Research Article | Open Access

Volume 2018 |Article ID 2969854 | https://doi.org/10.1155/2018/2969854

Li Zhang, Hongli Gao, Dawei Dong, Guoqiang Fu, Qi Liu, "Wear Calculation-Based Degradation Analysis and Modeling for Remaining Useful Life Prediction of Ball Screw", Mathematical Problems in Engineering, vol. 2018, Article ID 2969854, 18 pages, 2018. https://doi.org/10.1155/2018/2969854

# Wear Calculation-Based Degradation Analysis and Modeling for Remaining Useful Life Prediction of Ball Screw

Revised02 Oct 2018
Accepted28 Oct 2018
Published29 Nov 2018

#### 1. Introduction

There are many degradation modeling methods in previous studies: physical mechanism-based method; data driven method; and hybrid method. The physical mechanism-based method attempts to build mathematical or physical models to describe the degradation process of system based on degradation mechanisms. Marble et al. [10] developed a physics-based model for bearing prognostics by computing the spall growth trajectory and time to failure based on operating conditions. Chen et al. [11] built the accelerated degradation model of aerospace electrical connector after researching its failure mechanism. Physical mechanism-based method is direct and convenience. It can provide accurate RUL prediction if the degradation mechanisms are clear. For complex system like ball screw, however, it is difficult to completely understand the failure mechanisms and establish precise degradation model only based on physical mechanism.

The data driven method is an approach to derive the degradation model based on the available observed data. It can be categorized into machine learning-based method and model-based method. Data driven approach is becoming more and more appealing in recent years. It does not need to know the exact failure mechanism during modeling. Machine learning-based method attempts to derive the degradation model from measured data using machine learning techniques. Zhang et al. [12] developed a performance degradation model of screw using quantum genetic algorithm and dynamic fuzzy neural network based on measured vibration signals. Maio et al. [13] proposed a method based on relevance vector machine to estimate the RUL of bearing. Liu et al. [14] proposed an enhanced recurrent neural network to predict the RUL of lithium-ion battery. Zhang et al. [15] presented a degradation recognition method based on deep belief networks and multisensor data fusion to monitor the degradation of ball screw. Machine learning-based method could be beneficial for complex machine whose mechanical principles are not straightforward so that developing an accurate model is impossible. However, shortcomings still exist: the accuracy of machine learning-based method is highly dependent on the quantity and quality of the measured signals. For component with high reliability and long useful life like ball screw, it is prohibitively expensive to collect enough degradation data to establish a machine learning-based degradation model.

Model-based method builds mathematical model at first and then estimates model parameters based on collected data to describe the degradation path. Elwany et al. [16] presented a stochastic degradation modeling framework of partially degraded components to compute the RUL. The Paris-Erdogan (PE) model is one of the most widely used models in the RUL prediction of machinery. Lei et al. [17] transformed the PE model into an empirical model for RUL prediction of machinery. Liao [18] employed the Paris model combined with a genetic programming method to predict the RUL of bearing. Wiener process models are a kind of the most commonly used stochastic process models. Wang et al. [19] developed a linear Wiener process model for RUL prediction of machinery. Si et al. [20] present a relatively general degradation model based on a Wiener process for RUL estimation by considering three-source variability. Paroissin et al. [21] established a randomly delayed Wiener process model considering the degradation starting at a random time. In addition, Liu et al. [22] proposed a degradation modeling approach for a system with multiple degradation patterns based on inverse Gaussian process. Tian et al. [23] proposed a proportional hazard model-based method for the RUL prediction of the systems consisting of bearings. Gebraeel et al. [24] established an exponential degradation model with random error terms and updated the model parameters using Bayesian approach and real-time condition monitoring information. The model-based method incorporates both expert knowledge and measured data, predicting the RUL of machinery with less data. Therefore, it has more advantages than the physical mechanism-based method and machine learning-based method for ball screw.

