S S symmetry

Article Energy Dissipation Characteristics and Parameter Identiﬁcation of Symmetrically Coated Damping Structure of Pipelines under Diﬀerent Temperature Environment

Feng Jiang 1, Zheyu Ding 1,2, Yiwan Wu 1,* , Hongbai Bai 1, Yichuan Shao 1 and Bao Zi 1

1 Engineering Research Center for Metal Rubber, School of Mechanical Engineering and Automation, Fuzhou University, Fuzhou 350116, China; [email protected] (F.J.); [email protected] (Z.D.); [email protected] (H.B.); [email protected] (Y.S.); [email protected] (B.Z.) 2 Dongfeng Motor Corporation, Wuhan 430110, China * Correspondence: [email protected]

Received: 10 July 2020; Accepted: 30 July 2020; Published: 3 August 2020

Abstract: In this paper, a symmetrically coated damping structure for entangled metallic wire materials (EMWM) of pipelines was designed to reduce the vibration of high temperature (300 ◦C) pipeline. A series of energy dissipation tests were carried out on the symmetrically coated damping structure at 20–300 ◦C. Based on the energy dissipation test results, the hysteresis loop was drawn. The eﬀects of temperature, vibration amplitude, frequency, and density of EMWM on the energy dissipation characteristics of coated damping structures were investigated. A nonlinear energy dissipation model of the symmetrically coated damping structure with temperature parameters was established through the accurate decomposition of the hysteresis loop. The parameters of the nonlinear model were identiﬁed by the least square method. The energy dissipation test results show that the symmetrically coated damping structure for EMWM of pipelines had excellent and stable damping properties, and the established model could well describe the changing law of the restoring force and displacement of the symmetrically coated damping structure with amplitude, frequency, density, and ambient temperature. It is possible to reduce the vibration of pipelines in a wider temperature range by replacing diﬀerent metal wires.

Keywords: symmetrically coated damping structure; entangled metallic wire material; porous material; parameter identiﬁcation; energy dissipation; high temperature

1. Introduction Pipelines exist widely in military, civilian, and other ﬁelds. The vibration control method of the pipeline includes active control and passive control. Compared with active control technology, passive control (e.g., rubber pipe clamp) has low cost, mature and reliable, and has been widely used. However, the damping performance of the traditional polymer material is poor at high temperature environment [1,2], the vibration reduction of high temperature pipeline is a great challenge for designers and researchers, especially when the temperature exceeds 200 ◦C. Entangled metallic wire materials (EMWM) is a promising porous material made of metallic wire helixes. It is sometimes referred to as “metal rubber (MR)” [3–5], “metal wire mesh (MWM)” [6], or “tangled metal wire (TMW) devices” [7]. Because of its special spatial network structure, EMWM has good elasticity and damping characteristics. Moreover, because EMWM is all-metal material, it has good environmental adaptability. Therefore, EMWM has been used in extreme environments [8], such as aircraft air cycle machines, turbo blowers, and micropower generators [9–11]. Some scholars

Symmetry 2020, 12, 1283; doi:10.3390/sym12081283 www.mdpi.com/journal/symmetry Symmetry 2020, 12, 1283 2 of 17 have made a preliminary exploration on the mechanical properties of EMWM at high temperature environment. Ding et al. performed a series of quasi-static compression experiments for plate-like EMWM in the temperature range of 20–500 ◦C and pointed out that the stiﬀness of EMWM will not fail with the increase of temperature [12]. Hou et al. proposed a new damping capacity measurement method for EMWM and conducted a damping performance test for EMWM in the temperature range of 70–300 C[13]. They reported that the damping capacity of EMWM exhibits good resistance of − ◦ high–low temperature. An EMWM insertion damping structure was proposed by Zhu et al. [14] and applied to the vibration reduction design of foundation under high temperature (300 ◦C). Zhu et al. reported that the maximum insertion loss could reach 15.37 dB. It is an interesting attempt to use EMWM to reduce the vibration of the pipeline. A novel coated damping structure for EMWM of pipelines was designed by replacing a viscoelastic damping material with multiple EMWM blocks, which were uniformly distributed on the outer wall of the pipeline [15]. Xiao et al. proposed a theoretical model of the cladding damping structure for EMWM and carried out the test of pipeline vibration reduction at room temperature [16]. Wu et al. designed a coated damping structure for metal rubber (MR) of bellows, in which metal rubber was coated on the bellows by wire rope [17]. They conducted a dynamically tested on their structure in the bending direction at normal temperature and reported that the coated damping structure for metal rubber of bellows has a strong damping energy dissipation ability. Ulanov et al. proposed a calculation method of pipeline vibration with damping supports made of the MR material by means of the ﬁnite element ANSYS software package [18]. However, the eﬀectiveness of the calculation method was not veriﬁed by experiments, and the eﬀect of temperature on the structure was not taken into account. Although Bai [15], Xiao [16] and Wu [17] have veriﬁed the eﬀectiveness of EMWM in pipeline vibration reduction, they did not take into account the eﬀect of temperature and only carried out experiments and analysis at room temperature. This paper designs a symmetrically coated damping structure for EMWM of pipelines, and investigates the energy dissipation characteristics of the designed structure in the temperature range of 20–300 ◦C. The eﬀects of vibration frequency, amplitude, density, and temperature on the energy dissipation characteristics of the structure are analyzed. To provide an eﬀective theoretical basis for predicting the energy dissipation characteristics of the symmetrically coated damping structure for EMWM of pipelines and guiding its design under high temperature, a model of nonlinear elastic restoring force is set up, which describes the dynamic characteristics of the coated damping structure for EMWM of pipelines under diﬀerent temperature.

2. Design of the Symmetrically Coated Damping Structure for EMWM of Pipelines To reduce the vibration of pipelines under diﬀerent temperature environment, a symmetrically coated damping structure for EMWM of pipelines was designed as shown in Figure1. The coated damping structure is composed of plate-like EMWMs, constraining rings, pipe section and bolts. This structure is a kind of elastic damping pipe clip. Symmetrically installed EMWMs are the main energy dissipation elements of the structure. Symmetry 2020, 12, x FOR PEER REVIEW 2 of 16

such as aircraft air cycle machines, turbo blowers, and micropower generators [9–11]. Some scholars have made a preliminary exploration on the mechanical properties of EMWM at high temperature environment. Ding et al. performed a series of quasi-static compression experiments for plate-like EMWM in the temperature range of 20–500 °C and pointed out that the stiffness of EMWM will not fail with the increase of temperature [12]. Hou et al. proposed a new damping capacity measurement method for EMWM and conducted a damping performance test for EMWM in the temperature range of −70–300 °C [13]. They reported that the damping capacity of EMWM exhibits good resistance of high–low temperature. An EMWM insertion damping structure was proposed by Zhu et al. [14] and applied to the vibration reduction design of foundation under high temperature (300 °C). Zhu et al. reported that the maximum insertion loss could reach 15.37 dB. It is an interesting attempt to use EMWM to reduce the vibration of the pipeline. A novel coated damping structure for EMWM of pipelines was designed by replacing a viscoelastic damping material with multiple EMWM blocks, which were uniformly distributed on the outer wall of the pipeline [15]. Xiao et al. proposed a theoretical model of the cladding damping structure for EMWM and carried out the test of pipeline vibration reduction at room temperature [16]. Wu et al. designed a coated damping structure for metal rubber (MR) of bellows, in which metal rubber was coated on the bellows by wire rope [17]. They conducted a dynamically tested on their structure in the bending direction at normal temperature and reported that the coated damping structure for metal rubber of bellows has a strong damping energy dissipation ability. Ulanov et al. proposed a calculation method of pipeline vibration with damping supports made of the MR material by means of the finite element ANSYS software package [18]. However, the effectiveness of the calculation method was not verified by experiments, and the effect of temperature on the structure was not taken into account. Although Bai [15], Xiao [16] and Wu [17] have verified the effectiveness of EMWM in pipeline vibration reduction, they did not take into account the effect of temperature and only carried out experiments and analysis at room temperature. This paper designs a symmetrically coated damping structure for EMWM of pipelines, and investigates the energy dissipation characteristics of the designed structure in the temperature range of 20–300 °C. The effects of vibration frequency, amplitude, density, and temperature on the energy dissipation characteristics of the structure are analyzed. To provide an effective theoretical basis for predicting the energy dissipation characteristics of the symmetrically coated damping structure for EMWM of pipelines and guiding its design under high temperature, a model of nonlinear elastic restoring force is set up, which describes the dynamic characteristics of the coated damping structure for EMWM of pipelines under different temperature.

2. Design of the Symmetrically Coated Damping Structure for EMWM of Pipelines To reduce the vibration of pipelines under different temperature environment, a symmetrically coated damping structure for EMWM of pipelines was designed as shown in Figure 1. The coated damping structure is composed of plate-like EMWMs, constraining rings, pipe section and bolts. This Symmetry 2020, 12, 1283 3 of 17 structure is a kind of elastic damping pipe clip. Symmetrically installed EMWMs are the main energy dissipation elements of the structure.

