Accumulated damage process of thermal sprayed coating under rolling contact by acoustic emission technique

Abstract

The accumulated damage process of rolling contact fatigue (RCF) of plasma-sprayed coatings was investigated. The influences of surface roughness, loading condition, and stress cycle frequency on the accumulated damage status of the coatings were discussed. A ball-on-disc machine was employed to conduct RCF experiments. Acoustic emission (AE) technique was introduced to monitor the RCF process of the coatings. AE signal characteristics were investigated to reveal the accumulated damage process. Result showed that the polished coating would resist the asperity contact and remit accumulated damage. The RCF lifetime would then extend. Heavy load would aggravate the accumulated damage status and induce surface fracture. Wear became the main failure mode that reduced the RCF lifetime. Frequent stress cycle would aggravate the accumulated damage status and induce interface fracture. Fatigue then became the main failure mode that also reduced the RCF lifetime.

Keywords

accumulated damage / spray coating / rolling contact fatigue / acoustic emission

1 1 Introduction

Underwater manipulators, especially hydraulic-driven manipulators, have been widely utilized in ocean exploration because of their high power-to-weight ratio [1–3]. The oil hydraulic proportional/electrohydraulic servo valves, as core control components of underwater hydraulic manipulators (UHMs), play a critical role in the motion control of these manipulators [4,5]. However, oil valves have a high risk of oil–seawater mutual penetration [6,7]. Seawater immersion into the hydraulic system can cause poor lubrication and severe corrosion of components, which impairs UHM functionality. Conversely, oil leakage into the marine environment presents a serious ecological threat. These challenges are notably absent in systems utilizing water hydraulic valves [8,9]. The low viscosity characteristics of water will result in leakage and wear when using water hydraulic valves with a spool/sleeve [10,11]. A high-speed on/off valve (HSV) with a simple switching function of “on” and “off” may be a good option due to its excellent sealing and high response capabilities [12,13].

The realization of accurate, continuous, and reliable operation of UHMs introduces challenging requirements for the comprehensive performance of HSV. HSV is controlled by the pulse width modulation (PWM) signal and outputs the discrete flow by switching on/off [14,15]. Therefore, the action of HSV is often accompanied by pressure and flow pulsation, which is not conducive to the precise control of UHM. Increasing the switching frequency of HSV can mitigate the abovementioned problems, which enables a rapid dynamic response. Many studies have been conducted to improve the dynamic performance of HSVs. Liu et al. [16] designed a driving circuit to output double voltage, including high and low voltages. High voltage is used to speed up the opening of the HSV, while low voltage maintains its opening at a low current, which effectively reduces the closing time. A three-voltage control method with a negative voltage is proposed to expedite HSV closure further [17]. Considering the intelligent drive of HSVs, Zhang et al. [18] integrated the current feedback into the driving circuit of HSV, which enables the automatic adjustment of the duration of different voltages.

The fast response of HSVs is bound to be accompanied by increased power losses [19]. Power losses can be converted into heat, which may conversely degrade its dynamic performance [20,21]. Excessive power losses can reduce the energy reserves of the overall power system, which decreases the continuous operation time of UHMs. Some researchers have studied the intrinsic correlation between dynamic performance and power losses of HSVs. Owing to complex working conditions and lack of accurate measuring equipment, obtaining the power losses of HSVs by experimental methods is challenging, especially in the moving process. Therefore, the finite element method is commonly used [22]. Cheng et al. [23] investigated the influence of driving strategies on the dynamic response and power losses of HSVs through finite element simulations. They indirectly validated power loss estimations by comparing the coil current of the simulation and experimental results. Wang et al. [24] observed that increasing the driving current can shorten the opening time of HSV, but it will result in excessive ohmic loss. Zhao et al. [25] documented the influence of driving voltage on the dynamic response and power losses of HSVs. They found that, as the amplitude of the driving voltage increases, the opening time first decreases rapidly and then decreases slowly due to magnetic saturation, while the power loss of the HSV keeps rising rapidly. Further investigations by Zhao et al. [26] into the implications of modulating holding current revealed that reducing holding current lowers power consumption, albeit at the expense of slower closing time of HSVs. By comparing the traditional single-voltage control method, Zhong et al. [27,28] found that the multi-voltage driving strategy effectively improves the dynamic performance of HSVs and reduces their power losses.

The fast dynamics of HSV also result in the collision of the moving components with the valve seat at high speed. Owing to the frequent impact, the surface material of the valve port in the valve seat falls off, which worsens the valve port sealing and causes leakage. Low viscosity characteristics of water exacerbate this problem, which poses a significant reliability concern for valve-controlled UHMs. In the instantaneous collision process, the kinetic energy of the moving components is converted into the strain energy of the impact components [29]. Therefore, considerable stress is generated in the impact area. Owing to the short duration of the impact and the complex stress state in the impact contact area, the finite element method based on explicit dynamics has been utilized to study the impact process [30]. Zhang et al. [31] utilized ANSYS Ls-Dyna to analyze the maximum impact equivalent stress of the moving components and valve seat. Liu et al. [32] simulated the impact process of the poppet valve and valve seat. They found that, compared with uniform impact, non-uniform impact significantly amplifies the maximum equivalent stress of the valve seat. Gao [33] investigated the effects of driving voltage on the impact characteristics of HSVs based on the transient dynamic simulation module in ANSYS. They demonstrated that, with the increase in driving voltage amplitude, the equivalent stress and wear volume of the valve seat increase accordingly. Frequent collisions subject the impact parts of the HSV to cyclic stress, which leads to their fatigue failure [34]. Increased cyclic stress accelerates fatigue failure, which affects the service life of the HSV [35].

