﻿ 基于传递函数法的水下消声层声学性能研究
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 力学学报  2016, Vol. 48 Issue (1): 213-224  DOI: 10.6052/0459-1879-15-087 0

### 引用本文 [复制中英文]

[复制中文]
Ye Changzheng, Meng Han, Xin Fengxian, Lu Tianjian. TRANSFER FUNCTION METHOD FOR ACOUSTIC PROPERTY STUDY OF UNDERWATER ANECHOIC LAYER[J]. Chinese Journal of Ship Research, 2016, 48(1): 213-224. DOI: 10.6052/0459-1879-15-087.
[复制英文]

### 文章历史

2015-03-18 收稿
2015-08-21 录用
2015-09-07 网络版发表

1. 西安交通大学航天航空学院, 多功能材料和结构教育部重点实验室, 西安 710049;
2. 西安交通大学航天航空学院, 机械结构强度与振动国家重点实验室, 西安 710049

1 理论分析 1.1 无消声层时潜艇壳体的声学性能

 图 1 无消声层时潜艇壁结构示意图 Fig.1 Schematic of submarine wall structure without sound anechoic layer

 $D\left( {\nabla ^4 - k_{\rm f}^4 } \right)w_{\rm w} \left( r \right) = p_1 \left( {r,0} \right) - p_2 \left( {r,0} \right) + \dfrac{F\delta \left( r \right)}{2π r}$ (1)

 $\left( {\nabla _r^2 + \dfrac{\partial ^2}{\partial z^2} + k_1^2 } \right)p_1 \left( {r,z} \right) = 0$ (2)
 $\left( {\nabla _r^2 + \dfrac{\partial ^2}{\partial z^2} + k_2^2 } \right)p_2 \left( {r,z} \right) = 0$ (3)

 $D\left( {\gamma ^4 - k_{\rm f}^4 } \right)\tilde {w}_{\rm w} (\gamma) = \tilde {p}_1 \left( {\gamma ,0} \right) - \tilde {p}_2 \left( {\gamma ,0} \right) + \dfrac{F}{2π }$ (4)
 $\left[{\dfrac{\partial ^2}{\partial z^2} + \left( {k_1^2 - \gamma ^2} \right)} \right]\tilde {p}_1 \left( {\gamma ,z} \right) = 0$ (5)
 $\left[{\dfrac{\partial ^2}{\partial z^2} + \left( {k_2^2 - \gamma ^2} \right)} \right]\tilde {p}_2 \left( {\gamma ,z} \right) = 0$ (6)

 $\tilde {w}_{\rm w} (\gamma) = \dfrac{{\rm i}F}{2π \omega } \cdot \dfrac{1}{\tilde {Z}_{\rm w} (\gamma) + \tilde {Z}_1 ( \gamma ) + \tilde {Z}_2 (\gamma)}$ (7)
 $\tilde {p}_1 \left( {\gamma ,z} \right) = {\rm i} \omega \tilde {Z}_1 ( \gamma )\tilde {w}_{\rm w} (\gamma){\rm e}^{{\rm i}z\sqrt {k_1^2 - \gamma ^2} }$ (8)
 $\tilde {p}_2 \left( {\gamma ,z} \right) = - {\rm i}\omega \tilde {Z}_2 \left( \gamma \right)\tilde {w}_{\rm w} (\gamma){\rm e}^{{\rm i}z\sqrt {k_2^2 - \gamma^2} }$ (9)

 $\tilde {Z}_{\rm w} (\gamma) = - {\rm i}\omega \rho _{\rm w} h\left( {1 - \dfrac{\gamma ^4}{k_{\rm f}^4 }} \right)$ (10)
 $\tilde {Z}_1 (\gamma) = \dfrac{\rho _1 \omega }{\sqrt {k_1^2 - \gamma ^2} }$ (11)
 $\tilde {Z}_2 (\gamma) = \dfrac{\rho _2 \omega }{\sqrt {k_2^2 - \gamma ^2} }$ (12)

