# Study on Damping Energy Dissipation Characteristics of Cylindrical Metal Rubber in Nonforming Direction

Authors: Zhi Ying Ren; Qisheng Chen; Hongbai Bai; Yiwan Wu
Journal: Advances in Materials Science and Engineering
 (2018)

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## Abstract

Metal rubber (MR) is an excellent damping material that is often used in practical engineering applications. The performance of MR in the nonforming direction is different from that in the molding direction, and research is rare; the anisotropy of MR is gradually revealed in many engineering applications because characterization and analysis of damping energy dissipation characteristics in the nonforming direction of cylindrical MR were carried out by the characterization method of damping materials. Based on the testing methods, dry friction and dynamic stiffness are taken as the main analysis points. By performing single-factor control and orthogonal tests, the parameters affecting the energy dissipation and damping characteristics of MR are studied, and the main factors that cause changes in energy consumption and loss factor are analyzed. It is found that the influence of loading amplitude on damping energy is much higher than that of loading frequency, and the damping characteristic of MR is not sensitive to frequency. Therefore, the loading amplitude level of the actual environment should be considered first followed by the knitting technology in design of MR components.


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## Body

## 1. Introduction

“Metal rubber” (MR) possesses the same macromolecular structure and elasticity as rubber, although it is a metal product [1–3]. On the microcosmic point of view, the viscoelastic damping characteristics of the macroscopic MR are observed through the elastic deformation force of the spiral metal wire and the contact sliding interaction force between the spiral metal wires. Therefore, it has great advantages of high bearing capacity, fatigue resistance, no aging phenomenon, high temperature resistance, large temperature difference, radiation resistance, and corrosion environment [4, 5]. In special conditions, it shows obvious advantages such as improved life, reliability and performance of aerospace, naval equipment, and sophisticated military products [6–8]. Figure 1 shows the sample and the microstructure of the MR under electron microscope.Figure 1
The sample (a) and microstructure (b) of the MR under electron microscope.
(a)
(b)Therefore, MR has been studied intensively and attentively worldwide. At present, most of the studies are based on the mechanical properties and damping characteristics of the MR in the forming direction with few analyses in the nonforming direction. Taking the analysis of damping characteristics of the MR as an example, Hao et al. [9] carried out the dynamic tests on the biaxial compression MR isolator and concluded that the damping ratio of noncircular section wire MR is larger than that of ordinary MR. Li and Huang [10] changed the test weight and the height of the MR. The loss factor is obtained by the analysis and calculation of the MR isolator, which provides a certain basis for the dynamic response analysis of the MR. Wang et al. [11] analyzed the influence of acceleration on material damping characteristics by applying different acceleration loads to the MR shock absorbers. In terms of mechanical properties, many mechanical models of the forming direction, such as the pyramid model [12], laminated cantilever beam model [13], small curved beam model [14], microelement spring model [15], and porous material model [16], have been put forward.However, the working direction of many MR components used in the engineering application is their nonforming direction, such as the MR ring in the pipeline system [17] and MR block in the multidirectional vibration isolation [18]. In addition, similar applications exist in some similar damping material applications. Kwon et al. proposed a novel pseudoelastic gear that utilized the ring-type SMA mesh washer as a spring blade to bridge the output shaft of the motor with an existing gear wheel [19], and Oh et al. proposed a whole antenna isolation system employing a ring-type SMA mesh washer that supports the gimbal-type antenna [20].Besides, our group previously found that the elastic damping characteristic of a nonforming direction MR is obviously different from that of the forming direction MR [21]. Figure 2 represents the hysteresis recovery force curve of MR at different directions. It can be seen that the average stiffness of the nonforming direction is obviously greater than that of the forming direction, which indicates that MR is an anisotropic material.Figure 2
Hysteresis loop of MR in the forming/nonforming direction.Therefore, it is necessary to further study the damping energy dissipation characteristics of an MR in the nonforming direction. However, works on the performance of the nonforming direction (nonbearing direction) are not comprehensive enough. Rua et al. [22] calculated the basic parameters of an MR material using experimental data and then simulated its performance using the finite element method. The results showed that the simulation method has a high solution precision and is suitable for the static mechanical calculation. Wu et al. [23] compared the radial force on the MR parameters with some of the MR parameters such as weight, density, and temperature. The mechanical model of MR is established, which can comprehensibly describe the load-deformation relationship of MR.However, these studies are based on statics, and there is almost no study on the dynamic damping energy dissipation characteristics of MR in the nonforming direction. Therefore, in this paper, the damping characteristic of a cylindrical MR in the nonforming direction is characterized. The engineering significance of different parameters of the studied MR, such as the knitting mode, the density, the test amplitude, and the test frequency, is also emphasized.