(a) (b)

FigureFigure 1. 1.(a )(a Schematic) Schematic diagram diagram of the of symmetricallythe symmetrically coated coated damping damping structure structure for EMWM for ofEMWM pipeline; of andpipeline; (b) physical and (b picture) physical of symmetricallypicture of symmetrically coated damping coated structure damping for structure EMWM for of EMWM pipeline. of pipeline. Symmetry 2020, 12, x FOR PEER REVIEW 3 of 16

Because of the special spatial structure of EMWM, its energy dissipation mechanism can be divided Because of the special spatial structure of EMWM, its energy dissipation mechanism can be into three kinds. The ﬁrst kind is that the internal wire helixes of EMWM slip, friction, and squeeze divided into three kinds. The first kind is that the internal wire helixes of EMWM slip, friction, and under dynamic load, resulting in friction energy dissipation at each contact point of the adjacent wire squeeze under dynamic load, resulting in friction energy dissipation at each contact point of the helixes.adjacent The wire second helixes. kind The is thatsecond the changekind is ofthat the the spatial change position of the ofspatial the internal position wire of helixesthe internal of EMWM wire underhelixes external of EMWM load under cannot external be fully load recovered, cannot resulting be fully inrecovered, viscoelastic resulting energy in dissipation. viscoelastic The energy third kinddissipation. is due to The the third pores kind of EMWM is due to so the that pores it will of squeezeEMWM outso that the it internal will squeeze air or out inhale the theinternal external air or air duringinhale deformation,the external resultingair during inenergy deformation, dissipation resu [8lting,19]. in When energy the deformationdissipation of[8,19]. EMWM When exceeds the thedeformation micron level, of EMWM the internal exceeds wire the helixes micron of level, EMWM the internal will slip wire and helixes lead to of energy EMWM dissipation will slip and [20]. Therefore,lead to energy the damping dissipation energy [20]. dissipation Therefore, mechanism the damping of the symmetricallyenergy dissipation coated mechanism damping structure of the forsymmetrically EMWM of pipelines coated damping is as follows: structure when for the EMWM pipeline of ispipelines excited byis as external follows: excitation when the or pipeline due to theis transmissionexcited by external of ﬂuid excitation inside the or pipeline, due to therethe transm will beission a relative of fluid displacement inside the changepipeline, between there will the be pipe a andrelative theconstraining displacement ring, change which between means the that pipe the deformationand the constraining of EMWM ring will, which be changed, means that then the the vibrationdeformation energy of EMWM will be dissipated.will be changed, then the vibration energy will be dissipated.

3.3. Specimen Specimen andand TestTest Design Design 3.1. EMWM Specimen 3.1 EMWM Specimen The plate-like EMWM specimen was made of 304 (06Cr19Ni10) austenitic stainless steel wire with The plate-like EMWM specimen was made of 304 (06Cr19Ni10) austenitic stainless steel wire a diameter of 0.3 mm. The manufacture of the plate-like EMWM was referring to [14,15]. The size with a diameter of 0.3 mm. The manufacture of the plate-like EMWM was referring to [14,15]. The of the plat-like entangled metallic wire material (EMEM) specimen is 175 mm 40 mm 4.5 mm. size of the plat-like entangled metallic wire material (EMEM) specimen is 175 mm× × 40 mm × 4.5 mm. ThreeThree groupsgroups ofof specimensspecimens werewere mademade andand tested,tested, each group contains three plate-like plate-like EMWM EMWM specimensspecimens with with same same parameters parameters andand properties.properties. The manufactured specimen specimen and and its its manufacturing manufacturing parametersparameters are are shown shown in in Figure Figure2 2and and Table Table1 respectively.1 respectively.

FigureFigure 2.2. Plate-like EMWM specimen.

Table 1. Manufacture parameters for plate-like EMWM specimens.

Specimen Weight Molding Pressure (kN/cm2) Molding Density (g/cm3) EMWM1 70g 5.71 2.0 EMWM2 80g 8.57 2.286 EMWM3 90g 14.29 2.571

3.2 Test System The design of the test device is shown in Figure 3. The outer surface of both ends of the pipe section was tapped with threads, which was used to nest the end cover. The two end covers fixed the two connecting rods through a threaded connection and were connected with the upper connecting rod through a crossbar. The two ends of the upper constraining ring and the lower constraining ring were locked by bolts and were covered with several layers of plate-like EMEM specimens in the middle of the pipeline. An arc joint was welded on the lower constraining ring, and the joint was fixed with the lower connecting rod through an internal thread. The upper and lower connecting rods were respectively connected with the upper and lower chucks of the dynamic testing machine. During the test, the lower chuck was fixed, and the upper chuck moved up and down to drive the pipe section vibration, resulting in compression deformation of the upper and lower parts of the plate-like EMWM specimens.

Symmetry 2020, 12, 1283 4 of 17

Table 1. Manufacture parameters for plate-like EMWM specimens.

Specimen Weight Molding Pressure (kN/cm2) Molding Density (g/cm3) EMWM1 70 g 5.71 2.0 EMWM2 80 g 8.57 2.286 EMWM3 90 g 14.29 2.571

Symmetry 2020, 12, x FOR PEER REVIEW 4 of 16 3.2. Test System TheThe design high temperature of the test device environment is shown was in Figuregenerated3. The by outera self-made surface quartz of both lamp ends heating of the system, pipe sectionwhich was is shown tapped in with Figure threads, 3. The which heating was system used toconsists nest the of end12 quartz cover. lamps The two with end a coverslength ﬁxedof 400 the mm twoand connecting a diameter rods of 15 through mm as a a threaded heat source. connection Each quartz and were lamp connected has a rated with power the upperof 1500 connecting W and was rodinstalled through in aparallel crossbar. on the The quartz two ends lamp of fixing the upper plate. constrainingThe quartz lamp ring fixing and the plate lower was constraining made of 310 s ringstainless were locked steel and by boltscould and withstand were covered a high with temperature several layers of up of to plate-like 1300 °C. EMEMThe surface specimens of the in quartz the middlelamp offixing the pipeline.plate was An polished arc joint to wasreflect welded the quar on thetz lamplight lower constraining and could ring, to heat and the jointtest device was ﬁxedquickly. with A the layer lower of connectingaluminum rodsilicate through insulation an internal layer thread.was sandwiched The upper between and lower the connectingquartz lamp rodsfixing were plate respectively and the connectedmetal thermal with insulation the upper andboard, lower which chucks could of theplay dynamic a better testing role in machine. thermal Duringinsulation. the test, the lower chuck was ﬁxed, and the upper chuck moved up and down to drive the pipe sectionAn vibration, electro-hydraulic resulting in servo compression dynamic deformation testing ma ofchine the upper(SDS-200, and Sinotest lower parts Equipment of the plate-like Co., ltd., EMWMChangchun, specimens. China) was the excitation system, which could apply sinusoidal excitation of different frequencies to the symmetrically coated damping structure through its upper and lower chucks.

Figure 3. Test device and quartz lamp heating system. Figure 3. Test device and quartz lamp heating system. The high temperature environment was generated by a self-made quartz lamp heating system, which3.3 Test is shown Methods in Figure3. The heating system consists of 12 quartz lamps with a length of 400 mm and a diameterAfter all the of 15equipment mm as a was heat assembled source. Each and quartz adjust lamped, the has high a rated temperature power ofdynamic 1500 W mechanical and was installedexperiments in parallel of symmetrically on the quartz coated lamp ﬁxingdamping plate. struct Theure quartz with lamp different ﬁxing densities plate was were made carried of 310 out s stainlessrespectively. steel andThe couldtemperature withstand of the a high coated temperature damping structure of up to 1300is adjusted◦C. The to surface 20 °C, of100 the °C, quartz 200 °C, lampand ﬁxing300 °C plate by controlling was polished the to power reﬂect on the and quartz off of lamplight the quartz and lamp. could To to ensure heat the the test standardization device quickly. of Athe layer experiment, of aluminum the silicate temperature insulation was layer kept was for sandwiched 30 min after between reaching the quartzthe set lamptemperature, ﬁxing plate and andsinusoidal the metal displacement thermal insulation excitation board,s with which different could playamplitudes a better and role frequencies in thermal insulation.were applied to the symmetricallyAn electro-hydraulic coated damping servo dynamicstructure. testing The amplit machineude of (SDS-200, sinusoidal Sinotest displacement Equipment excitation Co., ltd., was Changchun,set to 0.2 mm, China) 0.5 mm, was 0.8 the mm, excitation 1 mm, system,and the whichfrequency could value apply was sinusoidal set to 1 Hz, excitation 2 Hz, 3 Hz, of di 4ﬀ Hz,erent and frequencies5 Hz. The tocontroller the symmetrically of the dynamic coated testing damping machine structure (SDS-200) through could its uppercollect andthe test lower data chucks. in real time and draw the force-displacement hysteresis loop. 3.3. Test Methods In this research, structural loss factor η, dissipated energy (ΔW) and maximum elastic potential energyAfter (W) all thewere equipment used to wascharacterize assembled the and energy adjusted, dissipation the high characteristics temperature dynamic of the symmetrically mechanical experimentscoated damping of symmetrically structure for coatedEMWM damping of pipelines. structure These withcould di beﬀ erentderived densities from the were experiment carried outdata. Figure 4 is the sketch of the force-displacement hysteresis loop.