The abovementioned analysis reveals that dynamic performance, power losses, and impact performance are critical metrics to evaluate the comprehensive performance of HSVs. These characteristics need to be optimized to improve the comprehensive performance of HSVs. However, this approach is contradictory and challenging, which necessitates effective solutions. Fortunately, multi-objective optimization techniques have emerged as a potent tool to reconcile these competing demands and identify optimal configurations. Wu et al. [36] used the particle swarm optimization algorithm to refine some key structure parameters for improving the temperature characteristics and dynamic performance of HSVs based on an equivalent magnetic circuit model. The inherent nonlinearity of HSVs during operation complicates the development of a precise mathematical model [37]. In response, the surrogate model has gained attention as a strategy to approximate the relationship between the input variables and objective values with minimal data [38]. Incorporating surrogate models into optimization algorithms has markedly improved efficiency [39]. Liu et al. [40] introduced a radial basis surrogate model to bridge HSV response times with structural parameters. This model employs the non-dominated sorting genetic algorithm II (NSGA-II) to optimize the surrogate model for obtaining Pareto solutions. Beyond structural considerations, control parameters have also been subjected to optimization efforts. Li et al. [41] optimized the switching and holding voltages of HSVs based on NSGA-II supported by the response surface model to improve their dynamic performance. Yu et al. [42] developed a surrogate model correlating the opening time, driving energy, Joule energy, and driving current parameters of HSVs. They employed NSGA-II for solution generation and used the minimum distance method for optimal selection from the Pareto set, which demonstrates a comprehensive approach to resolving the optimization contradiction.

Despite the extensive research on HSVs, previous studies have largely ignored optimizing their impact performance, not to mention achieving a synergistic optimization with other characteristics. Optimally applying a water hydraulic HSV (WHSV) in the UHM requires achieving the best balance among the dynamic performance, power losses, and impact performance of the WHSV. Given that negative voltage is closely related to the closing dynamic performance, power losses, and impact performance of the WHSV, this study focuses on enhancing the comprehensive performance of the WHSV by optimizing the negative voltage. For this purpose, we first establish the simulation models of the electromagnetic field and impact dynamics of the WHSV using Maxwell 3D and Ls-Dyna within the ANSYS software, respectively. Subsequent analyses determine the effects of the equivalent amplitude and duration of the negative voltage on the comprehensive performance of the WHSV. Based on the analysis results, a multi-objective optimization algorithm combined with the optimal Latin hypercube sampling method (OLHS), universal Kriging surrogate model, NSGA-II, and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method is proposed to refine the negative voltage, which enhances the comprehensive performance of the WHSV.

2 2 Structure and mathematical models

2.1 2.1 Structure and working principle

Fig.1 shows a closed 2-position and 2-way WHSV for UHM, which is designed for use in terrestrial and marine environments. This WHSV mainly comprises two components, namely, electromagnetic and valve body components, as shown in Fig.1(a). Considering the corrosive characteristics of water/seawater, components in contact with the water/seawater must withstand the corrosion. Therefore, the iron core and armature of the WHSV are made from 1J117 soft magnetic material, which has high saturation magnetic flux density, permeability, and resistivity. The sealing mechanism of the valve port features a Si₃N₄ ceramic sealing ball and a 17-4PH stainless steel valve seat, while the remainder of the valve body is constructed from 316L stainless steel.

Fig.1 Schematic of WHSV. (a) Structure diagram of WHSV. (b) Working diagram of WHSV in the marine environment. 1. Iron core; 2. Coil; 3. Spring; 4. Spring seats; 5. Magnetic isolation ring; 6. Supporting ring; 7. Armature; 8. Push rod; 9. Sealing ball; 10. Valve seat; 11. Locking screw; 12. Valve shell.

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To accommodate the operation of the WHSV at various ocean depths, it can be equipped with an oil-based pressure compensator, such as a flexible hose, attached to the upper end of the electromagnetic components, as shown in Fig.1(b) [43,44]. This compensator is easily deformed under external ocean pressure. Thus, it can allow for the equalization of external seawater pressure with the internal oil pressure, which prevents the large wall thickness and size of the WHSV [45]. In addition, a combination of a rubber ring and two retaining rings guarantees the effective separation of the oil and the water/seawater medium, which provides a reliable solution for static seals [46]. This configuration enhances the versatility of the WHSV. The pressure compensator is removable for applications on land, which further demonstrates the adaptability of the WHSV to different operational scenarios.

Without external voltage excitation, the sealing ball is tightly pressed against the flow port under the spring force from pre-compression deformation, which disconnects ports P and T. When the coil is energized, the electromagnetic force acting on the armature will gradually increase and exceed the spring force. As a result, the moving parts, including the armature, push rod, and sealing ball, will move upward, which reconnects ports P and T. Thereafter, when external excitation stops, the electromagnetic force acting on the armature will gradually decrease and be smaller than the spring force. Consequently, the moving parts will move downward, which disconnects ports P and T again. Some key structure parameters of the WHSV are listed in Tab.1. In the following sections, the meaning of the letters with the same symbols as in Tab.1 will not be repeated.

Tab.1 Structure parameters of WHSV

Parameter	Symbol	Value	Unit
Resistance of coil	R	0.55	Ω
Turns of coil	N	53	–
Pre-compression of spring	l0	3.5	mm
Stiffness of spring	Kt	39	N·mm−1
Maximum displacement	xm	0.2	mm
Mass of moving parts	M	0.048	kg
Maximum opening area	Af	0.79	mm2
Area of inlet port	As	2.22	mm2
Flow coefficient	Cd	0.85	–
Fluid velocity coefficient	Cv	0.9	–
Flow angle	θ	45	°
Pressure of inlet port	p1	5	MPa

2.2 2.2 Multiple physical field models

Within a cycle, the dynamic response process of the WHSV is divided into opening stage, opening maintenance stage, closing stage, and closing maintenance stage, as shown in Fig.2.

Fig.2 Dynamic response process diagram of WHSV.

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In the operation process of the WHSV, multiple physical fields, including the electric, magnetic, mechanical, and fluid fields, are involved, and its energy conversion process is shown in Fig.3.

Fig.3 Energy conversion process of WHSV.

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When the external voltage energizes the coil, the electrical behavior of the WHSV can be written as

(1)$U=iR\text{+}i\frac{\text{d}L}{\text{d}t}\text{+}L\frac{\text{d}i}{\text{d}t},$

where U is the external voltage, i is the coil current, and L is the coil inductance.

The increase in the coil current gradually strengthens the magnetic field around the coil. The magnetic field model of the WHSV can be expressed as follows:

(2)$F_{\text{m}}=Ni=\varphi R_{\text{m}},$

where F_m is the magnetic potential, ϕ is the magnetic flux, and R_m is the total magnetic reluctance.