## 2. Compression Characteristics and Deformation Mechanism of MR

The studied MR is made of a stainless steel wire that is only 0.1–0.4 mm in diameter produced through the process of wire selection, wires winding, stretching, blank weaving, molding, cleaning, and so on. Figure3 displays a roughcast of the MR wrapped around and the molding diagram of the MR. The loading and unloading tests are carried out; thus, the hysteresis recovery force curve of the MR in different directions is obtained, as shown in Figure 4.Figure 3
A roughcast of MR wrapped around (a) and the molding diagram of the MR (b).
(a)
(b)Figure 4
Different forming directions of MR.This property of MR is related to the spatial distribution and orientation of internal metal spiral coils of MR. After the MR is formed, the relative position between the spiral rolls is not completely random, and the plane of most spiral rolls is parallel to the normal plane in the direction of the pressure formed. When the MR bears load, the spiral coil is extruded and deformed, with increasing deformation, the contact between the spiral rolls will slip, and the sliding contact points are restricted by the change in the geometric position of the spiral coil and the restriction of the other spiral rolls. The spiral rolls are divided into multiple sections of curved beams at the contact point. These curved beams are interlaced and interlocked together. The elastic damping property of MR is mainly formed by the elastic deformation force of curved beams and the interaction between curved beams.In the nonforming direction, the deformation of the curved beam is not the compression deformation in the forming direction but the deformation of the spiral wire itself. Because the stiffness of the curved beam in the nonforming direction is generally greater than that in the forming direction, the average stiffness of the MR in the nonforming direction is greater than that in the forming direction.At present, more and more nonforming directions of the MR have been applied in the field of engineering. However, researchers mainly studied the mechanical properties of MR in the forming direction, while a little has been done on the mechanical properties in the nonforming direction. Therefore, in this study, the relationship between the parameters of the MR that affect the nonforming direction properties is obtained, analyzing the stiffness parameters. Finally, the structural design of MR which guides the nonforming direction is realized, and the energy dissipation performance in the nonforming direction of MR has also been improved.


## 3. Damping Parameters and Testing Method of the MR Material in Nonforming Direction

### 3.1. Damping Parameters Characterization

Damping materials are designed to reduce the impact of large vibration on operation accuracy and stability. It mainly decreases the magnitude of vibration by transforming the vibration energy of external excitation into other forms of energy. There are many parameters to characterize the damping properties of materials. The most commonly used measurement parameters are the damping ratio, the phase difference angle tangent, the loss factor, the logarithmic attenuation rate, and the reciprocal of the quality factor. Due to the nonlinear characteristics of the material, some testing methods based on the linear principle cannot be used; thus, the loss factor is used as the damping characterization of the MR material in this study [24].


### 3.2. Calculation Method of Damping Test Parameters for the MR Material

In order to measure the loss factor, the sinusoidal excitation method is used to collect and calculate the damping material of the MR. The sinusoidal excitation method is directly used to calculate the energy dissipation and maximum elastic potential energy in a period using an electrohydraulic servo material tester. Then, the MR element restoring force response signal is measured applying the sinusoidal displacement load to the MR material. In this experiment, an SDS-200 high- and low-temperature dynamic and static material testing machine (Figure5) was used to load the MR under sinusoidal force, and thereafter, the response was collected. The maximum working load of the tester was 200 kN, and the test frequency was 0.01–40 Hz.Figure 5
Dynamic (a) and static (b) material testing machine used in the present study.The sinusoidal alternating displacement in the following equation is applied to the tested parts using the servo-controlled material testing machine:(1)X=X0cosωt+α,where α is the initial phase, X0 is the amplitude of the test input, and ω is the loading cycle.The energy consumed by an MR specimen in one cycle isΔW, that is, the area surrounded by a hysteresis loop:(2)ΔW=∮Fdx,ΔW=∮FdX0cosωt+α,ΔW=−ωX0∫0TFsinωt+αt.According to the experimental conditions, the integral formula in Equation (2) is discretized and then the new formula is obtained [25]:(3)ΔW=−2πX0N∑i=1NFisin2πiN+α,N=f0f,where N is the number of discrete points in a cycle; f0 is the sampling frequency for the system, f0 = 2500 Hz; and f is the loading frequency.In order to calculate the maximum elastic potential energy storage of materials, the dynamic average stiffnessK is defined using the following equation:(4)K=Fmax−Fmin2X0.Therefore, the maximum elastic potential energy stored in the materialW would be(5)W=12K¯X02=Fmax−Fmin2X0,where Fmax and Fmin are the maximum and minimum values of restoring force collected in the sampling system, respectively.The loss factorη is the ratio between the loss energy and the maximum elastic energy in the system, and the loss factor can be calculated using Equations (3) and (5). In general, the larger the loss factor is, the better the damping performance is:(6)η=ΔW2πW=−4ff0Fmax−Fmin∑i=1NFisin2πiN+α.


