Wavelet analysis of multiple tonal phenomena generated from a simplified nose landing gear and a ring cavity

Journal of Sound and Vibration 464 (2020) 114980

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Wavelet analysis of multiple tonal phenomena generated from a simplified nose landing gear and a ring cavity Ling Li a,c , Peiqing Liu b,c , Yu Xing b,c , Hao Guo b,c, ∗ a

School of Energy and Power Engineering, Beihang University, Beijing, 100191, People’s Republic of China School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, People’s Republic of China c Key Laboratory of Aeroacoustics (Beihang University), Ministry of Industry and Information Technology, Beijing 100191, People’s Republic of China b

article info

abstract

Article history: Received 15 November 2018 Revised 29 August 2019 Accepted 23 September 2019 Available online 24 September 2019 Handling Editor: P. Joseph

Discrete tones exist in the airframe noise spectra. For landing gear components, some cavities may generate multiple tones. The tonal frequencies can be accurately predicted and two different tonal noise generation mechanisms have been already proposed, namely fluid-acoustic feedback loop and acoustic resonance. However, the inherent temporal features and the relationships between multiple tones are rarely researched. This paper introduces the continuous wavelet transform method to analyze the acoustic signals from a simplified nose landing gear model and a ring cavity model, aiming at revealing the temporal features and excitation rules of these multiple tones. For the landing gear model, the wavelet analysis results show the mid-high frequency tones generated from the acoustic resonance phenomenon are randomly excited in time and the excitation states are independent of one another. For the ring cavity model, the low frequency tones generated from fluid-acoustic feedback loop are excited alternately and the primary acoustic energy switches from one to another in time, namely the mode switching mechanism. While among the high frequency tones generated from acoustic resonance, the second to the fourth tones satisfy the amplitude modulation mechanism and the other tones are randomly excited. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Multiple tonal phenomenon Landing gear Ring cavity Temporal feature Wavelet analysis

1. Introduction Aircraft noise is one of the main pollution in the modern society. With the application of modern high-bypass turbofans, the engine noise has been reduced by 20–30 dB in the last 40 years and the airframe noise has accounted for a large proportion during the phases of approach and landing, when the engines are operated at low power and the airframe components are fully deployed. Generally, airframe components can be categorized into two major parts, namely landing gear and high-lift device. Among these, the landing gear represents one important airframe noise source for large commercial aircrafts [1–4]. The noise spectra of landing gear components are normally characterized by broadband noise which is mainly generated from some complex flow phenomena such as flow separation and the unsteady interaction of turbulent wake with the downstream components [3,4]. However, the appearance of multiple tones is another significant acoustic phenomenon for landing gears with some cavity configurations such as joint pin hole, wheel hub and inner-wheel cavity. These multiple tones have great influence on the overall noise of landing gears and have been widely investigated before. Dobrzynski et al. [5] detected the tonal noise in the far-field noise spectra of A340 nose and main landing gears and identified that these tones were generated from

∗ Corresponding author. School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, People’s Republic of China. E-mail address: [email protected] (H. Guo).

https://doi.org/10.1016/j.jsv.2019.114980 0022-460X/© 2019 Elsevier Ltd. All rights reserved.

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L. Li et al. / Journal of Sound and Vibration 464 (2020) 114980