#### 2. Wear Type Analysis of Ball Screw

In order to further study the wear type of ball screw, the microstructure of its raceway is observed using electron microscope. Properly loaded and lubricated ball screw that is installed on a test bench is selected. For comparison, the micrographs of new screw raceway and degraded screw raceway are both photographed as shown in Figure 1. It can be learned by comparing Figures 1(a) and 1(b) and Figures 1(c) and 1(d) that metallic particles flaking and spalled pit are generated on the raceway of degraded screw, which is judged to be fatigue spalls. Fatigue spalls typically occur at microstructural discontinuities such as inclusions and carbide clusters where the resultant stress exceeds the local microyield limit at that fatigue cycle.

Repeated cyclic stress is considered as the main cause of fatigue spalls. Cracks initiate at the surface stress concentrators and branch up toward the free surface when they reach a critical length or depth, removing a piece of surface material, and form a pit as shown in Figure 1(d) [31]. Fatigue crack growth will finally cause rolling contact fatigue and results in metallic particles flaking from the surface of the balls and raceway. This process that is caused by rolling contact fatigue is the fatigue wear, which is the predominant degradation mode for properly loaded, lubricated, and installed ball screw.

#### 3. Fatigue Wear Volume Calculation

##### 3.1. Normal Contact Force Analysis of Ball Screw

Single nut ball screw with single arc raceway is chosen to analyze the normal contact force in contact surface between ball and raceway. The force analysis diagram of ball is shown in Figure 2(a). Contact points between ball and guide screw, ball and nut are H and G, respectively. Normal contact forces of ball at points H and G are expressed as and , respectively, when the axial working load is . Points H, G and ball center O are collinear. According to the equilibrium condition of two forces, normal contact forces in H and G are equal in magnitude but opposite in direction, which can be expressed as .

Coordinate system Oxyz is established to facilitate the normal contact force analysis and calculation as shown in Figure 2(b), where x axis is in the axial direction of screw, y axis is in the radial direction of screw, and z axis is in the tangent direction of screw in point O. According to the geometrical relationship shown in Figure 2(b), the included angle between y axis and the HG connecting line is the contact angle , and mapping of included angle between x axis and the projection of HG connecting line in xOz surface is the helix angle .

Hence the relationship between axial working load and normal contact forces , iswhere is the number of working balls.

The normal contact forces , can be derived by formula (1).

##### 3.2. The Wear Volume Calculation

It is found that the guide screw wear is more serious than the nut in practice [32], and thus the wear volume of guide screw is calculated to indicate the performance degradation of ball screw in this paper. Researchers in International Business Machines Corporation (IBM) have put forward a model for calculating the wear, which is the function of two variables: the stroke number and energy [33, 34], and can be expressed as a differential equationwhere is the measurable wear volume, E is the energy consumed during each stroke, and N is the number of strokes used to express the useful life of ball screw.

Formula (3) can be rewritten as a differential equation about fatigue wear volume and stroke number N for fatigue wear [33, 35]where is the fatigue wear constant of screw, is the maximum shear stress that ball screw suffers from, and S is the sliding distance in each stroke.

In order to simplify the calculation, it is assumed that the axial working load of ball screw is constant. Hence the maximum shear stress in contact surface between ball and raceway iswhere is the normal contact force that equals and in magnitude and A is the area of contact surface between ball and raceway.

The sliding distance of ball screw in each stroke can be calculated according to the movement principle where is the nominal diameter of guide screw and is the turning laps of ball screw in each stroke.

Considering the initial wear volume and the initial stroke number of ball screw, take the integral to both sides of formula (4), and the calculation formula of fatigue wear volume of ball screw can be derived by substituting into formula (2), (5), and (6)

It can be learned from the fatigue wear volume formula of ball screw that once the type of ball screw is determined (the intrinsic parameters of ball screw including , , , , , are fixed), the fatigue wear volume is in proportion to stroke number N and nonlinearly related to axial load , which is in accordance with that proposed in [33]. Stroke number N is used to express the useful life of ball screw in this paper.

##### 4.1. Derivation of Degradation Model Based on Wear Volume Formula

There are many factors affecting the instantaneous degradation, including initial degradation-level, amount of harmful material, material properties, operating conditions, and environmental conditions (temperature and humidity) [36]. The initial degradation-level, material properties, and environmental conditions are fixed for ball screw with decided type and operating environment. Therefore, the degradation of ball screw is affected by the amount of harmful material and operating condition. Amount of harmful material can be measured by total degradation for ball screw. The wear rate of ball screw is related to axial load according to formula (7), which is a representation of operating condition. Therefore, both the total degradation and wear rate will affect the instantaneous degradation rate of ball screw [37].

