1.Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, China 2.National Key Laboratory on Ship Vibration and Noise, China Ship Development and Design Center, Wuhan 430064, China
Fund Project:Project supported by the National Natural Science Foundation of China (Grant No. 11572137).
Received Date:28 January 2019
Accepted Date:03 April 2019
Available Online:01 June 2019
Published Online:20 June 2019
Abstract:To obtain excellent sound reduction performance, in this paper we introduce a novel periodic poroelastic composite structure, which combines poroelastic material and periodic structure and aims at using the remarkable acoustic performance of these two. This periodic composite structure comprises three parts, i.e. the poroelastic domain, the elastic domain (thin plate), and the periodic resonators, which can be simple single-degree-of-freedom resonators (SRs) or composite two-degree-of-freedom resonators (CRs). A theoretical model is established by using Biot theory for the poroelastic domain, and by using the effective medium method for the resonator-plate coupling system, which is considered as an isotropic plate with an equivalent dynamic density. This method is validated with degenerated model in the literature; the results obtained by this method are in excellent consistence with the results in the literature. Parameter analyses are performed to test the influences of poroelastic addition and periodic resonator on the sound transmission loss (STL) of this periodic composite structure under two kinds of boundary conditions. The poroelastic addition is found to increase the STL while the influences of resonators are complicated. The STL increases notably in the frequency range bounded by the characteristic frequencies of these resonators, however, a decrease just follows when it exceeds these frequencies, which can be observed in both SR case and CR case under the two boundary conditions. In the meantime, when multiple SR is placed in a periodic lattice, it is found that different resonators with ascending mass and characteristic frequencies have superior STL to those with ascending characteristic frequencies but have equal mass. The case with CR, which is more complicated as expected, shows less STL decrease than the case with SR, but wider frequency range where the STL increases than a poroelastic composite structure without resonators. This results from the fact that the frequency band of vibration suppression in the CR case is wider than in the SR case. As a result, to achieve the desired STL performance in a frequency range, the proposed composite structure using SR with tuned characteristic frequencies is enough; however, if a wider frequency band is expected even if there is a slight STL tradeoff, the CR case is a better option. Though the method proposed is only valid in the low-to-medium frequency range, the results obtained can benefit theoretical development of low-to-medium sound modulation applications, they are also valuable and illuminating for investigating the broadband sound modulation. Keywords:poroelastic materials/ periodic structures/ sound transmission
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2.复合结构及其相关理论和求解图1(a) 为含多孔介质周期复合结构示意图, 此结构由多孔介质、薄板和周期分布的弹簧振子系统构成; 弹簧振子系统沿$x$方向周期分布, 周期间距为$a$; 在每个周期内, 假设含有${N_s}$个振子(或振子系统), 第$i$个振子的刚度和质量分别为${k_i}$和${m_i}$($i = 1,2, \cdots,{N_s}$). 假定复合结构在$y$方向无限延伸, 单位平面波以仰角$\varphi $入射, 其速度势为${\varPhi _i} = \exp ( {{\rm j}\omega t -} {{\rm j}{{kr}}} )$, 其中${{k}} = \left( {{k_x},{k_z}} \right)$为入射波矢, ${{r}} = (x,z)$为位置矢量, ${\rm{j}} = \sqrt { - 1} $. 下文中, 简谐时间项$\exp \left( {{\rm{j}}\omega t} \right)$都不再显式写出. 图 1 含多孔介质复合结构及其子结构示意图 (a) 含多孔介质复合结构; (b) 等效模型; (c) OU边界; (d) OB边界; (e) 板受力情况(OU边界); (f) 板受力情况(OB边界); (g) 弹簧振子受力 Figure1. Schematic of the poroelastic composite structure and its substructures: (a) The poroelastic composite structure; (b) the equivalent model; (c) the OU boundary connection; (d) the OB boundary connection; (e) the forces in OU boundary case; (f) the forces in OB boundary case; (g) the forces in a simple spring-mass resonator.