Symmetry 2020, 12, 1283 5 of 17

respectively. The temperature of the coated damping structure is adjusted to 20 ◦C, 100 ◦C, 200 ◦C, and 300 ◦C by controlling the power on and oﬀ of the quartz lamp. To ensure the standardization of the experiment, the temperature was kept for 30 min after reaching the set temperature, and sinusoidal displacement excitations with diﬀerent amplitudes and frequencies were applied to the symmetrically coated damping structure. The amplitude of sinusoidal displacement excitation was set to 0.2 mm, 0.5 mm, 0.8 mm, 1 mm, and the frequency value was set to 1 Hz, 2 Hz, 3 Hz, 4 Hz, and 5 Hz. The controller of the dynamic testing machine (SDS-200) could collect the test data in real time and draw the force-displacement hysteresis loop. In this research, structural loss factor η, dissipated energy (∆W) and maximum elastic potential energy (W) were used to characterize the energy dissipation characteristics of the symmetrically coated damping structure for EMWM of pipelines. These could be derived from the experiment data. SymmetryFigure 20204, 12is, thex FOR sketch PEER ofREVIEW the force-displacement hysteresis loop. 5 of 16

F

W 0 X ∆W

Figure 4. Sketch of the hysteresis loop. Figure 4. Sketch of the hysteresis loop.

In the experiment, the sampling frequency and loading frequency were set to f 0 and f respectively, In the experiment, the sampling frequency and loading frequency were set to f0 and f in which the number of sampling points in a vibration period is N = f 0/f. In this research, f 0 = 2500 Hz. respectively, in which the number of sampling points in a vibration period is N = f0/f. In this research, Deﬁne the measured restoring force as Fi, the displacement as Xi, I = 1, 2 ... N, the displacement excitationf0 = 2500 Hz. could be expressed as Define the measured restoring force as Fi, the displacement as Xi, I = 1, 2 ... N, the displacement Xi = X0 cos(ωt + ψ0) (1) excitation could be expressed as where ψ was the initial phase angle of displacement change, X was the vibration displacement 0 =+()ω ψ 0 amplitude, and ω was the angular frequency.XXi 00cos t (1) By substituting the measured restoring force and displacement values into Formula (1), where ψ0 was the initial phase angle of displacement change, X0 was the vibration displacement the displacement dispersion value could be expressed as amplitude, and ω was the angular frequency. By substituting the measured restoring2 πforcei and displacement values into Formula (1), the X = X cos + α , i = 1, 2, ... , N (2) displacement dispersion value couldi be0 expressedN as π The area of the hysteresis loop=+=… (∆W) represents2 i α the dissipated energy in one loading and unloading X i XiN0 cos，， 1 2，， (2) cycle, and could be obtained by the integral ofN the restoring force in the displacement direction as follow: H The area of the hysteresis loop∆W (Δ=W) Fdxrepresents the dissipated energy in one loading and unloading cycle, and could be obtained byH the integral of the restoring force in the displacement = Fd[X0 cos(ωt + α)] direction as follow: RT = ωX Fsin(ωt + α)dt (3) Δ=WFdx∮ 0 − 0 N =+∮ 2πX ()ωα Fd X0 cos0 P t 2πi = N Fi sin N + α − T i=1 =−ωωαXFsint() + dt 0 (3) 0 π N π =−2 X 0 2 i +α Fi sin NNi=1

The hysteresis loop was symmetrical to the origin, thus the maximum elastic energy storage (W) of the symmetrically coated damping structure for EMWM of pipeline in a vibration period could be expressed as:

==112 WkXFX000 (4) 22

F − F k = max min (5) 2

where k was the dynamic average stiffness, Fmax was the maximum elastic restoring force, Fmin was the minimum elastic restoring force, and X0 was the vibration amplitude. Therefore, the loss factor was defined as:

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The hysteresis loop was symmetrical to the origin, thus the maximum elastic energy storage (W) of the symmetrically coated damping structure for EMWM of pipeline in a vibration period could be expressed as: 1 1 W = kX 2 = F X (4) 2 0 2 0 0

Fmax F k = − min (5) 2

Symmetrywhere k 2020was, 12 the, x FOR dynamic PEER REVIEW average stiﬀness, Fmax was the maximum elastic restoring force, Fmin6 ofwas 16 the minimum elastic restoring force, and X0 was the vibration amplitude. Therefore, the loss factor was deﬁned as: ΔW η = (6) 2πW∆W η = (6) 2πW

4.4. Energy Energy Dissipation Dissipation Test Test Results and Discussion

4.1.4.1. The The Influence Inﬂuence of of the the Vibration Vibration Frequency Frequency ToTo investigate the effect eﬀect of vibration frequency on the energy dissipation characteristics of the symmetricallysymmetrically coatedcoated dampingdamping structure structure for for EMWM EMWM of of pipelines, pipelines, a series a series of sinusoidal of sinusoidal excitation excitation with withan amplitude an amplitude of 0.8 of mm 0.8 andmm di andﬀerent different frequencies frequencies (1 Hz, (1 2 Hz, 2 3 Hz, 3 4 Hz, 4 and Hz, 5 and Hz) 5 was Hz) applied was applied to the tocoated the coated damping damping structure structure with EMWM2 with EMWM2 at room attemperature. room temperature. The experimental The experimental data were data drawn were drawnas force-displacement as force-displacement curves, curves as shown, as shown in Figure in 5Figure. The energy 5. The dissipationenergy dissipation characteristics characteristics of coated of coateddamping damping structure structure at diﬀerent at different frequencies frequencies are shown are inshown Table in2. Table 2.

1.8 1Hz 1.2 2Hz

kN 3Hz 0.6 4Hz 0.0 5Hz

-0.6 Restoring force/ -1.2

-1.8 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 Displacement/mm

FigureFigure 5. 5. HysteresisHysteresis loop loop at at different diﬀerent frequenc frequenciesies (amplitude 0.8 mm, 20 °C,◦C, EMWM2).

Table 2. Energy dissipation characteristics of coated damping structure at diﬀerent frequencies. Table 2. Energy dissipation characteristics of coated damping structure at different frequencies.

FrequencyFrequency//HzHz Loss Loss Factor Factor ηη DissipatedDissipated Energy Energy ΔW∆ W Maximum Maximum Elastic Elastic Potential Potential Energy Energy W 1 0.2404 0.2404 0.9813 0.9813 0.6497 0.6497 2 0.2402 0.9744 0.6456 2 0.2402 0.9744 0.6456 3 0.2442 0.9696 0.6318 34 0.2442 0.2480 0.9696 0.9729 0.6318 0.6243 45 0.2480 0.2515 0.9729 0.9790 0.6243 0.6195 5 0.2515 0.9790 0.6195 In Figure5, the area of the hysteresis loop (Dissipated energy ∆W) of the structure was In Figure 5, the area of the hysteresis loop (Dissipated energy ΔW) of the structure was approximately unchangeable. It indicated that the inﬂuence of vibration frequency on the energy approximately unchangeable. It indicated that the influence of vibration frequency on the energy dissipation characteristics of the coated damping structure was very small under low-frequency dissipation characteristics of the coated damping structure was very small under low-frequency vibration (1–5 Hz). It could be seen from Table2 that the loss factor η of the structure increases with vibration (1–5 Hz). It could be seen from Table 2 that the loss factor η of the structure increases with the increase of frequency at room temperature, but the increase was small. the increase of frequency at room temperature, but the increase was small. With the increases of the vibration frequency, the slip of the wire helixes inside the EMWM would be insufficient, resulting in the reduction of friction energy dissipation. However, as the vibration frequency increase, the spatial position of the internal wire helixes would be more difficult to recover, and the second kind of energy consumption would gradually increase. When the vibration frequency was lower than 10 Hz, the pumping effect of metal rubber on air was not obvious, which means that the third kind of energy consumption has no significant change. With the increase of vibration frequency, the proportion of the first energy dissipation mode would gradually decline, the proportion of the second energy dissipation mode would gradually enhanced. Therefore, the dissipated energy ΔW decreased at first and then increased with the increase of frequency.