## 3.1. Damping Parameters Characterization

Damping materials are designed to reduce the impact of large vibration on operation accuracy and stability. It mainly decreases the magnitude of vibration by transforming the vibration energy of external excitation into other forms of energy. There are many parameters to characterize the damping properties of materials. The most commonly used measurement parameters are the damping ratio, the phase difference angle tangent, the loss factor, the logarithmic attenuation rate, and the reciprocal of the quality factor. Due to the nonlinear characteristics of the material, some testing methods based on the linear principle cannot be used; thus, the loss factor is used as the damping characterization of the MR material in this study [24].


## 3.2. Calculation Method of Damping Test Parameters for the MR Material

In order to measure the loss factor, the sinusoidal excitation method is used to collect and calculate the damping material of the MR. The sinusoidal excitation method is directly used to calculate the energy dissipation and maximum elastic potential energy in a period using an electrohydraulic servo material tester. Then, the MR element restoring force response signal is measured applying the sinusoidal displacement load to the MR material. In this experiment, an SDS-200 high- and low-temperature dynamic and static material testing machine (Figure5) was used to load the MR under sinusoidal force, and thereafter, the response was collected. The maximum working load of the tester was 200 kN, and the test frequency was 0.01–40 Hz.Figure 5
Dynamic (a) and static (b) material testing machine used in the present study.The sinusoidal alternating displacement in the following equation is applied to the tested parts using the servo-controlled material testing machine:(1)X=X0cosωt+α,where α is the initial phase, X0 is the amplitude of the test input, and ω is the loading cycle.The energy consumed by an MR specimen in one cycle isΔW, that is, the area surrounded by a hysteresis loop:(2)ΔW=∮Fdx,ΔW=∮FdX0cosωt+α,ΔW=−ωX0∫0TFsinωt+αt.According to the experimental conditions, the integral formula in Equation (2) is discretized and then the new formula is obtained [25]:(3)ΔW=−2πX0N∑i=1NFisin2πiN+α,N=f0f,where N is the number of discrete points in a cycle; f0 is the sampling frequency for the system, f0 = 2500 Hz; and f is the loading frequency.In order to calculate the maximum elastic potential energy storage of materials, the dynamic average stiffnessK is defined using the following equation:(4)K=Fmax−Fmin2X0.Therefore, the maximum elastic potential energy stored in the materialW would be(5)W=12K¯X02=Fmax−Fmin2X0,where Fmax and Fmin are the maximum and minimum values of restoring force collected in the sampling system, respectively.The loss factorη is the ratio between the loss energy and the maximum elastic energy in the system, and the loss factor can be calculated using Equations (3) and (5). In general, the larger the loss factor is, the better the damping performance is:(6)η=ΔW2πW=−4ff0Fmax−Fmin∑i=1NFisin2πiN+α.


## 4. Experimental Design and Result Analysis

### 4.1. Loading Pretest

Plastic deformation may occur when the metal material is subjected to excessive load [26], and the memory alloy is very useful to overcome the plastic deformation problem of metal materials [27]. Therefore, in this paper, the scope of loading capacity of MR is obtained by the pretest method, and the force loading test of MR with different densities is carried out. The experimental results are shown in Figure 6. From the graph, the force loading curves of MR with different densities are almost linear; that is, the MR materials satisfy the generalized Hooke’s law. It is indicated that MR exhibits linear elasticity within a certain load range. At the load level of 1 kN, the load-displacement curve of MR does not change; that is, there is no plastic deformation [28]. Therefore, we can ensure that the plastic deformation does not become a variable affecting the test accuracy through this preliminary experiment.Figure 6
Force loading tests of MR with different densities.


### 4.2. Experimental Design

The tooling is sandwiched between upper and lower heads of the material test machine, the cylindrical MR specimen is put into the clamp, and the clamp is trimmed on it, so as to prevent the loading process from breaking out of the tooling. The pretightening amount is represented by the displacement of the beam,Xa:(7)Xa=2X0,where X0 is the amplitude. The position after pretightening is assumed as the equilibrium position of sinusoidal displacement loading, and sinusoidal excitation is applied to the specimen (displacement control). The force-displacement signal is sampled by means of a data acquisition device, at the frequency f0 of 2500 Hz.In order to investigate the influence of different parameters on the damping property of MR and the degree of influence of various parameters, the experiment is divided into two parts. The first part comprised a single-factor controlled trial. The effect of single factor on the damping characteristic of the nonformed direction MR is compared to the results of the test. The second part comprised the orthogonal test using various factors, and the degree of interaction between the parameters was obtained by range analysis.