 $\begin{array}{l} {w_{\rm{w}}}\left( r \right) = \frac{{{\rm{i}}F}}{{4\pi \omega }}\int_{ - \infty }^{ + \infty } {\frac{1}{{{{\tilde Z}_{\rm{w}}}(\gamma ) + {{\tilde Z}_1}\left( \gamma \right) + {{\tilde Z}_2}(\gamma )}}} \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} J_0^{(1)}\left( {\gamma r} \right)\gamma d\gamma \end{array}$ (13)
 $\begin{array}{l} {p_1}\left( {r,z} \right) = \frac{F}{{4\pi }}\int_{ - \infty }^{ + \infty } {\frac{{{{\tilde Z}_1}(\gamma )}}{{{{\tilde Z}_{\rm{w}}}(\gamma ) + {{\tilde Z}_1}(\gamma ) + {{\tilde Z}_2}(\gamma )}}} \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {e^{iz\sqrt {k_1^2 - {\gamma ^2}} }}J_0^{(1)}(\gamma r)\gamma d\gamma \end{array}$ (14)
 $\begin{array}{l} {p_2}\left( {r,z} \right) = \frac{F}{{4\pi }}\int_{ - \infty }^{ + \infty } {\frac{{{{\tilde Z}_2}(\gamma )}}{{{{\tilde Z}_{\rm{w}}}\left( \gamma \right) + {{\tilde Z}_1}(\gamma ) + {{\tilde Z}_2}\left( \gamma \right)}}} \cdot \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{\rm{e}}^{{\rm{i}}z\sqrt {k_2^2 - {\gamma ^2}} }}J_0^{\left( 1 \right)}\left( {\gamma r} \right)\gamma d\gamma \end{array}$ (15)

 $p_2 \left( {R,\theta } \right) =\\ \qquad - \dfrac{{\rm i}\omega \rho _2 F{\rm e}^{{\rm i}k_2 R}}{2π R}\dfrac{1}{\tilde {Z}_{\rm w} \left( {\gamma _{\rm s} } \right) + \tilde {Z}_1 \left( {\gamma _{\rm s} } \right) + \tilde {Z}_2 \left( {\gamma _{\rm s} } \right)}$ (16)

1.2 有消声层时潜艇壁面结构的声学性能

 图 2 有消声层时潜艇壁面结构 Fig.2 Schematic of submarine wall structure with sound anechoic layer

 图 3 消声层的剖面图 Fig.3 Profile of anechoic layer
 图 4 消声层胞元 Fig.4 Unit cell in anechoic layer

 图 5 消声层胞元剖面 Fig.5 Cross-section of unit cell in anechoic layer

 图 6 有消声层时潜艇壁面结构 Fig.6 Schematic of submarine wall structure with sound anechoic layer

 $\tilde {Z}_{\rm t} (\gamma) = \tilde {Z}_{\rm w} (\gamma) + \tilde {Z}_{\rm r} (\gamma) + \tilde {Z}_1 (\gamma)$ (17)

 $\tilde {Z}_{\rm r} (\gamma) = \dfrac{\tilde {p}^{\rm ( f )}(\gamma)}{\tilde {v}^{\rm (f)}\left( \gamma \right)}$ (18)

 $\tilde {w}^{(f)}(\gamma) = \dfrac{{\rm i}F}{2π \omega } \cdot \dfrac{1}{\tilde {Z}_{\rm w} (\gamma) + \tilde {Z}_{\rm r} \left( \gamma \right) + \tilde {Z}_1 (\gamma)}$ (19)

 $\tilde {p}'_2 \left( {\gamma ,z} \right) = -{\rm i}\omega \tilde {Z}_2 \left( \gamma \right)\tilde {w}^{\rm ( b )}(\gamma){\rm e}^{{\rm i}\left( {z - H} \right)\sqrt {k_2^2 - \gamma ^2} }$ (20)

 $\left\{ \!\! \begin{array}{c} {\tilde {p}^{\rm (f)}(\gamma)} \\ {\tilde {v}^{\rm (f)}(\gamma)} \end{array} \!\! \right\} = {\pmb T}(\gamma) \left\{ \!\! \begin{array}{c} {\tilde {p}^{\rm (b)}(\gamma)} \\ {\tilde {v}^{\rm (b)}(\gamma)} \end{array} \!\! \right\}$ (21)

 ${\pmb T} (\gamma) _{2\times 2} = \left[\!\! \begin{array}{cc} {T_{11} (\gamma)} & {T_{12} (\gamma)} \\ {T_{21} (\gamma)} & {T_{22} (\gamma)} \end{array} \!\! \right]$ (22)

 ${\pmb T} (\gamma) = \prod\limits_{i = 1}^N {\pmb T}_i (\gamma)$ (23)

 ${\pmb T}_i (\gamma) = \left[\!\! \begin{array}{cc} {\cos \left( {\xi _i h_i } \right)} & { - {\rm i}\tilde {Z}_i \sin \left( {\xi _i h_i } \right)} \\ { - {\rm i}\dfrac{\sin \left( {\xi _i h_i } \right)}{\tilde {Z}_i }} & {\cos \left( {\xi _i h_i } \right)} \end{array} \!\! \right]$ (24)

