Active Control of Sound Transmission in Ship Cabins Through Multiple Independently Supported Flexible Subplates

Abstract

The vibration and noise produced by the powertrain and waves inside ship cabins limit working efficiency and crew and passengers’ accommodation quality. This paper simplifies ship cabins as cavities and explores active control techniques to attenuate sound transmission via multiple parallel-supported flexible subplates. The theoretical formulations of the interaction between multiple subplates and cavities were performed and the coupling relationships between them were analyzed. Based on the multiple subplates and the cavity coupling models, numerical simulations were performed using the derived optimal controller to minimize the transmission of sound into the cavities through two and nine parallel-supported subplates. The various control strategies were explored to minimize the coupling system’s acoustic potential energy, and the control performances were compared and discussed. The mechanism of reducing sound transmission through multiple supported subplates into a cavity is revealed. The simulation results showed that the vibration pattern of the controlled subplate is changed after it is regulated, which increases its radiation to subdue the other subplates’ radiation, while increasing vibration of the controlled subplate. The more subplates a cavity has, the more kinetic energy the controlled subplate possess. Furthermore, the noise reduction performance of a cavity with fewer subplates is better than that with more subplates.

Appendix

The coupling matrixes between the two subplates and the cavity for model 1 are:

$$ {\boldsymbol{B}}_1\left(l,i\right)={\left(-1\right)}^n\frac{2{uvL}_x{L}_y\left[1-{\left(-1\right)}^u\cos \left(l\uppi /2\right)\right]\left[1-{\left(-1\right)}^{v+m}\right]}{\left(4{u}^2-{l}^2\right)\left({v}^2-{m}^2\right){\uppi}^2} $$

(A1)

$$ {\boldsymbol{B}}_2\left(l,i\right)={\left(-1\right)}^n\frac{2{uvL}_x{L}_y\left[\cos \left(l\uppi /2\right)-{\left(-1\right)}^{l+u}\right]\left[1-{\left(-1\right)}^{v+m}\right]}{\left(4{u}^2-{l}^2\right)\left({v}^2-{m}^2\right){\uppi}^2} $$

(A2)

The coupling matrixes between the nine subplates and the cavity for model 2 are:

$$ {\boldsymbol{B}}_1\left(l,i\right)={\left(-1\right)}^n\frac{9{uvL}_x{L}_y\left[1-{\left(-1\right)}^u\cos \left(l\uppi /3\right)\right]\left[1-{\left(-1\right)}^v\cos \left(m\uppi /3\right)\right]}{\left(9{u}^2-{l}^2\right)\left(9{v}^2-{m}^2\right){\uppi}^2} $$

(A3)

$$ {\boldsymbol{B}}_2\left(l,i\right)={\left(-1\right)}^n\frac{9{uvL}_x{L}_y\left[\cos \left(l\uppi /3\right)-{\left(-1\right)}^u\cos \left(2l\uppi /3\right)\right]\left[1-{\left(-1\right)}^v\cos \left(m\uppi /3\right)\right]}{\left(9{u}^2-{l}^2\right)\left(9{v}^2-{m}^2\right){\uppi}^2} $$

(A4)

$$ {\boldsymbol{B}}_3\left(l,i\right)={\left(-1\right)}^n\frac{9{uvL}_x{L}_y\left[\cos \left(2l\uppi /3\right)-{\left(-1\right)}^{u+l}\right]\left[1-{\left(-1\right)}^v\cos \left(m\uppi /3\right)\right]}{\left(9{u}^2-{l}^2\right)\left(9{v}^2-{m}^2\right){\uppi}^2} $$

(A5)

$$ {\boldsymbol{B}}_4\left(l,i\right)={\left(-1\right)}^n\frac{9{uvL}_x{L}_y\left[1-{\left(-1\right)}^u\cos \left(l\uppi /3\right)\right]\left[\cos \left(m\uppi /3\right)-{\left(-1\right)}^v\cos \left(2m\uppi /3\right)\right]}{\left(9{u}^2-{l}^2\right)\left(9{v}^2-{m}^2\right){\uppi}^2} $$

(A6)

$$ {\boldsymbol{B}}_5\left(l,i\right)={\left(-1\right)}^n\frac{9{uvL}_x{L}_y\left[\cos \left(l\uppi /3\right)-{\left(-1\right)}^u\cos \left(2l\uppi /3\right)\right]\left[\cos \left(m\uppi /3\right)-{\left(-1\right)}^v\cos \left(2m\uppi /3\right)\right]}{\left(9{u}^2-{l}^2\right)\left(9{v}^2-{m}^2\right){\uppi}^2} $$

(A7)

$$ {\boldsymbol{B}}_6\left(l,i\right)={\left(-1\right)}^n\frac{9{uvL}_x{L}_y\left[\cos \left(2l\uppi /3\right)-{\left(-1\right)}^{u+l}\right]\left[\cos \left(m\uppi /3\right)-{\left(-1\right)}^v\cos \left(2m\uppi /3\right)\right]}{\left(9{u}^2-{l}^2\right)\left(9{v}^2-{m}^2\right){\uppi}^2} $$

(A8)

$$ {\boldsymbol{B}}_7\left(l,i\right)={\left(-1\right)}^n\frac{9{uvL}_x{L}_y\left[1-{\left(-1\right)}^u\cos \left(l\uppi /3\right)\right]\left[\cos \left(2m\uppi /3\right)-{\left(-1\right)}^{v+m}\right]}{\left(9{u}^2-{l}^2\right)\left(9{v}^2-{m}^2\right){\uppi}^2} $$

(A9)

$$ {\boldsymbol{B}}_8\left(l,i\right)={\left(-1\right)}^n\frac{9{uvL}_x{L}_y\left[\cos \left(l\uppi /3\right)-{\left(-1\right)}^u\cos \left(2l\uppi /3\right)\right]\left[\cos \left(2m\uppi /3\right)-{\left(-1\right)}^{v+m}\right]}{\left(9{u}^2-{l}^2\right)\left(9{v}^2-{m}^2\right){\uppi}^2} $$

(A10)

$$ {\boldsymbol{B}}_9\left(l,i\right)={\left(-1\right)}^n\frac{9{uvL}_x{L}_y\left[\cos \left(2l\uppi /3\right)-{\left(-1\right)}^{u+l}\right]\left[\cos \left(2m\uppi /3\right)-{\left(-1\right)}^{v+m}\right]}{\left(9{u}^2-{l}^2\right)\left(9{v}^2-{m}^2\right){\uppi}^2} $$

(A11)