⇦Chinese Physics Letters, 2017, Vol. 34, No. 7, Article code 076202 An Analysis of Structural-Acoustic Coupling Band Gaps in a Fluid-Filled Periodic Pipe * Dian-Long Yu(郁殿龙)1**, Chun-Yang Du(杜春阳)1, Hui-Jie Shen(沈惠杰)2, Jiang-Wei Liu(刘江伟)1, Ji-Hong Wen(温激鸿)1 Affiliations 1Laboratory of Science and Technology on Integrated Logistics Support, College of Mechatronics Engineering and Automation, National University of Defense Technology, Changsha 410073 2College of Power Engineering, Naval University of Engineering, Wuhan 430033 Received 28 February 2017 *Supported by the National Natural Science Foundation of China under Grant No 11372346.
**Corresponding author. Email: dianlongyu@vip.sina.com Citation Text: Yu D L, Du C Y, Shen H J, Liu J W and Wen J H 2017 Chin. Phys. Lett. 34 076202 Abstract A periodic pipe system composed of steel pipes and rubber hoses with the same inner radius is designed based on the theory of phononic crystals. Using the transfer matrix method, the band structure of the periodic pipe is calculated considering the structural-acoustic coupling. The results show that longitudinal vibration band gaps and acoustic band gaps can coexist in the fluid-filled periodic pipe. The formation of the band gap mechanism is further analyzed. The band gaps are validated by the sound transmission loss and vibration-frequency response functions calculated using the finite element method. The effect of the damp on the band gap is analyzed by calculating the complex band structure. The periodic pipe system can be used not only in the field of vibration reduction but also for noise elimination. DOI:10.1088/0256-307X/34/7/076202 PACS:62.30.+d, 43.40.-r, 46.40.Cd © 2017 Chinese Physics Society Article Text Fluid-filled piping systems are widely applied in many fields such as nuclear plants, chemical industries, and the shipping industry, where there is a high demand for vibration and noise reduction. For a fluid-filled pipe, three are three interaction mechanisms: the friction coupling, the Poisson coupling and the junction coupling.[1] The most important is the Poisson coupling, and the longitudinal waves in the pipe wall are coupled to the sound waves inside according to Poisson's ratio.[2] The dynamic behavior of pumps and associated piping systems is of practical interest due to its applications in noise and vibration control.[3] Many vibration reduction measures[4] and many kinds of mufflers[5] are applied for noise and vibration control of fluid-filled pipes. Rubber hoses have been used for the transport of fluids to vibrating isolate or mufflers.[6] Phononic crystals (PCs) are periodic structures made of materials with different acoustic impedances.[7,8] The band gaps in PCs can effectively control the propagation of vibration and sound, which provides a new basis for vibration isolation[9] and noise reduction.[10] On the basis of the PC theory, the propagation of vibration or acoustic waves in various periodic pipe structures has been studied theoretically and experimentally. For vibration band gaps, Yu et al.[11] calculated the longitudinal vibration property of a PC pipe system conveying fluid with the plane wave expansion method, while the propagation of the acoustic wave in the periodic pipe was neglected. Yu et al.[12] investigated the flexural vibration band gap with the Bragg gaps, locally resonant gaps, and the coupling between the two band gap mechanisms. Wen et al.[13] validated the vibration attenuation of fluid-filled periodic pipes experimentally. Sorokin[14] researched the power flow suppression in straight elastic pipes using equally spaced eccentric inertial attachments. Wei et al.[15] investigated the flexural vibration transfer properties of high-pressure periodic pipe theoretically and experimentally. For acoustic band gaps, Wang et al.[16] analyzed one-dimensional acoustic waveguides containing subwavelength-sized Helmholtz resonators. Li et al.[17] studied the acoustic wave propagation and sound transmission in a metamaterial-based piping system with the Helmholtz resonator attached periodically. Shi et al.[18] investigated the wave propagation in a periodic array of micro-perforated tube mufflers. In this Letter, we present a detailed theoretical study of the vibro-acoustic coupling band gaps in a fluid-filled periodic pipe. The periodic pipe system is composed of steel pipes and soft hoses with the same inner radius. The band structure is calculated using the transfer matrix method (TMM), and the finite element method (FEM) is also used to validate the correctness. The sound transmission loss (STL) and frequency response function (FRF) for a finite periodic pipe are calculated to discuss the acoustic propagation and vibration propagation, and the effect of the damp is considered by calculating the complex band structure. Figure 1 shows the configuration of the material composite periodic pipe, which is designed on the basis of the Bragg scattering mechanism of PCs.[7] The system consists of an infinite repetition of alternating pipe A with length $a_{1}$ and pipe B with length $a_{2}$. Thus the periodic pipe has a lattice constant of $a=a_{1}+a_{2}$. Moreover, pipes A and B are composed of different A and B materials, respectively. A coordinate system is established, where the $z$-axis is parallel to the pipe axis.