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With the increases of the vibration frequency, the slip of the wire helixes inside the EMWM would be insuﬃcient, resulting in the reduction of friction energy dissipation. However, as the vibration frequency increase, the spatial position of the internal wire helixes would be more diﬃcult to recover, and the second kind of energy consumption would gradually increase. When the vibration frequency was lower than 10 Hz, the pumping eﬀect of metal rubber on air was not obvious, which means that the third kind of energy consumption has no signiﬁcant change. With the increase of vibration frequency, Symmetrythe proportion 2020, 12, x of FOR the PEER ﬁrst REVIEW energy dissipation mode would gradually decline, the proportion of7 of the 16 second energy dissipation mode would gradually enhanced. Therefore, the dissipated energy ∆W decreasedThe maximum at ﬁrst and elastic then increasedpotential withenergy the (W) increase was proportional of frequency. to the maximum restoring force and theThe maximum elasticdisplacement, potential when energy the (W) amplitud was proportionale was constant, to the maximum it was only restoring related force to andthe maximumthe maximum restoring displacement, force. With when the theincrease amplitude of the was frequency, constant, the it slip was of only internal related wire to thehelixes maximum of the EMWMrestoring became force. Withmore the and increase more ofinsufficient, the frequency, which the made slip of the internal internal wire friction helixes force of the decrease EMWM gradually.became more With and the more decrease insuﬃ ofcient, internal which friction made, the the maximum internal friction restoring force force decrease of metal gradually. rubberWith became the smaller,decrease which of internal led to friction, a decrease the maximum of the maximum restoring elastic force of potential metal rubber energy became (W) with smaller, the whichincrease ed of to vibrationa decrease frequency. of the maximum elastic potential energy (W) with the increase of vibration frequency. ItIt can can be seen from Equation (6) (6) that that the the loss loss factor factor η isis directly proportional proportional to to the the energy energy consumption Δ∆WW and and inversely inversely proportional proportional to to the the maxi maximummum elastic elastic potential potential energy energy W. W. When the energy dissipation Δ∆WW and the maximum elastic potential energy energy W W did not change much, the loss factorfactor η would not change much.

4.2. The The Influence Inﬂuence of of the the Vibration Vibration Amplitude Amplitude A series ofof sinusoidalsinusoidal excitation excitation with with a a frequency frequenc ofy of 3 Hz3 Hz and and diﬀ differenterent amplitudes amplitudes (0.2 (0.2 mm, mm, 0.5 mm, 0.5 mm,0.8 mm, 0.8 1mm, mm) 1 were mm) applied were toapplied the coated to the damping coated structuredamping withstructure EMWM2 with at EMWM2 room temperature. at room temperature.The experimental The experimental data were drawn data as were force-displacement drawn as force-displacement curves, as shown curves, in Figure as shown6. The in loss Figure factor 6. Theη, dissipated loss factor energy η, dissipated∆W and energy maximum ΔW elastic and maximum potential energyelastic Wpotentialcould be energy derived W fromcould Figure be derived5 and fromare listed Figure in Table5 and3 are. listed in Table 3.

2.4 0.2mm 1.6 0.5mm

kN 0.8mm 0.8 1.0mm

0.0

-0.8 Restoring force/ Restoring

-1.6

-2.4 -1.2 -0.8 -0.4 0.0 0.4 0.8 1.2 Displacement/mm

FigureFigure 6. 6. HysteresisHysteresis loop loop at at different diﬀerent amplit amplitudesudes (amplitude 0.8 mm, 20 °C,◦C, EMWM2). Table 3. Energy dissipation characteristics of coated damping structure at diﬀerent amplitudes. Table 3. Energy dissipation characteristics of coated damping structure at different amplitudes. Amplitude/mm Loss Factor η Dissipated Energy ∆W Maximum Elastic Potential Energy W Amplitude/mm Loss Factor η Dissipated Energy ΔW Maximum Elastic Potential Energy W 0.20.2 0.37080.3708 0.1281 0.1281 0.0550 0.0550 0.5 0.3126 0.4905 0.2498 0.5 0.3126 0.4905 0.2498 0.8 0.2442 0.9696 0.6318 0.8 0.2442 0.9696 0.6318 1 0.2004 1.3499 1.0723 1 0.2004 1.3499 1.0723

As illustrated in in Figure Figure 66,, whenwhen thethe amplitudesamplitudes werewere didifferent,ﬀerent, the curves did not coincide with each otherother atat the the same same displacement. displacement. It indicated It indica thatted whenthat when the coated the dampingcoated damping structure structure was subjected was subjected to dynamic load, the restoring force of the EMWM was related to the deformation history of the specimen. As could be seen from Table 3, with the increase of amplitude, the loss factor η decreased, while the energy dissipation ΔW and the maximum elastic potential energy W increased gradually. With the increase of the amplitude, the maximum deformation of the EMWM specimen became larger, which led to the increase of the number of wire helix contact points in the EMWM, resulting in the increase of energy consumption ΔW. At the same time, the increase of internal friction would lead to the increase of the maximum restoring force, while the simultaneous increase of the maximum restoring force and the maximum displacement would lead to the increase of the maximum elastic potential energy W. However, the growth rate of the maximum elastic potential

Symmetry 2020, 12, 1283 8 of 17 to dynamic load, the restoring force of the EMWM was related to the deformation history of the specimen. As could be seen from Table3, with the increase of amplitude, the loss factor η decreased, while the energy dissipation ∆W and the maximum elastic potential energy W increased gradually. With the increase of the amplitude, the maximum deformation of the EMWM specimen became larger, which led to the increase of the number of wire helix contact points in the EMWM, resulting in the increase of energy consumption ∆W. At the same time, the increase of internal friction would lead to the increase of the maximum restoring force, while the simultaneous increase of the maximum Symmetryrestoring 2020 force, 12, x and FOR the PEER maximum REVIEW displacement would lead to the increase of the maximum elastic8 of 16 potential energy W. However, the growth rate of the maximum elastic potential energy was less than energythat of was the energyless than consumption that of the energy∆W, thusconsumption the energy ΔW, dissipation thus the energy factor dissipationη became smaller factor η with became the smallerincrease with of the the amplitude. increase of the amplitude.

4.3. The The Influence Inﬂuence of of the the Density Density A sinusoidal excitation excitation of of frequency frequency 3 3 Hz Hz and and amplitude 0.8 0.8 mm mm was was applied to to EMWM1, EMWM2 and and EMEM3 EMEM3 respectively, respectively, and and the the experimental experimental results results are are shown shown in in Figure Figure 77 and and Table Table 4.4.

2.0

1.5 EMWM1 EMWM2 1.0 kN EMWM3 0.5

0.0

-0.5

Restoring force/ -1.0

-1.5

-2.0 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 Displacement/mm Figure 7. HysteresisHysteresis loop loop of of symmetry symmetry coated coated damping damping structure structure with different-density diﬀerent-density EMWM (amplitude(amplitude 0.8 mm, frequency 3 Hz, 20 ◦°C).C).

Table 4. Energy dissipation characteristics of coated damping structure with diﬀerent-density EMWM. Table 4. Energy dissipation characteristics of coated damping structure with different-density SpecimenEMWM. Loss Factor η Dissipated Energy ∆W Maximum Elastic Potential Energy W SpecimenEMWM1 Loss Factor 0.2732 η Dissipated Energy 0.4938 ΔW Maximum Elastic Potential 0.2876 Energy W EMWM2 0.2442 0.9696 0.6318 EMWM1 0.2732 0.4938 0.2876 EMWM3 0.2741 1.2217 0.7092 EMWM2 0.2442 0.9696 0.6318 EMWM3 0.2741 1.2217 0.7092 It could be seen from Figure7 and Table4, at the same vibration amplitude and frequency, as the densityIt could of the beEMWM seen from increased, Figure 7 and the lossTable factor 4, at theη decreased same vibration at ﬁrst amplitude and then and increased, frequency, while as the densityenergy dissipationof the EMWM∆ W increased, and the the maximum loss factor elastic η decreased potential at energyfirst and W then increased increased, gradually. while the When energy the dissipationvolume of theΔ W EMWM and the was maximum the same, elastic the greater potential the energy density, W the increased more internal gradually. wire When helixes. the Itvolume means ofthat the the EMWM number was of contactthe same, points the greater of the wirethe density, increases th withe more the internal increase wire of density. helixes. When It means the EMWMthat the numberwas subjected of contact to the points same of external the wire excitation, increases the wi greaterth the theincrease density of ofdensity. EMWM, When the morethe EMWM wire helixes was subjectedinvolved into friction,the same resulting external in excitation, the increase the ofgr frictioneater the force, density energy of EMWM, dissipation the∆ moreW and wire maximum helixes involvedelastic potential in friction, energy resulting W. However, in the increa the growthse of friction rate of theforce, energy energy dissipation dissipation∆ W Δ wasW and diﬀ maximumerent from elasticthat of potential the maximum energy elastic W. However, potential the energy growth W, rate which of the led energy to the ﬂuctuationdissipation ofΔ W the was loss different factor η .from that of the maximum elastic potential energy W, which led to the fluctuation of the loss factor η.

4.4. The Influence of Temperature A sinusoidal excitation of frequency 3 Hz and amplitude 0.8 mm was applied to EMWM2 under different environment temperatures (20 °C, 100 °C, 200 °C, 300 °C), and the experimental results are shown in Figure 8 and Table 5.