### 4.3. Influence of Single Factor on Damping Characteristics of MR in Nonforming Direction

The characterization parameters of MR specimens are shown in Table1. The size of the specimen is D30 × d10 × h30 ± 1 mm. As shown in Figure 7, the density of MR specimens is 2.5 and 3.5 g·cm−3.Table 1
Single-factor control test of MR test parameters.
Number
Frequency (Hz)
Density (g·cm−3)
Amplitude (mm)
d (mm)
Methods
1
1, 2, 3, and 4
2.5 and 3.5
0.5
0.15
Twining
2
1
2.5, 3.0, 3.2, and 3.5
0.5
0.15
Twining
3
1
2.5
0.5, 0.8, 1.0, and 1.2
0.15
Twining
4
1
2.5
0.5
0.15 and 0.3
Netting and twiningFigure 7
MR specimen with different densities. (a)ρMR = 2.5 g·cm−3. (b) ρMR = 3.5 g·cm−3.
(a)
(b)


### 4.3.1. Influence of Different Loading Frequencies on Energy Dissipation Characteristics

Figures8 and 9 are the dynamic hysteresis loops in the nonforming direction, respectively, which are realized by different densities of MR under different loading frequencies.Figure 8
Dynamic hysteresis loop of MR with the density of 2.5 g·cm−3 under different loading frequencies.Figure 9
Dynamic hysteresis loop of MR with the density of 3.5 g·cm−3 under different loading frequencies.From the diagram, it is evident that the hysteresis loop shape of the MRs obtained after the sine excitation loading test shows a sickle type. Due to the nonlinear characteristics of MR materials, the pores between metal wires are filled during loading, and the friction between wires begins to rub. Both dynamic average stiffness and energy consumption increase gradually. In addition, the left half area of the hysteresis loop in the diagram is larger than the right half area of they = 0 axis, which means that the energy consumption of the compressed part is greater than that of the restorer in the loading process (i.e., compression process).The hysteresis loop diagram shows that the hysteresis loop of MR formed by different loading frequencies is mostly in the state of reclosing. It indicates that the energy dissipation of MR components at different frequencies and in the same amplitude is approximately the same. It is shown that the damping performance of MR components is not significantly affected by different excitation; in other words, the damper prepared by MR has good stable damping performance. It can be found that the restoring force of the lower density specimen decreases with increasing loading frequency, and the hysteresis loop has a relatively large fluctuation (Figures8 and 9). When the density is higher, the restoring force increases with the increase of loading frequency, and the hysteresis loop is found to be smooth relatively. The reason for this phenomenon is as follows: when the density is relatively small, the gap between the metal wires is relatively large, and the slip distance between the wires is larger. Then, the dry friction between the wires is changed from static friction to dynamic friction, which will form a big jump in the curve.The actual change in the material is difficult to obtain from the hysteresis loop, so the loss factor, energy consumption, and dynamic average stiffness calculated by the MATLAB program are listed in Table2. And the change law is shown in Figure 10.Table 2
Loss factor, energy dissipation, and dynamic average stiffness variation.
f (Hz)
1
2
3
4
η
ρ
MR
2.5
0.3461
0.3311
0.3224
0.3171
Δ
W
ρ
MR
2.5
0.0837
0.0817
0.0808
0.0812
K
ρ
M
R
2.5
0.32
0.32
0.31
0.29
η
ρ
MR
3.5
0.2693
0.2670
0.2640
0.2629
Δ
W
ρ
MR
3.5
0.2251
0.2402
0.2521
0.2583
K
ρ
MR
3.5
1.12
1.16
1.17
1.21Figure 10
The loss factor, energy dissipation, and dynamic average stiffness variation at different frequencies. (a) MR with the density of 2.5 g·cm−3. (b) MR with the density of 3.5 g·cm−3. η is the dissipation factor.
(a)
(b)Comparing the changes in the parameters of MR components with two different densities under the sinusoidal excitation, it is found that the low-density MR has larger porosity, and the pores are filled first under the condition of low loading amplitude. However, the energy consumption changes little with density as the energy consumption is the elastic energy of the metal wire bending which does not change with density. In addition, the wire did not return to the original space pore in the process of returning, which will lead to a decrease in dynamic average stiffness and maximum elastic potential energy and a small increase in loss factor. While the high-density MR itself has a small pore space, the friction energy between the wires is completed during the loading process. Under the continuous loading, the friction is more intense. Therefore, both energy consumption and the dynamic average stiffness show a rising trend. And the loss factor has a very small downward trend.


### 4.3.2. Influence of Structure Density on Energy Dissipation Characteristics

In order to find the effect of density on energy consumption of materials, four groups of specimens with a density of 2.5, 3, 3.2, and 3.5 g·cm−3 were designed. As shown in Figure 11, the hysteresis characteristics of the MR with different densities are different, and the graphic area increases by increasing the density of the specimen. Therefore, with higher density, the stronger energy dissipation capacity of the MR is obtained. However, the change in loss factor shows an opposite character because the loss factor is affected by maximum elastic energy storage, and the maximum energy storage is related to the stiffness (Figure 12). The maximum energy storage increases with stiffness. The dynamic stiffness of MR increases with the increase of density, while the decrease rate of loss factor increases with the increase of stiffness. Therefore, the energy dissipation characteristics of MR are highly correlated with materials density; i.e., MR elements with low density possess better damping characteristics.Figure 11
Hysteresis curve of MR with different densities.Figure 12
Loss factor and dynamic average stiffnessK with changing density. η is the dissipation factor.