The electromagnetic force F_e acted on the armature can be calculated by

(3)$F_{\text{e}}=\frac{{\varphi }^2}{2{\mu }_0{A}_{\text{g}}}=\frac{N^2{i}^2}{2R_{\text{m}}{\mu }_0{A}_{\text{g}}},$

where μ₀ is the magnetic permeability of air, and A_g is the effective area of working air gaps.

Under the action of electromagnetic force, hydraulic force, damping force, and flow force, the moving parts of the WHSV begin to move. In the opening process of the WHSV, its dynamic equilibrium equation can be expressed as follows:

(4)$m\frac{\text{d}{x}^2}{\text{d}{t}^2}=F_{\text{e}}+F_{\text{f}}+p_1{A}_{\text{s}}-K_{\text{t}}\left(l_0+x\right)-B_{\text{v}}\frac{\text{d}x}{\text{d}t},$

where x is the displacement, F_f is the flow force, and B_v is the damping coefficient.

When the WHSV is closing, its dynamic behavior can be written as follows:

(5)$m\frac{\text{d}{x}^2}{\text{d}{t}^2}=K_{\text{t}}\left(l_0+x\right)+F_{\text{f}}-F_{\text{e}}-B_{\text{v}}\frac{\text{d}x}{\text{d}t}-p_1{A}_{\text{s}}.$

Flow force consists of the steady and transient flow forces. However, considering that the displacement and moving time of the WHSV are very short, the transient flow force is very small compared with the steady flow force [14]. Thus, only the steady flow of the WHSV is considered in this study, and it can be obtained by

(6)$F_{\text{f}}=2C_{\text{v}}{C}_{\text{d}}{A}_{\text{f}}(p_1-p_0)\cos \theta .$

where p₀ is the pressure of the outlet port, and its value is 0.

2.3 2.3 Impact models

In the opening and closing stages of the WHSV, the moving parts of the WHSV will collide with the iron core and the valve seat. In this study, the impact behavior between the moving parts and the valve seat is investigated as a typical example. Fig.4 shows the process of the sealing ball hitting the valve seat, including the impact, rebound recovery, and underdamped vibration processes until the final stability. As shown in Fig.4, the moving parts are replaced by a sealing ball to simplify the impact model. In addition, v_i and F_i represent the impact velocity and force of the sealing ball, respectively, where i = 1,2,…,N.

Fig.4 Impact process schematic of sealing ball and valve seat.

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The impact model between the sealing ball and valve seat can be expressed as follows [29]:

(7)$\{\begin{array}{l}m\ddot{x}+c_{\text{h}}\dot{x}+k_{\text{s}}x=F_{\text{c}},\\ x(0)=0,\dot{x}(0)={\nu }_1,\end{array}$

where c_h is the damping coefficient, k_s is the contact stiffness, and F_c is the binding force.

For underdamped motion, the solution of Eq. (7) is

(8)$x(t)=A\exp (-\xi w_{\text{n}}t)\sin \left(w_{\text{d}}t\right)+\frac{F_{\text{c}}}{k_{\text{s}}},$

(9)$A=\frac{v_1}{w_{\text{d}}},$

(10)$w_{\text{d}}=\sqrt{1-{\xi }^2}{w}_{\text{n}},$

(11)$w_{\text{n}}=\sqrt{\frac{k_{\text{s}}}{m}},$

(12)$\xi =\frac{c_{\text{d}}{w}_{\text{n}}}{2k_{\text{s}}},$

where w_n and ξ are the natural frequency and damping ratio of the impact system, respectively.

2.4 2.4 Power loss models

During the operation of the WHSV, electromagnetic energy will be converted into effective mechanical energy. However, more energy, mainly including the stranded loss, the core loss, and the solid loss, can be converted to useless heat, which results in performance degradation and even failure of the WHSV.

The stranded loss P_strandedloss of the WHSV is generated due to the current in the coil, which can be determined by

(13)$P_{\text{strandedloss}}=i^{2}R.$

The variation in the coil current changes the magnetic field, which generates the induced current in the magnetic elements and produces the core loss. The core loss P_coreloss of the WHSV mainly includes the eddy current loss P_c, hysteresis loss P_h, and excess loss P_e, which can be expressed as

(14)$P_{\text{coreloss}}=P_{\text{c}}+P_{\text{h}}+P_{\text{e}},$

(15)$P_{\text{c}}=k_{\text{c}}{f}^2{B}_{\text{m}}^2,$

(16)$P_{\text{h}}=k_{\text{h}}f{B}_{\text{m}}^2,$

(17)$P_{\text{e}}=k_{\text{e}}{f}^{1.5}{B}_{\text{m}}^{1.5},$

where k_c, k_h, and k_e are the coefficients of the eddy current loss, hysteresis loss, and excess loss, respectively; f is the magnetic field frequency; and B_m is the magnetic induction intensity.

When the induced current flows through the magnetic elements, solid loss P_solidloss is produced and can be written as

(18)$P_{\text{solidloss}}=\frac{1}{\sigma }{\int }_V{J}^2\text{d}V,$

where σ, V, and J are the conductivity, volume, and induced current density of magnetic elements, respectively.

Therefore, the total power loss P_t of the WHSV can be expressed as

(19)$P_{\text{t}}=P_{\text{strandedloss}}+P_{\text{coreloss}}+P_{\text{solidloss}}.$

3 3 Control strategy and numerical simulation

3.1 3.1 Control strategy

This study proposes a compound PWM (CPWM) control strategy to drive the WHSV. This strategy consists of three driving voltages: high positive voltage, high-frequency holding voltage, and high-frequency negative voltage, as shown in Fig.5. Each voltage plays a different role in the working process of the WHSV. Specifically, high positive driving voltage can promote the coil current to rise rapidly, which accelerates the opening of the WHSV. The high-frequency holding voltage maintains the opening state of the WHSV with a small current, which is beneficial to reducing power consumption and speeding up the closing of the WHSV. The high-frequency negative voltage can contribute to the rapid drop of the coil current, which accelerates the closing of the WHSV.

Fig.5 CPWM control strategy for WHSV.