the acoustic resonance in some pin cavities. Zhang et al. [6] also found two tones at 630 Hz and 1250 Hz generated from the hub cavity of a single landing gear wheel (the wheel diameter is 0.478 m) and the tonal frequencies were highly dependent on the physical dimensions of the hub cavity. The conclusions were later confirmed by Wang et al. [7] with high-order numerical simulation of the same wheel geometry that the mid frequency tones were remarkably suppressed when the hub cavity was covered. McCarthy and Ekmekci [8] found two distinct tones at 2100 Hz and 3300 Hz in the sideline noise spectra of a twowheel nose landing gear with the wheel diameter of 0.152 m. Moreover, they confirmed that these tones were generated from the acoustic resonance in the inner-wheel cavity because these tones were successfully suppressed by installing a splitter plate on the rear of the main strut to disturb the acoustic resonance between the two facing inner-wheel cavities. In contrast to the acoustic resonance mechanism, the fluid-acoustic feedback loop mechanism inside cavity may also generate multiple tones. Li et al. [9] investigated a 25% scaled A340 main landing gear model (the wheel diameter is 1.017 m) with fairings and observed several tones between 700 Hz and 4000 Hz. These tones were assumed to be caused by a combination of the fluid-acoustic feedback mechanism in the gap between the leading edges of the leg door and the hinge door and the acoustic resonance in the open-ended cavity. In addition, a simplified two-wheel nose landing gear model has been experimentally tested by the LAGOON project (LAnding Gear NOise database for CAA validatiON) [10,11] and further numerically simulated by numerous researchers [12–16]. The model approximately corresponds to 40% of a nose gear for an Airbus A320 aircraft (the wheel diameter is 0.3 m). All the results show that the multiple tonal phenomenon can be observed in the noise spectra, especially at sideline location. Two tones at about 1000 Hz and 1500 Hz are visible in the noise spectra at sideline location, and other weak tones below 1000 Hz and above 1500 Hz may also be observed. Among these, Liu et al. [12] predicted landing gear noise with high-order finite difference schemes and showed good agreement with the acoustic measurement for the LAGOON two-wheel landing-gear configuration. They found that the wheels were the dominant noise source and the strong vortex shedding from the axle was the second major contributor to the landing gear noise. Furthermore, Casalino et al. [15,16] gave a deep investigation to study the resonance modes taking place in the volume between the two facing wheels. They confirmed that three tones were generated from the acoustic resonance in the inner-wheel cavity and analyzed their acoustic mode behaviors, suggesting that the tone at 1000 Hz was related to a plane mode (second-order transversally and zeroth-order circumferentially) for the floor-to-floor cavity distance, whereas another tone at 1500 Hz was related to an azimuthal mode (second-order transversally and first-order circumferentially) for the edge-to-edge cavity distance. Besides, they also paid attention to the temporal feature of landing gear tones and found that two modes with opposite mode numbers satisfied the transient switch feature and had the same probability to generate the second tone. As mentioned above, either acoustic resonance mechanism or fluid-acoustic feedback loop mechanism may generate multiple tones, and the spectral shape and tonal frequency range are similar. Thus, the problems arise what is the relationships between the tones from the same noise generation mechanism and how to distinguish different types of multiple tones appeared in the noise spectra. Solving these problems will play an important role in revealing the inherent features of multiple tones. Up to now, most investigations of landing gears are mainly focused on the tonal noise generation mechanisms and noise control methods, but the research on the temporal features of the multiple tonal phenomenon is inadequate. As is known to all, the traditional power spectral method is based on the time-averaged Fourier transform, so it is ineffective in revealing the temporal information of acoustic signals for the multiple tonal phenomenon. In order to overcome these disadvantages, some joint time-frequency analysis methods such as continuous wavelet transform method and short-time Fourier transform method are proposed. Recently, these novel analysis methods have been successfully applied to reveal the inherent temporal features of multiple tones generated from airfoils [17–20], high-lift configurations [21,22] and rectangular cavities [23]. Pröbsting et al. [17–19] analyzed the velocity and acoustic signals from NACA0012 airfoil with the wavelet transform method and found that the multiple tones were generated from the amplitude modulation mechanism. In other words, the acoustic energy was concentrated on only one primary tone among all the discrete tones and the secondary tones were resulted from the modulation effect of acoustic pressure fluctuations from the primary tone. Similarly, Padois et al. [20] revealed that the multiple tones from a controlleddiffusion airfoil were also generated from the amplitude modulation mechanism. Recently, Li et al. [21,22] measured the farfield noise of 30P30 N high-lift device model and found two types of multiple tones when the model was operated in landing and cruise conditions, respectively. Although the noise generation mechanism of these two type of multiple tones generated from landing and cruise configurations were fluid-acoustic feedback loop mechanism, the wavelet results showed that the inherent temporal features were quite different. The multiple tones of the landing configuration were resulted from the mode switching mechanism, namely that the tones were excited alternately and the acoustic energy switched from one tone to another in time. While the multiple tones of the cruise configuration are resulted from the amplitude modulation mechanism, similar to the features in single-element airfoil cases. Kegerise et al. [23] studied the acoustic features of a rectangular cavity and revealed the mode switching mechanism of the cavity tones. They found that the tones could almost not be excited simultaneously and the dominant acoustic energy switched from one mode to another in time. The aims of this paper are to reveal the inherent temporal features and ascertain the relationships between multiple tones generated from a landing gear model and a more simplified ring cavity model, respectively. Both the landing gear model and the ring cavity model are tested in the D5 aeroacoustic wind tunnel, and the acoustic signals are analyzed with two analysis methods, namely the traditional power spectral method and the continuous wavelet transform method. The organization of the present paper is as follows. In Section. 2, the experimental setup is described in detail and the continuous wavelet transform method is briefly introduced. In Section. 3, the experimental results of the landing gear model is analyzed and displayed with the traditional