Symmetry 2020, 12, 1283 9 of 17

4.4. The Inﬂuence of Temperature A sinusoidal excitation of frequency 3 Hz and amplitude 0.8 mm was applied to EMWM2 under diﬀerent environment temperatures (20 ◦C, 100 ◦C, 200 ◦C, 300 ◦C), and the experimental results are Symmetryshown in2020 Figure, 12, x 8FOR and PEER Table REVIEW5. 9 of 16

2.5 2.0 20℃ 1.5 100℃

kN 1.0 200℃ 0.5 300℃ 0.0 -0.5 -1.0 Restoring force/ -1.5 -2.0 -2.5 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 mm Displacement/ FigureFigure 8. HysteresisHysteresis loop loop of of symmetry symmetry coated coated damping damping structure structure with with different-density diﬀerent-density EMWM (amplitude(amplitude 0.8 0.8 mm, mm, frequency frequency 3 Hz).

Table 5. Energy dissipation characteristics of coated damping structure with diﬀerent-density EMWM. Table 5. Energy dissipation characteristics of coated damping structure with different-density

TemperatureEMWM. /◦C Loss Factor η Dissipated Energy ∆W Maximum Elastic Potential Energy W Temperature/20°C Loss 0.2442 Factor η Dissipated 0.9696Energy ΔW Maximum Elastic 0.6318 Potential Energy W 100 0.2237 0.8675 0.6173 20 0.2442 0.9696 0.6318 200 0.2280 0.8543 0.5964 300100 0.2237 0.2075 0.8675 0.9415 0.6173 0.7223 200 0.2280 0.8543 0.5964 300 0.2075 0.9415 0.7223 It could be seen from Figure8 that at the same frequency and amplitude, when the temperature wasIt less could than be 200 seen◦C, from the maximumFigure 8 that restoring at the sa forceme frequency of the coated and dampingamplitude, structure when the decreases temperature with wasthe increaseless than of 200 temperature; °C, the maximum when the restoring temperature force of was the above coated 200 damping◦C, it increases structure with decreases the increase with theof temperature. increase of temperature; when the temperature was above 200 °C, it increases with the increase of temperature.Table5 shows that the dissipated energy ∆W and the maximum elastic potential energy W decreaseTable at 5 ﬁrstshows and that then the increase dissipated with energy the increase ΔW and ofambient the maximum temperature, elastic andpotential the loss energy factor Wη decreasedecreases at slightly. first and For then austenitic increase stainless with the steel incr wire,ease of from ambient room temperature, temperature toand 200 the◦C, loss the factor friction η decreasescoeﬃcient slightly. (µ) increases For austenitic with the increasestainless of steel temperature, wire, from and room the te elasticmperature modulus to 200 (E) °C, decreases the friction with coefficientthe increase (μ of) increases temperature. with This the increase will lead of to temperature, three results: andO1 the the contact elastic force modulus (F) between (E) decreases wire helixes with thedecreases increase with of temperature. the increase ofThis temperature, will lead toO2 threethe number results: of ① contactthe contact points force (N) increases(F) between with wire the helixesincrease decreases of temperature, with theO3 increasethe elastic of temperature, stiﬀness of the ② wirethe number helix (K Tof) contact decreases. points The (N) dissipated increases energy with is proportional to the coeﬃcient of friction, the contact force and the number of contact points (∆W µ the increase of temperature, ③the elastic stiffness of the wire helix (KT) decreases. The dissipated∝ energyFN). The is proportional maximum elastic to the potential coefficient energy of friction is proportional, the contact to force the coeandﬃ thecient number of friction, of contact thecontact points force,∝ the number of contact points and the elastic stiﬀness of the wire helix (W µ FNKT). While the (ΔW μ FN). The maximum elastic potential energy is proportional to the coefficient∝ of friction, the contactelastic modulusforce, the (E),number the contactof contact force points between and the wire elastic helixes stiffness (F) and of the wire elastic helix stiﬀ (Wness ∝ of μ the FNK wireT). Whilehelix (KtheT) elastic decrease modulus with the (E), increase the contact of temperature. force between Therefore, wire helixes the dissipated (F) and the energy elastic (∆ W)stiffness and theof themaximum wire helix elastic (KT) potential decrease energywith the (W) increase decrease of graduallytemperature. with Therefore, the increase the ofdissipated temperature. energy (ΔW) and theIn themaximum temperature elastic range potential of 200~300 energy◦C, (W) the decrease elastic modulus gradually of the with wire the tends increase to be of stable temperature. gradually, so thatIn thethe contacttemperature force betweenrange of wire 200~300 helixes °C, (F) the and elas thetic elastic modulus stiﬀ nessof the of wirewire helixestends to (K Tbe) changestable gradually,little with theso that increase the contact of temperature. force between Due to wire the oxidationhelixes (F) of and stainless the elastic steel wirestiffness at high of wire temperature, helixes µ (KanT) oxide change ﬁlm little will with be formedthe increase on the of surfacetemperature. of the Du wire,e to and the theoxidation friction of coe staiﬃnlesscient steel ( ) will wire decrease at high temperature,with the increase an oxide of temperature. film will be However, formed on with the the surface increase of ofthe temperature, wire, and the the friction thermal coefficient expansion (μ of) willmetal decrease rubber willwith increase, the increase and the of internal temperature. pores ofHowever, EMWM willwith become the increase smaller of due temperature, to the limitation the thermal expansion of metal rubber will increase, and the internal pores of EMWM will become smaller due to the limitation of the constraining ring, which leads to a rapid increase in the number of contact points between wire helixes in EMWM. Although the elastic modulus E, the contact load F between turns and the elastic stiffness KT change little with the increase of temperature, the dissipated energy ΔW and the maximum elastic potential energy W increase gradually with the increase of temperature due to the increase of the number of contact points caused by thermal expansion.

Symmetry 2020, 12, x FOR PEER REVIEW 10 of 16

5. Modeling and Parameter Identification Symmetry 2020, 12, 1283 10 of 17 5.1. Modeling The restoring force of the symmetrically coated damping structure consists of two parts: of the constraining ring, which leads to a rapid increase in the number of contact points between wire nonlinear elastic restoring force and nonlinear damping restoring force. The damping restoring force helixes in EMWM. Although the elastic modulus E, the contact load F between turns and the elastic consists of two parts: memorized damping force and non-memory damping force, which are related stiﬀness K change little with the increase of temperature, the dissipated energy ∆W and the maximum to hysteresisT loop and deformation velocity respectively. As shown in Figure 9, the nonlinear elastic potential energy W increase gradually with the increase of temperature due to the increase of restoring force of the symmetrically coated damping structure could be described by a dynamic the number of contact points caused by thermal expansion. hysteretic oscillator model. 5. ModelingThe incremental and Parameter constitutive Identiﬁcation relation of nonlinear restoring force z(t) with memory characteristics is 5.1. Modeling =+k s − dz() t1sgn{ zs z() t} dy() t The restoring force of the symmetrically2 coated damping structure consists of two parts: nonlinear z (7) elastic restoring force and nonlineark = dampings restoring force. The damping restoring force consists of s y two parts: memorized damping force ands non-memory damping force, which are related to hysteresis loopwhere and zs deformationwas the memory velocity restoring respectively. force Asof shownslippage; in Figure y(s) was9, the the nonlinear elastic deformation restoring force limit of the of symmetricallyslippage; y(t) was coated the dampingrelative displacement structure could between be described the two by ends a dynamic of the hysteresis hysteretic link. oscillator model.

C z(t)

Symmetry 2020, 12, x FOR PEER REVIEW 10 of 16

5. Modeling and Parameter Identification Figure 9. Hysteretic oscillator model of the symmetricallysymmetrically coated damping structure. 5.1. Modeling TheWhen incrementalrestoring the structure force constitutive was of thesubject relationsymmetrically to the of nonlinearsinusoidal coated restoring excitation, damping force set structurez (yt()t with) as memory consists characteristics of two parts: is nonlinear elastic restoring force andyt nonlinear( ) =+ yh dampsin n(ωing t restoringϕ )oi force. The damping restoring force ( ) ks m ( ) ( ) (8) consists of two parts: memorizeddz dampingt = 2 force1 + sgn andzs non-memory z t dy t damping force, which are related zs − (7) m k = towhere hysteresis y was loopthe amplitude and deformation of displacement;s yvelocitys ωrespectively. was the frequency As shown of sinusoidal in Figure excitation; 9, the nonlinear φ is the restoringinitial phase force angle of theof displacement. symmetrically coated damping structure could be described by a dynamic wherehystereticThen,zs was oscillator a thehyperbolic memory model. restoringfunctional force constitutive of slippage; relationshipy(s) was the was elastic formed deformation between limit the of slippage;memory yrestoring(t) wasThe the forceincremental relative z(t) and displacement theconstitutive displacement between relation y( thet), as twoof shown no endsnlinear in of Figure the restoring hysteresis 10. link.force z(t) with memory characteristicsWhen the is structure was subject to the sinusoidal excitation, set y(t) as k z(t) =+y(st)= ym sin(ω −t + ϕ) (8) dz() t1sgn{ zs z() t} dy() t 2 z (7) where ym was the amplitude of displacement;k = s ω was the frequency of sinusoidal excitation; ϕ is the s y initial phase angle of displacement. 4s 1 whereThen, zs was a hyperbolic the memory functional restoring constitutive force of relationship slippage; y was(s) was formed the between elastic deformation the memory restoringlimit of forceslippage;z(t) andy(t) was the displacement the relative displacementy(t), as shown between ino Figure the 10 two. ends of the(t) hysteresis link. 3 2

Figure 10. Double broken line model.