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DC power determines the amplitude of the high positive voltage. The equivalent amplitudes of the high-frequency holding and negative voltages can be calculated by

(20)$U_{\text{h}}=U{\tau }_{\text{h}},$

(21)$U_{\text{n}}=U{\tau }_{\text{n}},$

where U_h and U_n are the equivalent amplitudes of the high-frequency holding and negative voltages, respectively; and τ_h and τ_n are the duty ratio of the high-frequency holding and negative voltages, respectively. Moreover, the positive and negative voltage switching is implemented based on the H-bridge circuit, which is introduced comprehensively in Ref. [27].

In this study, we focus on the influence of the high-frequency negative voltage, including its equivalent amplitude and duration, on the comprehensive performance of the WHSV.

3.2 3.2 Simulation model and settings

Considering that WHSVs have highly nonlinear characteristics in practical applications, using the analytical method directly is challenging. Therefore, the finite element method is employed. ANSYS Maxwell 3D is used to solve the transient electromagnetic field, and ANSYS Ls-Dyna is utilized to resolve the transient impact. Based on the electromagnetic simulation results obtained from Maxwell 3D, the velocity and force of the sealing ball are processed and then loaded into Ls-Dyna as the initial conditions to obtain the impact stress of the valve seat. The simulation process coupled with Maxwell 3D and Ls-Dyna is shown in Fig.6.

Fig.6 Simulation process coupled with Maxwell 3D and Ls-Dyna for WHSV.

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3.2.1 3.2.1 Maxwell 3D

Fig.7 shows the electromagnetic field simulation model of the WHSV. In Fig.7(a), only the coil and magnetic components are considered to simplify the model and save calculation time. In addition, the calculation domain and motion domain, namely, region and band, are added to the model to solve the transient electromagnetic field. The adaptive mesh refinement method is adopted to mesh the model, which provides high simulation accuracy and low cost [24], as shown in Fig.7(b).

Fig.7 Finite element and meshing models of WHSV. (a) Finite element model. (b) Meshing model.

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Tab.2 presents other detailed parameter settings for simulation. The B–H magnetization curve of the iron core and armature is shown in Fig.8, which is fitted by MATLAB software.

Tab.2 Detailed settings of finite element simulation of WHSV

	Type	Parameter	Setting or value
Physical property	Iron core and armature	Material property	1J117
		Relative magnetic permeability	Magnetization curve (Fig.8)
		Bulk conductivity	2272727 S·m−1
	Coil	Material property	Copper
		Relative magnetic permeability	1.01 m·H−1
		Bulk conductivity	58000000 S·m−1
		Winding type	Stranded
		Excitation type	Voltage
Boundary	Region	Convergence region offset	100%
		Boundary assign	Zero tangential H field
	Band	Motion type	Translation
		Initial velocity	0
		Damping	0
Solver	Solution type	Magnetic	Transient
		Stop time	50 ms
	Solution setup	Time step	0.01 ms
		Nonlinear residual	0.05

Fig.8 B–H magnetization curve of 1J117.

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3.2.2 3.2.2 Ls-Dyna

Fig.9 shows the simulation model for the transient impact of the WHSV. As shown in Fig.9, the simulation model of the valve seat is simplified, and only its upper part is selected. Furthermore, a quarter of the sealing ball and valve seat model is used for simulation to save calculation time because of structural symmetry.

Fig.9 Impact simulation model between sealing ball and valve seat.

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The property parameters of the sealing ball and valve sea are listed in Tab.3. The Si₃N₄ ceramic has a higher yield strength than the 17-4PH stainless steel, which means that the valve seat is more prone to fatigue failure due to impact stress. Therefore, we only focus on the impact stress of the valve seat in this study. The sealing ball and the valve seat are defined as flexible bodies.

Tab.3 Material parameters in the impact simulation model

	Density/(kg·m−3)	Young’s modulus/GPa	Poisson’s ratio	Yield strength/MPa
Si3N4	3200	300	0.26	3000
17-4PH	7800	200	0.30	1000

Considering the small size of the impact model, the tetrahedral meshes are applied, and the maximum mesh length of the sealing ball and valve seat is set to 0.2 mm. The contact area between the sealing ball and the valve seat, that is, the chamfer surface of the valve port P, requires local mesh refinement to avoid stress concentration and ensure simulation accuracy [47]. The maximum mesh size of this position is initially set to 0.02 mm. Fig.10 shows the mesh of the impact model. The contact between the sealing ball and the valve seat is defined as body interaction, that is, automatic surface-to-surface contact. The static and dynamic friction coefficients between them are set to 0.1 and 0.05, respectively [48]. The bottom surface of the valve seat is fixed as support, as shown in Fig.10. In addition, the force and initial velocity in the Y-direction are imposed on the sealing ball. Finally, the end time of impact is set to 0.05 ms, and other simulation parameters are default.

Fig.10 Mesh model of sealing ball and valve seat.

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Given that the mesh size on the contact area between the sealing ball and the valve seat influences the simulation results, the mesh independence analysis of this part needs to be performed to minimize the influence of the mesh size on the simulation results. The local maximum mesh size on the abovementioned contact area changes from 0.01 to 0.04 mm at intervals of 0.005 mm, while that in other parts keeps constant at 0.2 mm. Based on the voltage excitation signal, the initial impact velocity and force are obtained by Maxwell 3D simulation and then applied to the sealing ball as initial conditions. The maximum impact equivalent stress of the valve seat is considered the monitoring variable. As shown in Fig.11, when the number of the mesh reaches 215730, the monitoring variable tends to be stable with the mesh change. Therefore, the maximum mesh size on the abovementioned contact area can be set to 0.02 mm to ensure the accuracy of the simulation results while simultaneously reduce the simulation cost.

Fig.11 Mesh independence verification of impact simulation.