L. Li et al. / Journal of Sound and Vibration 464 (2020) 114980

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Fig. 1. (a) Simplified two-wheel landing gear model. (b) Ring cavity model.

power spectral method and the continuous wavelet transform method. In the next Section. 4, the experimental results of the ring cavity model is analyzed and displayed with the traditional power spectral method and the continuous wavelet transform method. Finally, this paper is concluded in Section. 5. 2. Experimental setup and procedures 2.1. Experimental setup Experiments are conducted in the D5 aeroacoustic wind tunnel at Beihang University. The D5 aeroacoustic wind tunnel is an open-jet, closed-circuit wind tunnel, and the test section is 2.0 m in length with a square cross section of 1.0 m by 1.0 m. An anechoic chamber with 6.0 m in length, 6.0 m in width and 7.0 m in height is built surrounding the test section to provide a non-reflecting condition, and the cut-off frequency is 200 Hz. The wind speed can be continuously controlled from 0 to 80 m/s with the turbulence intensity less than 0.08% in the core of the jet [24]. The first experimental model is a simplified two-wheel landing gear referred from the LAGOON project [10,11]. As shown in Fig. 1(a), the model includes one segmented cylindrical strut, one cylinder axle and two wheels, without any part of the fuselage. The wheel diameter D is 150 mm and each wheel has a ring cavity in the inner wheel region with the outer diameter dout = 81 mm, the inner diameter din = 22 mm and the depth h = 18.5 mm. The landing gear is mounted horizontally in the test section, the symbol 𝜓 is defined as the polar angle and the symbol 𝜙 is defined as the azimuthal angle. The model is equipped with a support covered with foam to minimize the noise reflection effect and 80 mesh sandpapers with the thickness of about 0.18 mm are applied on the model as the tripping device to ensure that the flow over the model is turbulent. To verify the dependency of the tonal frequencies on the freestream velocity, the freestream velocity changes from U = 30 m/s to U = 60 m/s, corresponding to the Mach number from 0.09 to 0.18. The blockage coefficient of the present experiment is less than 4%, so no blockage corrections are applied to the experimental data. Two far-field microphones are placed at upstream direction 𝜓 = 60◦ to measure the far-field noise. The first microphone placed at 𝜙 = 0◦ and 2.0 m away from the model is used to measure the flyover noise, and the second microphone placed at 𝜙 = 35◦ and 1.5 m away from the model is used to measure the sideline noise. The same landing gear model has been already tested in the CEPRA19 wind tunnel, but the landing gear model in this paper is reduced by half because of the smaller scale of the D5 wind tunnel. In the experiments of CEPRA19 wind tunnel, the transition triggering devices have been applied to all the cylindrical elements. The cud-cut strips (cylindrical dots) of 0.25 mm thickness have been installed on the leg and the axle, whereas zig-zag strips of 0.2 and 0.4 mm thickness have been glued on the wheels. Although the tripping devices in the two wind tunnel experiments are different, all types of the tripping devices can ensure the flow around the experimental model is turbulent, such that the influence of different tripping devices on the primary noise features are negligibly small. Besides, there are differences in the operation conditions such as freestream velocity and measurement distance, but the power spectral density (PSD) can be translated from one operation condition to another with all the effect factors quantitatively determined and summarized into the equations as: [24,25].