Based on these theories, it is assumed that the degradation rate of ball screw is proportional to degradation and wear rate. The degradation model of ball screw will be derived based on this assumption in this paper. The rationality of this assumption will be validated through the derived degradation model, and the degradation model will be verified by analyzing degradation data collected from run-to-failure test of ball screw. The equation is listed according to the above assumptionwhere is the degradation rate of ball screw, D is the degradation, is the instantaneous wear rate for ball screw whose wear volume is W, and is a constant.

Consider the initial degradation of ball screw as . Take the integral to both sides of formula (8) to get the simplified formula.

The degradation model is derived by substituting the wear volume formula (7) into formula (9).

In order to make the structure clearer and easier to understand, measures are taken to simplify the derived degradation model.

Define and as follows.

Then, substitute and into formula (10) to simplify the degradation model as follows.

and are both constant once the type of ball screw is determined and the axial load is invariable. The degradation D is the monotonic function of stroke number N in this situation. Stochastic effect during degradation is considered in the degradation model [8, 28]. The degradation model of ball screw can be expressed as where is the standard Brownian motion and is the diffusion coefficient.

It can be learned from formula (13) that the degradation analysis-based degradation model is a kind of exponential degradation model. It is a typical model in representing the degradation process where cumulative damage has a particular effect on the degradation rate [25, 28]. Exponential model, which was first established by Gebraeel et al. in [24], has been widely used in modeling degradation processes as a kind of experience-based model. Many studies indicate that the exponential model works well in exponential-like degradation processes [25]. In addition, exponential model is also widely used as the degradation model to predict the RUL of bearing [28]. Bearing is similar with ball screw in structure and many mathematical description methods. The success of exponential model in those studies of bearing can preliminarily determine the correctness of the derived degradation model of ball screw.

A diagrammatic curve based on the derived degradation model is drawn to describe the degradation process of ball screw as shown in Figure 3. It is seen that the ball screw degrades rapidly after reaching the critical stroke , which is in accord with the degradation process of machine where cumulative damage exists. Hence the degradation process of ball screw can be divided into two stages based on the critical stroke, i.e., (I) the normal operation stage before and (II) the degradation stage after . It is speculated according to Figure 3 that the degradation will propagate quickly once the critical stroke is reached.

##### 4.2. Construction of Degradation Index

It is crucial to choose suitable degradation index when the condition of ball screw is detected by vibration signals. Degradation index attempts to construct a representative indicator from the acquired signals to reveal the degradation process [38, 39]. Excellent degradation index is usually characterized by monotonicity and correlation. Some degradation indexes like root mean square (RMS), variance, kurtosis, wavelet packet energy, and Weibull distribution shape parameter of signal envelope have received much attention in recent years. Previous studies found that using Hilbert transform to extract envelope is conducive to the early mechanical fault information extraction [40]. Chen et al. [41] treated the Weibull distribution shape parameter of vibration signal envelope as the degradation index of rolling bearing to reflect its incipient failure, and have achieved good results. The Weibull distribution shape parameter of vibration signal envelope is also selected as the degradation index of ball screw in this paper to reflect the degradation D of ball screw. Its effectiveness in reflecting the degradation D of ball screw will be validated in Section 6 by utilizing collected test data.

The calculation process of Weibull distribution shape parameter of vibration signal envelope can be divided into the following two steps:

(1) Envelope Extraction. Envelope of raw signal is extracted based on Hilbert transformation, and the formula of calculating the envelope signal iswhere is the envelope signal of raw signal, is the raw signal, and is the Hilbert transformation of raw signal. can be calculated by the following formula.

(2) Weibull Distribution Shape Parameter Calculation. Fit the calculated envelope signal into a two-parameter Weibull distribution model to get the shape parameter of envelop signal. The probability density function of the two-parameter Weibull distribution model iswhere is the shape parameter and is the scale parameter. The shape parameter of Weibull distribution model is calculated by utilizing maximum likelihood estimation and Newton iterative method.

