This paper presents a method to predict the acoustic characteristics and steady-state responses of a flexible plate strongly coupled with rectangular cavity based on energy principle theory and Legendre polynomial series. First, the displacement of the plate and the sound pressure in the cavity are constructed in the form of two-dimensional and three-dimensional Legendre polynomial series, respectively. The unknown expansion coefficients are obtained using the Rayleigh–Ritz technique based on the energy expressions for the strongly coupled plate-cavity system. The accuracy, convergence, and efficiency of the present method are verified by comparing with the results available in the FEM and literature. Finally, the effects of the structural boundary conditions, cavity depth, and structural length-width ratio on the coupling natural frequency and the steady-state responses under three excitation conditions are analyzed.

The structure-acoustic interaction system consisting of a flexible vibrating plate and a closed acoustic cavity is widely used in daily life and practical engineering, for instance, bedrooms, aircraft, ship cabins, and sonar platform cavity. In recent years, with the growing requirements for acoustical performance of products as well as comfortable working and living conditions, a clear understanding of acoustic characteristics of structure-acoustic interaction system is the key to acoustical product design and noise control. However, most research is carried out with the assumption of weak coupling between structure and acoustics. With the development of engineering field, the strong coupling situation where, for example, the contained acoustic medium has large mass density, the plate is thin, and the cavity is shallow is becoming more and more common. Therefore, it is of great value to deeply study the acoustic characteristics of the strongly coupled structure-acoustic system.

In practice, a flexible vibrating plate backed by a closed rectangular cavity is a classical representation of numerous engineering backgrounds and has been extensively studied for a long time. In 1963, Lyon [

In addition to the research on weakly coupled panel-cavity system, the study of strongly coupled panel-cavity system has attracted the interest of many scholars. Tournour and Atalla [

In the view of the existing literatures, different from the research on the weakly coupled panel-cavity system, the research on the strongly coupled plate-cavity system focuses on the system characteristics and less on the steady-state response. Meanwhile, the computational cost of IFM and CPSM is high [

As shown in Figure

Schematic of the considered flexible plate strongly coupled with a rectangular cavity.

The strongly coupled plate-cavity system can be divided into the acoustic cavity system and flexible plate system. For the acoustic cavity system, the internal sound pressure satisfies the Helmholtz equation and boundary conditions:

For the flexible plate system, the effect of sound pressure at the interface should be considered as well as the point force source excitation. Based on the theory of structural dynamics, the differential equation for the elastic structural plate takes the following form:

The velocity continuity will not be satisfied at the coupling interface with the strong coupling between plate and cavity if the rigidly walled cavity mode is used to represent the sound pressure as well as the vacuum structure mode is used to represent the displacement. In order to overcome the problem, the structural displacement and sound pressure are expanded by Legendre polynomial series; in addition, coordinate transformation is required since the Legendre polynomials define the interval as [−1, 1]:

The sound pressure and structural displacement functions can then be expressed as

It should be noted that although the infinite polynomial expansion terms are included in equations (

Since the strongly coupled plate-cavity system can be divided into the acoustic cavity system and the flexible plate system, based on the energy principle, the Lagrange functions [

For the flexible plate system, the Lagrange function expression [

Substitute equation (

The linear algebraic equations of the expansion coefficient vectors

The overall form can be developed by combining equations (

According to the principle of action and reaction, we can get the equality equation of

The accuracy and convergence of the method are verified by comparing with the results available in the FEM and literature. The FEM model is established in COMSOL, and the sound-shell coupling module is used to model the strongly coupled plate-cavity system. To obtain reliable results, the maximum mesh size is set to 1/6 of the minimum wavelength of the analysis band. A water-filled cavity bounded by a simply supported plate is considered as an analysis model which has been studied in Ref [

The model parameters.

Cavity | Length (m) | |

Width (m) | ||

Depth (m) | ||

Water density ( | ||

Sound velocity ( | ||

Plate | Length (m) | |

Width (m) | ||

Thickness (m) | ||

Poisson’s ratio | ||

Young’s modulus (Pa) | ||

Mass density ( |

Table

The first eight natural frequencies of the strongly coupled plate-cavity system.

Mode number | |||||||||
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ||

33.564 | 46.523 | 78.589 | 112.209 | 163.658 | 171.812 | 207.975 | 269.727 | ||

33.548 | 46.580 | 78.575 | 85.232 | 125.997 | 137.665 | 162.189 | 175.863 | ||

33.547 | 46.579 | 78.572 | 84.802 | 125.100 | 137.282 | 161.111 | 172.606 | ||

33.547 | 46.579 | 78.572 | 84.797 | 125.090 | 137.275 | 161.095 | 172.552 | ||

33.547 | 46.579 | 78.572 | 84.797 | 125.090 | 137.275 | 161.094 | 172.552 | ||

33.564 | 46.622 | 78.587 | 112.179 | 163.590 | 171.759 | 207.902 | 269.588 | ||

33.548 | 46.580 | 78.574 | 85.211 | 125.963 | 137.635 | 162.141 | 175.742 | ||

33.547 | 46.578 | 78.571 | 84.764 | 125.033 | 137.231 | 160.992 | 172.268 | ||

33.547 | 46.578 | 78.571 | 84.757 | 125.020 | 137.220 | 160.965 | 172.175 | ||

33.547 | 46.578 | 78.571 | 84.757 | 125.019 | 137.220 | 160.965 | 172.174 | ||

33.564 | 46.622 | 78.587 | 112.176 | 163.582 | 171.754 | 207.894 | 269.571 | ||

33.547 | 46.580 | 78.573 | 85.211 | 125.962 | 137.634 | 162.140 | 175.741 | ||

33.547 | 46.578 | 78.571 | 84.764 | 125.032 | 137.229 | 160.991 | 172.265 | ||

33.547 | 46.578 | 78.571 | 84.757 | 125.018 | 137.218 | 160.962 | 172.167 | ||

33.547 | 46.578 | 78.570 | 84.757 | 125.018 | 137.218 | 160.962 | 172.166 | ||

FEM | 33.509 | 46.513 | 78.413 | 84.577 | 124.670 | 136.810 | 160.420 | 171.570 | |

Ref [ | 33.574 | 46.596 | 78.608 | 84.102 | 124.680 | 137.112 | 161.082 | 172.334 |