(

)

(

)

(

PSDn = PSDm − N × 10log10 Um ∕Un − 10log10 D2m ∕D2n + 20log10 Rm ∕Rn fn = fm

Dm Dn

)

(1)

(2)

where D is the wheel diameter, R is the distance between the microphone and the model center, and N is respectively chosen as 6 and 7 in low and high frequency range [24,25]. The subscripts m and n denote the variables in two operation conditions. Referring to previous literatures [15,16], three tones are generated from the inner-wheel cavity. Due to the complex geometry and shielding effect, only the tones with specialized resonance modes and strong intensities can radiate to the far field and other excited tones can only be captured inside the inner-wheel cavity. Hence, a ring cavity model which represents the dominant noise source of landing gears is tested as the second model for further study and analysis. As shown in Fig. 1(b), the ring cavity model is composed of a circular cavity and a circular cylinder in the middle of the cavity. Because the equivalent depth-todiameter ratio of the first model’s cavity is 1.02, the outer and inner diameter of the ring cavity are chosen as 0.19 m and 0.05 m

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with the depth of 0.1 m, corresponding to the equivalent depth-to-diameter ratio of 1.04. The model is flush mounted to the wind tunnel nozzle and a surface microphone is flush mounted at the middle of the cavity wall to measure the acoustic pressure fluctuations. The freestream velocity varies from 30 m/s to 55 m/s in order to distinguish two groups of multiple tones from different noise generation mechanisms. The noise is measured in the anechoic chamber by 1/2 inch microphones with the sensitivity of 50 mV/Pa, the frequency of 6.3 Hz ∼20 kHz and the dynamic range of 14.6 ∼146 dB. The acquisition and storage of the acoustic signal is acquired at 65536 samples/s with the record time of 50 s and the frequency span of 25.6 kHz. The measured data is divided into 1200 blocks with the overlapping ratio of 66.7%. Each block is treated by a Hanning window prior to a Fast Fourier Transform, then the final spectrum is averaged over these blocks to smooth the curve. In this paper, the reference pressure in the noise spectra is 2 × 10−5 Pa and all the narrow band spectra displayed have a frequency bin width of 8 Hz. 2.2. Continuous wavelet transform method In contrast to the traditional power spectral method, the continuous wavelet transform method is a joint time-frequency analysis method which can decompose a time series into time and frequency spaces simultaneously. The continuous wavelet transform can be defined as [23,26]: Wx (𝜏, a) =

∞

∫−∞

x(t )Ψ∗a,𝜏 (t ) dt

(3)

where Wx is the wavelet coefficient, x (t ) is the time series of experimental signal, Ψa,𝜏 (t) is the wavelet function, and the symbol

denotes the complex conjugate. Normally, the square modules of the wavelet coefficient ||Wx || are used to represent the energy density distribution [27]. The wavelet function is obtained by varying the wavelet scale a and the time delay 𝜏 of the mother wavelet function Ψ (t ) as: 2

∗

Ψa,𝜏 (t ) = a−1∕2 Ψ

(

t−𝜏 a

) (4)

