frequencies are compared to those obtained from the Euler–Bernoulli beam theory and the thick shell theory. Since there is no specific assumption used for the HAFEM procedures, the HAFEM-predicted dispersion results match better with the experimental results and the thick shell results than those obtained from the Euler–Bernoulli theory that is only valid for thin beams. This finding can also be accounted for by the wave coupling effects. The dispersion relations of both the HAFEM model and the thick shell model include the effects of the coupling between the longitudinal, torsional, and flexural wave motions, while the dispersion curves of the Euler–Bernoulli model do not include the coupling effects. In addition, the 2-D HAFEM procedure with the solid elements is extended by developing the fluid elements and solid–fluid boundary conditions, enabling to predict the dispersion curves of fluid-filled pipes. In the ultrasonic frequencies, this HAFEM procedure is validated by comparing the HAFEM-predicted dispersion curves to both the ‘‘experimental’’ ones for an empty pipe and the ‘‘analytical’’ ones for a fluid-filled pipe. Although the HAFEM procedure is useful to understand the wave propagation characteristics of fluid-filled, multi-layered composite pipes with complex cross-sections, the cross-sectional shapes should remain unchanged in the axial direction. In order to consider a pipe system assembled with multiple pipe sections with different cross-sections, the ATF approach is
J.-H. Han et al. / Wave Motion 51 (2014) 1193–1208
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b
c
Fig. 16. Experimental and HAFEM-predicted FRF results of water-filled pipe for ‘‘flexural’’ excitation: (a) z = 0.15 m, (b) z = 0.9 m, and (c) z = 1.68 m.
Fig. 17. HAFEM-predicted FRF results of water-filled pipe system with one middle joint: (a) longitudinal excitation and (b) flexural excitation.
extended based on the HAFEM formation. By comparing the measured and predicated results, it is shown that the proposed HAFEM-based ATF approach can be used to accurately estimate all of the longitudinal, torsional, and flexural wave modes in the multi-sectional pipe system. For the two pipe sections connected by the single joint, the FRFs estimated by using both the HAFEM model with 160 nodes and the full 3-D ANSYS model with 15443 nodes agree well with the experimental FRF results. Acknowledgment This work has been sponsored by a research grant from the Qatar National Research Fund (Grant No.: NPRP 4-537-2-200).
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
Y.K. Cheung, Finite Strip Method in Structural Analysis, Pergamon Press, Oxford, United Kingdom, 1976, pp. 1–8. S. Dong, R. Nelson, On natural vibrations and waves in laminated orthotropic plates, Trans. ASME E 39 (1972) 739–745. R. Nelson, S. Dong, High frequency vibrations and waves in laminated orthotropic plates, J. Vib. Acoust. 30 (1973) 33–44. H. Taweel, S.B. Dong, M. Kazic, Wave reflection from the free end of a cylinder with an arbitrary cross-section, Int. J. Solids Struct. 37 (2000) 1701–1726. Y.-J. Kim, J.S. Bolton, Analysis of tire vibration by using a hybrid two-dimensional finite element based on composite shell theory, Proceedings of Inter-Noise 2003 (2003). Y.-J. Kim, J.S. Bolton, Effects of rotation on the dynamics of a circular cylindrical shell with application to tire vibration, J. Sound Vib. 275 (2004) 605–621. T. Hayashi, W.-J. Song, J.L. Rose, Guided wave dispersion curves for a bar with an arbitrary cross-section, a rod and rail example, Ultrasonics 41 (2003) 175–183. P. Shorter, Wave propagation and damping in linear viscoelastic laminates, J. Acoust. Soc. Am. 115 (2004) 1917–1925. Y.-J. Kim, Identification of sound transmission characteristics of honeycomb sandwich panels using hybrid analytical/one-dimensional finite element method, Proceedings of Inter-Noise 2006 (2006). Y.-J. Kim, J.H. Han, Identification of acoustic characteristics of honeycomb sandwich composite panels using Hybrid Analytical/Finite Element Method, J. Vib. Acoust. 135 (2013) 011006. F. Birgersson, S. Finnveden, C.-M. Nilsson, A spectral super element for modelling of plate vibration—part 1: general theory, J. Sound Vib. 287 (2005) 297–314. D.C. Gazis, Three-dimensional investigation of the propagation of waves in hollow circular cylinders. I. Analytical foundation, J. Acoust. Soc. Am. 31 (1959) 568–572. C.R. Fuller, F.J. Fahy, Characteristics of wave propagation and energy distributions in cylindrical elastic shells filled with fluid, J. Sound Vib. 81 (1982) 501–518. R. Long, P. Cawley, M. Lowe, Acoustic wave propagation in buried iron water pipes, Proc. R. Soc. Lond. Ser. A 459 (2003) 2749–2770. J.V. Ramakrishnan, L.R. Koval, A finite element model for sound transmission through laminated composite plate, J. Sound Vib. 112 (1987) 433–446. C.I. Papadopoulos, Development of an optimized, standard-compliant procedure to calculate sound transmission loss: design of transmission rooms, Appl. Acoust. 63 (2003) 1003–1029. C.I. Papadopoulos, Development of an optimized, standard-compliant procedure to calculate sound transmission loss: numerical measurements, Appl. Acoust. 64 (2003) 1069–1085. C.-M. Nilsson, S. Finnveden, Waves in thin-walled fluid-filled ducts with arbitrary cross-sections, J. Sound Vib. 310 (2008) 58–76. J.H. Han, Y.-J. Kim, Ultrasonic Lamb wave propagation characteristics in hollow cylindrical pipes, Proceedings of Inter-Noise 2012 (2012). J.H. Han, Y.-J. Kim, M. Karkoub, Modeling of wave propagation in drill strings using acoustic transfer matrix methods, J. Acoust. Soc. Am. 134 (2013) 1920–1931. J.N. Reddy, Introduction to the Finite Element Method, McGraw Hill, New York, NY, 1993, pp. 248–250, and 421–429. I.G. Currie, Fundamental Mechanics of Fluids, third ed., Marcel Dekker, Inc., New York, NY, 2003, pp. 23–26. K.F. Graff, Wave Motion in Elastic Solids, Dover Publications, Mineola, NY, 1991, pp. 75–77, 125–126, and 140–143. G. Cowper, The shear coefficient in Timoshenko’s beam theory, J. Appl. Mech. 33 (1966) 335–340. H. Sato, M. Lebedev, J. Akedo, Theoretical and experimental investigation of propagation of guided waves in cylindrical pipe filled with fluid, Japan. J. Appl. Phys. 45 (2006) 4573–4576. Leonard Meirovitch, Elements of Vibration Analysis, second edition, McGraw Hill, New York, NY, 1986, pp. 50–55.