图1(c)—(g) 给出了复合结构的连接边界条件和相应部分受力情况. 多孔介质和板结构连接类型采用文献[3]中的分类, 记O类型为多孔介质开放边界, 即多孔介质界面直接与外部声学域接触; U类型为多孔介质间隔边界, 即多孔介质界面与弹性体间存在间隙(例如, 薄空气层); B类型为多孔介质固定边界, 即多孔介质界面与弹性体固定粘连. OU和OB类型为几种类型边界条件的组合, 分别如图1(c) 和图1(d) 所示. 当多孔介质与弹性体间为固定边界时, 由于多孔介质固相骨架中正应力和剪应力同时存在, 此时弹性体还会受到面内外力和面外力矩作用, 图1(e) 和图1(f) 给出了OU, OB两种边界情况下薄板法向的受力示意图. 除采用图1(a)所示简单振子系统外, 复合结构中的周期振子单元也可由图2(a)和图2(b) 所示的组合振子系统构成, 其相应的刚度、阻尼和质量如图所示 (其中, 参数$k_n^i$, $\zeta _n^i$和$m_n^i$中的$n = 1,2$为振子系统部件编号, $i$为周期单元序号). 采用周期振子系统排布时, ${N_s} \geqslant 1$, 可行的排布情况如图3(暂不考虑三类振子系统混合分布的情况). 为便于说明和分析, 下文采用表1中缩写形式描述图3所示振子系统分布情况. 图 2 组合振子系统示意图 (a) 串联弹簧振子系统A; (b) 复合弹簧振子系统B Figure2. Schematic of the composite-resonator-structure: (a) Composite resonator type A, two resonators placed in serial connection; (b) composite resonator type B, two resonators placed in composite connection.
振子系统分布
含义
N1SR
周期间隔内分布多个简单振子系统, 各振子系统${m_i}$和${f_i}$均相等
NNSR
周期间隔内分布多种简单振子系统, 各振子系统${m_i}$或${f_i}$不同
N1CR
周期间隔内分布多个组合振子系统, 各振子系统$m_n^i$和$f_n^i$均相等
NNCR
周期间隔内分布多种组合振子系统, 各振子系统$m_n^i$或$f_n^i$不同
表1振子系统分布情况简称及其对应含义 Table1.Abbreviations of the distribution of resonator systems and their meanings.
图 3 周期振子排布方式示意图 (a) 简单振子周期分布, 按各个振子质量${m_i}$和特征频率${f_i}$分为多个振子情况 (N1SR, ${m_i}$和${f_i}$均保持恒定) 和多种振子情况 (NNSR, ${m_i}$或${f_i}$不相同); (b) 组合振子周期分布, 按振子部件质量$m_n^i$和特征频率$f_n^i$分为多个振子情况 (N1CR, $m_n^i$和$f_n^i$均保持恒定) 和多种振子情况 (NNCR, $m_n^i$或$f_n^i$不相同); 图中虚线框内部分为单个振子单元, (b)中虚线框部分可替换为 图2中B类组合振子 Figure3. Schematic of the arrangement of periodic resonators: (a) An array of simple resonators, denoted as multiple resonators (N1SR, with constant ${m_i}$ and ${f_i}$) or multiple kinds of resonators (NNSR, with different ${m_i}$ and ${f_i}$); (b) an array of composite resonators, denoted as multiple resonators (N1CR, with constant $m_n^i$ and $f_n^i$) or multiple kinds of resonators (NNCR, with different $m_n^i$ or $f_n^i$). The area in the dash-line denotes the periodic lattice, in panel (b), the composite resonator can be type B in Fig. 2
为验证本文的理论模型, 分别将其退化为文献[35]中附加亚波长周期振子的均匀薄板结构模型($\epsilon = 1$, ${h_{\rm{p}}} = {h_{\rm{a}}} = 0$, 图4(a)为随机入射情况, 图4(b)为斜入射情况)及文献[3]中多孔介质复合结构模型(${m_{{\rm{sum}}}} = 0$, 图4(c)), 并选用相同计算参数与相应结果进行对照, 验证算例模型示意图及其结果在图4中给出. 图 4 不同类型隔声结构验证算例 (a) 文献[35]随机入射情况; (b) 文献[35]斜入射情况; (c) 文献[3]含多孔介质复合结构; 其中, 各曲线为本文结果, 各标记为文献中结果 Figure4. Validation of the results here with previous results: (a) The diffuse case in Ref. [35]; (b) the oblique incident cases in Ref. [35]; (c) the composite poroelastic structure without resonator in Ref. [3]. The lines are results obtained here, while the marks are the results in the references.
保持振子系统质量比$\gamma = 0.2$, 取复合结构中周期简单振子质量块${m_0} = 27\;{\rm{g}}$ (N1SR情况), 有无多孔介质时两种边界条件下的STL情况见图5. 图 5 有无多孔材料对含不同特征频率振子系统复合结构STL的影响 (a) OU边界情况; (b) OB边界情况; 有无多孔介质分别与相应实线和虚线对应 Figure5. Influence of porous material on the STL of the multiple-single-type-resonator composite structure with different characteristic frequencies: (a) OU case; (b) OB case. The solid lines correspond to cases with porous materials.
