Damping Characteristics of Metal Rubber in the Pipeline Coating System

,e reduction of vibration in submarine pipeline systems at high temperatures has always been a difficult problem. ,is paper aims to reduce the vibration of pipeline systems by using coated metal rubber. A theoretical model of the cladding damping structure, formed on the basis of the bilinear hysteresis model, is established. ,e mechanical model of the single degree of freedom hysteretic system is linearly equivalent to using the linearization method. ,e theoretical analysis indicates that the regularity of the stiffness of metal rubber decreases, and the damping increases, with the increase of the excitation amplitude. Experimental verification confirmed this analysis after an experimental system for pipelines coated by metal rubber was developed. Amethod for preparing the thin sheet of metal rubber, which is the damping layer, was introduced. At the same time, the force transfer rate and the structural loss factor were proposed to characterize the damping performance of the cladding damping structure. ,e vibration absorption characteristics of the cladding damping structure, along with its forming density, number of coating layers, and excitation magnitude, are investigated by means of the experimental method. ,e results indicate that the damping performance of metal rubber decreases with the increase of forming density, and the damping performance of metal rubber increases with the increase of the number of cladding layers and the magnitude of excited vibration.


Introduction
Pipelines are widely used in defense equipment and the marine industry, especially in warships, where a pipeline is often directly connected to the power device [1,2].Vibrations significantly affect this pipeline, for instance, the noise created by vibrations can affect the hidden nature of ship work.Eventually, this can lead to the breakdown of the pipeline or inestimable losses to the warship.
Generally, there are two methods of vibration and noise control in pipelines: the suppression of the vibration source and the disconnection of the vibration transmission.At present, there are two kinds of connection modes in the narrow space of pipelines: rigid connections and damping connections.ere are also suspension damper connections and cladding damping pipe connections in pipe vibration damping.Metal rubber is made by twisting wire into a spiral coil, winding the spiral coil according to a certain process, and finally cold stamping it [3].Metal rubber is a typical nonlinear material, and it has an obvious hysteresis characteristic [4,5].It is a good damping material because of its vibration absorption performance.Its energy dissipation and vibration absorption characteristics are not only affected by its own internal properties [6][7][8][9], including wire diameter and mass density and the blank winding mode and angle, but are also related to external conditions such as the vibration magnitude, loading amplitude, ambient temperature, and so on [10,11].Cai et al. [12] studied the influence of the rubber damping structure on the vibration absorption characteristics of a ship pipeline.eir results showed that rubber damping has an obvious damping performance compared with a rigid connection.Under high temperatures, the properties of rubber will be seriously affected, which directly reduces its vibration absorption. is is because rubber is a natural polymer material which cannot be made resistant to high temperatures.Ao et al. [13] undertook corresponding efforts to investigate the damping performance of a metal rubber damper supporting an engine pipeline.e results indicated that the metal rubber damper had a good dry friction damping performance.
e vibration of the warship pipeline has an inestimable effect on the ship hull, so studying pipeline vibration reduction is particularly important.In this paper, a theoretical model of the cladding damping structure, on the basis of the bilinear hysteresis model, is established.
e nonlinear, variable stiffness, and variable damping properties of metal rubber materials are proposed and verified by experiments.
e energy dissipation characteristics of metal rubber and the effects of the coating layer number, the exciting force, and the density of the cladding layer on the vibration absorption performance of the pipeline cladding damping structure are analyzed and discussed.

Modeling of the Cladding Damping Structure
Many previous researchers have reported that metal rubber is a nonlinear material.Its dynamic model includes a memory link and a nonmemory link characterized by the change of damping and stiffness during the vibration damping process.Based on the bilinear hysteresis model [3], the mechanical model of the single degree of freedom hysteresis system can be made linearly equivalent by using the equivalent linearization method involved in the change law of stiffness and damping of metal rubber.