It should be mentioned that the choice of the wavelet function plays an important role in analyzing signals and highly depends on the feature of analyzed signals. The Morlet function is a complex wavelet function which can return both amplitude and phase information, so that it is more appropriate to capture the oscillatory behaviors and analyze the acoustic signals [22,28]. Hence, the Morlet wavelet function is used in this paper and defined as:

Ψ (t ) = ei𝜔0 t e−t

2 ∕2

(5)

where the parameter 𝜔0 is the non-dimensional frequency and usually taken to be 6 to satisfy the admissibility condition [29]. Moreover, the wavelet scale a and the equivalent frequency of Fourier transform f have the relationship as:

f =

𝜔0 +

√

2 + 𝜔20

4𝜋 a

=

0.968 a

(6)

With this relationship, it is easy to make a direct comparison between wavelet spectrograms and noise spectra. 3. Multiple tones from landing gear model The far-field noise spectra measured at sideline location and flyover location of the landing gear model at four freestream velocities are displayed in Fig. 2. The landing gear noise is approximately broadband at flyover location, but some visible peaks are observed in the noise spectra at sideline location. Referring to the acoustic database provided by the LAGOON project [10,11], discrete tones can be clearly observed at sideline location, but are weak and less prominent at flyover location. This is mainly because the corresponding acoustic resonance modes cannot be effectively radiated in the flyover plane [15]. Since the characteristics of the tones at sideline location are approximately similar to the characteristics of the tones at flyover location, only the sideline noise signals are analyzed in this paper to clearly reveal the noise generation mechanisms and the temporal features of the multiple tonal phenomenon. As shown in Fig. 2(a), three mid-high frequency tones in the landing gear noise spectra at sideline location are labelled as T1 , T2 and T3 . These tonal frequencies are respectively f1 = 2120 Hz, f2 = 3040 Hz and f3 = 7328 Hz in all velocity cases. This important feature indicates that the tonal frequencies are independent of freestream velocities. Hence, the tonal frequencies can be normalized using the non-dimensional parameter Helmholtz number (He), which is based on the wheel diameter D and the sound speed c as He = 2𝜋 f × D∕c. The normalized noise spectra using the non-dimensional Strouhal number (St = f × D∕U) are plotted in the insets of Fig. 2(a), it is clear that the tonal frequencies cannot individually coincide in the four cases. These facts provide a strong evidence that the three tones satisfy the He scaling law instead of the St scaling law. Moreover, the He scaling law also implies that the dominant tonal noise generation mechanism is the acoustic resonance mechanism, which has been verified by Casalino et al. [15]. It should be mentioned that the tones T1 and T2 were also observed in the LAGOON project at around 1000 Hz and 1500 Hz, respectively. Considering the fact that the present landing gear model is 50% scaled from the

L. Li et al. / Journal of Sound and Vibration 464 (2020) 114980

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Fig. 2. Far-field noise spectra of the landing gear model at four velocities. (a) At sideline location. (b) At flyover location.

Fig. 3. Contour plot of the wavelet coefficients for the landing gear noise at sideline location. (a) In mid frequency range. (b) In high frequency range.

model of LAGOON project, the He values are the same in the two cases. This evidence again supports the conclusion mentioned above. From the time-averaged noise spectra shown in Fig. 2(a), it may give a false indication that the tones are always appeared with constant amplitude. However, the wavelet analysis results show quite different temporal features. The time-frequency spectrograms of the landing gear noise are shown in Fig. 3. It should be noted here that the region in the high frequency range is separately displayed in Fig. 3(b), since the magnitude of T3 is much lower than the broadband noise nearby and will be over-