C Regard memory link and non-memory link as parallel relationship.z(t) Then, the nonlinear restoring force of the structure could be expressed as =+ g n {yt ( ), yt ( ), t } g0 { yt ( ), yt ( )} zt ( ) (9)

Figure 9. Hysteretic oscillatorFigure model 10. Double of the broken symmetrically line model. coated damping structure.

When the structure was subject to the sinusoidal excitation, set y(t) as =+ω ϕ yt( ) ym sin ( t ) (8) where ym was the amplitude of displacement; ω was the frequency of sinusoidal excitation; φ is the initial phase angle of displacement. Then, a hyperbolic functional constitutive relationship was formed between the memory restoring force z(t) and the displacement y(t), as shown in Figure 10.

z(t)

4 1

o (t) 3 2

Figure 10. Double broken line model.

Regard memory link and non-memory link as parallel relationship. Then, the nonlinear restoring force of the structure could be expressed as =+ g n {yt ( ), yt ( ), t } g0 { yt ( ), yt ( )} zt ( ) (9)

Symmetry 2020, 12, 1283 11 of 17

Regard memory link and non-memory link as parallel relationship. Then, the nonlinear restoring force of the structure could be expressed as

n . o n . o gn y(t), y(t), t = g0 y(t), y(t) + z(t) (9)

n . o where the constitutive relation of the non-memory link g0 y(t), y(t) was taken as a linear model, which was expressed as

n . P1 n 1 g0 y(t), y(t) = a0sgn y(t) + an y(t) − y(t) { } { } n=1 n (10) . P2 . n 1 . +b0sgn y(t) + bn y(t) − y(t) { } n=1 where an and bn were the undetermined coeﬃcients of the linear model. The hysteresis loop could be decomposed into the upper and lower half branches, in which . the upper half corresponds to the velocity y(t) > 0 part and the lower half corresponds to the . velocity y(t) < 0 part. The upper and lower half hysteresis loops were ﬁtted by power series polynomials, respectively.

(n+1) (n 1) 2 −2 X 2i 1 X 2i . Qu y(t) = a2i 1 y(t) − + a2i y(t) y(t) > 0 (11) { } − i=1 i=0

(n+1) (n 1) 2 −2 X 2i 1 X 2i . Ql y(t) = a2i 1 y(t) − a2i y(t) y(t) < 0 (12) { } − − i=1 i=0 where Qu was the upper half of the hysteresis loop; Ql was the lower half of the hysteresis loop; ai was the polynomial coeﬃcient of the power series.

(n+1) (n 1) − . P2 2i 1 P2 2i . gn y(t), y(t), t = a2i 1 y(t) − + a2i y(t) sgn(y(t)) (13) { } i=1 − i=0 = Q1(t) + Q2(t)

The dynamic hysteresis loop was decomposed into two parts: Q1(t) and Q2(t), in which Q1(t) was a single-valued nonlinear function and Q2(t) was a double-valued nonlinear closed curve. Ignoring the high-order damping force related to the second power of velocity and the complex damping force related to deformation and deformation velocity at the same time. The double-valued . nonlinear closed curve could be decomposed into linear viscous damping force cy(t) and hysteretic damping force z(t). At the same time, the high-order nonlinear elastic restoring force was ignored. Therefore, the nonlinear constitutive relation of pipeline coated damping structure can be described as

n . o 3 . gn y(t), y(t), t = k1 y(t) + k3 y(t) + cy(t) + z(t) (14)

As shown in Figure 11, the nonlinear functional constitutive relation of the symmetrically coated damping structure could be obtained by accurately decomposing the force-displacement curve obtained from the test. Symmetry 2020, 12, x FOR PEER REVIEW 11 of 16

where the constitutive relation of the non-memory link g{(),()} was taken as a linear model, which was expressed as n =+1 n −1 g 00{(),()}yt yt a sgn{()} yt an yt () yt () n =1 (10) n2 ++n −1 bytbytyt0 sgn{ ( )} n ( ) ( ) n =1

where an and bn were the undetermined coefficients of the linear model. The hysteresis loop could be decomposed into the upper and lower half branches, in which the upper half corresponds to the velocity () >0 part and the lower half corresponds to the velocity () <0 part. The upper and lower half hysteresis loops were fitted by power series polynomials, respectively.

(1)nn+− (1) 22 =>21ii− 2 (11) Qytu { ( )} a21ii− yt ( ) + ayt 2 ( ) yt ( ) 0 ii==10

(1)nn+− (1) 22 =−21ii− 2 < (12) Qytl { ( )} a21ii− yt ( ) ayt 2 ( ) yt ( ) 0 ii==10

where Qu was the upper half of the hysteresis loop; Ql was the lower half of the hysteresis loop; ai was the polynomial coefficient of the power series.

(1)nn+− (1) 22 = 21ii− 2 g n {yt (),(),} yt t a21ii− yt () + a 2 yt () sgn(()) yt ii==10 (13) =+ Qt12 ( ) Q ( t )

The dynamic hysteresis loop was decomposed into two parts: Q1(t) and Q2(t), in which Q1(t) was a single-valued nonlinear function and Q2(t) was a double-valued nonlinear closed curve. Ignoring the high-order damping force related to the second power of velocity and the complex damping force related to deformation and deformation velocity at the same time. The double-valued nonlinear closed curve could be decomposed into linear viscous damping force () and hysteretic damping force (). At the same time, the high-order nonlinear elastic restoring force was ignored. Therefore, the nonlinear constitutive relation of pipeline coated damping structure can be described as =+3 ++ g n {yt (),(),} yt t kyt13 () k yt () cyt () zt () (14) As shown in Figure 11, the nonlinear functional constitutive relation of the symmetrically coated Symmetrydamping2020 structure, 12, 1283 could be obtained by accurately decomposing the force-displacement 12curve of 17 obtained from the test.

Figure 11.11. Exact decomposition of hysteresis loop.

It could be seen from Figure 11 that the nonlinear damping force was surrounded by a double-valued nonlinear closure curve, which reﬂects the complex damping composition of EMWM and was described as . α n . o Fc = c y(t) sgn y(t) (15) where c was the damping coeﬃcient and α was the damping component factor. The larger α is, the more sensitive the damping force is to the change of velocity. Considering the variation of elastic restoring force and damping force with deformation amplitude and frequency, and ignoring the high-order nonlinear elastic restoring forces that are above tertiary force, the nonlinear constitutive function of symmetrically coated damping structure for EMWM of pipelines can be described as [13]

n . o 3 . α(A, f ) n . o gn y(t), y(t), t = k1(A)y(t) + k3(A)y(t) + c(A, f ) y(t) sgn y(t) (16) where k1 was the primary linear stiﬀness coeﬃcient; k3 was the tertiary nonlinear stiﬀness coeﬃcient; A was the amplitude, f was the frequency, y(t) was the vibration displacement, The results of the energy dissipation tests shown that the dynamic characteristics of EMWM were aﬀected by temperature, especially the dynamic stiﬀness. Therefore, the elastic restoring force could be expressed as a function related to both amplitude and temperature.

n . o 3 . α(A, f ) n . o gn y(t), y(t), t = k1(A, T)y(t) + k3(A, T)y(t) + c(A, f ) y(t) sgn y(t) (17)

5.2. Parameter Identiﬁcation The displacement and its corresponding restoring force test data under various test conditions were extracted from the energy dissipation experimental data, and the hysteresis loop was ﬁtted by the least square method, and the power series polynomial was obtained.

(n+1) (n 1) 2 −2 . X 2i 1 X 2i . gn y(t), y(t), t = a2i 1 y(t) − + a2i y(t) sgn y(t) (18) { } − { } i=1 i=0 where the number of terms n (odd) taken by the power series polynomial was selected according to the ﬁtting accuracy, and the odd term coeﬃcient was the nonlinear elastic restoring force stiﬀness coeﬃcient to be identiﬁed. The curves of the ﬁrst-order stiﬀness coeﬃcient k1(A,T) with amplitude and temperature and the third-order stiﬀness coeﬃcient k3(A,T) with amplitude and temperature were drawn respectively, as shown in Figure 12. Symmetry 2020, 12, x FOR PEER REVIEW 12 of 16

It could be seen from Figure 11 that the nonlinear damping force was surrounded by a double- valued nonlinear closure curve, which reflects the complex damping composition of EMWM and was described as = α Fc cyt() sgn{()} yt (15) where c was the damping coefficient and α was the damping component factor. The larger α is, the more sensitive the damping force is to the change of velocity. Considering the variation of elastic restoring force and damping force with deformation amplitude and frequency, and ignoring the high-order nonlinear elastic restoring forces that are above tertiary force, the nonlinear constitutive function of symmetrically coated damping structure for EMWM of pipelines can be described as [13] α =+3 + (,)Af g n{yt (),(),} yt t k13 ( Ayt ) () k ( Ayt ) () cA ( , f ) yt () sgn{()} yt (16) where k1 was the primary linear stiffness coefficient; k3 was the tertiary nonlinear stiffness coefficient; A was the amplitude, f was the frequency, y(t) was the vibration displacement, The results of the energy dissipation tests shown that the dynamic characteristics of EMWM were affected by temperature, especially the dynamic stiffness. Therefore, the elastic restoring force could be expressed as a function related to both amplitude and temperature. α =+3 + (,)Af g n{yt (),(),} yt t k13 ( AT , ) yt () k ( AT , ) yt () cA ( , f ) yt () sgn{()} yt (17)

5.2. Parameter Identification The displacement and its corresponding restoring force test data under various test conditions were extracted from the energy dissipation experimental data, and the hysteresis loop was fitted by the least square method, and the power series polynomial was obtained.