cpl-34-7-076202-fig1.png
Fig. 1. (a) A sketch of the infinite fluid-filled periodic pipe. (b) A single straight pipe element conveying fluid.
If the pipe wall is assumed to be linearly elastic, isotropic circular, and thin-walled, and the fluid is assumed to be one-dimensional, linear, and homogeneous, with isotropic flow and uniform pressure and fluid velocity over the cross section, the well-known Poisson coupling governing equations of the pipe wall longitudinal vibration and fluid acoustic pressure will be[11,19,20] $$\begin{alignat}{1} &E\delta \frac{\partial ^2u}{\partial z^2}+r\sigma \frac{\partial p}{\partial z}-\rho _{\rm P} \delta \frac{\partial ^2u}{\partial t^2}=0,~~ \tag {1a}\\ &K^\ast \frac{\partial ^2p}{\partial z^2}-\rho _{\rm f} \frac{\partial ^2p}{\partial t^2}+2K^\ast \rho _{\rm f} \sigma \frac{\partial ^3u}{\partial z\partial t^2}=0,~~ \tag {1b} \end{alignat} $$ where $u=u(z,t)$ is the pipe wall longitudinal displacement in the $z$-axial direction, and $E$, $\rho _{\rm P}$, $\sigma$, $r$ and $\delta$ are Young's modulus, density, Poisson's ratio, inner radius, and thickness of the pipe wall, respectively. Additionally, $p=p(z, t)$ is the fluid acoustic pressure in the $z$-axial direction, $K^\ast =\frac{K}{1+2rK(1-\sigma ^2)/\delta E}$, and $K$ and $\rho _{\rm f}$ are the fluid bulk modulus and density, respectively. For the fluid-filled periodic pipe, because of the continuity conditions of the axial pipe wall displacement, axial pipe wall force, fluid acoustic pressure, and fluid displacement at the interfaces between cell $n-1$ and $n$, one can obtain the transfer matrix ${\boldsymbol T}$.[19] In accordance with the Bloch theorem, the eigenvalues of the infinite fluid-filled periodic pipe structure are the roots of the determinant $$\begin{align} |{\boldsymbol T}-e^{ika}{\boldsymbol I}|=0,~~ \tag {2} \end{align} $$ where ${\boldsymbol I}$ is a $4\times 4$ unit matrix, and $k$ is the Bloch wave vector in the $z$-direction. For given $\omega$, Eq. (2) gives the values of $k$. Depending on whether $k$ is real or has an imaginary part, the corresponding wave propagates through the pipe (pass bands) or is prohibited (band gaps). For the periodic pipe illustrated in Fig. 1(a), we calculate the band structure of the periodic pipe with rubber hose pipe A and steel pipe B. The elastic parameters employed in the calculations are listed in Table 1. The fluid in the pipe is water with density $\rho _{\rm w}=1000$ kg/m$^{3}$. The lattice constant is chosen to be $a=1$ m, and $a_{1}=a_{2}=0.5$ m. The internal and external diameters are chosen as $d_{\rm in}=0.09$ m and $d_{\rm ex}=0.10$ m, respectively.
Table 1. The material parameters of the pipes.