Contents lists available at ScienceDirect
Wave Motion journal homepage: www.elsevier.com/locate/wavemoti
Wave propagation modeling of fluid-filled pipes using hybrid analytical/two-dimensional finite element method Je-Heon Han a , Yong-Joe Kim a,∗ , Mansour Karkoub b a
Acoustics and Signal Processing Laboratory, Department of Mechanical Engineering, Texas A&M University, 3123 TAMU, College Station, TX 77843-3123, USA b
Department of Mechanical Engineering, Texas A&M University at Qatar, Texas A&M Engineering Building, P.O. Box 23874, Doha, Qatar
highlights • The first hybrid approach combining the HAFEM and ATF methods is proposed. • Long, fluid-filled, composite pipes with multiple cross-sectional shapes are modeled. • Experimental and numerical data are used to validate the proposed hybrid method.
article
info
Article history: Received 8 November 2013 Received in revised form 7 July 2014 Accepted 16 July 2014 Available online 23 July 2014 Keywords: Acoustic transfer function (ATF) Fluid-filled, composite pipes Hybrid Analytical/Finite Element Method (HAFEM) Structural wave propagation Fluid–structure interactions
abstract In this paper, a Hybrid Analytical/Two-Dimensional Finite Element Method (2-D HAFEM) is proposed to analyze wave propagation characteristics of fluid-filled, composite pipes. In the proposed method, a fluid-filled pipe with a constant cross-section is modeled by using a 2-D finite element approximation in the cross-sectional area while an analytical wave solution is assumed in the axial direction. Thus, it makes possible to use a small number of finite elements even for high frequency analyses in a computationally efficient manner. Both solid and fluid elements as well as solid–fluid interface boundary conditions are developed to model the cross-section of the fluid-filled pipe. In addition, an acoustical transfer function (ATF) approach based on the 2-D HAFEM formulation is suggested to analyze a pipe system assembled with multiple pipe sections with different cross-sections. An ATF matrix relating two sets of acoustic wave variables at the ends of each individual pipe section with a constant cross-section is first calculated and the total ATF matrix for the multisectional pipe system is then obtained by multiplying all individual ATF matrices. Therefore, the HAFEM-based ATF approach requires significantly low computational resources, in particular, when there are many pipe sections with a same cross-sectional shape since a single 2-D HAFEM model is needed for these pipe sections. For the validation of the proposed method, experimental and full 3-D FE modeling results are compared to the results obtained by using the HAFEM-based ATF procedure. © 2014 Elsevier B.V. All rights reserved.
1. Introduction For the purpose of nondestructively evaluating the structural health of a long pipe system, e.g., used to transport a fluid such as petroleum or natural gas, it is important to scan a long pipe section with a single measurement. Thus, it has gained
∗
Corresponding author. Tel.: +1 979 845 9779; fax: +1 979 845 3081. E-mail addresses: [email protected] (J.-H. Han), [email protected] (Y.-J. Kim), [email protected] (M. Karkoub).
http://dx.doi.org/10.1016/j.wavemoti.2014.07.006 0165-2125/© 2014 Elsevier B.V. All rights reserved.
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significant interest to identify the characteristics of guided waves that can propagate long distances with small spatial decay rates, making the nondestructive evaluation (NDE) of long pipe systems feasible. In this article, dispersion curves of fluid-filled pipes with constant cross-sections are identified by using a Hybrid Analytical/Two-Dimensional Finite Element Method (2-D HAFEM) to understand the structural wave propagation characteristics of the pipes, e.g., in terms of frequency-dependent phase speeds and wavenumbers. Although the HAFEM models are computationally efficient, requiring a small number of finite element (FE) meshes to analyze the wave propagation characteristics of fluid-filled, multi-layered composite pipes even at high frequencies (e.g., ultrasonic frequencies), the cross-sectional shapes should not be changed in the axial direction. In order to address this limitation, an acoustical transfer function (ATF) approach based on the 2-D HAFEM procedure is proposed for analyzing a long, fluid-filled, composite pipe with multiple cross-sections, while previous waveguide FE methods [1–11] cannot be applicable to these multi-cross-sectional pipe systems. When considering the only 2-D HAFEM separated from the ATM approach, it is similar to the waveguide FE methods in that all of these methods are based on 2-D FE approximations in the cross-sectional directions. However, the formulation of the 2-D HAFEM includes a general solution in the axial direction, allowing the application of any boundary conditions along the axial direction. Regarding previous analytical approaches, Gazis [12] obtained the dispersion curves of an isotropic, hollow, circular pipe and Fuller et al. [13] studied wave propagation characteristics in fluid-filled, thin circular shells. Long et al. [14] investigated the dispersion curves of fluid-filled pipes surrounded by soil using an analytical approach that was implemented in a computer program referred to as DISPERSE. However, for pipes with complicated cross-sectional configurations such as multiple composite layers with non-circular cross-sections, it is almost impossible to obtain analytical dispersion relations and associated mode shapes. In order to analyze these complex pipes, it is required to apply numerical methods such as finite element methods (FEMs). However, these approaches are not always applicable to relatively high frequency analyses that require a large number of FE meshes, resulting in high computational costs [15–17]. In order to reduce computational costs and perform high frequency analyses with a small number of FE meshes, hybrid approaches combining analytical and numerical procedures have been developed. For example, a hybrid finite strip element formulation was presented by Cheung in 1976 [1]. Nelson and Dong presented a hybrid analytical/1-D FE model to obtain the displacements