 $\xi _i = \sqrt {k_i^2 - \gamma ^2}$ (25)
 $\tilde {Z}_i = \dfrac{\rho _i \omega }{\sqrt {k_i^2 - \gamma ^2} }$ (26)

 $c = c_{\rm r} \left( {1 - {\rm i}\eta _{\rm c} } \right)$ (27)

 $\tilde {p}^{\rm (f)}(\gamma) = T_{11} \left( \gamma \right)\tilde {p}^{\rm (b)}(\gamma) + T_{12} \left( \gamma \right)\tilde {v}^{\rm (b)}(\gamma)$ (28)
 $\tilde {v}^{\rm (f)}(\gamma) = T_{21} \left( \gamma \right)\tilde {p}^{\rm (b)}(\gamma) + T_{22} \left( \gamma \right)\tilde {v}^{\rm (b)}(\gamma)$ (29)
 $\dfrac{\tilde {p}^{\rm (f)}(\gamma)}{\tilde {v}^{\rm ( f)}(\gamma)} = \dfrac{T_{11} \left( \gamma \right)\dfrac{\tilde {p}^{\rm (b)}(\gamma)}{\tilde {v}^{\rm (b)}(\gamma)} + T_{12} \left( \gamma \right)}{T_{21} (\gamma)\dfrac{\tilde {p}^{\rm ( b )}(\gamma)}{\tilde {v}^{\rm (b)}\left( \gamma \right)} + T_{22} (\gamma)}$ (30)

 $\tilde {Z}_2 (\gamma) = \dfrac{\tilde {p}^{\rm ( b )}(\gamma)}{\tilde {v}^{\rm (b)}\left( \gamma \right)}$ (31)

 $\tilde {Z}_{\rm r} (\gamma) = \dfrac{T_{11} (\gamma) \tilde {Z}_2 (\gamma) + T_{12} \left( \gamma \right)}{T_{21} (\gamma) \tilde {Z}_2 \left( \gamma \right) + T_{22} (\gamma)}$ (32)

 $\dfrac{\tilde {p}^{\rm (f)}(\gamma)}{\tilde {p}^{\rm ( b )}(\gamma)} = T_{11} (\gamma) + \dfrac{T_{12} (\gamma)}{\tilde {Z}_2 (\gamma)}$ (33)

 $\dfrac{\tilde {v}^{\rm (f)}(\gamma)}{\tilde {v}^{\rm ( b )}(\gamma)} = T_{21} (\gamma)\tilde {Z}_2 (\gamma) + T_{22} (\gamma)$ (34)

 $\tilde {w}^{\rm (b)}(\gamma) = \tilde {w}^{\rm ( f )}(\gamma) \cdot \dfrac{1}{T_{21} \left( \gamma \right)\tilde {Z}_2 (\gamma) + T_{22} (\gamma)}$ (35)

 $\tilde {w}^{\rm (b)}(\gamma) = \dfrac{{\rm i}F}{2π \omega } \cdot \dfrac{1}{\tilde {Z}_{\rm w} (\gamma) + \tilde {Z}_{\rm r} \left( \gamma \right) + \tilde {Z}_1 (\gamma)} \cdot \\ \qquad \dfrac{1}{T_{21} (\gamma)\tilde {Z}_2 (\gamma) + T_{22} (\gamma)}$ (36)

 ${\tilde {p}}'_2 \left( {\gamma ,z} \right) = \dfrac{F}{2π } \cdot \dfrac{\tilde {Z}_2 (\gamma)}{T_{21} (\gamma)\tilde {Z}_2 (\gamma) + T_{22} (\gamma)} \cdot \\ \qquad \dfrac{1}{\tilde {Z}_{\rm w} (\gamma) + \tilde {Z}_{\rm r} (\gamma) + \tilde {Z}_1 (\gamma)} \cdot {\rm e}^{{\rm i}\left( {z - H} \right)\sqrt {k_2^2 - \gamma ^2} }$ (37)