whelmed. As shown in Fig. 3, the maximum regions of ||Wx || appear around three frequency regions whose central values are about 2100 Hz, 3000 Hz and 7300 Hz, respectively. These values are close to the peak frequencies of the three tones observed in the noise spectra of Fig. 2(a). Besides, it is clear that the ranges of these maximum regions are respectively 1800–2300 Hz, 2600–3600 Hz and 6800–8000 Hz, consistent with the bandwidths from the landing gear noise spectra. Moreover, by extracting 2 the time series of ||Wx || at each tonal frequency, the averaged values are equal to the magnitudes in the noise spectra at corresponding frequencies. All these facts confirm that the wavelet analysis method can capture the primary spectral features of the tonal noise generated from landing gear components. The intermittent excitation feature can be qualitatively determined through the wavelet spectrogram. The detail excitation 2 states of these mid-high frequency tones can be exhibited through a more quantitative way. The time series of ||Wx || at three 2

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L. Li et al. / Journal of Sound and Vibration 464 (2020) 114980

Fig. 4. Spectra of the wavelet coefficients at three tonal frequencies.

tonal frequencies are extracted for more than 9 s and then the auto-spectrum is made to verify whether the underlying periodicity exists. As shown in Fig. 4, no obvious peaks can be observed and the spectral curves are flat up to 600–1000 Hz, providing 2 a strong evidence that the variations of ||Wx || at the three tonal frequencies are irregular and random in nature. Thus, it is hard to predict the excitation time and the duration time of these tones. The relationships between these tones are very weak and the excitation processes of these tones are independent of one another. This feature can be easily understood from the viewpoint of tonal noise generation mechanism. As discussed by Casalino et al. [15], the resonance loops of these tones can be either floor-to-floor rim or edge-to-edge rim, and the acoustic resonance modes are the complex combination of azimuthal mode and transversal mode. Hence, the acoustic resonance of the three tones are different. Thus, it can be conjectured that the excitation states of the three tones are independent of one another. This can be verified through additional experimental results that these tones can be individually controlled by some partly covering treatments. Two covering treatments, namely first-half covering treatment and one-side covering treatment, are applied on the landing gear model to investigate the effect of partly covering treatments on these tones. As shown in Fig. 5, the first and second tones can be suppressed effectively by the first-half covering treatment, while the second and third tones can be suppressed by the one-side covering treatment. The insets of Fig. 5(a) and (b) show that these treatments can break some resonance loops with little impact on other loops, such that only the associated tones can be individually suppressed by these partly covering treatments as expected. All these facts confirm that the tones resulted from different resonance loops and modes are excited independent of one another. In summary, the three mid-high frequency tones generated from the landing gear model at sideline location are intermittently excited and the amplitudes are not constant during the excitation process. Besides, the excitation states of the three tones seem to be independent of each other, even though they have the same acoustic resonance generation mechanism. 4. Multiple tones from ring cavity model The near-field noise spectra measured at freestream velocities from 30 m/s to 55 m/s are displayed in Fig. 6. Two groups of multiple tones are evident in the noise spectra with different frequency scaling laws, implying that the noise generation mechanisms are also different. The first group of multiple tones contains two low frequency tones TL1 and TL2 . The tonal frequencies increase with the freestream velocity increasing, but the Strouhal numbers of TL1 and TL2 are constant in the six cases, respec-

Fig. 5. Effect of partly covering treatments on the tonal noise suppression. (a) First-half covering treatment. (b) One-side covering treatment.

L. Li et al. / Journal of Sound and Vibration 464 (2020) 114980

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Fig. 6. Near-field noise spectra of the ring cavity model at six velocities.

