Issue 
yabo亚博
Volume22, 2021 


文章Number  29  
Number of page(s)  9  
DOI  https://doi.org/10.1051/meca/2021028  
网上发布  14 April 2021 
Regular Article
Wear assessment model for cylinder liner of internal combustion engine under fuzzy uncertainty
^{1}School of Mechanical and Precision Instrument Engineering, Xi'an University of Technology, Xi'an 710048, PR China
^{2}School of Printing, Packaging Engineering and Digital Media Technology, Xi'an University of Technology, Xi'an 710048, PR China
^{*}email:yanjunlu@xaut.edu.cn
Received:28 September 2020
Accepted:2021年3月29日
The wear of the piston ringcylinder system is inevitable in the operation of the internal combustion engines (ICEs). If wear exceeds the maximum, the piston ringcylinder system will be failure. A novel wear assessment model is proposed based on the support vector regression, and the fuzzy uncertainty is modeled to describe the random behavior under small sample. To verify the proposed model, the sample data of cylinder liner wear is applied. For best results, the particle swarm optimization (PSO) algorithm is used to optimize the model parameters. A back propagation neural network (BPNN) is employed to verify the effectiveness of the proposed model. The results show that the novel support vector regression has better prediction accuracy than other methods for cylinder wear in this paper, the proposed model can evaluate the cylinder liner wear of the ICEs effectively. The work provides a technical support for evaluating the service performance of the piston ringcylinder liner and a reference for regular maintenance of the ships.
Key words:Wear assessment / cylinder liner / support vector regression / fuzzy uncertainty / particle swarm optimization algorithm
© J. Kang et al., Published by EDP Sciences 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0.), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
yaboapp 介绍
The piston ringcylinder liner system is the core component of highperformance internal combustion engines. The wear of the piston ringcylinder liner system effects the energy transformation, and reduces the reliability of the ICEs. Therefore, how to effectively reduce the wear rate, monitor and assess the wear of cylinder liner are great significance to service performance of the ICEs.
In the operational process of the ICEs, the wear of piston ringcylinder liner system has a significant influence on energy conversion. According to the references, the friction of piston ringcylinder liner could account about 50 percent of the total mechanical friction of an internal combustion engine [yaboapp –yaboapp ]。In order to obtain the better tribological performance, the scholars are focusing on establishing friction and wear model to investigate the relationship between friction and energy consumption of the ICEs [yaboapp –yaboapp ], and the effect of lubrication on friction of piston ringscylinder liner was studied [yaboapp ,yaboapp ]。To monitor the wear state, some scholars analyzed the wear information of abrasive particles in the oil, and the wear of the ICEs was evaluated [yaboapp –yaboapp ]。在上面的作品中，最多的学者专注于系统的摩擦和活塞环缸衬里的磨损。汽缸衬里的磨损直接从活塞环到气缸衬里的间隙增加，并且它也影响润滑，密封，表面形貌，活塞环和气缸衬套之间的活塞二次运动。这些因素对冰的效率，生活和可靠性产生了强烈影响。因此，需要监测气缸衬套的磨损以提高活塞环缸衬里系统的可靠性。然而，很难在短期内收集大量磨损数据。2007年，Giorgio等人。[yaboapp ] presented a method to calculate the reliability of cylinder liner, the cumulative damage model was established to describe the wear process. By estimating the reliability of cylinder liner, the inspection and replacement of cylinder liner can be punctually conducted to reduce the extra expense loss. Subsequently, Giorgio et al. [yaboapp –yaboapp ] established a state dependent wear model, age and state dependent Markov model and Bayesian estimation model to predict the degradation process of the cylinder liner. They have done a lot of outstanding work in the degradation performance and reliability of the cylinder liner wear. Meanwhile, the fuzzy set and the failure mode, effects, and criticality analysis (FMECA) methods were used to analyze the reliability of the diesel engine turbocharger [yaboapp ], the expert knowledge was introduced into their model. Through analysis and calculation, the authors provided a new method to predict the reliability of the diesel engine turbocharger. In the references [yaboapp –yaboapp ], the most scholars established the model of cylinder liner wear to research the degradation process. However, the uncertainty factors are not considered into the operational process of diesel engine, which have great influence on wear of cylinder liner. Chang et al. [yaboapp ] proposed a calculation model to predict the life distribution of pneumatic cylinders using the performance degradation data. To predict the wear process of cylinder liner, a stochastic model was established, and a maintenance plan was developed by predicting model [yaboapp ]。Zhang et al. [yaboapp ]提出了一种具有非均相复合泊松过程的跳跃扩散过程，以利用随机发生的跳跃来模拟劣化过程，使用数值例子来验证所提出的方法的有效性。基于磨损的小样本数据，Wiederkehr等人。[yaboapp ] presented a new pointbased approach for modeling the grain wear of tool.