 ${p}'_2 \left( {{R}',{\theta }'} \right) = - \dfrac{{\rm i}\omega \rho _2 F{\rm e}^{{\rm i}k_2 {R}'}}{2π {R}'} \Bigg\{ \left[{T_{11} \left( {{\gamma }'_{\rm s} } \right)\tilde {Z}_2 \left( {{\gamma }'_{\rm s} } \right) + } \right. \\ \qquad \left. {T_{12} \left( {{\gamma }'_{\rm s}} \right)} \right] + \left[{\tilde {Z}_1 \left( {{\gamma }'_{\rm s} } \right) + \tilde {Z}_{\rm w} \left( {{\gamma }'_{\rm s} } \right)} \right] \cdot \\ \qquad \left[{T_{21} \left( {{\gamma }'_{\rm s} } \right)\tilde {Z}_2 \left( {{\gamma }'_{\rm s} } \right) + T_{22} \left( {{\gamma }'_{\rm s} } \right)} \right]\Bigg\} ^{ - 1}$ (38)

1.3 有/无消声层时潜艇壁面结构的声学性能比较

 $\dfrac{{p}'_2 \left( {{R}',{\theta }'} \right)}{p_2 \left( {R,\theta } \right)} = \left[{\tilde {Z}_{\rm w} \left( {\gamma _{\rm s} } \right) + \tilde {Z}_1 \left( {\gamma _{\rm s} } \right) + \tilde {Z}_2 \left( {\gamma _{\rm s} } \right)} \right] \cdot \\ \qquad \Bigg\{ \left[{T_{11} \left( {{\gamma }'_{\rm s} } \right)\tilde {Z}_2 \left( {{\gamma }'_{\rm s} } \right) + T_{12} \left( {{\gamma }'_{\rm s} } \right)} \right] + \left[{\tilde {Z}_1 \left( {{\gamma }'_{\rm s} } \right) + }\right. \\ \qquad \left.{ \tilde {Z}_{\rm w} \left( {{\gamma }'_{\rm s} } \right)} \right] \cdot \left[{T_{21} \left( {{\gamma }'_{\rm s} } \right)\tilde {Z}_2 \left( {{\gamma }'_{\rm s} } \right) + T_{22} \left( {{\gamma }'_{\rm s} } \right)} \right] \Bigg \}^{ - 1}$ (39)

 $IL_{\rm p} = 20\lg \left| {\dfrac{{p}'_2 \left( {{R}',{\theta }'} \right)}{p_2 \left( {R,\theta } \right)}} \right|$ (40)

 $\dfrac{\tilde {{p}'}_2 \left( {\gamma ,z} \right)}{\tilde {p}_2 \left( {\gamma ,z} \right)} = \dfrac{\tilde {w}^{\rm (b)}\left( \gamma \right)}{\tilde {w}_{\rm w }(\gamma)} = \dfrac{\tilde {w}^{\rm ( f )}(\gamma)}{\tilde {w}_{\rm w }(\gamma)} \cdot \dfrac{\tilde {w}^{\rm (b)}(\gamma)}{\tilde {w}^{\rm ( f )}(\gamma)}$ (41)

 $IL_{\rm p} = 20\lg \left| {\dfrac{\tilde {w}^{\rm (f)}\left( {{\gamma }'_{\rm s } } \right)}{\tilde {w}_{\rm w } \left( {\gamma _{\rm s } } \right)}} \right| + 20\lg \left| {\dfrac{\tilde {w}^{\rm (b)}\left( {{\gamma }'_{\rm s } } \right)}{\tilde {w}^{\rm (f)}\left( {{\gamma }'_{\rm s } } \right)}} \right|$ (42)

 $\dfrac{\tilde {w}^{\rm (f)}(\gamma)}{\tilde {w}_{\rm w} (\gamma)} = \dfrac{\tilde {Z}_{\rm w} (\gamma) + \tilde {Z}_1 (\gamma) + \tilde {Z}_2 (\gamma)}{\tilde {Z}_{\rm w} (\gamma) + \tilde {Z}_{\rm r} (\gamma) + \tilde {Z}_1 (\gamma)}$ (43)

 $\dfrac{\tilde {w}^{\rm (b)}(\gamma)}{\tilde {w}^{\rm ( f )}(\gamma)} = \dfrac{1}{T_{21} \tilde {Z}_2 \left( \gamma \right) + T_{22} }$ (44)

