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

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.

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)” [3,4,5], “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 [9,10,11]. 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 structure is a kind of elastic damping pipe clip. Symmetrically installed EMWMs are the main energy dissipation elements of the structure.

Because of the special spatial structure of EMWM, its energy dissipation mechanism can be divided into three kinds. The first kind is that the internal wire helixes of EMWM slip, friction, and squeeze under dynamic load, resulting in friction energy dissipation at each contact point of the adjacent wire helixes. The second kind is that the change of the spatial position of the internal wire helixes of EMWM under external load cannot be fully recovered, resulting in viscoelastic energy dissipation. The third kind is due to the pores of EMWM so that it will squeeze out the internal air or inhale the external air during deformation, resulting in energy dissipation [8,19]. When the deformation of EMWM exceeds the micron level, the internal wire helixes of EMWM will slip and lead to energy dissipation [20]. Therefore, the damping energy dissipation mechanism of the symmetrically coated damping structure for EMWM of pipelines is as follows: when the pipeline is excited by external excitation or due to the transmission of fluid inside the pipeline, there will be a relative displacement change between the pipe and the constraining ring, which means that the deformation of EMWM will be changed, then the vibration energy will be dissipated.

3. Specimen and Test Design

3.1. EMWM Specimen

The plate-like EMWM specimen was made of 304 (06Cr19Ni10) austenitic stainless steel wire with a diameter of 0.3 mm. The manufacture of the plate-like EMWM was referring to [14,15]. The size of the plat-like entangled metallic wire material (EMEM) specimen is 175 mm × 40 mm × 4.5 mm. Three groups of specimens were made and tested, each group contains three plate-like EMWM specimens with same parameters and properties. The manufactured specimen and its manufacturing parameters are shown in Figure 2 and

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

respectively.

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.

The high temperature environment was generated by a self-made quartz lamp heating system, which is shown in Figure 3. The heating system consists of 12 quartz lamps with a length of 400 mm and a diameter of 15 mm as a heat source. Each quartz lamp has a rated power of 1500 W and was installed in parallel on the quartz lamp fixing plate. The quartz lamp fixing plate was made of 310 s stainless steel and could withstand a high temperature of up to 1300 °C. The surface of the quartz lamp fixing plate was polished to reflect the quartz lamplight and could to heat the test device quickly. A layer of aluminum silicate insulation layer was sandwiched between the quartz lamp fixing plate and the metal thermal insulation board, which could play a better role in thermal insulation.

An electro-hydraulic servo dynamic testing machine (SDS-200, Sinotest Equipment Co., ltd., Changchun, 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.

3.3. Test Methods

After all the equipment was assembled and adjusted, the high temperature dynamic mechanical experiments of symmetrically coated damping structure with different densities were carried out 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 (ΔW) 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.

Figure 4 is the sketch of the force-displacement hysteresis loop.

In the experiment, the sampling frequency and loading frequency were set to f₀ and f respectively, in which the number of sampling points in a vibration period is N = f₀/f. In this research, f₀ = 2500 Hz.

Define the measured restoring force as F_i, the displacement as X_i, I = 1, 2 … N, the displacement excitation could be expressed as

$$X_i=X_0\cos \left(\omega t+{\psi }_0\right)$$

(1)

where ψ₀ was the initial phase angle of displacement change, X₀ was the vibration displacement amplitude, and ω was the angular frequency.

By substituting the measured restoring force and displacement values into Formula (1), the displacement dispersion value could be expressed as

$$X_i=X_0\cos \left(\frac{2\pi i}{N}+\alpha \right),i=1,2,\dots ,N$$

(2)

The area of the hysteresis loop (ΔW) represents the dissipated energy in one loading and unloading cycle, and could be obtained by the integral of the restoring force in the displacement direction as follow:

$$\begin{array}{ll}\Delta W& =\oint Fdx\\ & =\oint Fd\left[X_0\cos \left(\omega t+\alpha \right)\right]\\ & =-\omega X_0{\displaystyle \underset{0}{\overset{T}{{\displaystyle \int }}}}Fsin\left(\omega t+\alpha \right)dt\\ & =-\frac{2\pi X_0}{N}{\displaystyle {\displaystyle \sum }_{i=1}^N}{F}_i\sin \left(\frac{2\pi i}{N}+\alpha \right)\end{array}$$

(3)

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:

$$W=\frac{1}{2}k{X}_0{}^2=\frac{1}{2}{F}_0{X}_0$$

(4)

$$k=\frac{F_{\max }-F_{\min }}{2}$$

(5)

where k was the dynamic average stiffness, F_max was the maximum elastic restoring force, F_min was the minimum elastic restoring force, and X₀ was the vibration amplitude.