and dispersion relations in orthotropic composite plates with 2-D nodal displacements [2,3]. Taweel et al. [4] investigated the wave propagation characteristics of rectangular bars and a circular cylinder by modeling them using a hybrid analytical/2-D FE approach. In order to study tire vibration and its radiated noise, Kim and Bolton [5,6] modeled a tire using 2-D shell elements in tire cross-sectional directions and an analytical solution in the circumferential direction. Hayashi et al. [7] obtained the dispersion curves of a railroad by using a hybrid method referred to as the Semi Analytical Finite Element Method (SAFEM). Shorter [8] improved the previous Nelson and Dong’s approach by considering 3-D nodal displacements and calculated the dispersion relations of infinite-size composite plates. Kim and Han [9,10] studied the acoustic characteristics of honeycomb sandwich panels by developing the Hybrid Analytical/One-Dimensional Finite Element Method (1-D HAFEM) and considering the boundary conditions of finite-sized panels. In order to consider damping effects in a structure, a damping loss factor was considered by using complex Young’s modulus for analyzing dynamic responses of plates [11] and acoustic and structural wave propagation characteristics of the honeycomb sandwich panels [9,10]. When a pipe system is used to transport a fluid, the dispersion curves obtained from a ‘‘hollow’’ pipe model can mislead NDE results of the pipe system. Therefore, the dispersion curves including fluid loading effects are essential for obtaining the accurate NDE results. The aforementioned hybrid approaches may be used to analyze the fluid loading effects by modifying their formulations for analyzing viscoelastic materials to fluid materials. In Ref. [18], Nilsson et al. succeeded to obtain the dispersion curves of fluid-filled ducts for the first time by using platestrip elements for shell structures and fluid-strip elements in the triangular coordinates for fluid with linear shape functions. However, the coarse approximation used in this fluid element limits the frequency range of the dispersion curves up to 1 kHz, which may make it not suitable for a NDE application. In this paper, the existing 2-D HAFEM procedure with the solid elements in Ref. [19] is extended by developing fluid elements and solid–fluid boundary conditions, resulting in the highfrequency dispersion curves of fluid-filled pipes. In the proposed procedure, quadratic interpolation functions are used to improve the computational efficiency and accuracy. Similar to the hybrid solid elements in Ref. [19], the hybrid fluid element uses a 2-D finite element approximation in the cross-section, while an analytical wave solution is used in the axial direction. In order to consider a pipe system assembled with multiple pipe sections with different cross-sections, an ATF matrix is derived from the HAFEM formation. Although an ‘‘analytical’’ ATF matrix was formulated previously from the thick shell vibration equations in Ref. [20] to model pipe systems, this analytical approach cannot be used to consider fluid loading effects and multi-layered composite pipes with complex cross-sectional shapes. In the proposed ATF procedure, each pipe section with a constant cross-section is modeled by using an ATF matrix relating two sets of acoustic variables at the two ends of this pipe section. The total ATF matrix is then calculated by multiplying all of the individual ATF matrices. After the matrix multiplication, thus, only two sets of acoustic variables at the ends of the multi-cross-sectional pipe system are related by the total ATF matrix as the final form of the model equation. Although there are existing investigations on the dispersion curves and vibration responses of solid elastic pipes using one of hybrid modeling approaches or acoustical/vibrational transfer function approaches, this paper contributes to the modeling of long, fluid-filled, composite pipes with different cross-sectional shapes by combining both the HAFEM and ATF approaches, creating a novel, efficient approach for the modeling of the pipes.
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Fig. 1. Coordinate transformation of quadratic element.
Fig. 2. Illustration of 2-D HAFEM model for analyzing hollow, cylindrical pipe: (a) finite element approximation on cross-section and (b) analytical solution in axial direction.
For the validation of the proposed method, the dispersion curves and frequency response functions (FRFs) obtained by using the proposed method are compared to experimental or full 3-D FE analysis results for both hollow and fluid-filled pipes. 2. Hybrid analytical/two-dimensional finite element method for modeling fluid-filled pipes 2.1. Governing equation As shown in Fig. 1, a coordinate transformation from the x–y to ξ –η coordinates in the domain of ξ = −1 to 1 and η = −1 to 1 is defined to simplify a spatial integral in the x–y domain [21]. Fig. 2 illustrates a 2-D HAFEM model of a pipe or joint. Here, an analytical solution is assumed in the axial direction (i.e., z-direction in Fig. 2), while FE approximations are applied to the cross-section of the pipe or joint. Then, the vibration displacements of the HAFEM element in the ξ –η coordinates shown in Fig. 1 are approximated by multiplying the interpolation functions, N and the analytical nodal displacement vector, u: i.e.,
u(ξ , η, z , t ) v(ξ , η, z , t ) w(ξ , η, z , t )
ψ(ξ , η, z , t ) =
= N(ξ , η)u(z , t ) =
N1 0 0
0 N1 0
0 0 N1
···
Nn 0 0
0 Nn 0
u 1 v1 0 w1 . 0 .. , Nn un vn wn
(1)
where n is the number of nodes in an element. Then, a strain vector can be expressed by using the strain–displacement relation as
∂u e= ∂x
∂v ∂y
∂w ∂z
∂v ∂u + ∂x ∂y
∂w ∂v + ∂y ∂z
∂w ∂u + ∂x ∂z
T
.
(2)
For a solid element, the stress vector is related to the strain vector as s = Ce.
(3)
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The matrix C for the case of orthotropic materials can be expressed as
C
C12 C22 C23 0 0 0
11
C12 C13 C= 0 0 0
C13 C23 C33 0 0 0
0 0 0 C44 0 0
0 0 0 0 C55 0
0 0 0 . 0 0 C66
(4)
For a fluid element, the constitutive equation for a Newtonian fluid [22] is
σij = −pδij + µ
∂ u˙ i ∂ u˙ j + ∂ xi ∂ xj
+ λδij e˙ kk .
