1 Introduction

The vibration problems of aircraft pi** system have always been the focus of aircraft design [1]. Moreover, with the continuous development of modern aircraft pipeline systems toward large flow and high pressure, the problem of pipeline failure due to vibration is becoming increasingly prominent. According to the statistics made by the scholars from the United States, the failures of aircraft pi** systems, such as fuel, air, and hydraulic systems account for 50–60% of the aircraft component failures. The statistical data of Russian aircraft also shows that the failures of aircraft pipeline systems account for more than half of the aircraft failures [2, 3]. The main forms of failures in aircraft pi** systems include excessive vibration and vibration fatigue [4]. Excessive vibration is mainly caused by the resonance of the pipeline due to fluid or foundation excitation which directly causes large displacement of the joint surface between the pipeline fastener and the fixed part and results in surface friction or damage. The vibration fatigue of pipeline systems generally refers to the cumulative damage caused by the long-term operation of components, such as pipe bodies, pipe joints, and clamps in the working environment, and the crack growth and fracture after a certain number of cycles.

In general, there are two existing methods to control the vibration of the pipeline, that is, optimizing the shape of the pipeline and applying clamps to the pipeline [5]. The optimization of the shape of the conduit should be implemented mainly in the initial design stage, otherwise, it may consume a lot of resources to redesign and manufacture if the pipe is found to vibrate violently in actual use. The application of clamp is the most widely used method, but in the meantime, it will inevitably increase the weight of the pi** system. And more importantly, it is difficult to find a suitable position for the clamp in some narrow spaces. The recent development of dam** coating provides an option for engineering vibration reduction [6,7,8,9,10]. Koshy and Alva et al. [11, 12] explored the effect of functional fillers on dam** performance. Gao and Long [13] used a variety of inorganic materials as fillers, and found that when the material with layered structure is used, the dam** performance and sound insulation performance of water-based dam** coating can be significantly improved. Through the comparison of different fillers of the coating, including heavy calcium carbonate, mica powder, glass microbeads, silica, etc., Hu et al.[14] further found that the water-based dam** coating with mica powder as the filler has the best dam** performance. Zhang [15] used the free vibration attenuation method to evaluate the dam** performance of the material. Gao et al. [16,17,18] provided theoretical support for the design and vibration control of complex pipeline systems, and pointed out that the passive vibration reduction technology of pipelines still needs to be further researched. Shlykov et al. [19] found the most appropriate percentage of organic fiber, and that there is a correlation between moduli and bending strength and the increase of fiber fraction. Yu et al. [20] discussed the dam** efficiency of the coating system based on the Reuss model and Hashin-shtrickman equation. In addition, CHIBA et al. [21,22,23,24] proposed that sticking viscous dam** material onto the outside the pipeline can effectively suppress vibration.

In this paper, a superior dam** coating was proposed, of which the effectiveness and practicability were proved by simulation and experiment. The research results are of great application significance in reducing the vibration of aircraft pipelines. The novelty of this paper lies mainly in the fact that we obtained the optimal filler ratio and coating thickness by discussing the preparation of dam** coatings. In addition, we calculated the dam** ratio by using the half-power bandwidth method, and performed mutual verification based on the finite element simulation results, experimental results, etc., which are different from the previous efforts.

2 Experimental details

In this paper, 2A12 is selected as the substrate of coating with the dimension of 150 × 60 × 1 mm. The acrylic polyurethane emulsion, filler and functional additives, such as defoamer and leveling agent are mixed in turn by using the high-speed dispersing machine, as shown in Fig. 1. And the materials required for this experiment are listed in Table 1. At last, the dam** coating is sprayed onto the substrate with a spray gun, and cured in an oven. As an important part of dam** coating, the size and content of filler are important factors affecting the performance. In this paper, mica powder is selected as the main filler, with the sizes of 10 mesh, 40 mesh and 400 mesh, and the contents of 20%, 40%, and 60%, respectively. The coating thickness is set to less than 2 mm, because in practical engineering, the increase of coating thickness will lead to the increase of pipeline weight, resulting in reduction of economy.

Fig. 1
figure 1

The preparation process of dam** coating

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Table 1 Materials
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In the vibration experiment, the clam** length of sample is controlled at 44 mm. Besides, the sine excitation method is used to sweep the frequency, and the sweep frequency ranges from 0 to 80 Hz with the frequency interval of 1 Hz. Then, the basic excitation is applied by using the digital vibration control system, and the vibration acceleration of the sample is monitored by the acceleration sensor, so as to actively control the vibration. The above experimental principle is shown in Fig. 2.

