ZHANG Renqi(张仁琪), ZHU Congyun(朱从云)*, DING Guofang(丁国芳), HUANG Qibai(黄其柏)
1 School of Mechatronics Engineering, Zhongyuan University of Technology, Zhengzhou 450007, China2 School of Mechanical Science & Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
Abstract: The theory of active absorption of the perforated plate is proposed in this paper. The perforated plate is used as the material of active absorption and the depth of the cavity behind the perforated plate is changed according to the resonant frequency of the perforated plate. The rigid wall is moved to produce resonance so that the absorption coefficient can reach the maximal level. It is shown from the numerical calculation that when the perforated plate resonates, the moving distance is large at low frequencies, and the absorption coefficient is low under certain conditions. Perforated plate resonance is effective for single frequency of incident sound wave, which is difficult for the wide frequency, so active absorption based on airflow is posed, and the numerical calculation and experiment are carried out. The results denote that this method of active absorption is practical.
Key words: airflow; perforated plate; resonance; absorption coefficient; active absorption
Sound absorption in the acoustic field is one of the most basic and widely used measures to control the noise and improve the indoor sound quality. The sound absorption mainly includes the material sound absorption, the perforated plate sound absorption and the impedance compound sound absorption and so on. The sound absorption mechanism of perforated plates mainly involves three physical processes. First, when the sound wave is in the porous material, the viscous effect between the solid frame and numerous air cavities will attenuate part of the sound energy and convert it into heat. Then, heat transfer will happen due to temperature distinction between different parts caused by friction, which is an isothermal process. And this process will further dissipate sound energy. Finally, the vibration of air in the bulk materials will also lead to the vibration of fibers. Usually the sound absorption property in high frequencies is better than that in low frequencies. The moving gas has a certain mass that resists the change in velocity caused by the action of sound waves. Meanwhile, when sound waves enter the hole, part of the sound energy converts into heat energy due to friction and damping of the wall[1-4]. Perforated plate sound absorption has been developed for more than 20 years. It is a sound absorption structure with light weight and high sound resistance. In addition, the materials for perforated plates are available, light transparent， and pollution-free. The general perforated plate is arranged as shown in Fig. 1.
Fig. 1 Schematic diagram of the perforated plate layout
The perforated plate and the air behind it form a resonant sound absorption structure. The perforated plate, rigid wall, and the corresponding imaginary planes together can be regarded as a Helmholtz resonator. The perforated plate resonator can be interpreted as Helmholtz resonators arranged together. The resonance sound absorption structure is widely used in the internal combustion engine muffler, the axial flow fan muffler, the ventilating air conditioner muffler, the automobile engine exhaust pipe,etc. Perforated plate resonance sound absorption is applied in the industrial field. For example, because the perforated plate resonance structure can absorb large amplitude pressure wave generated by the oscillating combustion, it can be installed in the combustion chamber of the engine, and can restrain the oscillating combustion at a certain frequency. Researchers in the United States underwater acoustic communication (UAC) laboratory first tried using the perforated plate resonance structure in a combustion chamber of a ram, and successfully suppressed the oscillations that occurred at the 2 000 Hz frequency. Since then, the resonator installed on the wall surface of the combustion chamber with stable combustion measures has been increasingly widely used[5].
In 1975, Deanetal.[6]first proposed the concept of the use of ventilation to achieve the wall acoustic impedance rate control. They designed a kind of ventilation lining with a double layer structure, which could change the sound resistance by adjusting the airflow rate and the cushion of the middle layer. Based on the theory of eddy current interference, Howe[7]derived Rayleigh acoustic admittance model for a single circular hole with the airflow. Based on Howe’s theory, Hughesetal.[8]studied on the plate and cylindrical perforated plate in a large number of experiments. The experimental results were in good accordance with Howe’s theoretical model, which showed that such mode was reliable. But they did not consider the effect of the airflow on the sound absorption coefficient of the perforated plate during the resonance of the perforated plate. Fanetal.[2]used one-dimensional approach to investigate the effects of perforations of inlet and outlet tubes and flow on the acoustic attenuation performance of the mufflers. In 2020, Chenetal.[9]carried out a detailed study of the effects of airflow on perforated plates. They measured the acoustic impedance of a single circular hole at different air velocities. The experimental results showed that with the increase of air velocity, the acoustic resistance of the perforated plate increased linearly, while the acoustic impedance decreased slightly.
