Introduction

In order to ensure the lasting reliability of key structures, it is necessary to perform vibration reduction design in aerospace, ships and other major engineering fields. Metal vibration absorber has not only the inherent characteristic of the selected metal material, but also the elastic and damping characteristics similar to rubber. It can reduce the structural vibration through the dry friction of artificially wound metal wire components, and its shape can be customized according to different service environments. Therefore, it has been widely used to reduce the structural vibration under various complex environmental conditions [1,2,3,4] because of its designable stiffness, high and low temperature resistance, and ageing resistance etc. Research on the metal vibration absorbers originated in the United States in 1950s [5, 6]. After years of development, the manufacturing technique of metal vibration absorber has been relatively mature, which can be mainly divided into 6 steps [7]: wire material selection, wire winding, wire tension, blank weaving, cold pressing forming and post-treatment. The mainly control parameters involve wire material [8,9,10], wire diameter [11], cross-section shape [12], braiding method [13], forming pressure [14] etc. The change of each parameter will lead to the different performance of metal vibration absorber. Therefore, the dispersibility of metal vibration absorber is general larger than the traditional metal material. Zhang et al. [15]. conducted a CT scan on the internal microstructure of metal vibration absorbers. It can be seen that there are three typical contact states (separation, slip and adhesion) among different parts of the metal wire component [16, 17], and the vibration damping performance of the metal vibration absorber is mainly achieved through the slip state. Because metal vibration absorber is usually suffered alternating fatigue load in actual service environment, it is of great significance to study the fatigue performance of the metal vibration absorber. Jiang et al. [18, 19]. performed the fatigue experiments of metal vibration absorbers. The results indicate that the fatigue life will increase when the metal wire diameter, prestress and relative density of the metal vibration absorbers increase. Li [20] carried out sinusoidal constant-frequency fatigue tests of metal vibration absorbers. The change law between inherent frequency of vibration absorber and fatigue life is obtained. Then, the fatigue failure life of the metal vibration absorber is successfully determined according to the failure criterion of inherent frequency variation of 10%. However, this method requires extra resonance frequency test to obtain the real-time inherent frequency of metal vibration absorber, and the fatigue test must be interrupted several times. In addition, the fatigue failure life of the metal vibration absorber will be affected to some extent during the resonance frequency test. Cao et al. [21]. proposed a coupling damage factor of stiffness and damping as the failure criterion to determine the fatigue failure life of metal vibration absorber, but the effectiveness needs to be further verified. Therefore, although some researchers have carried out fatigue research on the metal vibration absorbers, the current research results are not mature and have not been recognized by everyone.

In this paper, the axial high-cycle fatigue experiment of the metal vibration absorber is conducted. Based on the fatigue displacement amplitude and residual preload displacement of the metal vibration absorber, a new fatigue failure criterion is put forward. Further, the fatigue failure life of the metal vibration absorber can be determined. In addition, a fatigue failure assessment diagram is successfully established on the basis of the displacement amplitude versus life curves and the residual preload displacement versus life curves of metal vibration absorbers, which can be used to assess the residual fatigue life of in-service metal vibration absorber. Finally, in order to research the microscopic fatigue failure behaviors, the microcosmic wear and fracture of metal wires in metal vibration absorbers are analyzed by the scanning electron microscopy (SEM) and three-dimensional tomography (µCT) devices.

Experimental Conditions

The configuration of the metal vibration absorber adopted in this study is shown in Fig. 1, which is suitable for the situation where axial fatigue load is mainly suffered in the service environment. The axial direction of vibration absorber is connected using a bolt with the diameter of 6 mm. The thickness of the mounting plate in the middle of the vibration absorber is 8 mm. The material of metal wire is 1Cr18Ni9Ti, and the diameter of the metal wire is 0.3 mm. Since the vibration absorption function of metal vibration absorber is mainly depended on the metal wire component at both ends, therefore, the metal wire component is the core part of the whole vibration absorber. The high-cycle fatigue test of vibration absorber is conducted by the high frequency electromagnetic fatigue testing machine with maximum load of 1 kN and maximum test frequency of 200 Hz. As the metal vibration absorber is a non-standard structure, in order to connect the vibration absorber to the testing machine, special transition connection fixture needs to be designed, as shown in Fig. 2. In order to reflect the fatigue loosening phenomenon that may occur in the actual use of the metal vibration absorber, the whole metal vibration absorber is adopted to carry out the fatigue test. The upper fatigue loading fixture adopts hollow design. It is easy to tighten the sample through hex bolts. The upper and lower fixtures are fastened on the test machine through the drill chuck of the testing machine.