由图5可以看出, 在OU和OB两种边界条件下, 附加多孔介质对复合结构STL影响情况基本一致. 采用某一特征频率振子时, 多孔材料的引入可以在远离振子特征频率频域提升其STL, 这是由多孔介质的声学性能决定的. 周期振子特征频率附近, 有无多孔介质对STL提升并不明显, 此区域STL主要由周期振子决定, 但由于多孔介质的引入, 在振子系统特征频率后的STL降低趋势被削弱. 因此, 多孔介质的引入, 总体上增大了结构的隔声能力, 这与文献[1]中对多孔介质的特性描述一致. 图6给出了两种边界条件下含相同简单振子系统复合结构(fr = 300 Hz)有无多孔介质情况, 及相应不含振子复合结构的STL. 由图6可知, 采用某一特征频率振子时, 多孔材料的引入可以在远离振子特征频率频域提升其STL, 最终趋近于相应不含振子复合结构的STL. 采用多孔介质和简单周期振子复合结构, 可以使得结构的隔声性能在保有多孔介质宽频优势情况下, 在特定频点或频段有一定提升; 或保有简单周期振子系统特定频点或频段优势情况下, 在宽频域有一定提升. 图 6 含相同简单振子系统复合结构(fr = 300 Hz)有无多孔介质及相应不含振子复合结构的STL (有多孔介质, Porous + Resonator; 无多孔介质, Resonator; 相应不含振子复合结构, Porous) (a) OU边界情况; (b) OB边界情况. Figure6. The STL of multiple-single-type-resonator composite structure (fr = 300 Hz) with/without porous, and composite structure without resonators: (a) OU case; (b) OB case. Composite structure here with porous material: Porous + Resonator. Without porous material: Resonator. Composite structure without resonators: Porous.
23.3.周期间隔内分布多个振子系统时STL情况 -->
3.3.周期间隔内分布多个振子系统时STL情况
33.3.1.N1SR-简单振子系统情况 -->
3.3.1.N1SR-简单振子系统情况
保持振子系统质量比$\gamma = 0.2$, 取复合结构中周期简单振子质量块${m_0} = $27 g, 采用不同特征频率简单振子系统时, 复合结构在OU和OB两类边界条件下的STL如图7(a)和图7(b). 由图7可知, 采用不同特征频率的周期振子系统, 都使得在此特征频率附近STL先显著提升然后降低, 最后恢复到不含振子时STL的水平. 这表明, 合适引入和布置周期振子, 可以在一定频域内达到提升STL的目的. 图 7 采用不同特征频率简单振子系统对复合结构STL的影响 (a) OU边界; (b) OB边界 Figure7. Influences of resonators with different characteristic frequencies on the STL: (a) OU case; (b) OB case.
图8(a)和图8(b)给出了两种边界条件下, 由特征频率fr = 300 Hz的简单振子构成的复合结构的STL及其位移传递率Ti、振子动态质量meq和板等效动态密度${\rho _{{\rm{eq}}}}$. 从图8可以看出, 在振子特征频率附近Ti, meq和${\rho _{{\rm{eq}}}}$变化趋势一致, 都由极大(f = 290 Hz)下降为极小(f = 307 Hz); STL也呈由极大(f = 298 Hz)到极小(f = 325或335 Hz)的变化趋势, 但STL变化与Ti, meq和${\rho _{{\rm{eq}}}}$变化特征并不同步, 存在一定的频率滞后. 这是由于局域共振板在特征频率附近存在一个范围约为${f_{\rm{r}}}$到${f_{\rm{r}}}{( {1 + }}$${{ \gamma } )^{1/2}}$的带隙[34,35], 而${f_{\rm{r}}}$处板等效动态密度呈极大值(无阻尼时为无穷大), ${f_{\rm{r}}}{\left( {1 + \gamma } \right)^{1/2}}$处呈极小; 此频域为质量定律控制区域[42], 其隔声量(即STL)与结构面密度呈正相关, 故该频域内STL呈极大过渡到极小趋势. 图 8 单一类型简单振子周期排布时 (a) OU, OB情况下STL及其位移传递率Ti; (b) 振子的等效质量meq和板等效动态密度${\rho _{{\rm{eq}}}}$ Figure8. (a) STL of OU and OB case in periodically-arranged single simple resonator case, and its displacement transmissibility Ti; (b) equivalent mass meq of a single resonator and the dynamic density ${\rho _{{\rm{eq}}}}$ of the equivalent plate.