Figure

The first five structural modes of the strongly coupled panel-cavity system. (a) FEM. (b) Present method.

In contrast to Ref [

It can be seen from equations (

In this section, the effects of the structural boundary conditions, cavity depth, and structural length-width ratio on the natural frequency of simply supported strongly coupled plate-cavity system are studied. The model parameters are shown in Table

In this section, the natural frequencies under different boundary conditions are analyzed by changing the spring stiffness coefficient. “

Natural frequencies under different boundary conditions.

Boundary conditions | Results | Mode number | |||||||
---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ||

SSSS | Present method | 33.547 | 46.578 | 78.571 | 84.764 | 125.033 | 137.231 | 160.992 | 172.268 |

FEM | 33.509 | 46.513 | 78.413 | 84.577 | 124.670 | 136.810 | 160.420 | 171.570 | |

SSSC | Present method | 40.389 | 48.410 | 85.340 | 96.514 | 127.324 | 150.292 | 168.131 | 191.673 |

FEM | 40.309 | 48.330 | 85.109 | 96.178 | 126.930 | 149.650 | 167.440 | 190.370 | |

SSCC | Present method | 48.625 | 50.755 | 93.262 | 108.600 | 130.969 | 164.836 | 175.427 | 213.522 |

FEM | 48.480 | 50.651 | 92.932 | 108.070 | 130.500 | 163.880 | 174.590 | 211.580 | |

SCCC | Present method | 51.439 | 62.058 | 104.229 | 111.989 | 150.289 | 175.091 | 194.538 | 217.899 |

FEM | 51.270 | 61.869 | 103.770 | 111.400 | 149.510 | 173.960 | 193.280 | 215.930 | |

CCCC | Present method | 55.198 | 75.789 | 115.662 | 117.261 | 172.088 | 187.087 | 210.256 | 228.636 |

FEM | 54.993 | 75.459 | 115.010 | 116.620 | 170.820 | 185.720 | 208.430 | 226.560 |

The natural frequencies in various cavity depths in the range of 0.01–1.0 m are studied. It should be noted that since the cavity depth is variable, the truncation value in the direction of cavity depth is selected as follows: when the cavity depth is less than 0.5 m,

Variation of the first eight-order coupling frequency with various cavity depths.

Keeping cavity depth and the length of plate constant, Figure

Variation of the first eight-order coupling frequency with various length-width ratios.

On the basis of the above results, the steady-state responses of the plate-cavity system are studied in this section under three excitation conditions, which are point force source excitation, internal point acoustic source excitation, and a combination of both. The model parameters are shown in Table

Response under point force source excitation. (a) Velocity response at (0.10, 0.12, 0.14). (b) Velocity response at (0.15, 0.30, 0.14). (c) Sound pressure response at (0.08, 0.11, 0.07). (d) Sound pressure response at (0.20, 0.33, 0.06).

Response under internal point acoustic source excitation. (a) Velocity response at (0.10, 0.12, 0.14). (b) Velocity response at (0.15, 0.30, 0.14). (c) Sound pressure response at (0.08, 0.11, 0.07). (d) Sound pressure response at (0.20, 0.33, 0.06).

Response under the combined excitation of point force source and internal point acoustic source. (a) Velocity response at (0.10, 0.12, 0.14). (b) Velocity response at (0.15, 0.30, 0.14). (c) Sound pressure response at (0.08, 0.11, 0.07). (d) Sound pressure response at (0.20, 0.33, 0.06).

In the paper, a strongly coupled structure-acoustic model for a rectangular cavity and its flexible wall is proposed, and the displacement of the plate and the sound pressure in the cavity are developed based on energy principle theory and Legendre polynomial series. The acoustic characteristics and the forced response of the coupled system are obtained using the Rayleigh–Ritz technique. In addition, the influence of geometric parameters on the acoustic characteristics of the coupled system is analyzed. The results show the following:

The acoustic characteristics of a strongly coupled plate-cavity system can be obtained accurately and quickly, and since the L2 inner product of the Legendre polynomials satisfies orthogonality, the efficiency of the calculation is greatly enhanced.

The change of the structural boundary from clamped supported to simply supported, the reduction of the cavity depth, and the increase in the length-width ratio of the structure all result in a shift of the coupled natural frequencies towards lower frequencies. The property can guide the design of acoustic products. Meanwhile, when the cavity depth is shallow, the coupled natural frequency is sensitive to the change of cavity depth, Therefore, special attention should be paid to cavity depth when designing and applying cavity structures.

The steady-state response under the point force excitation, internal acoustic excitation, and a combination of both can be predicted accurately by the present method, which can provide a new idea for active noise control of the strongly coupled plate-cavity system.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

The authors declare that they have no conflicts of interest.

This study was supported by the National Natural Science Foundation of China (no. 51675529).