As the degradation index, the Weibull distribution shape parameter of vibration signal envelope can be used to measure the degradation of ball screw. It can be treated as the degradation D in the derived degradation model (13) to express the degradation model of ball screw. The Weibull distribution shape parameter of vibration signal envelope is described as . Therefore, the derived degradation model (13) of ball screw can be rewritten as the degradation index-based degradation model, which is expressed as follows.

Correctness of the derived degradation model (17) and the selected degradation index will be validated in following paragraphs by proposing validation method and collecting degradation data from run-to-failure test.

#### 5. Degradation Data-Based Cross-Validation Method

The degradation model of ball screw has been established according to degradation analysis in previous sections. The cross-validation method is proposed to verify the exponential model in this section. This method validates the derived exponential degradation model using cross-validation theory based on experimental data, alternately calculating the goodness of fit between the derived degradation model and real degradation path formed by collected degradation data.

##### 5.1. A Brief Introduction to Cross-Validation Theory and Determination Coefficient

Cross-validation is a kind of statistical analysis method that can be used to verify model by calculating the goodness of fit between model and real process data. It divides the original process dataset into n groups and then alternately treats n-1 groups as training set while treating the remaining group as validation set [43]. The cross-validation method used for model verification usually consists of two steps: firstly, estimate unknown parameters of model using n-1 groups of training set to obtain a known model; secondly, calculate the goodness of fitting between the obtained known model and the remaining validation set, and treat the goodness of fitting as the performance evaluation index. Therefore, the data collected in degradation test of ball screw can be used to verify the derived degradation model according to the cross-validation method.

Determination coefficient, which is expressed as , is usually treated as the measurement of fitting [44]. ranges from 0 to 1. Bigger value of means better fitting degree. The goodness of fit between degradation model and degradation process data is measured by determination coefficient . It is calculated aswhere m is the number of degradation data, is the degradation of the ith group of degradation data, is the average degradation of all the performance degradation data, and is the degradation calculated by degradation model in the ith group of degradation process data.

##### 5.2. Cross-Validation Method Based on Experimental Data

Experimental data collected in the degradation test of ball screw can be utilized to calculate the degradation path to validate the derived degradation model. Validation method is proposed based on cross-validation theory and experimental degradation data. This method divides the degradation data of ball screw into n groups and alternatively selects n-1 groups as training set to estimate unknown parameters of the derived degradation model. Then, the goodness of fitting (measured by determination coefficient) between the estimated model and the remaining validation set is calculated to evaluate the derived model.

As shown in Figure 4, the degradation data-based cross-validation method mainly consists of data preprocessing and cross-validation. The overall process of the proposed method is described as follows.

Step 1. Divide total raw degradation signal set into n groups and the grouped vibration signal sets are expressed as , and . The raw degradation signals can be grouped according to the repetition times of data collection, which means the number of clusters is the same as the number of repeats.

Step 2. Calculate corresponding degradation sets of the n groups of raw vibration signal sets based on constructed degradation index. The calculated degradation sets, which are respectively expressed as , and , can be used to form real degradation path and also can be used to estimate unknown parameters of degradation model.

Step 3. Select n-1 groups of degradation sets from , and for model training, and the remaining degradation set is used for model validation. The selection is alternate. It repeats for n times to ensure that each degradation set can be used for both model training and validation.

Step 4. Estimate unknown parameters of derived degradation model using the selected n-1 groups of degradation sets. The obtained known degradation model sets can be expressed as , and , respectively.

Step 5. Calculate determination coefficient sets between the obtained degradation model sets and real degradation path formed by the remaining group of degradation validation set. Cross-validation is required for n times, and n groups of determination coefficient sets are obtained. The calculated determination coefficient sets are, respectively, expressed as , and .

Step 6. Calculate the average of all determination coefficient sets, and the average determination coefficient set is expressed as .

The average determination coefficient set is treated as the index of the derived degradation model in describing the performance degradation process of ball screw. Compared with validating the derived degradation model directly using the degradation data, the cross-validation method makes full use of the collected degradation data through cross-validating among multiple groups of data. This method validates the derived degradation model by calculating the determination coefficient between this model and the real degradation path of ball screw.