Fig. 1
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Picture of the metal vibration absorber (a) entire state (b) split state

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Fig. 2
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Axial loading fixture of high-cycle fatigue test of the metal vibration absorber

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High Cycle Fatigue Test of the Metal Vibration Absorber

Analysis of the Fatigue Testing Result

The central metal hollow cylinder has a limiting action (see Fig. 1). There is an initial interval between the leftmost metal plate and the central metal hollow cylinder in the natural state. And the interval will close in the preload state. Therefore, the initial preload displacement can be determined according to the initial interval of metal vibration absorber in the natural state. Table 1 lists the testing results of the initial preload displacements of the metal vibration absorbers. It can be observed that the initial preload displacements of the metal vibration absorbers have some dispersity, ranging from 1.410 mm to 1.635 mm. Because the initial state of the metal vibration absorber is pre-lock state, the initial preload displacement is measured after the preload is removed. However, some metal vibration absorbers recover well, while the others may have a large permanent deformation when the preload is removed, which cause the dispersity of the initial preload displacement measurement of the metal vibration absorber.

Table 1 Initial preload displacement measurements of the metal vibration absorbers
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Figure 3 shows the relations of the fatigue displacement amplitude versus fatigue life of 16 metal vibration absorbers. The results indicate that the initial displacement amplitude of each vibration absorber has a rapid decline stage, and the minimum displacement amplitude will occur at about 0.1 million of fatigue life. Then the displacement amplitude increases slowly along with the fatigue life. Under the same load level, the displacement amplitudes of different metal vibration absorbers have a certain dispersion, and it can be observed that there is no obvious consistency between the preload displacement and the fatigue displacement amplitude. Because the larger of the displacement amplitude, the worse of the carrying capacity for metal vibration absorber, and the fatigue failure will occur more easily. Therefore, Figs. 4, 5, 6, 7 and 8 show the evolutions of load-displacement cyclic hysteresis curves of the samples with largest displacement amplitudes at various load levels. It can be observed that the load-displacement hysteresis curves show good shapes when the fatigue life of each vibration absorber is 0.05 million cycles, which indicate that the vibration damping performances are also good. When the maximum fatigue load is 100 N, the load-displacement hysteresis curve of metal vibration absorber changes a little after 10 million cycles of fatigue test. Under different fatigue load levels, the displacement amplitude of each metal vibration absorber does not exceed the initial preload displacement when the fatigue life reaches 10 million cycles. Further analysis shows that the load-displacement hysteresis curve of the metal vibration absorber also has some vibration damping performance when the fatigue load is less than 400 N and the life reaches 10 million cycles. However, the load-displacement hysteresis curve of the vibration absorber almost coincides near the zero load when the maximum fatigue load is 600 N and the fatigue life reaches 10 million cycles, which can be considered that the vibration damping performance has been lost. Consequently, it is unreasonable to adopt the initial preload displacement as the fatigue failure criterion of the metal vibration absorber.

Fig. 3
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Displacement amplitude versus fatigue life curves of the metal vibration absorbers under different fatigue load levels

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Fig. 4
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Compression load-displacement hysteresis curves of 3# specimen under different fatigue lifes

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Fig. 5
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Compression load-displacement hysteresis curves of 4# specimen under different fatigue lifes

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Fig. 6
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Compression load-displacement hysteresis curves of 7# specimen under different fatigue lifes

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Fig. 7
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Compression load-displacement hysteresis curves of 13# specimen under different fatigue lifes

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Fig. 8
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Compression load-displacement hysteresis curves of 15# specimen under different fatigue lifes

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The secant stiffness of the load-displacement hysteresis curve represents the bearing capacity of the metal vibration absorber. The larger the stiffness of vibration absorber is, the stronger the bearing capacity is. The analysis result shows that the maximum secant stiffness Smax of each metal vibration absorber occurs at the position of the smallest initial displacement amplitude, which can be calculated by Eq. (1)

$${S_{\max }} = \frac{{\Delta P}}{{\Delta {h_{\min }}}}$$
(1)

in which ΔP represents the peak and valley load amplitude, namely ΔP = Pmax-Pmin, Δhmin is the minimum displacement amplitude in the initial fatigue loading stage. The secant stiffness change rate λ10m after 10 million cycles of fatigue test can be calculated by Eq. (2).

$${\lambda _{10{\text{m}}}} = \frac{{100({S_{\max }} - {S_{10{\text{m}}}})}}{{{S_{\max }}}}$$
(2)

where S10m represents the secant stiffness of the metal vibration absorber after 10 million cycles of fatigue test. Table 2 lists the secant stiffness change rates of the 16 metal vibration absorbers. The results indicate that the stiffness change rates of the metal vibration absorbers after10 million cycles of fatigue tests have no obvious law under different fatigue load levels. Therefore, the fatigue failure life of metal vibration absorber cannot be judged by the secant stiffness change rate.