(5)
By approximating the volume changing as dV /V ∼ = exx + eyy + ezz and using Eq. (5), the stress vector for the fluid is related to the strain vector as s = C1 e + C2 e˙ .
(6)
That is,
σ
κ σyy κ σzz κ = τxy 0 τyz 0 τzx 0 xx
κ κ κ
κ κ κ
0 0 0
0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
exx 0 2µ + λ 0 eyy λ 0 ezz + λ 0 γxy 0 0 γyz 0 0 γzx 0
λ 2µ + λ λ
λ λ 2µ + λ
0 0 0
0 0 0
0 0 0
µ 0 0
0 0 0 0
µ 0
e˙ xx 0 0 e˙ yy 0 e˙ zz , 0 γ˙xy γ˙yz 0
µ
(7)
γ˙zx
where κ is the bulk modulus (p = −κ dV /V ), µ is the dynamic viscosity coefficient, and λ is the bulk viscosity coefficient. By applying the Variational Principle in Eq. (8), the 2-D HAFEM equation of motion (EOM) for the single element can be obtained as in Eq. (13).
δ(U − T − W ) = 0,
(8)
where the potential energy U, the kinetic energy T , and the work W are represented, respectively, as Usolid = Ufluid =
T =
Kzz
2 1
t
x
y
2
eH Cedxdydzdt ,
2
t
x
t
x
y
1
2
t
(9)
z
y
1
W =
1
(eH C1 e + eH C2 e˙ )dxdydzdt ,
(10)
z
ρ z
∂ uH H ∂ u N N dxdydzdt , ∂t ∂t
uH f dz dt .
(11)
(12)
z
∂ 2u ∂u ∂ 2u + K + K u + M = fi + fe , ( x + y ) z x + y ∂ z2 ∂z ∂t2
(13)
where K is the element stiffness matrix, M is the element mass matrix, u is the nodal displacement vector, and fi and fe represent the internal and external force vectors, respectively. The detailed information on the stiffness and mass matrices for the solid element is presented in Ref. [19]. The stiffness and mass matrices of the fluid element can be expressed as Kzz (ω) = −
1
ξ =−1
(BHz C1 Bz − iωBHz C2 Bz )|J|dξ dη,
(14)
1
η=−1
(BHξ +η C1 Bz − BHz C1 Bξ +η − iωBHξ +η C2 Bz + iωBHz C2 Bξ +η )|J|dξ dη,
(15)
1
η=−1
(BHξ +η C1 Bξ +η − iωBHξ +η C2 Bξ +η )|J|dξ dη,
(16)
1
M= ξ =−1
ξ =−1 1
η=−1
1
1
ξ =−1
K(x+y)z (ω) = Kx+y (ω) =
1
η=−1
ρ NH N|J|dξ dη,
(17)
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where Bξ +η = B(ξ +η),1 B(ξ +η),2 B(ξ +η),3 B(ξ +η),4 B(ξ +η),5 B(ξ +η),6 B(ξ +η),7 B(ξ +η),8 B(ξ +η),9 ,
(18)
Bz = Bz ,1 Bz ,2 Bz ,3 Bz ,4 Bz ,5 Bz ,6 Bz ,7 Bz ,8 Bz ,9 ,
(19)
∂ Ni
B(ξ +η),i
(J−1 )11 +
∂ Ni −1 (J )12 ∂η
∂ξ 0 0 ∂ N ∂ Ni −1 i − 1 = (J )21 + (J )22 ∂ξ ∂η 0
0
0
∂ Ni −1 ∂ N i −1 (J )22 + (J )21 ∂η ∂ξ
0
0
0 0 0
B z ,i =
∂x ∂ξ J= ∂x ∂η
0 0 0
Ni 0 0
0 0 0
0 Ni 0
0 0 Ni
0
∂ Ni −1 ∂ N i −1 (J )11 + (J )12 ∂ξ ∂η 0 0
0 0
∂ Ni −1 (J )21 + ∂ξ ∂ Ni −1 (J )11 + ∂ξ
, ∂ Ni −1 (J )22 ∂η ∂ Ni −1 (J )12 ∂η
(20)
T ,
T ∂ N (ξ , η) ∂y x ∂ξ ∂ξ = ∂ y ∂ NT (ξ , η) x ∂η ∂η
(21)
∂ NT (ξ , η) y ∂ξ . T ∂ N (ξ , η) y ∂η
(22)
For a multi-element system, a global HAFEM EOM is obtained by assembling the local stiffness matrices, the local mass matrices, and the force vectors in the global coordinate. At the interface of solid and fluid elements, the shear forces of the solid element are balanced with the viscous shear forces of the fluid element. Likewise, the normal forces on the solid element have the same amplitudes but opposite directions to those of the neighbored fluid element per the Newton’s Third Law. In the same way, the internal forces at the interface of same type elements are canceled out during this global assembly process. For convenience, the same notation is used for the global matrices after dropping the underlines in Eq. (13): i.e., Kzz
∂u ∂ 2u ∂ 2u + K(x+y)z + Kx+y u + M 2 = f. 2 ∂z ∂z ∂t
(23)
In order to identify the wave propagation characteristics of a pipe modeled by using the proposed HAFEM procedure, wavenumbers are calculated by substituting a wave solution into Eq. (23) at each frequency. Here, it is assumed that there is no reflection wave in the z-direction for identifying the wave propagation characteristics of a HAFEM modeled pipe. Then, the wave solution with the only positive z-direction propagating wave component can be represented as u = u0 expi(kz −ωt ) .
(24)
By substituting the assumed solution into Eq. (23) for the free vibration condition, an eigenvalue problem can be derived as
(−Kzz k2 + ikK(x+y)z + Kx+y − ω2 M)u0 = 0.
(25)
For a non-trivial solution, the determinant of the coefficient matrix in Eq. (25) should be zero and the characteristic equation can be then obtained as det(−Kzz k2 + ikK(x+y)z + Kx+y − ω2 M) = 0.