#### 6. Experimental Data-Based Degradation Model Validation

In this section, run-to-failure test of ball screw is designed to collect experimental degradation data to generate the real degradation path. The constructed degradation index is validated by using the collected experimental data. Cross-validation method proposed in Section 5 is then utilized to verify the derived exponential degradation model. Besides, RUL of ball screw is predicted based on the proposed exponential model to further validate its correctness and rationality.

##### 6.1. Experiment Study Description and Degradation Data Acquisition

FFZD4010R-3 type of ball screw is chosen to do run-to-failure test by mounting on the acceleration performance degradation test bench, which is designed to simulate the whole performance degradation process of ball screw from new to failure during processing. As shown in Figure 5, this test bench consists of driving part and loading part and is composed of driving motor, slide guide, ball screw, rack and pinion unit, and magnetic power brake. In the driving part, drive torque and rotational speed of ball screw are provided by driving motor. In the loading part, the axial load of ball screw is provided by magnetic powder brake via rack and pinion drive and could be varied by changing the input current of magnetic powder brake. Industrial PC and PC-based control system are used in the test bench to carry out motion control, load control, and data collection as shown in Figure 5.

 Working condition Axial load (kN) Rotational speed (r/min) 1 0 100 2 1 100 3 2 100 4 0 300 5 1 300 6 2 300 7 0 800 8 1 800 9 2 800
##### 6.2. Degradation Index Verification and Data Preprocessing

The degradation index constructed in Section 4 needs to be verified before data preprocessing. After that, the degradation sets can be calculated based on the validated degradation index and raw degradation signals in the data preprocessing.

The degradation index of ball screw that has been constructed in Section 4 is the Weibull distribution shape parameter of vibration signal envelope. In order to verify whether the degradation index could well reflect the degradation of ball screw, data collected in the run-to-failure test is utilized. For reducing computation, working conditions 1, 5, and 9 are selected because these three working conditions contain all these 3 axial loads and 3 rotational speeds. Therefore, vibration signals collected in working condition 9 by accelerometer 1, working condition 5 by accelerometer 2, and working condition 1 by accelerometer 3 are selected to validate the degradation index in this section.

Figure 6 shows the temporal vibration signals of the tested ball screw at different degradation stages, including the normal operation stage, the degradation stage, and the failure stage. It is seen that the amplitude of collected temporal vibration signals increases over degradation. This indicates that the vibration signals can reflect the performance degradation process of ball screw and play a significant role in the performance degradation assessment.

Figure 7 is also compared with Figure 6 to further validate the effectiveness of the proposed degradation index. It is seen that the degradation index-based degradation path has higher monotonicity and tendency and is more direct and clear in describing the degradation process of ball screw than the temporal vibration signals. The constructed Weibull distribution shape parameter of signal envelope performs well in describing the degradation process of ball screw and is suitable to be treated as the degradation index.

###### 6.2.2. Degradation Index-Based Data Preprocessing

There are mainly two steps for calculating the degradation based on the proposed degradation index: firstly, use Hilbert transformation to extract envelope of vibration signal in each group; then, calculate Weibull distribution shape parameter of the extracted envelope, and treat it as the degradation.

##### 6.3. Cross-Validation for the Derived Degradation Model

In the proposed cross-validation method, the fitting degree between the degradation model and the real degradation path is used to evaluate the effectiveness of the derived degradation model. Vibration signals collected in degradation test are utilized to calculate the degradation path and update unknown parameters of degradation model.

In order to show the fitting process visually, degradation data collected in different working conditions by different accelerometers are selected in this section to calculate degradation path and corresponding degradation model. Fitting curves of the calculated degradation model and degradation path are drawn. Both the degradation model and the degradation path in these figures are normalized. Working conditions 1, 5, and 9 are selected to form fitting curves because these three working conditions contain all these 3 axial loads and 3 rotational speeds. Fitting curves of these 3 working conditions are, respectively, shown in Figures 8, 9, and 10. Figures 8(a), 8(b), and 8(c), respectively, represent fitting curves formed by signals collected in different repetitions by different accelerometers for working condition 1. To simplify graphics, Figure 8(a) represents fitting curves of cross-validation 1 and accelerometer 3, Figure 8(b) represents fitting curves of cross-validation 2 and accelerometer 2 and and Figure 8(c) represent fitting curves of cross-validation 3 and accelerometer 1. Figures 9 and 10 have the same structure as Figure 8 and, respectively, represent fitting curves of working condition 5 and 9.