Fig. 7. Contour plot of the wavelet coefficients for the ring cavity low frequency noise.

tively about 0.55 and 1.1 as shown in the inset of Fig. 6. In other words, the low frequency tones satisfy the Strouhal number scaling law. It suggests that the low frequency tones are generated from the fluid-acoustic feedback loop phenomenon, just like the mechanism of rectangular cavities [30]. The second group of multiple tones includes more than ten high frequency tones. As shown in Fig. 6, all the tonal frequencies approximately remain unchanged in all cases, suggesting that the high frequency tones satisfy the Helmholtz number scaling law. Thus, it can be conjectured that these high frequency tones are generated from the acoustic resonance mechanism of the ring cavity, similar to the generation mechanism of the mid-high frequency tones generated from the landing gear model at sideline location [15]. 4.1. Multiple tones generated from fluid-acoustic feedback mechanism After distinguishing two groups of the ring cavity multiple tones from the scaling law of tonal frequency, the next task is to reveal the temporal behaviors. For simplicity, only the acoustic signal acquired at U = 55 m/s is discussed in the following 2 analysis. The features of the low frequency tones are firstly examined. The square modulus of the wavelet coefficients ||Wx || are plotted in Fig. 7. It can be seen that both tones are intermittently excited. More importantly, the appearance of TL1 generally corresponds to the absence of TL2 instead of being excited simultaneously, except for some small time intervals when both tones are absent or present. This implies that the primary feature of two tones is the mode switching mechanism that the dominant acoustic energy switches from one mode to another in time. The reason why the low-frequency tones satisfy the mode switching mechanism is briefly discussed here. The noise generation mechanism of the two low frequency tones is the fluid-acoustic feedback loop, which is similar to the noise generation mechanism of rectangular cavities. In the rectangular cavity flow, these Strouhal number scaling tones are usually related to different Rossiter modes and the mode number is reported to be highly dependent on the number of vertical structures spanning along the cavity [23,31]. One important feature is that the number of vortex inside the rectangular cavity changes with time, which has been clearly confirmed through the Schlieren flow visualization method [23] and numerical simulation [31]. Considering the similar noise generation mechanism, it is considered that the mode switching mechanism of the low frequency tones comes from the temporal variations of vortical structures inside the ring cavity. 4.2. Multiple tones generated from acoustic resonance mechanism The inherent temporal features of the high frequency tones are more complex. As plotted in Fig. 6, twelve tones which satisfy the Helmholtz number scaling law are observed and named as TH1 to TH12 hereinafter, respectively. The tonal frequencies as well as the frequency spacings are summarized in Table 1. One interesting fact is that only the frequency spacing between TH2 and

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L. Li et al. / Journal of Sound and Vibration 464 (2020) 114980 Table 1 Frequencies and frequency spacings of the ring cavity high frequency multiple tones.

i

fHi (Hz)

Δf = fHi+1 − fHi (Hz)

1 2 3 4 5 6 7 8 9 10 11 12

1184 1912 2584 3256 3864 4408 5104 5728 6144 6768 7320 7960

728 672 672 608 544 696 624 416 624 552 640

Fig. 8. Contour plot of the wavelet coefficients for the ring cavity high frequency noise.

TH3 is the same as the frequency spacing between TH3 and TH4 , namely Δf23 = Δf34 = 672 Hz, while the adjacent frequency spacings of the other tones are different from each other. It implies that the inherent temporal features and the relationships between TH2 to TH4 may be different from the other tones. The wavelet spectrogram of the high frequency tones generated from the ring cavity model is shown in Fig. 8. It should be noted that the acoustic energy is much lower in the high frequency range, thus the tones at high frequencies will be over2 whelmed by the background noise at low frequencies. In order to highlight the high frequency tones, the magnitude of ||Wx || is 2.2 multiplied by f . As clearly shown in Fig. 8, the tones can be observed and the central frequencies as well as the frequency bin widths agree well with the spectral results in Fig. 6. However, the regions around TH2 and TH3 have much weaker energy than the regions around the other tones, which is quite different from the spectral results in Fig. 9. In order to explain the reason, 2 the time series of ||Wx || extracted at fH1 , fH2 and fH5 are analyzed with the auto-spectrum method and the results are shown in Fig. 9. A visible peak with the central frequency of Δf = 672 Hz can be seen at fH2 , indicating that the magnitude of ||Wx || at fH2 has an underlying periodicity. In other words, the TH2 is amplitude modulated in time and the modulation frequency is 672 Hz. Referring to the previous literature [17], the tonal frequencies from the amplitude modulation mechanism can be predicted as: 2

Fig. 9. Spectra of the wavelet coefficients at fH1 , fH2 and fH5 .