Since there is an inversely proportional relationship between the fatigue displacement amplitude of metal vibration absorber and the residual preload displacement, it is assumed that the displacement amplitude of metal vibration absorber has no preload displacement change before reaching the minimum initial displacement amplitude, and then the residual preload displacement of metal vibration absorber decreases with the increase of the displacement amplitude. Based on the initial preload displacement of metal vibration absorber and the remaining preload displacement after 10 million cycles of fatigue test, the evolution of the preload displacement can be estimated by Eq. (3)

$$\frac{{{\delta _{\text{i}}} - \delta }}{{{\delta _{\text{i}}} - {\delta _{\text{f}}}}} = \frac{{\nabla h}}{{\nabla {h_{\max }}}}$$
(3)

where δ is the real-time residual preload displacement under any fatigue life, and δi is the initial preload displacement of metal vibration absorber. δf is the residual preload displacement measured after 10 million cycles of fatigue test. ▽h represents the change of displacement amplitude under the current fatigue life compared with the initial minimum displacement amplitude Δhmin, namely ▽h = Δh-Δhmin. Analogously, ▽hmax represents the change of displacement amplitude after 10 million fatigue lives compared with Δhmin. Figure 9 shows the residual preload displacement versus fatigue life (δ-N) curve of each metal vibration absorber. When the fatigue displacement amplitude of metal vibration absorber exceeds the residual preload displacement, there will be a large fatigue gap between the vibration absorber and the damping structure during the fatigue process, which may cause fatigue loosening. The damping effect of the metal vibration absorber will also decrease significantly at the same time. Consequently, it can be taken as the fatigue failure criterion when the fatigue displacement amplitude of metal vibration absorber is equal to its residual preload displacement, which shows a clear physical significance. Further, the failure life of vibration absorber can be obtained after the test easily according to the fatigue failure criterion proposed in this paper. By comparing the displacement amplitude of each vibration absorber shown in Fig. 3 with the corresponding residual preload displacement shown in Fig. 9, it can be observed that only three samples with the maximum fatigue load of 600 N occur fatigue failure after 10 million cycles according to the failure criterion, and the critical fatigue lifes of 14#-16# metal vibration absorbers are 3.65 million, 3.73 million and 5.79 million, respectively. Figure 10 shows the load-displacement hysteresis curves corresponding to the critical failure lifes of the three metal vibration absorbers. It can be observed that the load-displacement hysteresis curves are relatively flat at this time, which represent the poor of energy absorption and vibration reduction effect. Therefore, it also confirms the validity of the residual preload displacement failure criterion. In conclusion, the residual preload displacement failure criterion is a macroscopical criterion, which is obviously superior to the initial preload displacement criterion or secant stiffness change rate criterion.

Fig. 9
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Residual preload displacement versus fatigue life curves of metal vibration absorbers

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Fig. 10
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Critical load-displacement hysteresis curves of metal vibration absorbers with maximum load of 600 N

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Since the increasing speed of the displacement amplitude decreases continuously during the process of the displacement amplitude of metal vibration absorber ranging from the initial minimum value to 10 million cycles of fatigue test (see Fig. 3). Similarly, the lowering speed of the residual preload displacement of metal vibration absorber also decreases continuously (see Fig. 9). Therefore, based on the variation ranges of displacement amplitude and preload displacement of each metal vibration absorber during the first 10 million cycles of fatigue test, it is assumed that the displacement amplitude and preload displacement of the vibration absorber will change the same results behind every 10 million cycles of fatigue test. Further, the fatigue failure life of each metal vibration absorber can be predicted when the displacement amplitude of the metal vibration absorber is equal to the residual preload displacement. Compared with the real fatigue life, this predicting life is a conservative result, which is more safety for engineering applications. The predicted fatigue lifes of metal vibration absorbers according to the residual preload displacement failure criterion are listed in Table 2. And Fig. 11 shows the corresponding P-N curve of metal vibration absorber based on the fatigue tests. It can be seen that the fatigue failure lifes of metal vibration absorbers predicted by the residual preload displacement failure criterion show a good law of power function, which can be described as.