Dynamic Analysis.
e cladding damping structure includes the metal rubber, the pipeline, and the cladding ring.e metal rubber is coated on the outside of the pipe through the cladding ring as the damping layer.e structure is composed of the damping layer and hanger and can be regarded as the shock absorber of the pipeline system.e scheme of metal rubber coating and its damping structure can be seen in Figure 1.
e pipe is subjected to radial force F when the pipe moves up and down and the direction of the force on the top is down; meanwhile, force on the bottom is opposite to this.Other forces can be decomposed into vertical forces F 2 and horizontal forces F 1 .e direction of vertical forces on both sides of the pipe is the same, and the horizontal forces are opposite, so the horizontal forces can counteract each other.In this case, the pipeline can be regarded as vertical force, and the pipeline in the cladding damping structure can be regarded as a lumped mass.
ere is a 5 mm gap between the two cladding rings, which is used to adjust the pretightening force.e damping layers can always closely contact both the pipe and the cladding rings because of the pretightening force, so the metal rubber layers are always in a state of force.e metal rubber layers can be seen to be closely contacting each other at all times, so there is no relative slip between the metal rubber layers.Furthermore, it can be estimated that there is no energy dissipation within the metal rubber layers.
e forces of the upper and lower damping layers are unequal due to the gravity of the pipeline, and the forces of the lower layers are greater than those of the upper layers.e pipe will vibrate up and down when it is affected by the external incentive.In this case, the stiffness of the whole damping coating structure can be regarded as K and the damping as C, and the nonlinear part of the cladding damping structure can be expressed by the memory link Z(t).Based on this, the cladding damping structure can be further simplified as a bilinear hysteretic oscillator model with viscous damping, and its dynamic model is established, as shown in Figure 2.
In Figure 2, F e is the external incentive, which is F e � P 0 sin ωt.K is the stiffness of the cladding damping structure, Z s is the restoring force of slip, and K s is the linear stiffness before slip.
On the basis of Newton's second law, the differential equation of motion of the cladding damping structure is established as follows: Substituting the variables: ( From Equations ( 1) and (2), the differential equation of motion can be given by

Variable Stiffness and Damping Characteristics.
A large number of previous experiments [14] prove that the highorder harmonics contained in the hysteretic oscillator response can be neglected, and the fundamental component is  Shock and Vibration dominant, so we assume the applied displacement excitation is e memory link model is shown in Figure 3(a), and its equivalent linear model related to the linear viscous damping and spring parallel structure is shown in Figure 3(b).
In Figure 3, the equivalent viscosity damping coefficient and stiffness coefficient can be shown as C eq and K eq in equivalent links, so the memory link Z(t) can be given by Z(X, _ X) � C eq _ X + K eq X. (5) From Equations ( 5) and ( 6), the memory link can be expressed by e coordinate transformation is given as follows: A simplification of Equation ( 7) is made, and after the averaging operation, the displacement excitation can be shown as follows: From the bilinear hysteresis model [3,15], the memory link Z(t) can be divided into In Equation ( 10): Furthermore, Equations ( 12) and ( 13) can be obtained after simplifying, integrating, and collating as follows: Assuming that Parameter n c and n k curves with X m /X s were drawn and delineated, as shown in Figures 4(a) and 4(b), respectively.
It can be seen from Figure 4(a) that the equivalent viscous damping coefficient is indicates that with the increase of excitation displacement, the damping property of metal rubber is enhanced and the stiffness characteristic of metal rubber is decreased.

Performance Characterization of the Cladding Damping Structure
In the cladding damping structure, metal rubber is mainly converted into thermal energy through kinetic energy to achieve energy consumption.e energy dissipation characteristics of the cladding damping structure are characterized by the force transfer rate and the structural loss factor.Shock and Vibration 3

e Force Transfer Rate.
e force transfer rate is the earliest evaluation index of the vibration isolation e ect, which is de ned as the ratio of excitation force to responsive force.e responsive force is the force that the excitation force transmits to the foundation [16].In the cladding damping structure, the excitation force applied to the pipe is de ned as F, the responsive force is F, and the system force transfer rate T A can be expressed as: e transfer rate is based on the assumption of a rigid foundation and is only applicable to low frequency bands.e cladding damping structure of the pipeline coating system is the vibration isolation system, and it is based on the support of a rigid foundation.e assumption stands if the sweep frequency interval is set in the low frequency range.e energy dissipation characteristics of metal rubber can be qualitatively characterized according to the force transfer rate T A .

e Structural Loss
Factor.e frequency-response curves of the pipeline system can be obtained through experiments, and the structural loss factor of the cladding damping structure can be calculated by using the half-power method.e half-power method is the most commonly used method to obtain damping values in the frequency domain.
e dynamic exibility curve (the ratio of dynamic displacement to excitation force), the admittance curve (ratio of velocity to excitation force), and inertial rate curve (ratio of acceleration to excitation force) can be used as the frequency-response curves [17][18][19].In this paper, according to the inertial rate curve shown in Figure 5, taking the frequencies ω 1 and ω 2 corresponding to (1/ 2 √ )|H| max value, the loss factor of the cladding damping structure can be obtained by the following equation:

Specimen Preparation and Preparation of the Test Bench
4.1.Specimen Preparation.e preparation process of the metal rubber specimens for the experiment is shown in Figure 6, including wire winding, stripping, and stamping.
e raw material of the specimen is austenitic stainless steel wire [20], because the ductility is very high in a hightemperature environment, so it can withstand coating the pipeline in such a high-temperature environment.
e metal rubbers need to meet special application requirements.
e di culties of preparation include the preparation of the blank from a thin sheet of metal rubber, especially the selection of the core-shaft winding method to make the blank.e wires should have a perfect connection between each layer of the spiral coil and the next layer after drawing to avoid a pitch that is too large or too small, which can be caused by an incomplete or loose connection, among other problems.After the spiral coil was wound, the blank was removed from the core axis and placed in the mold.e top and bottom layers of the spiral coil will be hooked together after compression and, because the center of the blank is hollow, the sheet metal rubber specimen can be obtained by cold stamping.Problems with the sheet metal rubber specimen are that the edge angle is not full, and the thickness is inconsistent.e solution to these kinds of problems can be addressed in the following ways: (1) e winding length of the blank should be greater than the length of the specimen.Because the blank e raw material of the specimen is 06Cr19Ni10 stainless steel wire, with a wire diameter of φ 0.3 mm. e stainless steel wire is processed into a spiral coil by a wire winding machine, and the spring diameter is controlled at 3 mm by adjusting the probe position and stretching quantity of the wire winding machine.e metal wire is installed into the blank winding machine after ensuring that the exact quality of the wire, the spindle speed, and the wire feeding speed are cooperating well.e forming spiral winding pitch is twined on the core shaft after drawing.e winding angle of the blank is 60 degrees, and the uniformity of the pitch can be guaranteed by the machine drawing the pitch to ensure the consistency of the forming process of the specimen.e blank is placed in the designed mold shown in Figure 7, after the blank winding is completed.
e mold includes the external mold, press mold, internal mold, and metal rubber roughcast.
By utilizing the 1000 kN cold stamping machine, which is produced by the Tianjin Huidian Company and used in the cold stamping process, the stamping uniformity should be guaranteed when punching.Firstly, pressing to the set pressure, then adding the pressure to the set pressure for 30 s, and then releasing the pressure was carried out, and the process was repeated three times.e metal wire in the blank was completed to ensure a better performance of the metal rubber parts.
e size of the metal rubber specimen is 175 mm × 40 mm × 4 mm, and the technological parameters are listed in Table 1.

Test Bench.
On the basis of the existing experimental conditions, the pipe parameters were the external diameter of the pipe (D 108 mm), the thickness of the pipe (H 15 mm), and the length of the pipe (L 5600 mm).e material of the pipe was 304 stainless steel, and the corresponding hanger and experimental stand were designed based on these measurements.e construction of the test bench is exhibited in Figure 8(a).
e cladding damping structure (Figure 8(b)), xed on the support, was formed by the pipelines, the hangers, and the dampers.e support was xed on the support platform by T-type bolts.e support platform can be considered as a clamping platform, as it is a solid steel platform of better quality than the support and the pipeline system.ere were two force sensors installed between the supports and the hangers to measure the response of the vibration from the pipe to the support.e vibration exciter was suspended above the center of the pipe by a crane hook and a rubber rope, and the exciting force was applied to the midpoint of the pipe.An acceleration sensor was installed directly below the exciting point of the pipeline to measure the response acceleration.e connecting rod of the exciter was equipped with a force sensor to measure the fundamental excitation force, and the measurement data were transmitted to the vibration control system and the computer.e test ow chart is displayed in Figure 8(c).
e E-JZK-50 type electric vibrator produced by Hangzhou Yiheng Science and Technology Co., Ltd., was   6 Shock and Vibration used to test the testing equipment.e maximum exciting force was 500 N, the maximum amplitude was ±10 mm, the maximum acceleration was 49.5 g, and the frequency range was DC-2 kHz.e E5874A power amplifier was used with the exciter.A YD-303 piezoelectric quartz force sensor with a charge sensitivity of 3.00 pC/N was used between the connecting rod and the exciter; its working temperature region was −40 to 150 °C, and the measuring range of force was ±2 kN. e model of the KD3000 quartz force sensor with a charge sensitivity of 3.408 pC/N was used at the response point of the support; its working temperature region was −40 to 200 °C, and the measuring range of force was 0-5 kN.A 1A102E type IEPE piezoelectric accelerometer was used at the response point of the pipeline; its sensitivity was 10.80 MV/g, and the measuring range was ±500 g.