Material Density Young's Poisson's
$\rho $ (kg/m$^{3}$) modulus $E$ (GPa) ratio $\mu$
Hose pipe A 1190 3.2 0.35
Steel pipe B 7850 200 0.30
The structural-acoustic coupling band structure of the fluid-filled periodic pipe is calculated using TMM, which is shown as black lines in Fig. 2. To validate the correctness, the band structure is also calculated using the finite-element method (FEM) with COMSOL Multiphysics software which has been successfully used to calculate the band gaps of PCs and acoustic metamaterials.[21,22] In the calculation, the Floquet periodic conditions are applied on both the pipe wall cross sections in the structural mechanics module and also the fluid domain cross sections in the acoustics module. The band structure calculated using FEM is shown by the blue lines in Fig. 2. It can be found that the results calculated using TMM and FEM show good agreement. At high frequencies, discrepancies between the two methods occur, which could be attributed to the plane wave assumption used in TMM not being accurate at high frequencies. It is also noticed that the band structure calculated using FEM shows more dispersion curves than those calculated using TMM. The additional curves correspond to flexural and torsional modes of the pipe wall, which are not the concern of this work.
cpl-34-7-076202-fig2.png
Fig. 2. The band structure calculated using TMM (black lines) and FEM (blue lines).
Since we are interested in the Poisson coupling, the structural-acoustic coupling band structure of pipe wall between longitudinal vibration and sound in fluid is illustrated in Fig. 2 as black lines. It can be observed that some structural-acoustic coupling band gaps exist in the periodic pipe system within 0–800 Hz. On account of this, the velocity of the longitudinal vibration wave in the pipe wall is different from that of the acoustic wave in water, and there are two branches from 0 Hz to 400 Hz. In the structural-acoustic coupling system, the longitudinal wave velocity in the pipe wall is faster than the acoustic wave velocity in water. We can determine that the first branch is induced by acoustic wave, and the second branch by longitudinal vibration wave. The third dispersion curve is coupled with the second dispersion curve at about $k=0.75$. To describe the coupling mechanism of the fluid-filled periodic pipe, we further use the FEM to calculate the mode shape of the eigenfrequency for a single unit cell with a periodic condition at different Bloch wave vectors $k=0.5$, 0.75 and 1, corresponding to points in Fig. 2.
cpl-34-7-076202-fig3.png
Fig. 3. The mode shape of a single unit cell at different Bloch wave vectors, corresponding to points in Fig. 2.
cpl-34-7-076202-fig4.png
Fig. 4. The effect of Poisson's ratio on the band gaps. The black, blue and red curves correspond to Poisson's ratio for the hose pipe as 0.35, 0.4 and 0.45, respectively.
In Fig. 3, the mode shapes of the single unit cell are illustrated as displacement deformation, and a typical color legend is chosen to describe the displacement value. From Fig. 4, the mode shapes of points A, B and C in the first dispersion curve indicate that the expansion displacement of the hose pipe is larger than that of the steel pipe. Thus the Bragg scattering gap for the acoustic wave will be generated. Furthermore, the mode shapes of points A, B and C are similar, which can verify that the first dispersion curve is an acoustic branch. Moreover, one can observe that the mode shapes of points F and G are similar, and the mode shapes of points D and I are similar. For the mode shapes of points F and G, the unit cell only has longitudinal displacement, which can verify that the second dispersion curve is a longitudinal vibration branch. In addition, the mode shapes of points E and H are complex, because the structural-acoustic coupling appears. Since Poisson's ratio is an important factor in the structural-acoustic coupling, we next consider the effect of Poisson's ratio on the band gaps of a fluid-filled periodic pipe. Figure 4 shows the band structure for different values of Poisson's ratio for hose pipe A. The black, blue, and red curves correspond to Poisson's ratio as 0.35, 0.4 and 0.45, respectively. The other parameters remain the same as above. From Fig. 4, one can find that the band gap around 400 Hz becomes wider as Poisson's ratio of the hose pipe increases, while there is slight change for the dispersion curves in other frequency ranges. STL is usually used to describe the acoustic attenuation performance, which is defined as[23] $$\begin{align} {\rm STL}=20\log\Big|\frac{p_{\rm i}}{p_{\rm o}}\Big|,~~ \tag {3} \end{align} $$ where $p_{\rm i}$ and $p_{\rm o}$ are the inlet and outlet acoustic pressures, respectively.