$${P_{\max }} = {\text{7340847}}{\text{.04}}{N^{ - 0.6089}}$$
(4)
Table 2 Fatigue life testing results of the metal vibration absorbers
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Fig. 11
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P-N curve determined from the fatigue tests of metal vibration absorbers

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The average curves of the displacement amplitude-life relations and residual preload displacement-life relations of the metal vibration absorbers under the same load levels can be obtained on the basis of the Figs. 3 and 9, respectively. And the predicted displacement amplitude and residual preload displacement under longer life can be determined by assuming that the displacement amplitude and preload displacement of the vibration absorber will change the same result behind every 10 million cycles of fatigue test. With reference to the classical J-integral crack driving force diagram of EPRI manual [22], a similar fatigue failure assessment diagram can be established to assess the fatigue life of the metal vibration absorber, as shown in Fig. 12. It can be seen that the average displacement amplitude-life curves and residual preload displacement-life curves of metal vibration absorbers under different load levels show certain laws, but the average residual preload displacement-life curves of metal vibration absorbers under the fatigue load of 300 N and 400 N are very close due to their own dispersions. According to the failure criterion that the preload displacement is equal to the fatigue displacement amplitude, the failure life obtained when the fatigue load is 300 N is still significantly higher than that obtained when the fatigue load is 400 N. With the help of the fatigue failure assessment diagram, the residual fatigue life of the in-service metal vibration absorber can also be predicted. For example, if the load level and residual preload displacement of the in-service metal vibration absorber are assumed to be at point “A” in Fig. 12, and the corresponding in-service fatigue life is 2.1 × 107. The average failure life of the fresh metal vibration absorber under the current load level is at point “B” in Fig. 12, the fatigue life of which is 3.05 × 107. Therefore, the predicted residual fatigue life of the in-service metal vibration absorber is about 9.5 × 106. For other service load levels that have not been tested, the displacement amplitude-life curves and residual preload displacement-life curves can be determined by interpolation method through the existing test data, so as to evaluate the failure life.

Fig. 12
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Fatigue failure assessment diagram of the metal vibration absorber

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Fatigue Wear Analysis of Metal Vibration Absorber

Metal vibration absorber dissipate energy mainly through the friction of metal wire, which has obvious fatigue wear and slagging phenomenon. For some specific service environments, this slagging phenomenon may cause short circuit of electronic devices, which has certain safety risks. Because the phenomenon of metal wire slagging is difficult to be described quantitatively, some simple observations and analyses are carried out in this study. It is assumed that the metal wire component of metal vibration absorber close to the nut is called the upper end, and the metal wire component close to the lower fixture is called the lower end. The fatigue slagging phenomenon of metal vibration absorber after 10 million cycles is photographed in the test. Because the slags are mostly left inside the metal vibration absorber, manually shake off the residue of the metal vibration absorber on the white paper and take photos. Figure 13 shows the slagging photos of metal vibration absorbers under different fatigue load levels. It can be found that the friction slagging phenomenon of lower end of metal vibration absorber is more serious because of the higher constraint of lower end. When the fatigue load is less than 100 N, the slagging phenomenon can not be observed obviously after 10 million cycles of fatigue test. When the fatigue load is greater than 200 N, the obvious slagging phenomenon can be observed after 10 million cycles, and the larger the fatigue load is, the more obvious the slagging phenomenon is.

Fig. 13
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Pictures of the slagging phenomenon for metal vibration absorbers after 10 million cycles of fatigue tests

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The fatigue wear condition of metal vibration absorbers is analyzed by means of scanning electron microscopy. Figure 14 shows the microscopic wear condition of metal vibration absorber after 10 million cycles under 200 N fatigue load. It can be observed that there is obvious wear debris on the local metal wire, but the metal wire fracture phenomenon does not appear in the metal wire component. Figure 15 shows the microscopic wear condition of the metal vibration absorber after 10 million cycles under 300 N fatigue load. The wear condition is significantly higher than that under 200 N fatigue load, and a fractured metal wire is found in the lower end metal wire component. Figure 16 shows the microscopic wear condition of the metal vibration absorber after 10 million cycles under 600 N fatigue load. A large number of wear slag and fracture phenomena of metal wires are observed in the metal vibration absorber. On the whole, the edge of the metal wire component near the center metal sheet is more likely to suffer from fatigue wear. The wear conditions of different parts of the metal wire component show obvious differences, which indicate that the carrying capacity of metal vibration absorber under different zones are also different. Moreover, the metal vibration absorber still has a good bearing capacity when some of the metal wires have broken, which only has a certain influence on the damping performance of the metal vibration absorber. However, the failure ratio of HCF and wear is different to distinguish, which will be considered in the further research.