In recent years, the machine learning algorithm and statistical methods are developed rapidly, and it is widely used to predict the wear of the equipments [yaboapp ,yaboapp ]。The fault tree analysis (FTA) and failure mode and effects analysis (FMEA) method was used to analyze the reliability of the engines, and the Artificial Neural Network (ANN) was used to predict the characteristic parameters of exhaust gas temperatures of main engine cylinders [yaboapp ]。Kong et al. [yaboapp ] presented a hidden semiMarkov model (HSMM) method to estimate the tool wear in milling process. The experiments showed that the proposed method can achieve higher accuracy in tool wear evaluation. In their works, the kernel principal component analysis (KPCA) technique was used to reduce the effect of noise. The Gaussian process regression (GPR) and relevance vector machine (RVM) were used for predicting the tool wear [yaboapp ,yaboapp ]。The above works provided the effective methods for wear prediction in industry, and the wear characteristics of the pistoncylinder liner system are similar to the tool wear, it can provide the reference for the cylinder liner. The support vector machine (SVM) has a great advantages in solving small sample, nonlinear and highdimensional problems. Considering the characteristics of the wear, the SVM is suitable for diagnosis and prediction the wear of the cylinder liner. In reference [yaboapp ], the condition monitoring method for onmachine tool was proposed and the support vector regression (SVR) was used to predict the cutting tool of the flank wear. Zhang et al. [yaboapp ] used SVR to predict wear volume at the runningin, the optimization method was used to obtain the optimal results of different parameters. For predicting the wear rate, an ANNSVR model was developed [yaboapp [结果表明，所提出的模型具有优异的性能而不是ANN模型。建立了集成模型，以预测基于SVR的工具的磨损和剩余寿命[yaboapp ], and the relationship between the signal characteristic quantity and the tool wear was also studied. Zhang et al. [yaboapp ] established a parameter prediction model of surface topography before and after runningin, the SVM was adopted to simulate the wear process.
In the ICEs, the wear information is closely related to the tribological, dynamic characteristics and operational condition of the pistoncylinder liner system. Therefore, the wear capacity shows the uncertainty characteristics. In this paper, a SVRbased model is proposed to predict the cylinder liner wear. In order to predict the wear effectively, a novel fuzzySVR model is proposed to assess the wear by incorporating the uncertainty information into the proposed model. To achieve the optimum results, the PSO algorithm is used to optimize the model parameters. BPNN is employed to compare with the proposed model for verification of the effectiveness. The numerical results showed that the proposed model can predict cylinder liner wear effectively.
2 Theoretical analysis
2.1 The SVR model
The SVM is based on the principle of structural risk minimization, it has a good generalization ability of learning model. More importantly, it can deal with the small sample data well. As a branch of the SVM, the main purpose of SVR is to fit a reasonable structural model by collecting data. The theories of the SVR are as follows:(1)(2)whereωis the weight vector,x_{i}is the input variable matrix,bis the bias.y_{i}是目标值，f(x_{i}) is the predicted value. Thef(x_{i}) is the unknown function, which depends on the sample data. If the sample data within two hyperplanes (see inyaboapp ), the error can be ignored. It can be expressed as
whereεis the precision. Through the above analysis, the regression problem can be transformed into minimizing problem of an empirical risk. It can be given byyaboapp
当。。。的时候εis given, the equationyaboapp can be solved by any proper algorithm. In order to solve practical problems, we had extended the above mathematical model, and the detailed derivation process can be seen as theyaboapp andyaboapp 。
图。1 SVR模型。 
3模糊不确定性模型
The wear is a gradual process, and it is related to operational performance of equipments. The service conditions and material parameters can affect on the wear rate, which lead to different wear capacity in the same time period. In other words, there is uncertain of the wear under the same conditions, and it has a negative effect on the wear capacity. However, it is unrealistic to describe this uncertainty quantitatively. In order to decrease the effect of sample data which is outside the permissive range, the membership function is designed to express fuzzy uncertainty factors (load, speed, lubrication state).