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3.3 3.3 Validation of simulation model

A test bench of the WHSV is designed and built to verify the accuracy of the simulation model, as shown in Fig.12. However, this test bench cannot test the impact stress of the valve seat due to the limitation of experimental conditions. This test rig is mainly used to obtain the coil current, opening time, and closing time of the WHSV, which includes the electric power system, the signal control and acquisition system, and the hydraulic power system. In the electric power system, the DC powers (30 V/60 A, 24 V/5 A) are supplied to the driver and pressure sensor, respectively. The driver is used to drive the WHSV. In the signal control and acquisition system, the host computer is employed to send the control signal to the driver. The current sensor (HCP8050, Shenzhen Zhiyong Co., Ltd., China) is used to measure the coil current. The armature displacement cannot be acquired directly due to the WHSV structure. However, when the WHSV opens and closes, the moving parts will collide with the iron core and the valve seat, which causes the shell to vibrate. The opening and closing times of the WHSV can be obtained by measuring the time interval between the vibration of the shell and the rise and fall of the control signal. Therefore, the acceleration sensor (4535-B-001, HBK Co., Ltd., UK) is utilized to obtain the vibration acceleration of the shell. The pressure sensor (HM90-H-2-A1-F0-W1-T, Nanjing Helm Sci-tech Co., Ltd., China) is used to monitor the pressure at port P. The oscilloscope is applied to acquire the data on the coil current, shell vibration acceleration, and pressure. The hydraulic power system is responsible for supplying pressure to the WHSV.

Fig.12 Test bench for the performances of WHSV. (a) Schematic of test bench. (b) Physical diagram of test bench.

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Based on the analysis in Section 2.4, the power losses, including the stranded, core, and solid losses of the WHSV, are closely related to the coil current. Therefore, the finite element simulation results of the power losses can be verified by comparing the current curves of the simulation and experiment. The comparison of the dynamic response between the simulation and experimental results of the WHSV is shown in Fig.13.

Fig.13 Comparison of the dynamic response between simulation and experimental results of WHSV. (a) Excitation signal of equivalent CPWM. (b) Coil current. (c) Armature displacement.

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Fig.13(a) is an equivalent CPWM control signal used in the experiment with a period of 50 ms and a duty cycle of 50%. As shown in Fig.13(b), the curves of the coil current between the simulation and experimental results of the WHSV agree well with each other. The small discrepancies between them can be attributed to the change in coil resistance due to temperature. As shown in Fig.13(c), the dynamic response times between the simulation and experimental results are also very close. The opening times between the simulation and experimental results are 2.16 and 2.23 ms, respectively, with an error of 3.1%. The closing times between the simulation and experimental results are 1.82 and 1.95 ms, respectively, with a deviation of 7.1%. The response times obtained by the simulation are slightly faster than those of the experimental results because the mechanical friction of the moving components is ignored in the simulation. Based on the aformentioned comparative analysis, the simulation results obtained from Maxwell 3D are considered valid and suitable for further analysis.

3.4 3.4 Simulation result analysis

In this section, the influence of the equivalent amplitude and duration of the negative voltage on the dynamic performance, power losses, and impact behavior of the WHSV is studied by finite element simulation.

3.4.1 3.4.1 Effects of equivalent amplitude of negative voltage

Fig.14 shows the variation in the dynamic response times and power losses of the WHSV with different equivalent amplitudes of the negative voltage U_n.

Fig.14 Effects of Un on the dynamic response and power losses of WHSV. (a) Coil current. (b) Response times. (c) Power losses.

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As shown in Fig.14(a), the coil current curves are completely coincident before 25 ms, but they change abruptly after 25 ms caused by the different values of U_n. This result indicates that the change in U_n only affects the coil current at the closing stage of the WHSV. When the WHSV is closing, the decrease rate of the current increases with the rise in U_n. The amplitude of the current can even become negative, which reverses its direction and influences the performance of the WHSV.

As shown in Fig.14(b), with the increase in U_n, the opening time of the WHSV remains unchanged at 2.16 ms, while the closing time decreases from 5.2 to 1.82 ms, which is a reduction of 65%. Therefore, the total response time decreases, which indicates that the dynamic performance of the WHSV is improved. The main reason is that increasing U_n promotes the current decrease, which leads to the rapid decline in electromagnetic force and the rise in the downward resultant force of moving parts. In addition, the closing time decreases sharply when U_n increases from 12 to 15 V. Then, the trend of the closing time decreasing with the increase in U_n is relatively stable.

However, as U_n increases, the power losses of the WHSV rise, as shown in Fig.14(c). When U_n increases from 12 to 24 V, the stranded loss, core loss, and solid loss rise from 10.7, 8.6, and 7.1 W to 11.9, 11.5, and 8.2 W, respectively. Their corresponding increases are 1.2, 2.9, and 1.1 W (11.2%, 33.7%, and 15.5%, respectively). This result can be explained by that, as U_n increases, the current drop and its change rate become faster, which generates a greater eddy current inside the magnetic elements. The total power loss increases from 26.4 to 31.6 W, which is a rise of 19.7%. This increase aggravates the energy loss. Moreover, Fig.14(c) shows a linear relationship between U_n and power losses. The abovementioned power losses can be converted into harmful heat, which will degrade the dynamic performance of the WHSV.

Fig.15 shows the effects of U_n on the impact characteristic of the WHSV. As shown in Fig.15(a), the increase in U_n also improves the initial impact velocity and force, which inevitably influences the impact stress of the valve seat of the WHSV. As shown in Fig.15(b), the maximum impact equivalent stress of the valve seat increases with the rise in U_n. When U_n changes from 12 to 24 V, the maximum impact equivalent stress increases from 384.1 to 536.6 MPa, which is an increment of 152.5 MPa (39.7%). After the valve seat is subjected to frequent high-speed impact, the surface material of the impact area may fall off and crack due to fatigue failure at certain time, which will degrade the sealing performance of the WHSV and further shorten its service life.

Fig.15 Effects of Un on the impact characteristic of WHSV. (a) Initial impact velocity and force. (b) Maximum equivalent stress.

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3.4.2 3.4.2 Effects of duration of negative voltage

Fig.16 shows the variation in the dynamic response times and power losses of the WHSV with different durations of the negative voltage t_n.

Fig.16 Effects of tn on the dynamic response and power losses of WHSV. (a) Coil current. (b) Response times. (c) Power losses.

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As shown in Fig.16(a), t_n only affects the coil current at the closing stage of the WHSV. With the extension of t_n, the amplitude of the current drop increases, but its decline rate does not change, which is different from the influence of the equivalent amplitude of the negative voltage U_n on the coil current in the closing stage of the WHSV.