Fig. 14
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SEM observed result of metal wire component with Pmax fixing at 200 N and N at 10 million

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Fig. 15
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SEM observed result of metal wire component with Pmax fixing at 300 N and N at 10 million

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Fig. 16
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SEM observed result of metal wire component with Pmax fixing at 600 N and N at 10 million

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Three-Dimensional Tomography Analysis of Metal Vibration Absorber

Three-dimensional tomography (µCT) equipment is adopted to conduct the microscopic analysis of the metal vibration absorber. Figure 17 demonstrates the µCT picture of metal wire component in the new metal vibration absorber. It can be seen that the integrity of the metal wire component is good. The metal wire fracture phenomenon has not been found even changing the viewing angle, which represents that no metal wire fracture occurs in the process of integrated braid extrusion forming for metal wire component. In addition, it can also be observed that the deformation in the different parts of the metal wire after extrusion forming is unordered, resulting in a large dispersion of the performance of the vibration absorber. Metal wire volumes in different height regions of the two new metal wire components are analyzed by µCT equipment, as shown in Table 3. The results indicate that the volume contents of metal wire in different height regions of metal wire components show small deviations.

Fig. 17
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Three-dimensional tomography picture of the new metal wire component

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Table 3 Metal wire volumes of the metal vibration absorbers in different height zones
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Figures 18, 19, 20 show the three-dimensional scanning pictures of the metal vibration absorbers after 10 million cycles under different fatigue loads, respectively. The results indicate that no metal wire fracture phenomenon is found at the upper end of the metal wire components after 10 million cycles when the maximum fatigue load is 200 N and 300 N, but a small amount of metal wire fracture occurs on the top surface of the lower end metal wire component in contact with the central metal sheet, which also indicates that the metal wire wear at the lower end of the metal vibration absorber is more serious than that at the upper end. When the maximum fatigue load is 600 N, both the upper and lower ends of the metal wire components appear more metal wire fracture phenomena, and some metal wire fracture phenomena are also observed on the lateral of the metal wire component.

Fig. 18
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Three-dimensional tomography picture of the metal wire component with Pmax fixing at 200 N

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Fig. 19
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Three-dimensional tomography picture of the metal wire component with Pmax fixing at 300 N

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Fig. 20
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Three-dimensional tomography picture of the metal wire component with Pmax fixing at 600 N

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Conclusions

In this paper, the macro and micro approaches are adopted to research the fatigue properties of the metal vibration absorbers. Conclusions of the current study are given below:

  1. (1)

    Both the initial preload displacement and the secant stiffness change rate of the fatigue load-displacement hysteresis curve are not suitable for the fatigue failure criterion of the metal vibration absorber. A new criterion is proposed to judge the fatigue failure of metal vibration absorber when the residual preload displacement of the metal vibration absorber is equal to the fatigue displacement amplitude;

  2. (2)

    The residual preload displacement prediction formula of metal vibration absorber is established. Further, a conservative fatigue failure life prediction method is put forward for the metal vibration absorber without fatigue failure. The experimental results demonstrate that the P-N curve of the metal vibration absorbers predicted by the new method shows a good power function relation.

  3. (3)

    Based on the experimental displacement amplitude-life curves and the residual preload displacement-life curves of the metal vibration absorbers, a corresponding fatigue failure assessment diagram is drawn successfully. Then the residual fatigue failure life of in-service metal vibration absorber can be predicted according to the fatigue failure assessment diagram.

  4. (4)

    The microanalyses of the metal vibration absorbers are performed by means of the scanning electron microscopy and three-dimensional tomography equipment. The results indicate that the integrity of the metal vibration absorber is very good when it is produced, and no metal wire fracture phenomenon is observed. Moreover, the metal wire located in the upper edge of the lower end metal wire component near the central metal sheet is the most prone to occur fatigue wear and fracture phenomenon.

Consequently, this study put forward a novel fatigue failure criterion of metal vibration absorber. Further, a set of fatigue life prediction and evaluation method of the metal vibration absorber is successfully established, so as to realize the residual fatigue life prediction of the in-service metal vibration absorber. Moreover, the microscopic failure law of the metal vibration absorber under fatigue loading is also obtained.