Results and Analysis
First of all, a sinusoidal sweep frequency excitation of 20 N was carried out for a rigid pipe system, and the sweep frequency range was 5-200 Hz. Figure 9 shows the force transfer rate curve of the rigid pipeline system.e selected frequency range is dependent on the peak numbers of the force transfer rate (T A ) by experimental test.Generally, the first three natural frequencies are enough for the analyses of force transfer rate evolution.In this study, the experimental test results have already three peak numbers of the force transfer rate under the testing frequency range from 5 Hz to 200 Hz.As the authors wrote in the paper, the natural frequency is mainly used for comparison, like "low-order frequency" and "high-order frequency."In fact, it would be too many external noises (e.g., electromagnetic interference) to identify the force transfer rate when the frequency was higher than 200 Hz.erefore, the authors only chose the frequency range from 5 Hz to 200 Hz for the analysis.
From the force transfer rate curve, it is obvious that the first-order natural frequency has the greatest influence on the vibration of the pipeline system.A great deal of research experience shows that the system vibration has the greatest influence on low-order natural frequencies.is is because the frequency of the general load is lower than that of the actual engineering vibration, and the resonance of the pipeline is often caused by a low-order natural frequency.
e high-order resonance energy is relatively low, and the influence on the structure is much smaller than that of the low-order resonance.erefore, only the vibration at the first-order natural frequency point can be considered, and the vibration at the first-order natural frequency can be considered as the object of study.In order to study the vibration of the pipeline system in a convenient way, the sweep frequency range was set to 10-20 Hz. e frequency sweep rate was 1 OCT/min.

Study of Stiffness and Damping
Characteristics.Before the experiment, the suspension height of the exciter was adjusted to give the exciter an appropriate pretightening force, so as to prevent the shock exciter from breaking away from the pipe due to the exciting force.e accuracy of the experiment will be affected by this.
e experiment conditions should remain unchanged after adjusting the pretightening force.
e different densities (specimen 1, specimen 2, and specimen 3) and the number of different cladding layers (one layer, two layers, and three layers) were applied to different exciting forces of different magnitudes (20 N, 50 N, and 80 N), and the experiment process for different parameters must ensure the consistency of other conditions in order to achieve standardization of the test.
According to the data obtained from the force transfer rate experiment of specimen 2 under different excitatory magnitudes (20 N, 50 N, and 80 N), the curves of the force transfer rate frequency are drawn (Figure 10).e other conditions of different exciting magnitudes are the same, and the number of coating layers is two.
According to the experimental data, the natural frequency and the peak values of the force transfer rate under different excitations are obtained, as listed in Table 2.
As can be seen from Figure 10 and Table 2, with the increase of the magnitude of excitation, the natural frequency ω n tends to decrease and the peak value of the force transfer decreases notably.Because the increase of the magnitude of excitation leads to the increase of displacement, the metal rubber produces a stiffness softening and damping energy dissipation enhancement effect.e experimental results agree with the theoretical analysis (Figure 4), which proves the rationality and accuracy of the theoretical analysis.

Different Densities.
According to the experimental data of different specimens (specimen 1, specimen 2, and specimen 3), under the conditions of 50 N excitation and two layers of cladding, the force transfer rate frequency curves and the inertia rate frequency curves are drawn (Figures 11(a According to the computing method (Equation ( 15)) of the force transfer rate, the peak of the force transfer rate T Am of different specimens is obtained, and the structural loss

Shock and Vibration
factor η is calculated based on the half-power method, as shown in Table 3.
From Figure 11 and Table 3, it can be seen that with the increase of specimen density, the peak of the force transfer rate T Am of the cladding damping structure increases and the structural loss factor η decreases.With the increase of the density of metal rubber, the number of wires inside increases and the number of contact points between wires will increase accordingly.e ability of metal rubber to overcome the external load will be enhanced as the sti ness of the metal rubber increases.Under the condition of constant excitation on the order of 50 N, the sti ness increases and the amplitude decreases with the increase of the density of the metal rubber specimen.e energy consumption and the potential energy of the metal rubber decrease because of the increase of the sti ness and the increase of the metal wire inside the metal rubber.
e force transfer rate increases and the structural loss factor decreases with the increase of the density of metal rubber, according to the formula of the loss factor [21].  e force transfer rate frequency curves and the inertia rate frequency curves are drawn, as can be seen in Figures 10 and 12, according to the experimental data measured from specimen 1 under the conditions of di erent excitatory magnitudes (20 N, 50 N, and 80 N) and two layers of cladding.
e peak value of the force transfer rate T Am of di erent excitatory magnitudes is obtained according to the computing method (Equation ( 15)) of the force transfer rate.e structural loss factor η is calculated based on the half-power method, as presented in Table 4.
From Figures 10 and 12 and Table 4, it can be seen that the peak force transfer rate T Am of the cladding damping structure tends to decrease, and the structural loss factor η increases gradually with the increase of the exciting magnitude.e amplitude response increases with the increase of the exciting magnitude, the slip amplitude between metal wires increases with the increase of amplitude, and the energy dissipation characteristic increases accordingly under the conditions of a certain density of metal rubber.is is because metal rubber energy dissipation is achieved by sliding dry friction between wires.e experimental results show that the force transfer rate decreases and the structural loss factor increases.e energy dissipation characteristics of metal rubber are enhanced with the decrease of the force transfer rate and the increase of the structural loss factor.e magnitude of excitation increases from 20 N to 80 N and the span is 30 N, and the force transfer rate clearly decreases.is shows that the magnitude of excitation has a great in uence on the energy dissipation characteristics.Furthermore, the structural loss factor can quantitatively describe the energy dissipation enhancement characteristics of metal rubber.