### 4.3.3. Influence of Different Amplitudes on Energy Dissipation Characteristics

The influence of different amplitudes on the energy dissipation characteristics of MR was examined by maintaining the pretightening amount, where the maximum amplitude of 1.2 mm is selected for this study. The same sine excitation origin for all four sets of data is ensured so that the test comparison becomes more accurate. The hysteresis loop of MR in very forming direction is shown in Figure13. The change in loss factor, energy consumption, and average dynamic stiffness under the effect of different amplitudes is shown in Table 3, and the trend of change is shown in Figure 14.Figure 13
Hysteresis loops of MR under different amplitude conditions.Table 3
Loss factor, energy dissipation, and dynamic average stiffness variation.
Amplitude (mm)
0.5
0.8
1.0
1.2
η
0.2881
0.2832
0.2658
0.2449
Δ
W
0.1238
0.2146
0.3033
0.3700
K
0.52
0.44
0.37
0.33Figure 14
The variation curves of each parameter under different amplitudes.η is the dissipation factor.From the graph, it is evident that the area surrounded by hysteresis loops increases with the increasing amplitude, meaning that the energy loss increases. In addition, with the increase of amplitude, the value of the maximum elastic potential energy increases. Simultaneously, a downward trend in loss factor and stiffness is found (Figure14).This phenomenon is because when the amplitude increases at a certain frequency, the compression degree of the MR increases and the number of contact points of the wire increases. However, in the course of returning, the wires are still in contact with each other because of the strong elastic properties of the wire so that the return speed is slower than that of the external load. The energy dissipation effect has a downward trend.


### 4.3.4. Influence of Different Braiding Process Parameters on Energy Dissipation Characteristics

Figure15 shows three kinds of specimens with different knitting processes to study the influence of different knitting processes on the energy dissipation characteristics of MR. Then, the sinusoidal excitation of MR with different process parameters was carried out to obtain the hysteresis loop (Figure 16), and the variation of the parameters on the hysteresis loop is also investigated (Figure 17). The summary of loss factor, energy dissipation, and dynamic stiffness is listed in Table 4.Figure 15
MR with different weaving process parameters.Figure 16
Hysteresis loop of MR under different processes.Figure 17
Parameter variation curves under different processes.η is the dissipation factor.Table 4
Loss factor, energy dissipation, and average dynamic stiffness.
Serial number
1
2
3
Methods
d = 0.15 and twining
d = 0.30 and twining
d = 0.30 and netting
η
0.3461
0.2801
0.2732
Δ
W
0.1647
0.1764
0.2001
K
0.32
0.46
0.56A progressive relationship between the energy dissipation capacities of MR under the three process parameters, which is due to the increase of stiffness, is visible (Table4). Therefore, according to the knitting process, with a larger wire diameter, the wire stiffness is higher that causes higher energy consumption. The netting MR has more contacts in the mesoscopic structure than the twisted MR, and the structure is more stable. The dry friction energy consumption of the netting MR sample is much earlier than that of the twisted MR sample, so that the rigidity is greater and thus the energy consumption is greater. From the abovementioned analysis, it is obvious that loss factor of the braided MR is more affected by the stiffness, giving a decreasing trend.


### 4.4. Orthogonal Test and Result Analysis of Various Parameters

The above experiment is based on control variables and explains the influence of various factors on the damping energy consumption of MR from the viewpoint of dry friction. In order to investigate the sensitivity of various factors to damping energy consumption, an orthogonal test was designed. According to the influence of the damping property parameters of MR, four test factors such as density (A), amplitude (B), test frequency (C), and molding process (D) have been selected. Each factor has 3 levels. Specific factors and factors level selected are shown in Table5. The loss factor is used as the experimental index. Therefore, we chose orthogonal arrays to carry out 9 orthogonal experiments. The specific factors are shown in Table 6.Table 5
Orthogonal test factors and their level.
Factor
Symbol
Levels
1
2
3
Density
A
2.5
3.0
3.5
Amplitude
B
0.5
0.8
1.0
Frequency
C
1
3
5
Condition of molding
D
0.15 forming of twining
0.3 forming of twining
0.3 forming of nettingTable 6
Orthogonal test results of the loss factor index.
Density
Amplitude
Frequency
Condition of molding
Loss factor
A
B
C
D
1
1
1
1
1
0.3461
2
1
2
2
2
0.2979
3
1
3
3
3
0.2366
4
2
1
2
3
0.2669
5
2
2
3
1
0.2795
6
2
3
1
2
0.1999
7
3
1
3
2
0.2843
8
3
2
1
3
0.2476
9
3
3
2
1
0.2670
K1
0.8806
0.8973
0.7936
0.8926
B > D > A >C
K2
0.7463
0.8250
0.8318
0.7821
K3
0.7989
0.7035
0.8004
0.7511
Extreme difference
0.1343
0.1938
0.0382
0.1415
Rank
3
1
4
2As shown in Table6, under different parameter levels, by increasing the loss factor, better damping performance is obtained. By comparing the extreme values at the level of each parameter, primary and secondary relationships of different factors (B > D > A >C) can be obtained, which means amplitude > forming condition > density > frequency.It can be seen that the effect of amplitude on the loss factor is most obvious, while the frequency is less affected. The difference between density and molding conditions is close, both of which are design parameters, and their effect on stiffness is obvious.