Fig.16(b) shows that, as t_n increases, the closing time decreases from 5.46 to 1.81 ms, which is a reduction of 66.8%. Accordingly, the total response time decreases, which means that the dynamic performance of the WHSV is improved. This result can be interpreted as extending t_n, which decreases the current further. Interestingly, when t_n changes from 0.6 to 0.8 ms, the closing time decreases sharply. However, within the range of 0.8–1.4 ms, the closing time slowly decreases and tends to be stable with minimal change.

Accordingly, with the increase in t_n, the power losses of the WHSV rise, as shown in Fig.16(c). When t_n increases from 0.6 to 1.4 ms, the stranded loss, core loss, and solid loss rise from 11, 10.4, and 7.2 W to 12.3, 11.6, and 8.6 W, respectively. Their corresponding increases are 1.3, 1.2, and 1.4 W (11.8%, 11.5%, and 19.4%, respectively). The total power loss increases from 28.6 to 32.5 W, which is a rise of 3.9 W (13.6%). This result indicates that the energy loss becomes serious. As shown in Fig.16(c), a linear relationship exists between the power losses and t_n, which is the same as the relationship between the power losses and U_n.

Fig.17 shows the effects of t_n on the impact performance of the WHSV. As shown in Fig.17(a), as t_n increases, the initial impact velocity and force rise rapidly and then slowly, and t_n = 1.2 ms is the demarcation point. Fig.17(b) shows that, as t_n increases, the maximum impact equivalent stress of the valve seat rises rapidly and then slowly, and t_n = 1.2 ms is the dividing point, which has the same law as when the initial impact conditions change with t_n. When t_n increases from 0.6 to 1.2 ms, the maximum impact equivalent stress rises from 377.8 to 536.6 MPa, which is an improvement of 158.8 MPa. However, when t_n increases from 1.2 to 1.4 ms, the maximum impact equivalent stress rises from 536.6 to 548.7 MPa, which is an enhancement of only 12.1 MPa. In general, when t_n increases from 0.6 to 1.4 ms, the maximum equivalent stress rises by 170.9 MPa (45.2%).

Fig.17 Effects of tn on the impact characteristic of WHSV. (a) Initial impact velocity and force. (b) Maximum equivalent stress.

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4 4 Multi-objective optimization method

4.1 4.1 Optimization problem description

The abovementioned analysis shows that the equivalent amplitude of the negative voltage U_n and duration of the negative voltage t_n significantly influence the performance of the WHSV, including dynamic performance, power losses, and impact performance. Therefore, U_n and t_n need to be optimized to improve the comprehensive performance of the WHSV.

Fig.18 shows the entire optimization process for the performance improvement in the WHSV. In this optimization process, the design of the experiment method is utilized to obtain the sampling points from design variables, namely, U_n and t_n. These sampling points are introduced to the finite element simulation to obtain results, such as closing time, total power loss, and maximum impact equivalent stress of the valve seat. Then, the surrogate model is used to build the mapping relationships between the simulation results and design variables. Based on the surrogate model, multi-objective optimization and decision-making methods are employed to obtain the solution with the best objective function value from the Pareto set and the best combination of U_n and t_n.

Fig.18 Flowchart of multi-objective optimization for WHSV.

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4.2 4.2 Design of experiments

The ranges of U_n and t_n, as the design variables, are presented in Tab.4. Considering the surrogate model accuracy and calculation time, the OLHS method is used to select the sampling points, which ensures that they remain representative even with fewer points [49]. Thus, a total of 30 sampling points are generated. Among them, 24 sampling points are randomly selected to build the surrogate model, and the remaining sampling points are used to verify the accuracy of the surrogate model.

Tab.4 Range of design variables

Design variable	Unit	Lower bound	Upper bound
Un	V	12.0	24.0
tn	ms	0.6	1.4

4.3 4.3 Surrogate model

The universal Kriging model has been widely used in engineering optimization due to its high accuracy and flexibility [50]. Therefore, this study uses the universal Kriging model to build the surrogate model between the objective functions and the design variables. The primary expression of the universal Kriging model is

(22)$y(\mathit{x})=\text{λ}\cdot \mathit{f}(\mathit{x})+\mathit{\text{z}}(\mathit{x}),$

where x = [x₁, x₂,…, x_m]^T is the collection of the sampling points, the subscript m is the number of sampling points, y(x) is the prediction collection, $\text{λ}$ is the vector of regression coefficients, f(x) is the collection of the regression function, and z(x) is the collection of modeling the deviation from the global model.

The accuracy of the surrogate model directly determines the subsequent optimization effect. A few evaluation indexes are used to level the precession of the surrogate model, such as coefficient of determination (R²), maximum absolute error (MAE), and root mean square error (RMSE) [51]. R², MAE, and RMSE are expressed as follows:

(23)$R^2=1-\frac{{\displaystyle \sum _{i=1}^n{\left(y_{\text{a}i}-{y_{\text{p}}}_i\right)}^2}}{{\displaystyle \sum _{i=1}^n{\left(y_{\text{a}i}-{\overline{y}}_{\text{a}i}\right)}^2}},$

(24)$\text{MAE}=max\left|y_{\text{a}i}-{y_{\text{p}}}_i\right|,$

(25)$\text{RMSE}=\sqrt{\frac{{\displaystyle \sum _{i=1}^n{\left(y_{\text{a}i}-y_{\text{pi}}\right)}^2}}{n}},$

where y_ai is the actual value of the ith sampling point, y_pi is the predicted value of the ith sampling point calculated by the surrogate model, and ${\overline{y}}_{\text{a}i}$ is the average value of all the sampling points. When R² approaches to 1 and MAE and RMSE reach 0, the accuracy of the surrogate model increases.