（）nn+−11（） 22 =+21ii− 2 g n {yt (),(),} yt t a21ii− yt() a 2 yt() sgn{()} yt (18) ii==10 where the number of terms n (odd) taken by the power series polynomial was selected according to the fitting accuracy, and the odd term coefficient was the nonlinear elastic restoring force stiffness coefficient to be identified.

Symmetry 2020, 12, 1283 13 of 17

(a) (b)

Figure 12. (a) the first-order ﬁrst-order stistiffnessﬀness coecoefficientﬃcient kk11(A,T) with amplitude and temperature; and (b) the third-order stistiffnessﬀness coecoefficientﬃcient kk33((A,,T) with amplitude andand temperature.temperature.

It can be seen from Figure 12a that the ﬁrst-order stiﬀness coeﬃcient k1(A,T) decreases gradually with the increase of vibration amplitude (A), while k1(A,T) increases with the increase of temperature. The k1(A,T) was obtained by parameter ﬁtting in the form of a power function.

0.2218 0.05131 k1(A, T) = 2.38A− T− (19)

It can also be seen from Figure 12b that the third-order stiﬀness coeﬃcient k3(A,T) decreases gradually with the increase of vibration amplitude (A), and then tends to be stable. With the increase of temperature, the third-order stiﬀness coeﬃcient k3(A,T) decreases at ﬁrst and then increases with the increase of temperature. The power function form was used to ﬁt these parameters.

1.228 0.1374 k3(A, T) = 1112A− T− (20)

Therefore, the nonlinear elastic restoring force could be expressed as

3 0.2218 0.05131 1.228 0.1374 3 F (y) = k y + k y = 2.38 A− T− y + 1112 A− T− y (21) k 1 3 × × × × × ×

By subtracting the nonlinear elastic restoring force Fk(y) from the test data of the hysteretic restoring force measured by the energy dissipation test gn(i), the nonlinear damping force Fc (yi) could be obtained. 3 Fc(y ) = gn(i) k y k y (22) i − 1 i − 3 i Through the parameter identiﬁcation of the hysteresis loop under each working condition, the damping coeﬃcient at room temperature could be obtained. Figure 13 shows the spatial surface diagram of the damping coeﬃcient c(A, f ) varying with amplitude (A) and frequency (f ). Symmetry 2020, 12, x FOR PEER REVIEW 13 of 16

It can be seen from Figure 12a that the first-order stiffness coefficient k1(A,T) decreases gradually with the increase of vibration amplitude (A), while k1(A,T) increases with the increase of temperature. The k1(A,T) was obtained by parameter fitting in the form of a power function. = −−0.2218 0.05131 kAT1 (,) 2.38 A T (19)

It can also be seen from Figure 12b that the third-order stiffness coefficient k3(A,T) decreases gradually with the increase of vibration amplitude (A), and then tends to be stable. With the increase of temperature, the third-order stiffness coefficient k3(A,T) decreases at first and then increases with the increase of temperature. The power function form was used to fit these parameters. = −−1.228 0.1374 kAT3 ( , ) 1112 A T (20) Therefore, the nonlinear elastic restoring force could be expressed as =+3 = × -0.2218 × -0.05131 ×+× -1.228 × -0.1374 × 3 Fk (ykyky )13 2.38 A T y 1112 A T y (21)

By subtracting the nonlinear elastic restoring force Fk(y) from the test data of the hysteretic restoring force measured by the energy dissipation test gn(i), the nonlinear damping force Fc (yi) could be obtained. =−−3 F cn()ygikykyiii () 13 (22) Through the parameter identification of the hysteresis loop under each working condition, the Symmetrydamping2020 coefficient, 12, 1283 at room temperature could be obtained. Figure 13 shows the spatial surface14 of 17 diagram of the damping coefficient c(A, f) varying with amplitude (A) and frequency (f).

(a) (b)

Figure 13. ((a) the the damping damping coefficient coeﬃcient c(A, f)) with with amplitude amplitude and and temperature; temperature; and ( b) the third-order

stistiffnessﬀness coecoefficientﬃcient kk33(A,T) with amplitude and temperature.

As illustratedillustrated inin FigureFigure 13 13 that that the the variation variation of of damping damping coe coefficientﬃcient c(A ,cf(A) with, f) with amplitude amplitude (A) and(A) frequencyand frequency (f ) is complex.(f) is complex. Therefore, Therefore, to ﬁt the to surfacefit the surf accurately,ace accurately, the damping the damping coeﬃcient coefficient was described was bydescribed the combination by the combination of the sine function,of the sine exponential function, exponential function and function power function. and power Then, function. the damping Then, coethe ﬃdampingcient could coefficient be obtained. could be obtained.

−⋅0.7084 cAf( , )=+⋅⋅+ e1.9971.997f f 0.7084 sin(0.01938 A2.7342.734 f ) 0.03811 c(A, f ) = e− · + sin(0.01938 A f ) + 0.03811 (23) · · The same method was used to identify the parameters of the damping component factor α (A, The same method was used to identify the parameters of the damping component factor α f). Figure11 shows the damping component factor α(A, f) at different amplitudes and frequencies was (A, f ). Figure 11 shows the damping component factor α(A, f ) at diﬀerent amplitudes and frequencies obtained. was obtained. It can be seen from Figure 13 that the variation of damping component factor α(A, f) with It can be seen from Figure 13 that the variation of damping component factor α(A, f ) with amplitude (A) and frequency (f) is also complex. Through analysis, it was found that the power amplitude (A) and frequency (f ) is also complex. Through analysis, it was found that the power function could be used to describe its variation with amplitude (A) and frequency (f) with high fitting function could be used to describe its variation with amplitude (A) and frequency (f ) with high ﬁtting accuracy. In addition, then α(A, f) could be expressed as follows: accuracy. In addition, then α(A, f ) could be expressed as follows:

0.4822 0.01232 α(A, f ) = 0.4718 A− f (24) · · As a consequence of the above, the nonlinear constitutive function of symmetrically coated damping structure for EMWM of pipelines could be obtained by substituting Equation (21), Equation (23) and Equation (24) into Equation (17), and could be expressed as

n . o 3 . α(A, f ) n . o gn y(t), y(t), t = k1(A, T)y(t) + k3(A, T)y(t) + c(A, f ) y(t) sgn y(t) 0.2218 0.05131 1.228 0..1374 3 = 2.38 A− T− y + 1112 A− T− y 1.997· f 0.7084 · · 2.734 · · · (25) +(e− · + sin(0.01938 A f ) + 0.03811) 0.4822 0.01232 · · . 0.4718 A− f n . o y(t) · · sgn y(t) · 5.3. Model Veriﬁcation To verify the accuracy of the nonlinear constitutive function of symmetrically coated damping structure for EMWM of pipeline and the accuracy of the parameter identiﬁcation algorithm, the nonlinear functional constitutive relation model was used to predict the hysteresis curve of restoring force under diﬀerent temperature (T), diﬀerent exciting amplitude (A) and diﬀerent exciting frequency (f ), and compared with the experimental data. The results of the comparison are shown in Figure 14. Symmetry 2020, 12, x FOR PEER REVIEW 14 of 16

− α(Af , )=⋅ 0.4718 A0.4822 ⋅ f 0.01232 (24)

As a consequence of the above, the nonlinear constitutive function of symmetrically coated damping structure for EMWM of pipelines could be obtained by substituting Equation (21), Equation (23) and Equation (24) into Equation (17), and could be expressed as α =+3 + (,)Af g n {(),(),}yt yt t k13 (, AT )() yt k (, AT )() yt cA ( , f ) yt () sgn{()} yt −− −− = 2.38⋅⋅AT0.2218 0.05131 ⋅+⋅⋅⋅ y 1112 AT 1.228 0..1374 y 3 0.7084 (25) +(e −⋅1.997 f + sin(0.01938⋅⋅+Af2.734 ) 0.03811) 0.4718⋅⋅Af− 0.4822 0.01232 ⋅ yt ( ) sgn{ yt ( )}

5.3. Model Verification To verify the accuracy of the nonlinear constitutive function of symmetrically coated damping structure for EMWM of pipeline and the accuracy of the parameter identification algorithm, the nonlinear functional constitutive relation model was used to predict the hysteresis curve of restoring force under different temperature (T), different exciting amplitude (A) and different exciting Symmetryfrequency2020 (f,),12 and, 1283 compared with the experimental data. The results of the comparison are shown15 of in 17 Figure 14.