(26)
2.2. Dispersion relation in low audible frequencies From the characteristic equation in Eq. (26), the wavenumbers associated with longitudinal, torsional, and flexural waves can be obtained at each frequency of interest: i.e., at a single frequency, two longitudinal wavenumbers, kL+ and kL− , two torsional wavenumbers, kT + and kT − , and four flexural wavenumbers, kFR+ , kFR− , kFI + , and kFI − can be obtained by solving Eq. (26). The wave types associated with the calculated wavenumbers can be determined by comparing their real part amplitudes. That is, at a frequency, the real parts of kL+ and kL− are smaller than those of kT + and kT − since the wave speed of a longitudinal wave is faster than that of a torsional wave. Similarly, in low frequencies, the real parts of kFR+ , kFR− , kFI + , and kFI − are larger than those of kT + and kT − . When there are N nodes in a 2-D HAFEM model, 3N wavenumbers are obtained from Eq. (26). Except the wavenumbers associated with the longitudinal, torsional, and flexural waves at low frequencies, other wavenumbers can be ignored due to their large imaginary parts. As shown in Fig. 3, the dispersion relations for the
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Fig. 3. Dispersion relations of longitudinal and torsional waves for empty and water-filled pipes in Table 1. Table 1 Material properties of hollow pipe. Young’s modulus (GPa) Density (kg/m3 ) Thickness (mm) Outer diameter (mm) Poisson’s ratio Structural damping coefficient
201 7400 3.55 42.2 0.285 0.0026
Table 2 Material properties of water filled inside of pipe in Table 1.
Water (at 15 °C)
Density (kg/m3 ) Bulk viscosity (Pa s) Dynamic viscosity (Pa s) Bulk modulus (GPa)
998.2 3.1 × 10−3 1.155 × 10−3 2.2
longitudinal and torsional waves can be obtained for the given material properties of a hollow pipe in Table 1. The HAFEM model of this pipe is built by using 5 nodes in the r-direction and 16 nodes in the θ -direction. This model is validated by comparing its dispersion relations to the analytical ones obtained by using ω = k(E /ρ)1/2 for the longitudinal waves and ω = k(G/ρ)1/2 for the torsional waves where G and ρ are the shear modulus and density, respectively. As shown in Fig. 3, the dispersion curves obtained from the HAFEM model match well with the analytical ones for the empty pipe. When the pipe is filled with water (see Table 2 for the water properties), the wavenumber slightly increases as the frequency increases for the longitudinal and torsional waves as shown in Fig. 3. In Fig. 4, the dispersion relations for the flexural waves are plotted. The real and imaginary wavenumbers are plotted in Figs. 4(a) and 4(b), respectively. The imaginary wavenumbers are associated with exponentially-decaying evanescent waves. Thus, the two real and two imaginary wavenumbers are required to describe the flexural waves properly at a single frequency. As the frequency increases, the discrepancy between the analytical and HAFEM results increases since the analytical solution obtained from the Euler–Bernoulli beam theory (i.e., k = (ω2 ρ A/E /I )1/4 ) is valid only for thin beams in low frequencies where the wavelength are much larger than the cross-sectional dimensions. Therefore, the dispersion curves obtained from both the analytical and HAFEM analyses agree with each other at only low frequencies. In addition to the dispersion relations obtained by using the Euler–Bernoulli beam theory, the dispersion curves obtained by using the thick shell theory are compared with the HAFEM dispersion results in Fig. 4 (see Ref. [20] for the thick shell theory and Refs. [23,24] for the Timoshenko beam theory). The HAFEM dispersion curves for the empty pipe match well with those obtained from the thick shell theory that includes the coupling effects between the propagating waves and the effects of shear deformation and rotational inertia as in the Timoshenko beam theory in the entire frequency range of Fig. 4. In addition, the dispersion relations of the water-filled pipe are presented in Fig. 4 (see Table 2 for the water properties). When the pipe is filled with water, the flexural wavenumber increases much more than the previous longitudinal and torsional wavenumbers at each frequency as shown in Fig. 4 and, therefore, the associated phase speed decreases. 2.3. Dispersion relations in ultrasonic frequencies In ultrasonic frequencies, an extensively large number of Lamb wave modes can be excited in a pipe. Thus, the HAFEM dispersion relation in Eq. (26) has many corresponding k solutions at a single ultrasonic frequency, making it extremely
J.-H. Han et al. / Wave Motion 51 (2014) 1193–1208
a
1199
b
Fig. 4. Dispersion relations of flexural wave for empty pipe and water-filled pipe in Tables 1 and 2: (a) real parts of wavenumbers and (b) imaginary parts of wavenumbers.
Fig. 5. Dispersion curves of axisymmetric longitudinal wave modes for empty and water-filled pipe in Ref. [14] (see also Table 3): (a) empty case and (b) water-filled case.
Table 3 Material properties of pipe in Fig. 5 and Ref. [14]. Young’s modulus (GPa) Density (kg/m3 ) Thickness (mm) Outer radius (mm) Poisson’s ratio
113.6 7100 16 143 0.28
difficult to study all of the individual Lamb wave modes. Therefore, in order to validate the HAFEM procedure in ultrasound frequency ranges, the dispersion curves for only axisymmetric longitudinal wave modes obtained from a HAFEM model are compared with the results presented in Ref. [14]. The material properties and geometry information of the water-filled pipe in Ref. [14] are presented in Table 3. The HAFEM model of this pipe is built by using 9 nodes in the r-direction for the pipe structure, 11 nodes in the r-direction for water, and 36 nodes for both the pipe and water in the θ -direction. Thus, the total of 684 nodes is used for this model. The dispersion curves of both the empty and water-filled pipe cases in an ultrasonic frequency range up to 25 kHz are presented in Fig. 5. It is shown that the dispersion curves obtained from the proposed HAFEM agree well with the analytical dispersion curves presented in Ref. [14]. In addition to the dispersion curves for the pipe in Ref. [14] (see also Table 3), the dispersion curves of the axisymmetric longitudinal wave modes for the water-filled pipe in Tables 1 and 2 are estimated from the HAFEM model up to 200 kHz in Fig. 6. When the pipe is filled with water, as shown in Fig. 6(b), the second mode of the hollow pipe is separated to several modes and an additional mode referred to as α mode3 can be observed. In low frequencies (e.g., below 30 kHz), the phase
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Fig. 6. Dispersion curves of axisymmetric longitudinal wave modes obtained by using HAFEM procedure for empty and water-filled pipes in Tables 1 and 2: (a) empty case and (b) water-filled case.