 $\left.\!\! 20\lg \left| {\dfrac{\tilde {w}^{\rm (f)}\left( {{\gamma }'_{\rm s} } \right)}{\tilde {w}_{\rm w} \left( {\gamma _{\rm s} } \right)}} \right| = 20\lg \left| {\dfrac{\tilde {Z}_{\rm w} \left( {\gamma _{\rm s} } \right) + \tilde {Z}_1 \left( {\gamma _{\rm s} } \right) + \tilde {Z}_2 \left( {\gamma _{\rm s} } \right)}{\tilde {Z}_{\rm w} \left( {{\gamma }'_{\rm s} } \right) + \tilde {Z}_r \left( {{\gamma }'_{\rm s} } \right) + \tilde {Z}_1 \left( {{\gamma }'_{\rm s} } \right)}} \right| \\ 20\lg \left| {\dfrac{\tilde {w}^{\rm (b)}\left( {{\gamma }'_{\rm s} } \right)}{\tilde {w}^{\rm (f)}\left( {{\gamma }'_{\rm s} } \right)}} \right| = 20\lg \left| {\dfrac{1}{T_{21} \left( {{\gamma }'_{\rm s} } \right)\tilde {Z}_2 \left( {{\gamma }'_{\rm s} } \right) + T_{22} \left( {{\gamma }'_{\rm s} } \right)}} \right| \!\!\right\}$ (45)

2 数值计算与结果分析

2.1 理论模型的检验

 图 7 介质1 不存在时潜艇壁面结构的声振性能 Fig.7 Vibro-acoustic properties of submarine wall structure in the absence of medium 1

2.2 不同介质1时的潜艇壁面结构声学性能

 图 8 介质1 对潜艇壁面结构声振性能的影响 Fig.8 Influence of medium 1 on vibro-acoustic properties of submarine wall structure

2.3 艇壳厚度的影响

 图 9 艇壳厚度对潜艇壁面结构声振性能的影响 Fig.9 Influence of submarine hull thickness on vibro-acoustic properties

2.4 消声层厚度的影响

 图 10 消声层厚度对潜艇壁面结构声振性能的影响 Fig.10 Influence of anechoic layer thickness on vibro-acoustic properties of submarine wall structure

2.5 孔腔形状的影响

 图 11 不同孔腔形状对插入损失的影响 Fig.11 Influence of hole topology on vibro-acoustic properties of submarine wall structure

3 结 论

(1)介质1为空气时，在低频段，消声层对结构振动的削弱量小于消声层加入后艇壳振动的增强量，故消声层的加入反而使潜艇壁面结构 的振动增大；而当介质1为水时，在低频段，覆盖消声层后，壁面结构的振动则明显减小. 在高频段，介质1为空气或者水这两种情况的区别不大，均比低频段的减振降噪效果更好.

(2)增加艇壳的厚度有利于潜艇壁面结构的减振降噪.

(3)增加消声层的厚度总体有利于潜艇壁面结构的减振降噪效果，尤其是在高频段.

(4)当消声层孔腔的内径分布沿前界面至后界面方向先减小再增大时，潜艇壁面结构的减振降噪效果在高频段最好，中频段的规律则与之相反.

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TRANSFER FUNCTION METHOD FOR ACOUSTIC PROPERTY STUDY OF UNDERWATER ANECHOIC LAYER
Ye Changzheng, Meng Han, Xin Fengxian, Lu Tianjian
1. MOE Key Laboratory for Multifunctional Materials and Structures, Xi'an Jiaotong University, Xi'an 710049, China;
2. State Key Laboratory for Mechanical Strength and Vibration, School of Aerospace, Xi'an Jiaotong University, Xi'an 710049, China
Abstract: This paper investigates theoretically the noise reduction property of a submarine wall structure consisting of submarine hull and underwater anechoic layers that contain inner holes. The case that both sides of the submarine wall structure are excited by fluids is considered. Built upon the transfer function method, an analytical anechoic model is developed to determine the pressure insertion loss induced by the anechoic layer. Numerical calculations are subsequently carried out to investigate the influence of fluid medium type, thickness of submarine hull, thickness of anechoic layer as well as shape of inner holes upon the vibro-acoustic properties of the structure. When the fluid medium inside is air, at low frequencies, the reduced vibration due to the anechoic layer is smaller than the increased vibration of submarine hull caused by the anechoic layer, and hence the vibration of the whole structure is increased; at high frequencies, however, the anechoic layer leads to significant attenuation in vibration of the whole structure. When the fluid inside is water, the anechoic layer attenuates the vibration of the whole structure at all frequencies, with the attenuation increasing with increasing frequency. Increasing the thickness of submarine hull or anechoic layer reduces the vibration of the whole structure. When the hole diameter first decreases and then increases in the direction from inside to outside the submarine, the anechoic layer exhibits the best attenuation e ect at high frequencies; when the hole diameter first increases and then decreases, the best attenuation is achieved at medium frequencies.
Key words: anechoic layer    transfer function method    vibration and noise reduction    acoustic property