The noise problem caused by the interaction between the turbulent boundary layer (TBL) pulsating pressure and aircraft side plates is one of the most representative problems in vibro-acoustics [1,2]. Many efforts carried out on the problem of TBL-induced structural noise can be summarized in three aspects. One is wavenumber frequency spectrum models quantifying the TBL excitations. Some famous semi-empirical formulations, such as Corcos [3], Efimtsov [4], Williams [5] and Chase [6,7], have been obtained by fitting a large amount of experimental data and statistical turbulence theory. The other one is about how to predict the vibration and radiated noise of a plate caused by TBL excitations. Graham [8,9] proposed a model to predict the TBL-induced noise for aircraft side and trim plates, in which the modal excitation terms are expressed analytically, and the advantages of different wavenumber frequency spectrum models induced by TBL are discussed. Liu et al. [10] predicted the TBL-induced noise of a stiffened plate using the receptance method. It was found that the stiffeners perpendicular to the direction of incoming flow have an obvious effect on the radiated noise. Rocha and Palumbo [11] investigated the sensitivity of sound power radiated by aircraft plates to TBL parameters, and discussed the findings by Liu [12] that ring stiffeners may increase TBL induced noise radiation significantly. Liu [13] further compared TBL-induced vibrations with the in-flight measured data of P180, where a simplified double integral for the calculation of the modal excitation term is provided. The third aspect is the passive methods for the control of the radiated noise. It has been reported that passive damping is always effective in controlling the vibration and noise caused by TBL. However, the reduction in vibration level is more significant in comparison with the radiated noise level, which implies that the radiation efficiency of the plate increases with increasing damping treatment. Kou et al. [14] described formulas to include the influence of structural damping on the radiation efficiency of finite and infinite plates. Thus, the phenomenon that the radiation efficiency of a plate increases with the increase in the damping treatment is explained. Kou et al. [15] also concluded that the modal averaged radiation efficiency increases significantly with the increase in the convection velocity below the hydrodynamic coincidence frequency, and the damping effect is more significant with the increase in the flow velocity.
In addition to passive methods, active methods have great potential for the control of TBL-induced plate noise. Among them, the control strategy based on distributed velocity feedback has received much attention for acoustically or TBL-induced noise [16,17]. The simulation results given by Elliott et al. [18] and Jayachandran et al. [19] show that the distributed velocity feedback is unconditionally stable in a large gain coefficient range, which is a relatively robust control method. However, the force driver needs some large mass to generate the reaction force, and when a large force is required in the low frequency range, the force driver will be relatively large and heavy. In practice, it is more convenient to use piezoelectric patch actuators integrated with plates. Gardonio et al. [20,21,22] used piezoelectric patch actuators and acceleration sensors to analyze in detail the control effect of distributed velocity feedback control and the existence of optimal gain coefficient from theoretical and experimental perspectives. These works further show that the distributed velocity feedback control is easy to implement and the control effect is approximately optimal. Since it is usually not convenient to obtain the physical information of the structure, it is difficult to obtain the optimal gain coefficient. To solve this problem, Cao et al. [23,24] proposed the concept of the virtual absorption energy of piezoelectric sheets, which uses the maximum virtual absorption energy to obtain the best gain coefficient and is easier to measure compared to kinetic energy or acoustic radiation power. Distributed velocity feedback control is not only applicable to diffused sound field excitation but also to random excitation and TBL excitation. Rohlfing et al. [25] specified the mesh density of finite cells on the plate and investigated the effectiveness of negative feedback control of uniform and light sandwich panels under random excitation and TBL excitation. The control effects of a series of ideal speed negative feedback control circuits on a homogeneous plate and a lightweight sandwich plate are compared. Alouf et al. [26] developed a new active control mechanism for aircraft cabin windows using an active structural acoustic control strategy that provides a significant improvement in acoustic attenuation performance at low frequencies. The effects of voltage, actuator position and number on the sound transmission characteristics were analyzed. Yuan et al. [27] investigated the dispersive velocity feedback control of thin plates under TBL excitation based on the newer TBL semi-empirical model, and the results showed that the pre-stress effect and hydrodynamic overlap have a large effect on plate vibration, which has an important influence on the plate vibration acoustic performance and the selection of the number of control channels. Ma et al. [28] investigated the dispersive velocity feedback control of a ribbed plate using inertial actuators and discussed the effect of feedback gain and number of actuators on control performance, further demonstrating the existence of an optimal gain for dispersive velocity feedback control.
Typical aircraft plates generally exhibit unidirectional curvature. A typical case is that when an aircraft plate is excited by TBL, the direction of air velocity is perpendicular to the curved direction of the plate. The sound radiation properties of curved and flat plates can be significantly different. As pointed out in reference [10], the curvature results in the convergence of resonance frequencies of the plate led by the interaction of bending forces and membrane tensions in the shell. The convergence not only increases the modal density of the curved plate around the ring frequency but also increases the sound radiation efficiency of these modes by shifting them to a relatively higher frequency. Although the active control of flat plates can be found in many works in the literature, there are few studies on the acoustic characteristics of active control of curved plates under TBL excitations. Graham [8] studied the induced noise of aircraft wall panels under TBL excitation, elucidating that the presence of panel membrane tension causes a shift in the lowest resonant frequency to high frequencies. Nourzad et al. [29] used inertial actuators to control the vibration and radiation of doubly-curved plates and analyzed the effect of curvature on the vibration response of doubly curved plates. In this paper, the control effect of a curved thin plate under TBL excitations is numerically investigated. Sixteen active control units are scattered on the plate, and each active control unit includes a piezoelectric actuator, an acceleration sensor, and a feedback actuator. The kinetic energy and radiated sound power of the plate are discussed in detail for different curved plate thicknesses, bending curvatures, and active control unit distribution.