cpl-34-7-076202-fig5.png
Fig. 5. The acoustic propagation through a finite periodic pipe with 5 unit cells. (a) The STL, and the black continuous and red dash-dotted lines correspond to the isotropic loss factor of the hose pipe at 0 and 0.05, respectively. (b) The sound pressure level at 100 Hz, 285 Hz and 600 Hz.
For the finite periodic pipe with 5 cells, we calculate the STL to describe the propagation property of the acoustic wave using FEM. In the calculation, the acoustic and structure couple model is applied. The incident sound wave is located at the inlet of the periodic pipe and modeled under the plane wave radiation boundary condition with an amplitude of $p_{0}=1$ Pa. If the inlet and the outlet of the finite periodic pipe are perfectly impedance matched, the STL of the finite periodic pipe can be easily obtained by calculating the sound power at the inlet and outlet boundary. The black continuous lines in Fig. 5(a) show the STL for the finite periodic pipe without material damping. It can be observed that there are two main sound suppression ranges in the STL curve, which correspond to the two acoustic band gaps in Fig. 2. From Fig. 5(a), we can determine that the structural-acoustic coupling is weak because the first acoustic attenuation range still exists despite the effect of the vibration. The attenuation ranges in STL correspond to the acoustic branch band gaps in Fig. 2. The STL range around 400 Hz is very narrow, and the sound suppression value is very small. Furthermore, there are two troughs in STL about 240 Hz and 336 Hz. Recently, the effect of damping in PCs has attracted a great deal of attention.[24,25] The material damping of the hose pipe is considered. In the calculation, the complex Young's modulus is chosen, $E_{\rm damp} =(1+i\eta)E$, where $E$ is Young's modulus of the hose pipe, and $\eta$ is the material isotropic damping loss factor. For $\eta =0.05$, the STL is illustrated as the red dash-dotted line in Fig. 5(a). One can find that the material damping can improve the STL, especially the two troughs in the first acoustic gap range. In Fig. 5(b), the sound pressure levels at 100 Hz, 285 Hz and 600 Hz for the finite periodic pipe with 5 cells are illustrated, and the unit for the color legend is in dB. One can find that the propagation of the acoustic wave in the band gap is suppressed although the radii of pipes A and B are the same. The FRF of longitudinal vibration for the finite periodic pipe can be calculated by[8,15] $$\begin{align} {\rm FRF}=20\log \frac{D_{\rm o} }{D_{\rm i}},~~ \tag {4} \end{align} $$ where $D_{\rm i}$ and $D_{\rm o}$ are the longitudinal vibration displacement of the inlet and the outlet, respectively.
cpl-34-7-076202-fig6.png
Fig. 6. The longitudinal vibration propagation through a finite periodic pipe with 5 unit cells. (a) The FRF of the longitudinal vibration, and the black continuous and red dash-dotted lines correspond to the isotropic loss factor of the hose pipe at 0 and 0.05, respectively. (b) The displacement deformation of the finite periodic pipe at 100 Hz, 285 Hz and 600 Hz.
The FRF of a finite periodic pipe with 5 cells is calculated by FEM, as illustrated in Fig. 6(a). In the calculation, the inlet pipe wall is excited in the $z$-direction with displacement whose amplitude is 0.01 m ranging from 0–800 Hz. There is no boundary limitation at both ends of the periodic pipe, i.e., the pipe vibrates freely. We can observe one attenuation frequency range in the FRF curve, which is due to the longitudinal vibration gap in the periodic pipe. Furthermore, the effect of the material damping of the hose pipe on the FRF curve is considered. The red dash-dotted line in Fig. 6(a) corresponds to the isotropic loss factor of the hose pipe $\eta=0.05$. We can see that the effect of the material damping on the FRF curve is not significant. Only the FRF peaks are attenuated. Furthermore, in Fig. 6(b), we calculate the displacement deformation at 100 Hz, 285 Hz and 600 Hz. We find that the propagation of the longitudinal wave in the band gap is suppressed. There is no vibration attenuation at 285 Hz. When comparing Fig. 5(a) and Fig. 6(a), we can see that the acoustic and vibration propagations are both suppressed in the range 500–700 Hz. From Figs. 5(a) and 6(a), we can find that the effect of the damping on STL and FRF is prominent. To analyze the mechanism, the complex band structures of the PC pipe with damping loss factors $\eta=0.01$ and $\eta=0.05$ are illustrated in Figs. 7(a) and 7(b), respectively.