Therefore, the loss factor was defined as:

$$\eta =\frac{\Delta W}{2\pi W}$$

(6)

4. Energy Dissipation Test Results and Discussion

4.1. The Influence of the Vibration Frequency

To investigate the effect of vibration frequency on the energy dissipation characteristics of the symmetrically coated damping structure for EMWM of pipelines, a series of sinusoidal excitation with an amplitude of 0.8 mm and different frequencies (1 Hz, 2 Hz, 3 Hz, 4 Hz, and 5 Hz) was applied to the coated damping structure with EMWM2 at room temperature. The experimental data were drawn as force-displacement curves, as shown in Figure 5. The energy dissipation characteristics of coated damping structure at different frequencies are shown in

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

Frequency/Hz	Loss Factor η	Dissipated Energy ΔW	Maximum Elastic Potential Energy W
1	0.2404	0.9813	0.6497
2	0.2402	0.9744	0.6456
3	0.2442	0.9696	0.6318
4	0.2480	0.9729	0.6243
5	0.2515	0.9790	0.6195

.

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 dissipation characteristics of the coated damping structure was very small under low-frequency 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.

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.

The maximum elastic potential energy (W) was proportional to the maximum restoring force and the maximum displacement, when the amplitude was constant, it was only related to the maximum restoring force. With the increase of the frequency, the slip of internal wire helixes of the EMWM became more and more insufficient, which made the internal friction force decrease gradually. With the decrease of internal friction, the maximum restoring force of metal rubber became smaller, which led to a decrease of the maximum elastic potential energy (W) with the increase of vibration frequency.

It can be seen from Equation (6) that the loss factor η is directly proportional to the energy consumption ΔW and inversely proportional to the maximum elastic potential energy W. When the energy dissipation ΔW and the maximum elastic potential energy W did not change much, the loss factor η would not change much.

4.2. The Influence of the Vibration Amplitude

A series of sinusoidal excitation with a frequency of 3 Hz and different amplitudes (0.2 mm, 0.5 mm, 0.8 mm, 1 mm) were applied to the coated damping structure with EMWM2 at room temperature. The experimental data were drawn as force-displacement curves, as shown in Figure 6. The loss factor η, dissipated energy ΔW and maximum elastic potential energy W could be derived from Figure 5 and are listed in

Table 3. Energy dissipation characteristics of coated damping structure at different amplitudes.

Amplitude/mm	Loss Factor η	Dissipated Energy ΔW	Maximum Elastic Potential Energy W
0.2	0.3708	0.1281	0.0550
0.5	0.3126	0.4905	0.2498
0.8	0.2442	0.9696	0.6318
1	0.2004	1.3499	1.0723

.

As illustrated in Figure 6, when the amplitudes were different, the curves did not coincide with each other at the same displacement. It indicated that when the coated damping structure 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 energy was less than that of the energy consumption ΔW, thus the energy dissipation factor η became smaller with the increase of the amplitude.

4.3. The Influence of the Density

A sinusoidal excitation of frequency 3 Hz and amplitude 0.8 mm was applied to EMWM1, EMWM2 and EMEM3 respectively, and the experimental results are shown in Figure 7 and

Table 4. Energy dissipation characteristics of coated damping structure with different-density EMWM.

Specimen	Loss Factor η	Dissipated Energy ΔW	Maximum Elastic Potential Energy W
EMWM1	0.2732	0.4938	0.2876
EMWM2	0.2442	0.9696	0.6318
EMWM3	0.2741	1.2217	0.7092

.

It could be seen from Figure 7 and Table 4, at the same vibration amplitude and frequency, as the density of the EMWM increased, the loss factor η decreased at first and then increased, while the energy dissipation Δ W and the maximum elastic potential energy W increased gradually. When the volume of the EMWM was the same, the greater the density, the more internal wire helixes. It means that the number of contact points of the wire increases with the increase of density. When the EMWM was subjected to the same external excitation, the greater the density of EMWM, the more wire helixes involved in friction, resulting in the increase of friction force, energy dissipation ΔW and maximum elastic potential energy W. However, the growth rate of the energy dissipation Δ W was different 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. Energy dissipation characteristics of coated damping structure with different-density EMWM.

Temperature/°C	Loss Factor η	Dissipated Energy ΔW	Maximum Elastic Potential Energy W
20	0.2442	0.9696	0.6318
100	0.2237	0.8675	0.6173
200	0.2280	0.8543	0.5964
300	0.2075	0.9415	0.7223

.

It could be seen from Figure 8 that at the same frequency and amplitude, when the temperature was less than 200 °C, the maximum restoring force of the coated damping structure decreases with the increase of temperature; when the temperature was above 200 °C, it increases with the increase of temperature.