Fig. 7. Experimental setup for measuring ultrasonic wave speeds in hollow pipe.
speed of this α mode converges to the non-dispersive leak noise propagation velocity [14] defined as
V = cL
1+
κ 2a
−1/2
Ed
,
(27)
where κ is the bulk modulus of water, a is the inner radius of the pipe, d is the thickness of the pipe, and cL is the wave speed in water. In order to validate the proposed HAFEM modeling procedure experimentally in this ultrasonic frequency range, eight piezoelectric transducers (APC-851 manufactured by APC International, Ltd.) are attached on the pipe in the circumferential direction to generate longitudinal wave modes as shown in Fig. 7. The material properties and geometric information of this pipe are the same as the previous simulation case in Table 1 and the size of each transducer is 7 × 7 mm. A National Instruments (NI) system equipped with a PXIe-5122 ultrasonic data acquisition (DAQ) module, a PXI-5421 signal generator, and an in-house LabView code is used to generate a burst sinusoidal excitation signal. This signal is fed to drive the eight transducers. The excited waves are then propagating and measured by using another piezoelectric transducer placed 40 cm apart from the eight excitation transducers. A Brüel & Kjær Type 2693 Nexus conditioning amplifier is used to amplify the measured ultrasonic wave signals before the signals are fed to the NI DAQ system. The measured ultrasonic wave signals are recorded for 0.05 s at the sampling frequency of 20 MHz. In order to compensate a time-lag caused by the signal conditioner, the excitation signal is also fed directly to the signal conditioner and measured with the NI DAQ system as described in Fig. 7. In order to determine the wave speeds of the longitudinal modes, the Hilbert transformation is applied to the excitation and measured signals as shown in Fig. 8(a) and (b) to extract the envelopes of the signals. Then, the peak envelope locations are used to calculate group velocities. The group velocity can be also obtained by using a finite difference approximation from predicted dispersion relations [25] as f ≃
f1 + f2 2
,
vg =
∂ω ω2 − ω1 ≃ , ∂k k2 − k1
(28)
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Fig. 8. Measured wave signals and group speeds: (a) excitation signal, (b) measured signal, and (c) comparison of experimental and HAFEM-predicted group speeds.
where the subscripts represent two adjacent data points. The HAFEM-predicted group velocities in Fig. 8(c) are obtained by applying Eq. (28) with ∆f = f2 − f1 = 5 kHz to the HAFEM results in Fig. 6(a). Fig. 8(c) shows that the measured group velocities match well with the HAFEM-predicted ones. 3. HAFEM-based acoustic transfer function approach for modeling multi-cross-sectional pipe systems Although the HAFEM modeling approach is useful to understand the wave propagation characteristics of fluid-filled, multi-layered composite pipes, the cross-sectional shapes of the pipes should not change in the axial direction. In order to consider a pipe system assembled with multiple pipe sections with different cross-sections, it is proposed that an acoustic transfer function (ATF) matrix is derived from the HAFEM formation. In particular, the wavenumbers in the analytical ATF matrix calculated in Ref. [20] are obtained from the 2-D HAFEM based characteristic equation in Eq. (26) of a fluid-filled, multi-layered composite pipe. By solving the characteristics equation in Eq. (26) at each frequency, the wavenumbers for the longitudinal, torsional, and flexural waves (i.e., kL+ , kL− , kT + , kT − , kFR+ , kFR− , kFI + , and kFI − ) are obtained for the fluid-filled pipe. Then, the ATF matrix between two axial locations in a constant cross-sectional pipe is derived by using the uncoupled three wave equations as in Ref. [20]. The acoustic variables for deriving the ATF matrix are described in Fig. 9. By using the procedure described in Ref. [20], the 8 × 8 ATF matrix, T can then be obtained as
− →
TL
− → 0 − →
0
− → − →
U z =L = 0
TT
0
− →
0
1 TF 1 TF 2 T− F1
0
Uz =0 = TUz =0 ,
(29)
where U = uz F u θ T u y θ M V
TL =
1 kL+ − kL−
(30)
kL− EAikL+ eikL+ z − kL+ EAikL− eikL− z
1 kT + − kT −
,
−kL− eikL+ L + kL+ eikL− L
TT =
T
−kT − eikT + L + kT + eikT − L
1
ikFR+ TF 1 = −EIFR k2FR+ iEIFR k3FR+
kT − JikT + eikL+ z − kT + JikT − eikT − z 1 ikFR−
−EIFR k2FR− iEIFR k3FR−
1 ikFI + −EIFR k2FI + iEIFR k3FI +
i
(eikL+ z − eikL− z )
,
(31)
(eikT + z − eikT − z ) , J kT + eikT + z + kT − eikT − z
(32)
EA kL+ eikL+ z + kL− eikL− z i
1 ikFI − , −EIFR k2FI − iEIFR k3FI −
(33)
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Fig. 9. Acoustic variables: (a) Longitudinal and torsional waves and (b) flexural waves. Table 4 Material properties and diameters of drill pipe system for experimental setup in Ref. [19]. Pipe Young’s modulus (Pa) Density (kg/m3 ) Structural damping coefficient Outer diameter (mm) Inner diameter (mm)
eikFR+ z 0 = 0 0
TF 2
θ=
0 eikFR− z 0 0
0 0 eikFI + z 0
0 0 0 e
ikFI − z
,
104.78 50.8
(34)
(35)
∂ 2 uy , ∂ z2
V = −EIFR
73.03 54.65
∂ uy , ∂z
M = EIFR