气缸套的磨损过程经历不同ent wear levels (the initial wear, stable wear, severe wear). When the wear is close to maximum wear stipulated in the technical documents, the wear shows the fuzzy uncertainty characteristics. To model this phenomenon, two hypotheses are given: (1) when the wear is at the primary and stationary stages, the wear capacity can't exceed permissive range, (2) when the wear is at the severe stage, the wear capacity may exceed permissive range. The purpose of the hypotheses is to eliminate the inaccurate prediction caused by the sudden changes of working conditions, that is, to define the applicable scope of the model. Based on the hypotheses, the fuzzy function is employed to describe the uncertainty of the wear capacity in wear process. If the sample data exceed the given threshold, the membership function needs to play a role to make it within the specified range (see inyaboapp ), and the membership function of fuzzy function can be expressed as:(5)wherea_{1}anda_{2}are the wear capacity of the cylinder liner. We defined that the system is safe when the wear capacity within 95% of the maximum wear, and the system has a potential risk when the wear capacity is between 95% and 110% of the maximum wear. The wear capacity is monotonic increase, when the maximum wear capacity is given,a_{2}anda_{1}are determined. Based on this,a_{2}minusa_{1}is always positive, and the convexity ofµ_{A}(x_{i}) does not changed. The system is failure when the wear capacity exceeds the specified range.
When considering the uncertainty factors, the membership function is introduced into the proposed SVR model, and a novel SVR model is established by combining SVR and membership function model. It can be given as(6)
图2 Schematic diagram of membership function. 
4数值应用
The cylinder liner wear has a large impact on performance of the piston ringcylinder system, and the wear can lead to the failure of the system. Therefore, the wear capacity of the cylinder liner is an important parameter for the ICEs. The top dead center of the cylinder liner is the worst working region due to the factors of the soot particles, wear particles, thermal loads and insufficient lubrication, and the maximum wear capacity always occurs in this region. Therefore, in order to prevent sudden failure, the wear capacity in the top dead center of cylinder liner is monitored to diagnose the operational state of the ICEs. In literature [yaboapp ),气缸套的磨损数据报告,and cylinder liners were equipped a fleet of three identical cargo ships of the Grimaldi Group under similar loads, environment and operating conditions. The data set were collected from January 1999 to August 2006, the measure accuracy is 0.05 mm, and wear data were accumulated with the operational time of cylinder liners, as shown inyaboapp 。
图3. Wear data of the 32 cylinder liners. 
5结果和讨论
5.1 Wear data analysis
In order to simulate the wear process and predict the trace of cylinder liner wear, the wear data are analyzed firstly. It can be seen fromyaboapp that the wear capacity increases nonlinearly with operational time, and the linear methods cannot achieve the assessment of cylinder liner wear. Polynomial fitting can deal with nonlinear problems, we try to use polynomial fitting method to process the collected data. The wear data are sorted from small to large according to the time sequence, and the relationship between cylinder liner wear and operational time are fitted by polynomial function, the results are shown inyaboapp 。The result inyaboapp a shows that polynomial function can fit the average wear path, it cannot accurately assess the wear capacity in the next sample point. The residual analysis inyaboapp b also shows that polynomial regression method is not suitable for evaluating of cylinder liner wear. (The closer the residual is to 0, the better the results.) The polynomial regression function ofyaboapp can be expressed by equationyaboapp 。我们可以看到等式yaboapp cannot satisfy the constraint of being null att_{i} = 0, and the regression function is only to find the average wear path from all the wear data. If it satisfies the constraint of being null att_{i} = 0, the fitting error will be greater. Through analyzing of theyaboapp ，不同的磨损能力表明系统磨损方面存在不确定性。因此，为了实现气缸衬套磨损的评估，我们需要找到更有效的回归方法来分析样本数据。(7)
图4. The polynomial fitting curve of the cylinder liner wear. (a) Polynomial fitting. (b) The residual analysis. 