## 4.1. Loading Pretest

Plastic deformation may occur when the metal material is subjected to excessive load [26], and the memory alloy is very useful to overcome the plastic deformation problem of metal materials [27]. Therefore, in this paper, the scope of loading capacity of MR is obtained by the pretest method, and the force loading test of MR with different densities is carried out. The experimental results are shown in Figure 6. From the graph, the force loading curves of MR with different densities are almost linear; that is, the MR materials satisfy the generalized Hooke’s law. It is indicated that MR exhibits linear elasticity within a certain load range. At the load level of 1 kN, the load-displacement curve of MR does not change; that is, there is no plastic deformation [28]. Therefore, we can ensure that the plastic deformation does not become a variable affecting the test accuracy through this preliminary experiment.Figure 6
Force loading tests of MR with different densities.


## 4.2. Experimental Design

The tooling is sandwiched between upper and lower heads of the material test machine, the cylindrical MR specimen is put into the clamp, and the clamp is trimmed on it, so as to prevent the loading process from breaking out of the tooling. The pretightening amount is represented by the displacement of the beam,Xa:(7)Xa=2X0,where X0 is the amplitude. The position after pretightening is assumed as the equilibrium position of sinusoidal displacement loading, and sinusoidal excitation is applied to the specimen (displacement control). The force-displacement signal is sampled by means of a data acquisition device, at the frequency f0 of 2500 Hz.In order to investigate the influence of different parameters on the damping property of MR and the degree of influence of various parameters, the experiment is divided into two parts. The first part comprised a single-factor controlled trial. The effect of single factor on the damping characteristic of the nonformed direction MR is compared to the results of the test. The second part comprised the orthogonal test using various factors, and the degree of interaction between the parameters was obtained by range analysis.


## 4.3. Influence of Single Factor on Damping Characteristics of MR in Nonforming Direction

The characterization parameters of MR specimens are shown in Table1. The size of the specimen is D30 × d10 × h30 ± 1 mm. As shown in Figure 7, the density of MR specimens is 2.5 and 3.5 g·cm−3.Table 1
Single-factor control test of MR test parameters.
Number
Frequency (Hz)
Density (g·cm−3)
Amplitude (mm)
d (mm)
Methods
1
1, 2, 3, and 4
2.5 and 3.5
0.5
0.15
Twining
2
1
2.5, 3.0, 3.2, and 3.5
0.5
0.15
Twining
3
1
2.5
0.5, 0.8, 1.0, and 1.2
0.15
Twining
4
1
2.5
0.5
0.15 and 0.3
Netting and twiningFigure 7
MR specimen with different densities. (a)ρMR = 2.5 g·cm−3. (b) ρMR = 3.5 g·cm−3.
(a)
(b)