Joint 2.08 × 1011 7856 0.0044
∂ 3 uy . ∂ z3
(36)
(37)
For a fluid-filled pipe system with multiple pipe sections and joints, the total ATF matrix can be determined by multiplying all individual ATF matrices, each is obtained from a single pipe or joint. Therefore, once a measurement is made at one end of the combined system, the acoustic wave variables at the other end can be estimated from the total ATF matrix without making another measurement at this end. Furthermore, by controlling the length of a pipe virtually, the acoustic wave variables at arbitrary axial locations can be also estimated once acoustic data are given at one end. 4. Experiments and finite element analyses for validation In order to validate the HAFEM-based ATF matrix approach, the existing experimental and full 3-D FE analysis results for the two empty pipes connected with one joint in Ref. [19] are reused in this article. Table 4 shows the dimensions and material properties of the pipe system. The damping coefficient in Table 4 is obtained by applying the half power method to the experimental data [26] and the complex Young’s modulus [20] is used to consider the structural damping. In addition, an experiment with a water-filled pipe is conducted as described below. An experimental setup with the water-filled pipe is shown in Fig. 10. The 1.83 m pipe is hanged by using two steel cables at z = 0.3 m and 1.53 m. Table 1 shows the material properties of the pipe. The density is determined by measuring its weight and dimensions. The Young’s modulus is decided by fitting predicted and measured natural frequencies for a longitudinal excitation case with the empty pipe and the damping coefficient is obtained by applying the half power method to the experimental data [26]. Similar to the experimental setup described in Ref. [20], longitudinal and flexural waves are generated by exciting the left pipe end with a Brüel & Kjær (B&K) Type 8206 impact hammer. For the purpose of generating the flexural waves, a vertical force is applied to the pipe end by using the impact hammer. In order to generate the longitudinal waves, a circular cover is
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Fig. 10. Experimental setup for fluid-filled single pipe.
a
b
c
d
Fig. 11. Experimental and HAFEM-predicted FRF results for ‘‘longitudinal’’ excitation (L = 9.74 m): (a) z = 0, (b) z = 0.219L, (c) z = 0.5L and (d) z = L.
glued to the left end of the pipe and the impact hammer is then used to excite the center of the glued circular cover. On the other side of the water-filled pipe, a thin plastic wrap is installed to hold the water inside the pipe. The acceleration data is measured by installing a PCB Piezotronics triaxial accelerometer (Model: 356A24) on the pipe for 2 s at a sampling frequency of 6400 Hz. A B&K PULSE system (Model: 3560-B-130) is connected to the accelerometer for acquiring three-directional acceleration data for the axial and transversal excitation cases at the three measurement points at z = 0.15 m, 0.9 m, and 1.68 m as shown in Fig. 10. Ten measurements are repeatedly made at each location and then linearly averaged. 5. Results and discussion 5.1. Existing experiment results with two hollow pipes connected by joint The HAFEM model is built with 160 nodes (i.e., 5 nodes in r-direction × 16 nodes in θ -direction × 2 pipe and joint sections). For the ‘‘longitudinal’’ excitation cases in Fig. 11, the FRFs estimated by using the proposed HAFEM-based ATF matrix method agree well with the experimental FRF results except for some anti-resonance locations at approximately 140, 160, and 380 Hz in Fig. 11(a) and (b). Due to the low signal-to-noise-ratio (SNR) and the high sensitivity of accelerometer position error on the FRF results at these anti-resonance locations, the measured amplitudes at these locations are expected to
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a
b
c
d
Fig. 12. Full 3-D ANSYS FE analysis and HAFEM-predicted FRF results for ‘‘torsional’’ excitation (L = 9.74 m): (a) z = 0, (b) z = 0.219L, (c) z = 0.5L, and (d) z = L.
a
b
c
d
Fig. 13. Experimental, HAFEM-predicted, and analytical transfer matrix results at for ‘‘flexural’’ excitation (L = 9.74 m): (a) z = 0, (b) z = 0.219L, (c) z = 0.5L, and (d) z = L.
be imprecise. For the FRFs at the accelerometer locations of z = 0 m, 0.219L, and L, there are the large longitudinal peaks approximately at 280 Hz while this peak becomes small at the joint (i.e., z = 0.5L) as shown in Fig. 11. For the ‘‘torsional’’ excitation case in Fig. 12, the FRFs obtained from the proposed HAFEM-based method are also well in line with the full 3-D ANSYS FE analysis results. For the ‘‘flexural’’ excitation case in Fig. 13, there is the discrepancy between the predicted and measured results at low frequencies (e.g., below 100 Hz). Since the pipe system is hanged by using the two steel cables as described in Ref. [19],
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b
Fig. 14. Experimental FRF results at z = 0.15 m: (a) longitudinal excitation and (b) flexural excitation.