This paper presents the theory of active sound absorption for perforated plates. The perforated plate is active sound absorption material. According to the resonance frequency of the perforated plate, the depth of the cavity behind the perforated plate was changed to meet the requirement of the resonance of the perforated plate. The sound absorption coefficient could reach the maximum value to achieve the goal of active sound absorption by changing the depth of the cavity behind the perforated plate. But after numerical calculation, it was found that the active sound absorption was not obvious in the case of the perforation rate increasing. Therefore, it was put forward that the optimum sound absorption effect could be achieved through airflow in the cavity due to the acoustic impedance rate changing with the velocity of the airflow when it entering the cavity. The basic principle is that the unsteady vortex ring forms when the pressure wave is in the flow field or the sound wave and the airflow are in the small hole. The energy transfer induced by wave and vortex interaction will change the impedance property of the wall surface, thereby affecting the sound absorption coefficient. Good sound absorption effect is obtained through numerical calculation in this study.
The perforated plate structure is shown in Fig. 2, whereDis the depth of the cavity behind the perforated plate,tis the thickness of the perforated plate,dis the hole diameter of the perforated plate,bis the spatial distance,Piis the incident sound pressure, andPris the reflected sound pressure. Rigid walls can move freely from left to right side.
Fig. 2 Perforated plate structure
(1)
whereZDis the acoustic impedance of the air contact with the cavity[10],ρ0is the density of air,c0is the speed of sound,ωis the circular frequency of the wave, andjis the imaginary part of the complex number.
The sound absorption coefficientαN[11]of the perforated plate can be expressed as
(2)
whereris relative acoustic resistance,ωmis the circular frequency of the incident wave, andmis the sound mass.
According to Eq. (2), if the depth of the cavity changes, the sound absorption coefficient of the perforated plate changes accordingly. At the same time, the resonant frequencyfrsof the perforated plate with a single hole can be expressed as[12]
(3)
(4)
(5)
wherepis the perforation rate, and
(6)
Equations (4)-(6) give the relationship betweenfrsandDas
(7)
IfDis changed to meet the requirements of the resonance,αNcan reach the maximum. This is the basic idea of the active sound absorption by changing the depth of the cavity.
Under the conditions oft=2 mm,b=8 mm, andfrs=500 Hz, whendis 1, 2, and 3 mm,Dcalculated by Eq. (7) is 50, 120, and 144 mm, respectively. The relationship betweenαNand the frequency of the incident sound wavefis shown in Fig. 3.
Fig. 3 Relationship between sound absorption coefficient and incident acoustic wave frequency when frs=500 Hz
Under the conditions oft=2 mm,b=8 mm, andfrs=1 000 Hz, whendis 1, 2, and 3 mm,Dcalculated by Eq. (7) is 12, 40, and 54 mm, respectively. The relationship betweenαNandfis shown in Fig. 4.
Fig. 4 Relationship between sound absorption coefficient and incident acoustic wave frequency when frs=1 000 Hz
Under the conditions oft=2 mm,b=8 mm, andfrs=1 500 Hz, whendis 1, 2, and 3 mm,Dcalculated by Eq. (7) is 4, 20, and 30 mm respectively. The relationship betweenαNand thefis shown in Fig.5.
Fig. 5 Relationship between sound absorption coefficient and incident acoustic wave frequency when frs=1 500 Hz
By analyzing the curve of the sound absorption coefficient in Figs. 3-5, it is found that (1) with increasing resonance frequency, the corresponding sound absorption coefficient increases; (2) with decreasing the diameter of the hole, the sound absorption coefficient of the resonance correspondingly increases, and the effect of active sound absorption is more obvious.
Therefore, when the hole diameter is large, the sound absorption coefficient of the active sound absorption is low, and the sound absorption effect is unsatisfactory. To solve this problem, a method of active sound absorption of the perforated plates with airflow is presented.
An active sound absorption method of the perforated plate with airflow is shown in Fig. 6.
Fig. 6 Layout of active sound absorption method with airflow
2.1 Active sound absorption principle of perforated plate with airflow
After detecting the frequency of the sound wave, the rigid wall moves a certain distance under the action of the driving motor. When the perforated plate meets the requirements of resonance, the airflow is injected into the cavity through the valve, and thus the sound absorption coefficient reaches the maximum.
When the airflow is injected, the reflection coefficientRof the perforated plate is[6]
(8)
wherek0is the wave number.
Strouhal number can be expressed as
(9)
γandδwith Sideluha coefficient are expressed as[7]
γ=
(10)
(11)
whereI1(x) andK1(x) are modified by the Bessel function and can be expressed as
(12)
(13)
whereJis the total inertia of the motor rotor itself with load.
The sound absorption coefficientαNof the perforated plate can be calculated by
(14)
Whent=2 mm,d=2 mm,b=8 mm,andD=120 mm, the relationship between the velocityvof the airflow and the theoretical calculated sound absorption coefficient is shown in Fig.7.
Fig. 7 Relationship between the sound absorption coefficient and the airflow velocity with different f
According to the above theoretical calculation, the relationship between the velocity of airflow and the frequency of the incident wave can be obtained, as shown in Fig.8.