Table 5 shows that the dissipated energy ΔW and the maximum elastic potential energy W decrease at first and then increase with the increase of ambient temperature, and the loss factor η decreases slightly. For austenitic stainless steel wire, from room temperature to 200 °C, the friction coefficient (μ) increases with the increase of temperature, and the elastic modulus (E) decreases with the increase of temperature. This will lead to three results: ① the contact force (F) between wire helixes decreases with the increase of temperature, ② the number of contact points (N) increases with the increase of temperature, ③ the elastic stiffness of the wire helix (K_T) decreases. The dissipated energy is proportional to the coefficient of friction, the contact force and the number of contact points (ΔW ∝ μ FN). The maximum elastic potential energy is proportional to the coefficient of friction, the contact force, the number of contact points and the elastic stiffness of the wire helix (W ∝ μ FNK_T). While the elastic modulus (E), the contact force between wire helixes (F) and the elastic stiffness of the wire helix (K_T) decrease with the increase of temperature. Therefore, the dissipated energy (ΔW) and the maximum elastic potential energy (W) decrease gradually with the increase of temperature.

In the temperature range of 200~300 °C, the elastic modulus of the wire tends to be stable gradually, so that the contact force between wire helixes (F) and the elastic stiffness of wire helixes (K_T) change little with the increase of temperature. Due to the oxidation of stainless steel wire at high temperature, an oxide film will be formed on the surface of the wire, and the friction coefficient (μ) will decrease with the increase of temperature. However, with the increase of temperature, 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 K_T 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.

5. Modeling and Parameter Identification

5.1. Modeling

The restoring force of the symmetrically coated damping structure consists of two parts: nonlinear elastic restoring force and nonlinear damping restoring force. The damping restoring force consists of two parts: memorized damping force and non-memory damping force, which are related to hysteresis loop and deformation velocity respectively. As shown in Figure 9, the nonlinear restoring force of the symmetrically coated damping structure could be described by a dynamic hysteretic oscillator model.

The incremental constitutive relation of nonlinear restoring force z(t) with memory characteristics is

$$\begin{array}{l}dz\left(t\right)=\frac{k_s}{2}\left[1+\mathrm{sgn}\left\{z_s-\left|z\left(t\right)\right|\right\}\right]dy\left(t\right)\\ k_s=\frac{z_s}{y_s}\end{array}$$

(7)

where z_s was the memory restoring force of slippage; y(s) was the elastic deformation limit of slippage; y(t) was the relative displacement between the two ends of the hysteresis link.

When the structure was subject to the sinusoidal excitation, set y(t) as

$$y\left(t\right)=y_m\sin \left(\omega t+\phi \right)$$

(8)

where y_m 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.

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

$$g_n\{y(t),\dot{y}(t),t\}=g_0\{y(t),\dot{y}(t)\}+z(t)$$

(9)

where the constitutive relation of the non-memory link $g_0\left\{y\left(t\right),\dot{y}\left(t\right)\right\}$ was taken as a linear model, which was expressed as

$$\begin{array}{ll}{g}_0\{y(t),\dot{y}(t)\}=& a_0\mathrm{sgn}\{y(t)\}+{\displaystyle {\displaystyle \sum _{n=1}^{n_1}}{a}_n{\left|y(t)\right|}^{n-1}y(t)}\\ & +b_0\mathrm{sgn}\{\dot{y}(t)\}+{\displaystyle {\displaystyle \sum _{n=1}^{n_2}}{b}_n{\left|\dot{y}(t)\right|}^{n-1}\dot{y}(t)}\end{array}$$

(10)

where a_n and b_n 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 $\dot{y}\left(t\right)>0$ part and the lower half corresponds to the velocity $\dot{y}\left(t\right)<0$ part. The upper and lower half hysteresis loops were fitted by power series polynomials, respectively.

$$Q_u\{y(t)\}={\displaystyle {\displaystyle \sum _{i=1}^{\frac{(n+1)}{2}}}{a}_{2i-1}}y{(t)}^{2i-1}+{\displaystyle {\displaystyle \sum _{i=0}^{\frac{(n-1)}{2}}}{a}_{2i}}y{(t)}^{2i}\dot{y}(t)>0$$

(11)

$$Q_l\{y(t)\}={\displaystyle {\displaystyle \sum _{i=1}^{\frac{(n+1)}{2}}}{a}_{2i-1}}y{(t)}^{2i-1}-{\displaystyle {\displaystyle \sum _{i=0}^{\frac{(n-1)}{2}}}{a}_{2i}}y{(t)}^{2i}\dot{y}(t)<0$$