cpl-34-7-076202-fig7.png
Fig. 7. The complex band structures of periodic pipe systems with damping loss factors (a) $\eta=0.01$ and (b) $\eta=0.05$.
Considering the damping loss factor, a non-zero imaginary part of the Bloch wave vector $k$ exists in both band gaps and pass bands, and increases with damping. There are two large acoustic attenuation zones and one large vibration attenuation zone, which agree with the SLT in Fig. 5(a) and FRF in Fig. 6(a). When the damping loss factor is small, the imaginary part of the Bloch wave vector for the acoustic wave and longitudinal vibration are the same in the attenuation zone around 400 Hz, which means that the formation mechanism of the band gap is due to the structural-acoustic coupling. However, the band gap frequency range and attenuation are small, which means that the structural-acoustic coupling is weak. The attenuation zones become wider as the increase of the loss factor of the hose pipe, and the acoustic attenuation is larger than the longitudinal vibration attenuation, which explains why the effect of the material damping on the STL is stronger than that on FRF vibration. Furthermore, when the damping loss factor increases, the two imaginary parts of Bloch wave vector separate in the coupling band gap. In conclusion, the structural-acoustic coupling band gap in a fluid-filled periodic pipe has been studied theoretically. The band structure illustrates that the acoustic and vibration coupling gaps can exist in the PC pipe system, but the structural-acoustic coupling is weak. The Bragg band gap formation mechanism is revealed by calculating the mode shapes of a single unit cell. The STL and vibration FRF for finite periodic cells are calculated to validate that the PC pipe system can be used as an acoustic muffler and vibration isolator. The acoustic and vibration propagations are both suppressed in certain frequency ranges. The effect of the material damping on the STL is stronger than that on FRF vibration. The complex band structure of damp periodic pipe systems is calculated to reveal the damp mechanism. This finding will provide a novel avenue for noise reduction and vibration attenuation in fluid-filled piping systems. References FLUID-STRUCTURE INTERACTION IN LIQUID-FILLED PIPE SYSTEMS: A REVIEWAnalysis of wave propagation in fluid-filled viscoelastic pipesA study of vibroacoustic coupling between a pump and attached water-filled pipesPrediction of the vibro-acoustic transmission loss of planar hose-pipe systemsBand stop vibration suppression using a passive X-shape structured lever-type isolation systemAcoustic metamaterial panels for sound attenuation in the 50–1000 Hz regimeVibration reduction by using the idea of phononic crystals in a pipe-conveying fluidTheoretical and Experimental Investigation of Flexural Wave Propagating in a Periodic Pipe with Fluid-Filled LoadingOn Power Flow Suppression in Straight Elastic Pipes by Use of Equally Spaced Eccentric Inertial AttachmentsTheoretical and Experimental Investigation of Flexural Vibration Transfer Properties of High-Pressure Periodic PipeAcoustic wave propagation in one-dimensional phononic crystals containing Helmholtz resonatorsControl of low-frequency noise for piping systems via the design of coupled band gap of acoustic metamaterialsSound attenuation of a periodic array of micro-perforated tube mufflersA lightweight low-frequency sound insulation membrane-type acoustic metamaterialUltra-thin smart acoustic metasurface for low-frequency sound insulationMaterial loss influence on the complex band structure and group velocity in phononic crystalsWave propagation in two-dimensional viscoelastic metamaterials
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