Fig. 8 Relationship between the optimal airflow velocity and the incident acoustic wave frequency with different d
It can be seen from Figs.7 and 8 that for perforated plates of the same specification, the airflow velocity increases with the expansion of the resonance frequency; for perforated plates with the same thickness and hole spacing, the airflow velocity increases with the expansion of the hole diameter at the same resonance frequency.
2.2 Active control unit and device
In this system, the stepper motor is used to control the position of moving rigid wall to adjust the depth of the cavity. The output of the stepper motor is the angular displacement, so the angular displacement of the stepper motor is converted to the displacement of the moving rigid wall by using the screw drive. Figure 9 shows the schematic diagram of the drive.
Fig. 9 Schematic diagram of stepper motor drive
The model of the stepper motor and its driver are selected. Subdivision number is 4. The length of the screw (M14-2) is 100 mm. Finished selecting the stepper motor and the stepper motor drive, the dynamic characteristics of the stepper motor can be determined. According to the voltage equations and motion equations of the stepper motor, the dynamic structure diagram of the stepper motor can be obtained, as shown in Fig.10.
Fig. 10 Dynamic structure of stepper motor
Here,sis the lift,Rwis the winding resistance,Bis the coefficient of viscous friction,Kmis the torque constant,Kcis the magnetic pressure drop coefficient,Keis the back electromotive force coefficient,θiis the standard step angle, andθ0is the actual step angle.
The transfer function of the angular displacement response is[15]
(15)
The above equations can be simplified as a first-order inertia link:
(16)
where the time constantTis 0.65 s.
The electric control valve is used to control the flow rate of the perforated plate. The structure diagram of electric control valve is shown in Fig. 11, in which FC is the servo amplifier, SD is the servo motor, WF is the position transmitter, Z is the reducer, VN is the single-seat control valve, and DKZ is the electric actuator.
Fig. 11 Structure diagram of electric control valve
According to the working principle of DKZ, the dynamic model can be represented by a second-order oscillation link:
(17)
whereKDis the valve opening coefficient,ξis the resistance coefficient of regulating valve, andωnis the angular velocity.
The transfer function of the control valve is obtained by Eq. (15).
(18)
So, the transfer function of the electric control valve can be expressed as
(19)
2.3 Control algorithm under the condition of optimal sound absorption coefficient
The control object of this control system is the perforated plate. The control variables are the depth of the cavity behind the perforated plate and the airflow velocity. The control objective is to change the depth of the cavity behind the perforated plate at the certain frequency of the incident sound wave, so that it can reach the resonant requirements of the perforated plate without airflow.
The velocity of the airflow is calculated under the condition of a certain incident frequency and a certain depth of the cavity, so that the perforated plate can maintain the optimum sound absorption coefficient. Therefore, the open loop control scheme is adopted to control the optimal sound absorption coefficient. The system block diagram is shown in Fig.12. The dynamic structure diagram is shown in Fig.13.
Fig. 12 Control block diagram of optimal sound absorption coefficient of the perforated plate
According to Fig.13, the control of cavity depth is a first-order inertial link, and the airflow velocity control is a second-order oscillation link. The system is stable without correction.
Fig. 13 Dynamic structure diagram of optimal sound absorption coefficient control
The active sound absorption coefficient of the perforated plate can be theoretically calculated by Eq. (14). Keepingtandbconstant (t=2 mm,b=8 mm), adjustingd, the relationships betweenαandfbefore and after airflow control are shown in Fig. 14.
Fig. 14 Relationships between α and f before and after airflow control: (a) d=1 mm; (b) d=2 mm; (c) d=3 mm
As shown in Fig. 14, the experimental sound absorption coefficient after the airflow control increases obviously and reaches nearly 1.0. And the sound absorption coefficient is high in the case of increasing the perforation rate.
In this paper, the theory of active sound absorption for perforated plates is presented. Under the condition of the perforated plate resonance, the active sound absorption of a certain airflow in the cavity is proposed. The theory and method of detecting the acoustic impedance and sound absorption coefficient of the perforated plate with two microphones are put forward. The sound absorption coefficient obtained by the method of changing the depth of the cavity is low in the case of increasing the perforation rate. The sound absorption effect is improved greatly after the airflow control, and the sound absorption coefficient reaches nearly 1.0. This kind of active sound absorption method is feasible.
In terms of active sound absorption based on the relationship between cavity depth and resonance frequency, the current research is to introduce a certain amount airflow under the perforated plate resonance for active sound absorption. In fact, the exhaust of airflow through the muffler, cavity depth and resonant frequency are integrated to influence the sound absorption, that is, the airflow through the muffler itself is used to achieve active sound absorption. Therefore, the key problem is how to control the exhaust flow and then effectively integrate the exhaust flow with the cavity depth and resonance frequency, which implies the direction for future research.
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