(12)

where Q_u was the upper half of the hysteresis loop; Q_l was the lower half of the hysteresis loop; a_i was the polynomial coefficient of the power series.

$$\begin{array}{ll}{g}_n\{y(t),\dot{y}(t),t\}& ={\displaystyle {\displaystyle \sum _{i=1}^{\frac{(n+1)}{2}}}{a}_{2i-1}}y{(t)}^{2i-1}+{\displaystyle {\displaystyle \sum _{i=0}^{\frac{(n-1)}{2}}}{a}_{2i}}y{(t)}^{2i}\mathrm{sgn}(\dot{y}(t))\\ & =Q_1(t)+Q_2(t)\end{array}$$

(13)

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 $c\dot{y}\left(t\right)$ and hysteretic damping force $z\left(t\right)$. 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

$$g_n\{y(t),\dot{y}(t),t\}=k_{1}y(t)+k_{3}y{(t)}^3+c\dot{y}(t)+z(t)$$

(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.

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

$$F_c=c{\left|\dot{y}(t)\right|}^{\alpha }\mathrm{sgn}\{\dot{y}(t)\}$$

(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]

$$g_n\{y(t),\dot{y}(t),t\}=k_1(A)y(t)+k_3(A)y{(t)}^3+c(A,f){\left|\dot{y}(t)\right|}^{\alpha (A,f)}\mathrm{sgn}\{\dot{y}(t)\}$$

(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.

$$g_n\{y(t),\dot{y}(t),t\}=k_1(A,T)y(t)+k_3(A,T)y{(t)}^3+c(A,f){\left|\dot{y}(t)\right|}^{\alpha (A,f)}\mathrm{sgn}\{\dot{y}(t)\}$$

(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.

$$g_n\{y(t),\dot{y}(t),t\}={\displaystyle {\displaystyle \sum _{i=1}^{\frac{(n+1)}{2}}}{a}_{2i-1}y{(t)}^{2i-1}}+{\displaystyle {\displaystyle \sum _{i=0}^{\frac{(n-1)}{2}}}{a}_{2i}y{(t)}^{2i}}\mathrm{sgn}\{\dot{y}(t)\}$$

(18)

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.

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.

$$k_1(A,T)=2.38A^{-0.2218}{T}^{-0.05131}$$

(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.

$$k_3(A,T)=1112A^{-1.228}{T}^{-0.1374}$$

(20)

Therefore, the nonlinear elastic restoring force could be expressed as

$$F_k(y)=k_{1}y+k_3{y}^3=2.38\times A^{-0.2218}\times T^{-0.05131}\times y+1112\times A^{-1.228}\times T^{-0.1374}\times y^3$$

(21)

By subtracting the nonlinear elastic restoring force F_k(y) from the test data of the hysteretic restoring force measured by the energy dissipation test g_n(i), the nonlinear damping force F_c (y_i) could be obtained.

$$F_c(y_i)=g_n(i)-k_1{y}_i-k_3{y}_i{}^3$$

(22)

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).

As illustrated in Figure 13 that the variation of damping coefficient c(A, f) with amplitude (A) and frequency (f) is complex. Therefore, to fit the surface accurately, the damping coefficient was described by the combination of the sine function, exponential function and power function. Then, the damping coefficient could be obtained.

$$c(A,f)=e^{-1.997\cdot f^{0.7084}}+\sin (0.01938\cdot A^{2.734}\cdot f)+0.03811$$

(23)

The same method was used to identify the parameters of the damping component factor α (A, f). Figure 11 shows the damping component factor α(A, f) at different amplitudes and frequencies was obtained.

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 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:

$$\alpha (A,f)=0.4718\cdot A^{-0.4822}\cdot f^{0.01232}$$

(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

$$\begin{array}{ll}{g}_n\{y(t),\dot{y}(t),t\}& =k_1(A,T)y(t)+k_3(A,T)y{(t)}^3+c(A,f){\left|\dot{y}(t)\right|}^{\alpha (A,f)}\mathrm{sgn}\{\dot{y}(t)\}\\ & =2.38\cdot A^{-0.2218}\cdot T^{-0.05131}\cdot y+1112\cdot A^{-1.228}\cdot T^{-\mathrm{0..1374}}\cdot y^3\\ & +(e^{-1.997\cdot f^{0.7084}}+\sin (0.01938\cdot A^{2.734}\cdot f)+0.03811)\\ & \cdot {\left|\dot{y}(t)\right|}^{0.4718\cdot A^{-0.4822}\cdot f^{0.01232}}\mathrm{sgn}\{\dot{y}(t)\}\end{array}$$

(25)

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.

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).

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 (ΔW) and the maximum elastic potential energy (W) increase gradually.

(3)

With the increase of temperature, the energy dissipation Δ 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.