5.2 The novel SVR for PSO
Based on the above analysis, the traditional regression methods are difficult to achieve the evaluation of cylinder liner wear process. Therefore, the new SVR model is used for analyzing the proposed problem. Due to the high nolinear, the kernel techniques are used to deal with nonlinear SVR. The Gaussian radial basis kernel function (GRBKF) is selected in this paper because the strong performance in handling nonlinear problems. In the new SVR model, the model parameters are vital important for the model, and they determine on the performance of the SVR. However, there is no effective way to determine the parameters value, so the optimization method is introduced into SVR model. The PSO algorithm is inspired from the rules by bird swarm activity [yaboapp ,yaboapp ]。It is an evolutionary computing technology which is established by using swarm intelligence, and it mainly uses the individual information sharing of the swarm to optimize the problem. In order to obtain the better consequence, the PSO algorithm is used for optimizing the model parameters. The mean square error (MSE）和平方相关系数（R^{2}）用于判断评估结果，可以给出(8)(9)
The wear data in literature [yaboapp ] is used to assess the model parameters. The material properties and structure of the cylinder liners are the same, and thus the influence of different cylinder liners on wear is ignored. Generally, the number of samples in the training set should be sufficient, and the number of training samples is at least greater than 50% of the total number. Therefore, the 70% of the sample data are selected for model training, and the remaining parts are used to test model in this paper.yaboapp shows the training results of cylinder liner wear by novel SVR.yaboapp a shows the comparison between the measured and the training value of cylinder liner wear on the training set. To observe the errors of proposed model, the 90% prediction interval is given in the figure. It can be seen from the figure that the errors are very small, and the training model can reflect the wear of cylinder liner. In order to understand the error between training and measured data more clearly, the comparison between measured and training data are shown inyaboapp 湾We can draw the conclusion fromyaboapp that the training model is reliable. To illustrate the availability of the training model, theMSEandR^{2}are calculated, the results are shown inyaboapp 。In order to discuss the influence of the parameters on the solution, we changed the particle speed to improve the particle global search ability. The optimal results at different speeds are listed inyaboapp 。Compared with other results the existing results are optimal.
yaboapp shows the testing results of cylinder wear using the training model. Inyaboapp a, the measured data almost coincide with the testing value, and the variation trend of wear capacity indicates that the predicted data are reliable. It can be seen that the error is small, which further illustrates the availability of the proposed model. The comparison between measured data and testing data on testing set are characterized inyaboapp b, the small offset distance reveals that the training model is accurate for evaluating cylinder liner wear. In the testing set, theMSEandR^{2}are 0.010 and 0.968 respectively, which can illustrate the effectiveness of the model parameters.
Through the analysis of the cylinder liner wear data, the proposed SVR model can evaluate cylinder liner wear capacity of the ICEs. Whether in the training or testing set, the results are slightly different from the real values; it can reflect the basic conditions of cylinder liner wear. Comparing the proposed method with polynomial regression, obviously, the proposed model can describe the wear capacity of cylinder liner more accurately at different times.
图5. The SVR training results of cylinder liner wear. (a) Comparison of training and measured data on the training set. (b) Performance of SVR on the training set. 
TheMSEandR^{2}values of the SVR model on training set.
TheMSEandR^{2}以不同的速度设定训练。
图6. 气缸衬套磨损的SVR测试结果。（a）测试集上的测试和测量数据的比较。（b）SVR在测试集上的性能。 
5.3 BPNN analysis
BPNN is a classic prediction algorithm, generally speaking, the consequences obtained by BPNN are reliable. To further verify the validity of the proposed model, the back propagation neural network algorithm is used to predict cylinder liner wear. In order to compare with proposed model, the sample data on training and testing set is set the same as the SVR model. The training network is obtained by processing the training data using BPNN. Based on the established training network, the predicted calculation is employed depend on the testing data.yaboapp shows the BPNN testing results of the cylinder liner wear. Inyaboapp a, we can see that the testing and measured data have the same varies trend, however, the error is relatively larger than SVR model. The performance of BPNN on the testing set is given inyaboapp 湾测量数据的相对距离显着大于图中的测试数据，这表示误差相对较大。yaboapp is a comparison of the testing results between SVR and BPNN on the testing set. It can be seen fromyaboapp that the prediction data obtained from the SVR model are better than BPNN.yaboapp lists theMSEandR^{2}测试集上SVR和BPNN模型的值。经过综合分析，发现SVR对汽缸衬套磨损预测具有更好的性能。
图7. The BPNN testing results of the cylinder liner wear. (a) Comparison of testing and measured data on the testing set. (b) Performance of BPNN on the testing set. 