由(21)和(22)式可知, 组合振子A和B二特征频率间的带宽为$\Delta {f^i} = {{\left| {\omega _1^i - \omega _2^i} \right|} / {2{\text{π}}}}$, i = A, B. 根据前述分析, 复合结构STL在振子特征频率附近呈由极大到极小变化趋势, 为获得较高STL, 组合振子二特征频率间带宽$ \Delta f ^i$应控制到合适水平, 相关研究还在进行中, 这里仅给出一些初步结果. 忽略弹簧质量, 保持振子系统质量比$\gamma = 0.2$, 取组合系统参数如表4所列.
参数
$\omega _1^i$
$\omega _2^i$
r
s
$\eta _1^i$
$\eta _2^i$
值
600${\text{π}}$
1088
0.075
0.0625
0.01
0.05
表4振子系统参数 Table4.Parameters of the composite resonators.
组合振子系统 A和B 的位移传递率${T_1}$, ${T_2}$和动态质量${m_{{\rm{eq}}}}$变化情况见图9. 图 9 两类组合振子系统中质量块的位移传递率${T_1}$, ${T_2}$和动态质量${m_{{\rm{eq}}}}$ (a1) 组合振子系统A中各质量块的位移传递率${T_1}$, ${T_2}$; (a2) 组合振子系统A的动态质量${m_{{\rm{eq}}}}$; (b1) 组合振子系统B中各质量块的位移传递率${T_1}$, ${T_2}$; (b2) 组合振子系统B的动态质量${m_{{\rm{eq}}}}$ Figure9. Displacement transmissibility and dynamic mass of the mass components in the two composite resonators: (a1) Displacement transmissibility ${T_1}$, ${T_2}$ of composite resonator type A; (a2) dynamic mass ${m_{{\rm{eq}}}}$ of composite resonator type A; (b1) displacement transmissibility ${T_1}$, ${T_2}$ of composite resonator type B; (b2) dynamic mass ${m_{{\rm{eq}}}}$ of composite resonator type B.
由图9(a1) 和图9(b1) 可知, 在组合振子系统A和B的共振频率附近, 系统中振子位移传递率均呈先增大至共振频率再降低的趋势, 且振子系统A和B两个共振频率之间形成了一段振动抑制频带, 此频带宽度与组合振子系统特征频率有关; 组合振子系统A和B均可以对主结构振动产生明显抑制, 但其抑制带宽不同. 由于声波通过此复合结构中薄板(主结构)向透射侧传播, 针对薄板向外部辐射声的振动模态采用合适的组合振子系统, 预计可以改善结构隔声性能. 图10显示了保持振子系统质量比$\gamma = 0.2$, 单周期内分布${N_{\rm{s}}} = 4$个相同简单振子, 组合振子A或B和不含振子时复合结构的STL情况. 由图10可知, 在两种边界条件下, 由于振子系统特征频率附近其STL呈先升高后降低趋势, 对单一共振频率的简单振子系统, 其STL提升频域较窄; 选用合适参数的组合振子系统A或B, 相对简单振子系统, 可以在较宽频域提升复合结构的STL, 同时不显著降低其STL值. 图 10 复合结构周期间隔内分布4个相同简单振子(Single resonator), 组合振子A或组合振子B时的STL和不含振子复合结构(Without resonator)的STL (a) OU边界情况; (b) OB边界情况 Figure10. STL of the proposed composite structure with 4 identical simple resonators (Single resonator), composite resonators of type A or B versus its STL without any resonators (Without resonator) in a periodic lattice: (a) OU boundary case; (b) OB boundary case.
为对照不同振子系统分布对复合结构STL的影响, 保持振子系统质量比$\gamma = 0.2$, 取${N_{\rm{s}}} = 4$, 各振子系统特征频率$ {f_r^i} $(简单振子)或$f_1^i$(组合振子)取为300+50(i–1) (Hz), i = 1, 2, ···, Ns. 不同周期振子系统分布时复合结构的STL情况如图13所示. 图 13 不同振子系统分布时STL对比 (a), (a1) OU边界情况; (b), (b1) OB边界情况; $\Delta m = 0$和$\Delta m > 0$对应简单振子情况NNSR; Type A和Type B对应组合振子情况NNCR Figure13. STL of different resonator system configuration: (a), (a1) OU case; (b), (b1) OB case. $\Delta m = 0$ and $\Delta m > 0$ correspond to simple resonator case NNSR. Type A and Type B correspond to composite resonator case NNCR.