L. Li et al. / Journal of Sound and Vibration 464 (2020) 114980

fn = fn,max ± kΔf

9

(7)

where fn, max is the frequency of primary tone, fn is the frequencies of secondary tones and k is the positive integer. By putting the values fH2 and Δf into Eq. (7), the frequencies of the third and the fourth tones can be accurately predicted with k = 1 and 2, respectively. These features give a strong evidence that among the second to the fourth tones, the tone TH2 is the primary tone with the dominant acoustic energy, and the other tones TH3 and TH4 are the secondary tones generated from the amplitude modulation mechanism of the primary tone. It should be noted that the symmetric left-side tones of TH2 , i.e., fLS1 = fH2 − Δf = 1240 Hz and fLS2 = fH2 − 2 × Δf = 568 Hz are not shown in the noise spectra. The absence of left-side tones may be due to the high background noise level and low tonal intensities in the low frequency range, similar to the study of single-element airfoil noise [20] and 30P30 N configuration noise [21,22]. In contrast to the spectrum of TH2 , the spectra of TH1 and TH5 are broadband without visible peaks. Thus, it can be concluded that all the other high frequency tones have no strong relationship and they are not excited by amplitude modulation mechanism. For other tones at higher frequencies than fH5 , the results are similar to TH1 and TH5 , such that they are omitted for brevity. 5. Conclusions The landing gear components can generate multiple tones, which may be resulted from the fluid-acoustic feedback loop or the acoustic resonance. The inherent temporal features may be independent of the tonal noise generation mechanisms. The continuous wavelet transform method can capture the frequency and temporal features simultaneously, thus it is introduced to reveal the temporal features and excitation rules of multiple tonal phenomena from a landing gear model and a ring cavity model. A two-wheel nose landing gear model is firstly analyzed with the traditional power spectral method and the continuous wavelet transform method. Three mid-high frequency tones generated from the acoustic resonance of inner-wheel cavity can be observed in the noise spectra at sideline location. The wavelet results show that they are randomly excited and the excitation states are independent of one another. Thus, with the application of the covering treatments, the mid-high frequency tones can be suppressed individually. Then the traditional power spectral method and the continuous wavelet transform method are applied to analyze the multiple tones generated from a ring cavity model. Two groups of multiple tones can be observed in the noise spectra and the tonal noise generation mechanism can be clearly revealed through the frequency scaling laws. Their temporal features are more complex and are slightly different from the landing gear multiple tones. The low frequency tones are generated from fluid-acoustic feedback loop and satisfy the mode switching mechanism, namely that the two tones are excited alternately and the primary acoustic energy switches from one to another in time. While the high frequency tones are generated from acoustic resonance, but there are two different inherent temporal features. For the second to the fourth 2 tones, the magnitudes of ||Wx || periodically modulate in time with the modulation frequency nearly the same as the frequency spacing among the second to the fourth tones, confirming that the three tones satisfy the amplitude modulation mechanism. That is, the primary acoustic energy concentrates on the second tone and the other two tones are generated from the strong 2 amplitude modulation process of the second tone. For other high frequency tones, the values of ||Wx || change irregularly in time, considering that they are randomly excited independent of one another. In summary, the continuous wavelet transform method can successfully reveal the temporal features and excitation rules of the multiple tones generated from the typical landing gear components. Acknowledgments This work was supported by the National Natural Science Foundation of China (11502012, 11850410440, 11772033 and 117221202) and the China Postdoctoral Science Foundation (2019M650417). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

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