Fig. 2
figure 2

The experimental principle

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Furthermore, in the simulation analysis, the finite element simulation model is established with the material parameters shown in Table 2. In addition, the material properties and acceleration sensor are both set in Hypermesh. After the model is completed, they are imported into Ansys.

Table 2 Simulation parameters
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The structure is unlikely to undergo plastic deformation under normal load, and it is even farther away from the plastic region in the test. Therefore, the linear elastic material constitutive model is used. During modeling, the substrate, dam** coating, and sensor are considered as a whole, ignoring the connecting surfaces between them. In the meantime, SOLID185 unit is used to divide the above whole, and the established finite element simulation model is shown in Fig. 3. In addition, the viscoelastic properties of materials are represented by dam** factor, which is related to the content and size of filler. The dam** factor introduced in the model is calculated from the experimental data in Sect. 3.2 by the half-power bandwidth method.

Fig. 3
figure 3

The finite element simulation model

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3 Results and discussion

3.1 Vibration experiment analysis

Based on the above vibration signal analysis system, firstly, the vibration acceleration of the uncoated sample at the resonance is obtained as 38.22436 g. Secondly, the coated samples prepared using mica powder as filler, with the size of 10 mesh, 40 mesh and 400 mesh, and the content of 20%, 40% and 60%, respectively, are subjected to vibration test. Figure 4a–i illustrates the frequency spectrum of the vibration acceleration which, at the resonance of the sample with dam** coating, is greatly reduced compared with that of the uncoated sample. The reason is that the addition of fillers increases the friction between polymer segments and fillers and that between fillers in the emulsion environment, thereby improving the dam** performance of the sample, and achieving great reduction of the vibration acceleration.

Fig. 4
figure 4

Spectrum comparison of vibration acceleration

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In addition, it can be found that the vibration acceleration response of the coating with 20% filler is better than that of the coating with 60% filler when using different sizes mica powder as filler. This may be because when the filler content is low, the mica powder can be well dispersed in the emulsion environment, increasing the energy consumption between the polymer and the filler. When the filler content is high, on the one hand, mica powder is difficult to disperse uniformly in the emulsion environment, on the other hand, the energy consumption between polymer and polymer is reduced because of the decrease of polymer proportion, therefore, the dam** properties of the coating decrease [15].

For the samples prepared with mica powder in the filler sizes of 10 mesh, 40 mesh and 400 mesh, respectively, the coating thickness and vibration acceleration are fitted, as shown in Fig. 5a–c. Seen from Fig. 5, when the size and content of mica powder are fixed, with the increase of coating thickness, the dam** performance of the sample first increases and then decreases.

Fig. 5
figure 5

The influence of the coating thickness on the vibration acceleration

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Furthermore, in Fig. 5a, when the coating thickness is low, the 60% content curve is between the 20 and 40% content curves, which differ from Fig. 5b, c. On the one hand, this is the result of the experiment, on the other hand, this result proves that the factors affecting the dam** performance of the coating include the content of polymer and filler, and the coating thickness, rather than a single factor.

In addition, Fig. 4 and Fig. 5 show that with the size of mica powder of 10 mesh, the content of 20%, and the coating thickness of 0.24 mm, the vibration reduction effect of the coating is the best, up to 63.23%. The reason is that the mica powder has a large particle size and a moderate content, which can better disperse in the emulsion, making the sliding of the lamella easier, converting more vibration energy into heat energy, and eventually achieving the superior vibration reduction effect [25].

3.2 Comparative analysis of experiment and simulation

The half-power bandwidth method is employed to analyze the spectrum obtained experimentally, and the dam** ratio is calculated. Besides, the principle is as follows, the severe vibration of the system occurs not only at the resonance frequency, but also within a frequency band around it. In general, the amplitude amplification factor \(\beta_{v}\) is decreased to \(1/\sqrt{2}\) of the peak value, and the interval corresponding to this frequency band is defined as the resonance region. To describe the intensity of resonance and the width of resonance region, the concept of system quality factor is introduced, as shown in Eq. (1):

$$Q^{def} = \frac{1}{2\zeta } = \left. {\beta_{d} } \right|_{\lambda = 1} = \left. {\beta_{v} } \right|_{\lambda = 1} = \left. {\beta_{a} } \right|_{\lambda = 1}$$
(1)

where \(Q^{def}\) represents the system quality factor, and \(\zeta\) stands for the dam** ratio.