(a) (b) (c)

(d) (e) (f)

(g) (h)

Figure 14. Comparison between the the measured measured curve curve and and estimated estimated curve: curve: (a (a) )AA == 0.20.2 mm, mm, f =f =2 Hz,2 Hz, T

T= 20= 20 °C;◦ C(b;() bA) =A 0.2= 0.2mm, mm, f = 3f Hz,= 3 Hz,T = 20T =°C;20 (c◦)C; A (=c )0.5A mm,= 0.5 f mm,= 4 Hz,f = T4 = Hz, 20 °C;T = (d20) A◦ C;= 0.5 (d )mm,A = f0.5 = 3 mm, Hz, fT= = 3100 Hz °C;, T (=e) 100A = ◦0.5C; mm, (e) A f = 30.5 Hz, mm, T = f200= 3°C; Hz (,f)T A= = 2000.5 mm,◦C;( f )= A3 Hz,= 0.5 T mm,= 300 f°C;= 3(g Hz,) A =T 0.8= mm,300 ◦ fC; = (3g Hz,) A =T 0.8= 20 mm, °C; f(h=) 3A Hz,= 1 mm,T = 20 f =◦ 5C; Hz, (h) TA == 201 mm,°C. f = 5 Hz, T = 20 ◦C.

When the amplitude is small (<0.8 mm), the estimated results were consistent with the measured results. With the increase of the amplitude (0.8 mm, 1 mm), the deviation between the estimated results and the measured results increases, but it could still eﬀectively predict the changing trend of load and displacement. It can be seen from Figure 14 that the proposed model can eﬀectively predict the eﬀect of temperature on the energy dissipation characteristics of the structure. Through the comparison of the measured data and the estimated data, we found that:

(1) The ﬁrst-order stiﬀness coeﬃcient k1(A,T) and the third-order stiﬀness coeﬃcient k3(A,T) could describe the stiﬀness variation for symmetrically coated damping structure for EMWM of pipelines, and their variation with amplitude (A) and temperature (T) were consistent with the experimental results. (2) The damping coeﬃcient c and damping component factor α could describe the energy dissipation characteristics of symmetrically coated damping structure for EMWM of pipelines, and their variation with amplitude (A) and frequency (f) were consistent with the experimental results. (3) The proposed nonlinear functional constitutive relation model of symmetrically coated damping structure for EMWM of pipelines could properly describe the variation of restoring force with temperature (T), amplitude (A), frequency (f), displacement (y). Symmetry 2020, 12, 1283 16 of 17

6. Conclusions In this paper, a symmetrically coated damping structure for EMWM of high temperature pipelines was designed and tested to investigate the eﬀect of the vibration amplitude, frequency, density, and temperature on the energy dissipation characteristics of the structure. A revised model of nonlinear elastic restoring force was set up, which could describe the energy dissipation characteristics of the structure. The main conclusions, which can be drawn from the conducted research, are as follows:

(1) In the rage of 1–5 Hz, the inﬂuence of vibration frequency on symmetrically coated damping structure for EMWM of pipelines can be ignored. (2) With the increase of density, the loss factor (η) decreases at ﬁrst and then increases, while the dissipated energy (∆W) and the maximum elastic potential energy (W) increase gradually. (3) With the increase of temperature, the energy dissipation ∆ W and the maximum elastic potential energy W decrease in the temperature range of 20–200 ◦C, and then increase in the temperature range of 200–300 ◦C, while the loss factor η decreases slightly in the temperature range of 20–300 ◦C. (4) Through the decomposition of the hysteretic loop, a revised energy dissipation model of symmetrically coated damping structure for EMWM of pipelines with temperature parameters was set up and was proved to properly describe the dynamic characteristics of the symmetrically coated damping structure for EMWM of pipelines by parameter identiﬁcation and simulation.

Author Contributions: Conceptualization and Methodology, F.J., Z.D., Y.W. and H.B.; Data curation, F.J., Z.D. and B.Z.; validation, F.J., Z.D. and Y.W.; writing—original draft preparation, F.J., Z.D., and Y.W.; writing—review and editing, Y.W., H.B. and Y.S. All authors have read and agreed to the published version of the manuscript. Funding: This work was supported by the National Natural Science Foundation of China (grant number 51805086) and the Natural Science Foundation of Fujian Province, China (grant number 2018J01763). Conﬂicts of Interest: The authors declare no conﬂict of interest.

References

1. Xie, Z.; Sebald, G.; Guyomar, D. Temperature dependence of the elastocaloric eﬀect in natural rubber. Phys. Lett. A 2017, 81, 2112–2116. [CrossRef] 2. Sakulkaew, K.; Thomas, A.G.; Busﬁeld, J.J.C. The eﬀect of temperature on the tearing of rubber. Polym. Test. 2013, 32, 86–93. [CrossRef] 3. Ma, Y.H.; Zhang, Q.C.; Wang, Y.F.; Hong, J.; Scarpa, F. Topology and mechanics of metal rubber via x-ray tomography. Mater. Des. 2019, 181, 108067. [CrossRef] 4. Zhang, Q.C.; Zhang, D.Y.; Dobah, Y.; Scarpa, F.; Fraternali, F.; Skelton, R.E. Tensegrity cell mechanical metamaterial with metal rubber. Appl. Phys. Lett. 2018, 113, 031906. [CrossRef] 5. Hu, J.L.; Du, Q.; Gao, J.H.; Kang, J.Y.; Guo, B.T. Compressive mechanical behavior of multiple wire metal rubber. Mater. Des. 2018, 140, 231–240. [CrossRef] 6. Zarzour, M.; Vance, J. Experimental evaluation of a metal mesh bearing damper. J. Eng. Gas Turbines Power 2000, 122, 326–329. [CrossRef] 7. Chandrasekhar, K.; Rongong, J.; Cross, E. Mechanical behaviour of tangled metal wire devices. Mech. Syst. Signal Process. 2019, 118, 13–29. [CrossRef] 8. Wu, Y.W.; Li, S.Z.; Bai, H.B.; Jiang, L.; Cheng, H. Experimental and constitutive model on dynamic compressive mechanical properties of entangled metallic wire material under low-velocity impact. Materials 2020, 13, 1396. [CrossRef][PubMed] 9. DellaCorte, C. Oil-free shaft support system rotordynamics: Past, present and future challenges and opportunities. Mech. Syst. Signal Process. 2012, 29, 67–76. [CrossRef] 10. Lee, Y.B.; Park, D.J.; Kim, C.H.; Ryu, K. Rotordynamic characteristics of a micro turbo generator supported by air foil bearings. J. Micromech. Microeng. 2007, 17, 297–303. [CrossRef] 11. Zhizhkin, A.M.; Zrelov, V.A.; Zrelov, V.V.; Ardakov, A.Y.; Osipov, A.A. Rotor sealings based on a metal–rubber elastic porous material for turbomachinery. J. Frict. Wear 2018, 39, 259–263. [CrossRef] Symmetry 2020, 12, 1283 17 of 17

12. Ding, Z.Y.; Bai, H.B.; Wu, Y.W.; Ren, Z.Y.; Shao, Y.C. A constitutive model of plate-like entangled metallic wire material in wide temperature range. Materials 2019, 12, 2538. [CrossRef][PubMed] 13. Hou, J.F.; Bai, H.B.; Li, D.W. Damping capacity measurement of elastic porous wire-mesh material in wide temperature range. J. Mater. Process. Tech. 2008, 206, 412–418. [CrossRef] 14. Zhu, Y.; Wu, Y.W.; Bai, H.B.; Ding, Z.Y.; Shao, Y.C. Research on vibration reduction design of foundation with entangled metallic wire material under high temperature. Shock Vib. 2019, 2019, 1–16. [CrossRef] 15. Bai, H.B.; Lu, C.H.; Cao, F.L.; Li, D.W. Material and Engineering Application of Metal Rubber; Science Press: Beijing, China, 2014. 16. Xiao, K.; Bai, H.B.; Xue, X.; Wu, Y.W. Damping characteristics of metal rubber in the pipeline coating system. Shock Vib. 2018, 2018, 1–11. [CrossRef] 17. Wu, K.A.; Bai, H.B.; Xue, X.; Li, T.; Li, M. Energy dissipation characteristics and dynamic modeling of the coated damping structure for metal rubber of bellows. Metals 2018, 8, 562. [CrossRef] 18. Ulanov, A.M.; Bezborodov, S.A. Calculation method of pipeline vibration with damping supports made of the MR material. Procedia Eng. 2016, 150, 101–106. [CrossRef] 19. Wu, Y.W.; Jiang, L.; Bai, H.B.; Lu, C.H.; Li, S.Z. Mechanical Behavior of Entangled Metallic Wire Materials under Quasi-Static and Impact Loading. Materials 2019, 12, 3392. [CrossRef] 20. Wang, Y.Y.; Bai, H.B.; Liu, Y.F. Study on Microscopic Characteristics of Compressive Properties of Metal Rubber Materials. Mech. Sci. Technol. 2011, 30, 404–407.

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