it cannot be assumed as the free–free boundary condition, in particular, in the low frequencies, while the predicted FRFs are based on the free–free boundary condition. However, at the high frequencies above 100 Hz, the discrepancy becomes insignificant and the predicted FRF results agree well with the experimental results. As shown in Fig. 4, the wavenumbers obtained from both the HAFEM model and the analytical solution (i.e., k = (ω2 ρ A/E /I )1/4 ) are slightly different. In order to investigate these differences in detail, the analytical FRF results obtained from an analytical transfer matrix method [20] based on the analytical solution are also plotted in Fig. 13. At high frequencies above 250 Hz, the HAFEM-predicted results are matched better to the experimental results than the analytical FRF results since the HAFEM-predicted wavenumbers are more accurate than the analytical solution based on the Euler–Bernoulli beam theory for thin pipes. As shown in Fig. 4(a), at high frequencies, the Euler–Bernoulli beam theory results in lower wavenumbers (i.e., higher stiffness) than the other two methods. Therefore, the resonance frequencies estimated by using this analytical approach are higher than the HAFEMpredicted ones in Fig. 13. Since the flexural wave equation used for the derivation of the ATF matrix is based on the pure bending deformation in low frequencies, the proposed ATF approach becomes inaccurate in high frequencies. In the near future, this limitation will be addressed by obtaining an ATF matrix from a general governing equation. Regardless of this limitation, the proposed modeling approach can be a power tool for the computationally efficient modeling of an extremely long, fluid-filled pipe with multiple cross-sectional shapes like a drill string system filled with mud, while any existing pure FE or hybrid FE approaches cannot be used to model this long, fluid-filled pipe. A pure FE approach requires an extensively large number of FE meshes to model this pipe, making it infeasible in terms of computational costs, while a hybrid FE approach cannot be used to model the fluid-filled, multi-cross-sectional pipe system. 5.2. Experiment results with water-filled single pipe In order to see the fluid loading effects, the measured FRFs for both the empty and water-filled pipe cases are compared in Fig. 14. The resonance frequencies, in particular, for the flexural wave excitation case decrease when the pipe is filled with water as shown in Fig. 14(b), although this frequency shift is insignificant for the longitudinal excitation case in Fig. 14(a). The FRFs of the water-filled pipe are estimated by using the proposed HAFEM-based ATF approach and compared with the experimental results in Figs. 15 and 16. Although the resonance frequencies are shifted significantly, in particular, for the flexural excitation case as shown in Fig. 14 (b), it is shown that in Figs. 15 and 16, the HAFEM can be used to estimate the FRFs of the water filled pipe precisely. Unlike the previous results for the flexural excitation case in Section 5.1, the FRF results match well even at low resonance frequencies. The mass of the multi-sectional pipe system in Section 5.1 is approximately 150 kg, while the mass of this single pipe is only 5.84 kg. Therefore, the reaction forces at the cable hanging positions in the multi-sectional pipe system cannot be ignored. Thus, the multi-sectional pipe system cannot be assumed to have the free–free boundary conditions which may cause the discrepancy in the low frequencies below 100 Hz. On the other hand, since the reaction forces in the single pipe system are much smaller than the multi-sectional pipe case, the assumption of the free–free boundary condition is valid even at the lowest resonance frequency. In addition, in order to see the fluid loading effects in a multi-sectional pipe, a 10 mm-thick joint with the same inner diameter is inserted to the center of the water-filled pipe. The length of the joint is 0.4 m and the total length of the pipe system is unchanged as 1.83 m. At x = 0.9 m, the FRFs of the water-filled pipe with the joint are estimated by applying the HAFEM and compared with the no joint case. As shown Fig. 17(a), for a longitudinal excitation case, the resonance frequencies are unchanged while peak levels become changed due to the wave interactions at the joint. For the flexural excitation case in Fig. 17(b), the resonance frequencies are shifted to the higher frequencies by inserting the joint in the
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a
b
c
Fig. 15. Experimental and HAFEM-predicted FRF results of the water-filled pipe for ‘‘longitudinal’’ excitation: (a) z = 0.15 m, (b) z = 0.9 m, and (c) z = 1.68 m.
middle of the pipe since the bending stiffness of the water-filled, multi-sectional pipe system becomes higher than that of the uniform cross-sectional pipe due to the thick joint in the middle. 6. Conclusion In this article, the wave propagation characteristics of pipes are investigated by developing the HAFEM procedure. The proposed HAFEM approach can be used to build computationally-efficient models by combining the finite element approximation and the analytical wave solution. For the validation of the proposed HAFEM procedure, the analytical solutions for the simple pipes are compared to the HAFEM-predicted results. The developed HAFEM procedure can be applied to model multi-layered composite pipes, identifying the wave speeds and cross-sectional mode shapes. The wave speeds predicted by using the HAFEM procedure match well with the analytical ones for the empty pipe case. In particular, the HAFEM-predicted dispersion curves in low frequencies are compared to those obtained from the Euler–Bernoulli beam theory and the thick shell theory. Since there is no specific assumption used for the HAFEM procedures, the HAFEM-predicted dispersion results match better with the experimental results and the thick shell results than those obtained from the Euler–Bernoulli theory that is only valid for thin beams. This finding can also be accounted for by the wave coupling effects. The dispersion relations of both the HAFEM model and the thick shell model include the effects of the coupling between the longitudinal, torsional, and flexural wave motions, while the dispersion curves of the Euler–Bernoulli model do not include the coupling effects. In addition, the 2-D HAFEM procedure with the solid elements is extended by developing the fluid elements and solid–fluid boundary conditions, enabling to predict the dispersion curves of fluid-filled pipes. In the ultrasonic frequencies, this HAFEM procedure is validated by comparing the HAFEM-predicted dispersion curves to both the ‘‘experimental’’ ones for an empty pipe and the ‘‘analytical’’ ones for a fluid-filled pipe. Although the HAFEM procedure is useful to understand the wave propagation characteristics of fluid-filled, multi-layered composite pipes with complex cross-sections, the cross-sectional shapes should remain unchanged in the axial direction. In order to consider a pipe system assembled with multiple pipe sections with different cross-sections, the ATF approach is
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b
c
Fig. 16. Experimental and HAFEM-predicted FRF results of water-filled pipe for ‘‘flexural’’ excitation: (a) z = 0.15 m, (b) z = 0.9 m, and (c) z = 1.68 m.
Fig. 17. HAFEM-predicted FRF results of water-filled pipe system with one middle joint: (a) longitudinal excitation and (b) flexural excitation.
extended based on the HAFEM formation. By comparing the measured and predicated results, it is shown that the proposed HAFEM-based ATF approach can be used to accurately estimate all of the longitudinal, torsional, and flexural wave modes in the multi-sectional pipe system. For the two pipe sections connected by the single joint, the FRFs estimated by using both the HAFEM model with 160 nodes and the full 3-D ANSYS model with 15443 nodes agree well with the experimental FRF results. Acknowledgment This work has been sponsored by a research grant from the Qatar National Research Fund (Grant No.: NPRP 4-537-2-200).
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