## 4.3.1. Influence of Different Loading Frequencies on Energy Dissipation Characteristics

Figures8 and 9 are the dynamic hysteresis loops in the nonforming direction, respectively, which are realized by different densities of MR under different loading frequencies.Figure 8
Dynamic hysteresis loop of MR with the density of 2.5 g·cm−3 under different loading frequencies.Figure 9
Dynamic hysteresis loop of MR with the density of 3.5 g·cm−3 under different loading frequencies.From the diagram, it is evident that the hysteresis loop shape of the MRs obtained after the sine excitation loading test shows a sickle type. Due to the nonlinear characteristics of MR materials, the pores between metal wires are filled during loading, and the friction between wires begins to rub. Both dynamic average stiffness and energy consumption increase gradually. In addition, the left half area of the hysteresis loop in the diagram is larger than the right half area of they = 0 axis, which means that the energy consumption of the compressed part is greater than that of the restorer in the loading process (i.e., compression process).The hysteresis loop diagram shows that the hysteresis loop of MR formed by different loading frequencies is mostly in the state of reclosing. It indicates that the energy dissipation of MR components at different frequencies and in the same amplitude is approximately the same. It is shown that the damping performance of MR components is not significantly affected by different excitation; in other words, the damper prepared by MR has good stable damping performance. It can be found that the restoring force of the lower density specimen decreases with increasing loading frequency, and the hysteresis loop has a relatively large fluctuation (Figures8 and 9). When the density is higher, the restoring force increases with the increase of loading frequency, and the hysteresis loop is found to be smooth relatively. The reason for this phenomenon is as follows: when the density is relatively small, the gap between the metal wires is relatively large, and the slip distance between the wires is larger. Then, the dry friction between the wires is changed from static friction to dynamic friction, which will form a big jump in the curve.The actual change in the material is difficult to obtain from the hysteresis loop, so the loss factor, energy consumption, and dynamic average stiffness calculated by the MATLAB program are listed in Table2. And the change law is shown in Figure 10.Table 2
Loss factor, energy dissipation, and dynamic average stiffness variation.
f (Hz)
1
2
3
4
η
ρ
MR
2.5
0.3461
0.3311
0.3224
0.3171
Δ
W
ρ
MR
2.5
0.0837
0.0817
0.0808
0.0812
K
ρ
M
R
2.5
0.32
0.32
0.31
0.29
η
ρ
MR
3.5
0.2693
0.2670
0.2640
0.2629
Δ
W
ρ
MR
3.5
0.2251
0.2402
0.2521
0.2583
K
ρ
MR
3.5
1.12
1.16
1.17
1.21Figure 10
The loss factor, energy dissipation, and dynamic average stiffness variation at different frequencies. (a) MR with the density of 2.5 g·cm−3. (b) MR with the density of 3.5 g·cm−3. η is the dissipation factor.
(a)
(b)Comparing the changes in the parameters of MR components with two different densities under the sinusoidal excitation, it is found that the low-density MR has larger porosity, and the pores are filled first under the condition of low loading amplitude. However, the energy consumption changes little with density as the energy consumption is the elastic energy of the metal wire bending which does not change with density. In addition, the wire did not return to the original space pore in the process of returning, which will lead to a decrease in dynamic average stiffness and maximum elastic potential energy and a small increase in loss factor. While the high-density MR itself has a small pore space, the friction energy between the wires is completed during the loading process. Under the continuous loading, the friction is more intense. Therefore, both energy consumption and the dynamic average stiffness show a rising trend. And the loss factor has a very small downward trend.


## 4.3.2. Influence of Structure Density on Energy Dissipation Characteristics

In order to find the effect of density on energy consumption of materials, four groups of specimens with a density of 2.5, 3, 3.2, and 3.5 g·cm−3 were designed. As shown in Figure 11, the hysteresis characteristics of the MR with different densities are different, and the graphic area increases by increasing the density of the specimen. Therefore, with higher density, the stronger energy dissipation capacity of the MR is obtained. However, the change in loss factor shows an opposite character because the loss factor is affected by maximum elastic energy storage, and the maximum energy storage is related to the stiffness (Figure 12). The maximum energy storage increases with stiffness. The dynamic stiffness of MR increases with the increase of density, while the decrease rate of loss factor increases with the increase of stiffness. Therefore, the energy dissipation characteristics of MR are highly correlated with materials density; i.e., MR elements with low density possess better damping characteristics.Figure 11
Hysteresis curve of MR with different densities.Figure 12
Loss factor and dynamic average stiffnessK with changing density. η is the dissipation factor.


## 4.3.3. Influence of Different Amplitudes on Energy Dissipation Characteristics

The influence of different amplitudes on the energy dissipation characteristics of MR was examined by maintaining the pretightening amount, where the maximum amplitude of 1.2 mm is selected for this study. The same sine excitation origin for all four sets of data is ensured so that the test comparison becomes more accurate. The hysteresis loop of MR in very forming direction is shown in Figure13. The change in loss factor, energy consumption, and average dynamic stiffness under the effect of different amplitudes is shown in Table 3, and the trend of change is shown in Figure 14.Figure 13
Hysteresis loops of MR under different amplitude conditions.Table 3
Loss factor, energy dissipation, and dynamic average stiffness variation.
Amplitude (mm)
0.5
0.8
1.0
1.2
η
0.2881
0.2832
0.2658
0.2449
Δ
W
0.1238
0.2146
0.3033
0.3700
K
0.52
0.44
0.37
0.33Figure 14
The variation curves of each parameter under different amplitudes.η is the dissipation factor.From the graph, it is evident that the area surrounded by hysteresis loops increases with the increasing amplitude, meaning that the energy loss increases. In addition, with the increase of amplitude, the value of the maximum elastic potential energy increases. Simultaneously, a downward trend in loss factor and stiffness is found (Figure14).This phenomenon is because when the amplitude increases at a certain frequency, the compression degree of the MR increases and the number of contact points of the wire increases. However, in the course of returning, the wires are still in contact with each other because of the strong elastic properties of the wire so that the return speed is slower than that of the external load. The energy dissipation effect has a downward trend.