Assuming that there are two endpoints, that is, A and B with the amplitude amplification factors of \(Q/\sqrt 2\) in the resonance region, their corresponding system power is exactly half of the corresponding power at the resonance frequency. Therefore, Point A and Point B are called half-power points. In addition, the square of the amplitude amplification factor at the half-power point can be expressed by using Eq. (2):

$$\beta_{v}^{2} = \frac{{\lambda_{2} }}{{(1 - \lambda_{2} ) + (2\zeta \lambda )^{2} }} = \frac{{Q^{2} }}{2} = \frac{1}{{8\zeta^{2} }}$$
(2)

where \(\beta_{v}\) refers to the amplitude amplification factor.

Integrating both sides of Eq. (2) to obtain Eq. (3) and Eq. (4):

$$\lambda_{A} = \sqrt {1 + \zeta^{2} } - \zeta$$
(3)
$$\lambda_{B} = \sqrt {1 + \zeta^{2} } + \zeta$$
(4)

The bandwidth of the resonance region, also known as the half-power bandwidth, can be obtained by Eq. (5):

$$\Delta \lambda^{def} = \lambda_{B} - \lambda_{A} = 2\zeta = \frac{1}{Q}$$
(5)

It can be seen from the above results that in the case that the dam** ratio is small, the quality factor is high, the resonance region is narrow, and the resonance peak is steep, vice versa. Then, we can get Eq. (6) from Eq. (5):

$$\zeta = \frac{\Delta \lambda }{2}$$
(6)

Draw a line in parallel with the frequency axis at 0.707 times of the formant. The straight line intersects the resonance curve at two points, and the abscissa values corresponding to these two points are recorded as \(f_{1}\) and \(f_{2}\), respectively.

Therefore, the dam** ratio calculated according to the half-power bandwidth method is shown in Eqs (6, 7):

$$\zeta = \frac{{f_{2} - f_{1} }}{2f} \times 100\% = \frac{\Delta f}{{2f}} \times 100\%$$
(7)

where \(f\) represents the actual resonance frequency, that is, the natural frequency.

Based on the half-power bandwidth method, at the particle size of mica powder of 400 mesh, the content is 20%, the coating thickness reaches 0.25 mm, 0.8 mm, 1 mm, 1.2 mm and 1.4 mm, and the corresponding dam** ratios are 0.03933, 0.03313, 0.03089, 0.02286 and 0.01989, respectively. We substitute the obtained dam** ratios into the finite element simulation model, and compare the results with the experimental data, as shown in Fig. 6.

Fig. 6
figure 6

Comparison between experimental data and simulation results

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As can be seen from Fig. 6, when the size and content of mica powder are fixed, the spectrum obtained by the simulation calculation is in perfect agreement with the experimental data, and the vibration amplitude errors are only 0.63%, 0.18%, 2.24%, 9.06%, and 3.99%, respectively. In a word, the above results prove the correctness of the experiment and the applicability of the half-power bandwidth method. In addition, these similar contents of the finite element methods and structural vibration are also mentioned in some recent additional pioneer works [26, 27].

4 Conclusions

In this paper, a superior dam** coating was prepared, and its dam** properties were discussed. During the research, firstly, the dam** coating was prepared with acrylic polyurethane emulsion, filler and functional additives. Secondly, the effects of filler content, size and coating thickness on dam** performance were analyzed, and the optimal formulation was obtained. Moreover, the dam** ratio was calculated by using the half-power bandwidth method, and substituted into the finite element simulation model. In addition, the simulation results were compared with the experimental data. The specific conclusions of the entire paper were detailed as follows:

  1. (1)

    The sample with dam** coating has an obvious vibration dam** effect compared with the uncoated sample. And when the size and content of mica powder are fixed, with the increase of the thickness of the dam** coating, the dam** performance of the sample will gradually reach the optimum.

  2. (2)

    At the size of mica powder of 10 mesh, the content of 20%, and the coating thickness of 0.24 mm, the vibration reduction effect of the coating is the best, i.e., up to 63.23%.

  3. (3)

    The error between the experimental data and the simulated data does not exceed 9.06%, proving the correctness of the experiment and the applicability of the half-power bandwidth method.

  4. (4)

    The superior dam** coating can effectively reduce the vibration of the pipeline system, which has significant theoretical significance and engineering practical value for effectively controlling the failure of the aircraft pipeline system and improving its safety and reliability.