Di erent Cladding Layers.
According to the experimental data measured from specimen 1, the force transfer rate frequency curves and the inertia rate frequency curves are drawn, as shown in Figures 13(a) and 13(b), under the conditions of di erent cladding layers (one layer, two layers, and three layers) and an exciting magnitude of 50 N.
e peak value of the force transfer rate T Am of di erent excitatory magnitudes is obtained according to the computing method (Equation ( 15)) of the force transfer rate.e structural loss factor η is calculated based on the half-power method, as demonstrated in Table 5.
From Figure 13 and Table 5, it can be seen that the peak of the force transfer rate, T Am , of the cladding damping structure tends to decrease and the structural loss factor η increases gradually with the increase of cladding layers.e sti ness of the metal rubber decreases with the increase of the number of cladding layers, according to Hooke's law.
is is manifested in the decrease of natural frequency, which can be con rmed from Figure 13.In addition, the damping increases with the increase of the number of cladding layers, which is because of the decrease of metal rubber sti ness with the increase of layers.As the number of cladding layers increases, the slip amplitude inside metal rubber will increase, which will lead to increased energy consumption.On the other hand, the number of wires and the friction of the contact point are increased with the increase of the number of layers, which undoubtedly enhances the energy dissipation characteristics.

Conclusions
(1) In this paper, the mechanical behavior of the metal rubber cladding damping structure was analyzed, and the dynamic model was established.e variable sti ness and variable damping characteristics of the metal rubber were obtained by using the equivalent linearization method.e results reveal that the sti ness of metal rubber softens, and the damping of metal rubber is enhanced, with the increase of the loading amplitude.e rationality and accuracy of the analysis are veri ed by experiments.(2) e force transfer rate and the structural loss factor were proposed to characterize the damping performance of the cladding damping structure.e variation of damping performance with density, the number of cladding layers, and the magnitude of excitation were studied experimentally.e results showed that the damping property of metal rubber increased with the increase of the number of cladding layers and the magnitude of excitation, and the energy dissipation characteristic of metal rubber decreased with the increase of density under the same exciting magnitude.e research in this paper provides a theoretical and experimental basis for the study of the vibration reduction of pipeline systems in engineering and has a clear guiding signi cance for future research work on the vibration damping of high-temperature pipeline systems.

1 F 2 Figure 1 :
Figure 1: Scheme of the metal rubber coating.

Figure 3 :Figure 2 :
Figure 3: (a) e memory link model and (b) its equivalent linear model.

Figure 4 : 2 Figure 5 :
Figure 4: (a) Parameter n c curve with X m /X s ; (b) parameter n k curve with X m /X s .

Figure 6 :
Figure 6: e preparation of metal rubber specimens.

Figure 9 :
Figure 9: e force transfer rate curve of the pipeline.

Figure 10 :Figure 11 :
Figure 10: e force transfer rate frequency curves of di erent vibration magnitudes.

Figure 13 :
Figure 13: (a) e force transfer rate frequency curves with di erent cladding layers; (b) the inertia rate frequency curves with di erent cladding layers.

Table 1 :
Process parameters for the metal rubber specimens.

Table 2 :
Natural frequencies and the peak values of the force transfer rate under di erent excitation levels.

Table 3 :
Vibration absorption characteristics at di erent densities.Figure12: e inertia rate frequency curves at di erent excitation levels.

Table 4 :
Characteristics of vibration reduction under di erent magnitudes of excitation.

Table 5 :
Vibration absorption characteristics with di erent cladding layers.