4.4 4.4 Multi-objective optimization method

The minimum value of the closing time, total power loss, and maximum impact equivalent stress of the valve seat are considered the optimization objectives. This typical multi-objective optimization problem can be written as

(26)$\begin{array}{ll}\text{Find}& x={[U_{\text{n}},t_{\text{n}}]}^{\text{T}},\\ \text{Minimize}& \{F_{\text{ct}}(x),F_{\text{tp}}(x),F_{\text{ms}}(x)\},\\ \text{s}\text{.t}\text{.}& \text{12 V}⩽U_{\text{p}}⩽24\text{V,}\\ & \text{0}\text{.6 ms}⩽t_{\text{n}}⩽1.4\text{ms},\end{array}$

where F_ct, F_tp, and F_ms are the objective functions of the closing time, total power loss, and maximum impact equivalent stress, respectively.

NSGA-II, with its excellent solution–set convergence and suitability for contradictory and extensive targets, is adopted to solve the multi-objective optimization problem [52]. The detailed optimization process of NSGA-II is shown in Fig.19. The specific parameters of the NSGA-II are also listed in Tab.5.

Fig.19 Optimization process of NSGA-II.

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Tab.5 Parameters setting of NSGA-II

Parameter	Value
Population size	100
Number of generations	100
Crossover probability	0.9
Mutation distribution index	20
Crossover distribution index	10

4.5 4.5 Decision-making method

In this optimization, a typical problem of conflict objectives exists in multi-objective optimization: the optimal solution for one objective function may not be ideal for other objective functions. This issue undoubtedly increases the selection difficulty for decision-makers. The TOPSIS method has been widely used as a multi-criteria decision method to provide decision-makers with the best solution to meet their goals and values [53]. The TOPSIS method identifies the most and least desirable objective value solutions as positive and negative solutions, respectively. Then, it ranks each solution according to the Euclidian distance from the positive and negative ideal solutions [53]. As a result, the top-ranked solution best meets the requirements of decision-makers. The operation steps of the TOPSIS method are detailed as follows:

(1) The optimal solutions are post-processed in the following matrix form:

(27)${\mathit{Z}}_{n\times m}=\left[\begin{array}{cccc}{\mathit{\text{z}}}_{11}& {\mathit{\text{z}}}_{12}& \dots & {\mathit{\text{z}}}_{1m}\\ {\mathit{\text{z}}}_{21}& {\mathit{\text{z}}}_{22}& \cdots & {\mathit{\text{z}}}_{2m}\\ ⋮& ⋮& \ddots & ⋮\\ {\mathit{\text{z}}}_{n1}& {\mathit{\text{z}}}_{n2}& \dots & {\mathit{\text{z}}}_{nm}\end{array}\right],$

where n and m are the number of evaluation sampling points and indicators, respectively.

(2) The normalization of the optimal solution is operated as follows:

(28)$r_{ij}=\frac{{\text{z}}_{ij}}{\sqrt{{\displaystyle \sum _{i=1}^n{\text{z}}_{ij}^2}}}\text{,}i=1,2,\dots ,n,j=1,2,\dots ,m.$

(3) Considering that the significance of different evaluation indicators differs, the weight W_j is assigned to the corresponding normalized solutions:

(29)$V_{ij}=r_{ij}\times W_j\text{,}i=1,2,\dots ,n,j=1,2,\dots ,m,$

(30)$\sum _{j=1}^m{W}_j=1.$

(4) For each evaluation index, their positive and negative ideal solutions P⁺ and N⁻ are recognized by

(31)$\begin{array}{rl}{\text{P}}^{+}=& \left[max\left\{V_{11},\dots ,V_{i1}\right\},\dots ,max\left\{V_{1m},\dots ,V_{im}\right\}\right],\\ & i=1,2,\dots ,n,\end{array}$

(32)$\begin{array}{rl}{\text{N}}^{-}=& \left[min\left\{V_{11},\dots ,V_{i1}\right\},\dots ,min\left\{V_{1m},\cdots ,V_{im}\right\}\right],\\ & i=1,2,\dots ,n.\end{array}$

(5) The distances $D_i^{+}$ and $D_i^{-}$ of the jth evaluation index from positive and negative ideal solutions $V_j^{+}$ and $V_j^{-}$ can be obtained from

(33)$D_i^{+}=\sqrt{\sum _{j=1}^m{\left(V_j^{+}-V_{ij}\right)}^2}\text{,}i=1,2,\dots ,n,$

(34)$D_i^{-}=\sqrt{\sum _{j=1}^m{\left(V_j^{-}-V_{ij}\right)}^2}\text{,}i=1,2,\dots ,n.$

(6) The relative closeness (RC) of each evaluation index to the ideal solution is calculated by

(35)$\text{R}{\text{C}}_i=\frac{D_i^{-}}{D_i^{+}+D_i^{-}}\text{,}i=1,2,\dots ,n.$

The value of RC_i is between 0 and 1. When the value of RC_i is closer to 1, the corresponding Pareto solution can meet the requirements of decision-makers.

5 5 Results and discussion

5.1 5.1 Validation of surrogate model

As shown in Fig.20, six random sampling points are selected to verify the accuracy of the universal Kriging model. For the surrogate models of the closing time, total power loss, and maximum impact equivalent stress, Fig.20 shows that their R² values are 0.95, 0.98, and 0.93, respectively. Their MAE values are 0.167, 0.081, and 0.193, and their RMSE values are 0.091, 0.045, and 0.099. All the R² values are greater than 0.9, and all the MAE and RMSE values are less than 0.2, which indicates that the universal Kriging model has high accuracy in predicting the closing time, total power loss, and maximum impact equivalent stress [40].

Fig.20 Verification of universal Kriging model. (a) Closing time. (b) Total power loss. (c) Maximum impact equivalent stress.

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5.2 5.2 Optimization results

After 10000 iterations, 637 Pareto solutions are obtained. However, finding the best solution is still difficult for decision-makers. The TOPSIS method is a good option to help design-makers select the best solution. Different weights of objectives will produce distinct best solutions. In this optimization, all the objectives, including closing time, total power loss, and maximum impact equivalent stress, are considered equally important. Therefore, the weights of the three objectives are the same. Fig.21 shows the best solution selected from the Pareto solutions.

Fig.21 Pareto solutions of multi-objective optimization.

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The comparison of the closing time, total power loss, maximum impact equivalent stress of the valve seat, and their corresponding equivalent amplitude of the negative voltage U_n and duration of the negative voltage t_n before and after optimization are presented in Tab.6. After optimization, the closing time is prolonged by 0.07 ms, while the total power loss and the maximum impact equivalent stress are reduced by 4.2 W and 112.5 Mpa, respectively.