图8 Comparing of SVR and BPNN on the testing results. 
TheMSEandR^{2}SVR和BPNN模型在测试集中的值。
6 Conclusion
A novel wear assessment model is established based on the SVR in this paper. In order to evaluate the influence of the uncertainty, the external factors can be considered into the proposed model under small sample conditions. The experiment data of cylinder liner wear are employed to evaluate the effectiveness of the proposed model. The particle swarm optimization (PSO) algorithm is used to optimize the parameters of the proposed model. To verify superiority of the proposed model, a comparison with the BPNN is employment. The results show that the novel SVR has a better evaluating performance of mean square error and squared correlation coefficient, and higher regression performance under fuzzy uncertainty conditions, it can assess the cylinder liner wear of the internal combustion engines effectively.
致谢
This work was supported by the National Natural Science Foundation of China (Grant No. 51775428), the Key Research and Development Program of Shaanxi Province of China (Grant No. 2020GY106) and the Open Project of State Key Laboratory for Manufacturing Systems Engineering (Grant No. sklms2020010).
Appendix A Model extension
在本文中，等式yaboapp gives the mathematical model of SVR, however, not all the sample points are within the ±ϵ范围。如果样本点超出±ϵrange, the relaxation factorξ_{i}andξ_{i}*(ξ_{i}_{,}ξ_{i}*≥ 0) must be introduced to satisfy the equationyaboapp 。Therefore, the equationyaboapp can be written as(A.1)
Thus, the minimizing problem of equationyaboapp can be written as(A.2)whereCis the penalty factor (C> 0), the purpose of the penalty factor is to control the penalty degree of the sample points. When consider the relaxation factor, the sample points outside the ±ϵ被称为“ε−insensitive loss function (see inyaboapp ). It can be expressed as [yaboapp ](A.3)
Fig. A.1 The εinsensitive loss function. 
The equationyaboapp is the convex optimization problem, the Lagrange multipliers are introduced to solve the equation. Thus, the Lagrange function can be described as follows(A.4)whereη_{i},η_{i}*,α_{i}, andα_{i}*are the Lagrangian multipliers, and theη_{i}≥ 0,η_{i}*≥ 0,α_{i}≥0，和α_{i}*≥ 0.
In order to obtain the optimal consequences of the proposing problem, the partial derivatives of the Lagrange function to the primal variables (ω,b,ξ_{i},) must be zero. It can be achieved by(A.5)(A.6)(A.7)(A.8)
In finally, the line regression function can be written as
当样本点存在非线性特性时，可以写入回归函数(A.10)where theK(x_{i},x) is the kernel function.
Appendix B Kernel function for SVR
In fact, in the SVR model, the most regression problems are nolinear rather than simple linear regression. In these cases, the computing method is required to map the sample data into high dimensional feature space. After introducing such a mapping, it is not necessary to solve the real mapping function, but only the kernel function needs to be known. In this way, only a specific kernel function needs to be given, which reduces the difficulty of solving regression equation greatly. The commonly used kernel functions are mainly classified into the following categories:
(2) Polynomial Kernel Function(B.2)wherec≥ 0, ifc> 0,K是非均匀多项式内核功能。如果c= 0,Kis the homogeneous polynomial kernel function.P是任意正整数，可以控制VAPnikChervonenkis（VC）尺寸的数量。
(3) Gauss Radial Basis Kernel Function (GRBKF)(B.3)
Gauss radial basis kernel function has a high flexibility, and its flexibility can be controlled byσ.whereσis the variance.
通常，不同的内核功能将产生不同的后果。因此，选择适当的内核功能是非常重要的。
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Cite this article as: J. Kang, Y. Lu, H. Luo, J. Li, Y. Hou, Y. Zhang, Wear assessment model for cylinder liner of internal combustion engine under fuzzy uncertainty, Mechanics & Industry22, 29 (2021)
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图4. The polynomial fitting curve of the cylinder liner wear. (a) Polynomial fitting. (b) The residual analysis. 

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图5. The SVR training results of cylinder liner wear. (a) Comparison of training and measured data on the training set. (b) Performance of SVR on the training set. 

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