## 4.3.4. Influence of Different Braiding Process Parameters on Energy Dissipation Characteristics

Figure15 shows three kinds of specimens with different knitting processes to study the influence of different knitting processes on the energy dissipation characteristics of MR. Then, the sinusoidal excitation of MR with different process parameters was carried out to obtain the hysteresis loop (Figure 16), and the variation of the parameters on the hysteresis loop is also investigated (Figure 17). The summary of loss factor, energy dissipation, and dynamic stiffness is listed in Table 4.Figure 15
MR with different weaving process parameters.Figure 16
Hysteresis loop of MR under different processes.Figure 17
Parameter variation curves under different processes.η is the dissipation factor.Table 4
Loss factor, energy dissipation, and average dynamic stiffness.
Serial number
1
2
3
Methods
d = 0.15 and twining
d = 0.30 and twining
d = 0.30 and netting
η
0.3461
0.2801
0.2732
Δ
W
0.1647
0.1764
0.2001
K
0.32
0.46
0.56A progressive relationship between the energy dissipation capacities of MR under the three process parameters, which is due to the increase of stiffness, is visible (Table4). Therefore, according to the knitting process, with a larger wire diameter, the wire stiffness is higher that causes higher energy consumption. The netting MR has more contacts in the mesoscopic structure than the twisted MR, and the structure is more stable. The dry friction energy consumption of the netting MR sample is much earlier than that of the twisted MR sample, so that the rigidity is greater and thus the energy consumption is greater. From the abovementioned analysis, it is obvious that loss factor of the braided MR is more affected by the stiffness, giving a decreasing trend.


## 4.4. Orthogonal Test and Result Analysis of Various Parameters

The above experiment is based on control variables and explains the influence of various factors on the damping energy consumption of MR from the viewpoint of dry friction. In order to investigate the sensitivity of various factors to damping energy consumption, an orthogonal test was designed. According to the influence of the damping property parameters of MR, four test factors such as density (A), amplitude (B), test frequency (C), and molding process (D) have been selected. Each factor has 3 levels. Specific factors and factors level selected are shown in Table5. The loss factor is used as the experimental index. Therefore, we chose orthogonal arrays to carry out 9 orthogonal experiments. The specific factors are shown in Table 6.Table 5
Orthogonal test factors and their level.
Factor
Symbol
Levels
1
2
3
Density
A
2.5
3.0
3.5
Amplitude
B
0.5
0.8
1.0
Frequency
C
1
3
5
Condition of molding
D
0.15 forming of twining
0.3 forming of twining
0.3 forming of nettingTable 6
Orthogonal test results of the loss factor index.
Density
Amplitude
Frequency
Condition of molding
Loss factor
A
B
C
D
1
1
1
1
1
0.3461
2
1
2
2
2
0.2979
3
1
3
3
3
0.2366
4
2
1
2
3
0.2669
5
2
2
3
1
0.2795
6
2
3
1
2
0.1999
7
3
1
3
2
0.2843
8
3
2
1
3
0.2476
9
3
3
2
1
0.2670
K1
0.8806
0.8973
0.7936
0.8926
B > D > A >C
K2
0.7463
0.8250
0.8318
0.7821
K3
0.7989
0.7035
0.8004
0.7511
Extreme difference
0.1343
0.1938
0.0382
0.1415
Rank
3
1
4
2As shown in Table6, under different parameter levels, by increasing the loss factor, better damping performance is obtained. By comparing the extreme values at the level of each parameter, primary and secondary relationships of different factors (B > D > A >C) can be obtained, which means amplitude > forming condition > density > frequency.It can be seen that the effect of amplitude on the loss factor is most obvious, while the frequency is less affected. The difference between density and molding conditions is close, both of which are design parameters, and their effect on stiffness is obvious.


## 5. Conclusion

(1)
According to the single-factor control test results, with the increase of dynamic stiffness, the level of energy consumption is significantly improved. The low-density MR energy dissipation is achieved by overcoming the bending of the steel wire, so it loses less energy. By increasing the density of MR, the main form of energy dissipation changes to the friction between the wires, which causes the energy consumption rate to increase rapidly. At different loading frequencies, the motion of the metal wire only changes with speed, which has little effect on stiffness and a small change in dissipative energy. The damping property of MR is characterized by loss factor. The loss factor is proportional to energy consumption and affected by the maximum elastic energy storage.
(2)
The influence of different factors on the damping property is studied by performing the orthogonal test. The order of the factors that affect the damping performance is found to be amplitude > molding condition > density > frequency. The result shows that the effect of amplitude on damping energy is much greater than that of frequency in practical application, so the damping characteristic of the MR element is not sensitive to loading frequency. Therefore, when the MR components are designed, the amplitude level of the actual environment should be considered first followed by a suitable technology.



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*Source: 5014789-2018-11-05.xml*