Tab.6 Results of the best compromise solution.

Parameter	Unit	Initial value	Best solution	Improvement
Uh	V	24	20.89	–
tn	ms	1.2	0.947	–
Fct	ms	1.82	1.89	−3.8%
Ftp	W	31.6	27.4	13.3%
Fms	MPa	536.6	424.1	20.9%

Considering operational feasibility, U_n and t_n are slightly adjusted to 21 V and 0.95 ms, respectively. This combination is referred to as the best combination. Furthermore, the adjusted U_n and t_n will serve as CPWM input parameters for further comparative simulation analysis.

The comparison of coil current, dynamic response, and total power loss before and after optimization is depicted in Fig.22. As shown in Fig.22(a), a reduction in the change rate and amplitude of the coil current occurs during the closing process of the WHSV after optimization. As a result, the closing time of the WHSV extends from 1.82 to 1.88 ms, which is an increase of 0.06 ms (3.3%). Before and after optimization, the total power loss curve of the WHSV also changes, as shown in Fig.22(b). After optimization, the total power loss of the WHSV decreases from 31.6 to 28.5 W, which is a reduction of 3.1 W (9.8%). This decrease saves energy and extends the operational duration of the WHSV.

Fig.22 Comparison of simulation results for WHSV before and after optimization. (a) Dynamic response curve. (b) Total power loss curve.

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Fig.23 illustrates the stress contours of the valve seat caused by impact before and after optimization. As evident in Fig.23(a) and Fig.23(b), the maximum impact equivalent stress after optimization is significantly reduced from 536.6 to 458.7 MPa, which is a decrease of 77.9 MPa (14.5%). This reduction is beneficial to enhancing the fatigue life of the valve seat, which extends the operational lifespan of the WHSV.

Fig.23 Stress contours of valve seat before and after optimization. (a) Equivalent stress distribution before optimization. (b) Equivalent stress distribution after optimization.

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Accordingly, the deformation contours of the valve seat before and after optimization are also shown in Fig.24. As shown in Fig.24(a) and Fig.24(b), the maximum total deformation of the valve seat after optimization decreases from 8.6 × 10⁻⁴ to 6.9 × 10⁻⁴ mm, which is a reduction of 1.7 × 10⁻⁴ mm (19.8%).

Fig.24 Deformation contours of valve seat before and after optimization. (a) Total deformation distribution before optimization. (b) Total deformation distribution after optimization.

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5.3 5.3 Discussion

The simulation results after optimization, with U_n set at 21 V and t_n at 0.95 ms, agree well with the best solution obtained from the optimization algorithm, and the maximum deviation is 7.5%. This level of error is deemed acceptable and validates the optimization methodology introduced in this study. The primary source of this discrepancy is attributed to the employment of a surrogate model instead of a precise mathematical model for establishing the relationships between the optimization objectives (closing time, total power loss, and maximum impact equivalent stress) and the design variables (amplitude and duration of negative voltage), which inevitably introduces some level of deviation from the actual values.

Further analysis reveals that the multi-objective optimization method proposed herein significantly enhances the comprehensive characteristics of the WHSV through the optimization of the negative voltage. Specifically, the optimization reduces 9.8% in total power loss, 14.5% in maximum impact equivalent stress, and 19.8% in maximum total deformation, with a minor increase in closing time by 0.06 ms (3.3%). Consequently, this optimization guarantees an optimal balance among the dynamic performance, power losses, and impact performance of the WHSV.

In selecting the best solution, equal importance is assigned to the different optimization objectives according to our actual requirements. However, this balance can be recalibrated by adjusting the weights assigned to each objective, which allows for the attainment of alternative optimal solutions tailored to specific performance priorities of the WHSV. For instance, when prioritizing total power loss reduction, the weight of the total power loss can be appropriately increased. This flexibility in the optimization approach highlights the adaptability of the WHSV to meet varying operational demands within the UHMs and even other hydraulic systems.

6 6 Conclusions

In this study, we develop and assess a WHSV designed for UHM. We achieve an optimal synergy among the dynamic performance, power losses, and impact performance of the WHSV by applying a multi-objective optimization methodology and optimizing negative voltage. This optimization significantly enhances the comprehensive performance of the WHSV, which makes it more suitable for diverse applications within UHM contexts. The main conclusions of our study can be summarized as follows:

(1) The negative voltage positively affects the improvement in the dynamic response of the WHSV, which is manifested in reduced closing time. Specifically, an increase in the amplitude and duration of the negative voltage from 12 V and 0.6 ms to 24 V and 1.4 ms leads to a substantial reduction in closing times by 65% and 66.8%, respectively.

(2) The negative voltage negatively affects the reduction in the power losses of the WHSV, which is evidenced by increased total power loss. When the amplitude and duration of the negative voltage increase from 12 V and 0.6 ms to 24 V and 1.4 ms, the total power loss is improved by 20.1% and 13.1%, respectively. Interestingly, a linear relationship is observed between the power losses and the negative voltage.

(3) The impact characteristics of the WHSV also deteriorate with higher negative voltage settings. As the amplitude and duration of the negative voltage enhance from 12 V and 0.6 ms to 24 V and 1.4 ms, the maximum impact equivalent stresses of the valve seat are increased by 39.7% and 45.2%, respectively.

(4) The multi-objective optimization method proposed in this study can substantially improve the comprehensive performance of the WHSV. After optimization, the WHSV exhibits a slight increase in closing time (3.3%) but gains substantial reductions in total power loss (9.8%), maximum impact equivalent stress (14.5%), and maximum total deformation (19.8%).

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Acknowledgment

This study was financially supported by the Distinguished Young Scholars of National Natural Science Foundation of China (Grant No. 51125023), the 973 Project (Grant No. 2011CB013405), the National Natural Science Foundation of China (Grant Nos. 51305397 and 51375457), and the Open Foundation of the State Laboratory of Fluid Power Transmission and Control (GZKF-201411) (Grant Nos. 2014C31099 and EM2015042003).

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2016 Higher Education Press and Springer-Verlag Berlin Heidelberg