In addition, determination coefficients of working conditions 1, 5, and 9 are also calculated as listed in Table 2, to measure the fitting degree between degradation model and fitting curves. Determination coefficients that correspond to fitting curves in Figures 8, 9, and 10 are marked in bold black font as shown in Table 2. Each group of vibration signals collected in one working condition by one accelerometer could get three determination coefficients because three repetitive cross-validations are applied. The average of these three determination coefficients is calculated to measure the goodness of fit between the derived degradation model and the real degradation path in this working condition. Accelerometer is abbreviated as Acc in Table 2.

 Working condition 1 Working condition 5 Working condition 9 Acc 1 Acc 2 Acc 3 Acc 1 Acc 2 Acc 3 Acc 1 Acc 2 Acc 3 Cross-validation 1 0.6546 0.8029 0.7763 0.7382 0.8322 0.7587 0.8180 0.7950 0.8448 Cross-validation 2 0.6683 0.8157 0.7944 0.7309 0.8278 0.7607 0.8538 0.8074 0.8452 Cross-validation 3 0.6670 0.8090 0.7861 0.7347 0.8294 0.7576 0.8386 0.8099 0.8474 Average 0.6633 0.8092 0.7856 0.7346 0.8298 0.7590 0.8368 0.8041 0.8458

It is seen from these figures that the calculated degradation path and the derived degradation model increase relatively slow in early stage but rise rapidly in later stage, both showing an exponential increase trend. In Table 2, all the average determination coefficients are over 0.6546 and the maximum of these determination coefficients is 0.8538. All of these figures and the table indicate that the derived degradation model fits well with the real degradation path of ball screw.

For comparison, the linear model is also selected to fit the degradation path of ball screw, which is described aswhere is the degradation of ball screw, is the number of stroke, and are model parameters, is the standard Brownian motion, and is the diffusion coefficient.

Taking working conditions 1, 5, and 9 as examples, the determination coefficients between linear model and degradation path of ball screw are calculated and listed in Table 3.

 Working condition 1 Working condition 5 Working condition 9 Acc 1 Acc 2 Acc 3 Acc 1 Acc 2 Acc 3 Acc 1 Acc 2 Acc 3 Cross-validation 1 0.4992 0. 5355 0. 5478 0. 4881 0. 5657 0. 5671 0. 6976 0. 7106 0. 7645 Cross-validation 2 0.4848 0. 5041 0. 5166 0. 4783 0. 5751 0. 5781 0. 7211 0. 7262 0. 7635 Cross-validation 3 0.4929 0. 5213 0. 5336 0. 4839 0. 5713 0. 5733 0. 7167 0. 7290 0. 7659 Average 0. 4923 0. 5203 0. 5327 0. 4835 0. 5707 0. 5728 0. 7118 0. 7219 0. 7647

It can be seen from Table 3 that most of the determination coefficients obtained by linear model in working conditions 1 and 5 are around 0.5, and the maximum value is 0.5781, which is less than 0.6. In working condition 9, the determination coefficients are relatively large and most of them are greater than 0.7. However, it is also less than the value of the proposed exponential model. The results show that the fitting between linear model and degradation path is worse than that of the proposed model, which means the proposed exponential model performs much better than the linear model in describing the real degradation process of ball screw. The comparison further validates the correctness and rationality of the proposed degradation model.

###### 6.3.2. The Results of Cross-Validation

The derived degradation model of ball screw is validated by calculating determination coefficients based on the proposed cross-validation method and collected data in this section. Cross-validation was carried out for 81 times because 81 groups of degradation data are collected in 9 working conditions by 3 accelerometers for 3 repetitions. One cross-validation will calculate one determination coefficient; hence 81 determination coefficients are generated to measure the goodness of fit between degradation model and degradation path. The 81 determination coefficients can be divided into 3 sets according to the 3 repetitive data collections, and each set contains 27 determination coefficients. The average of these three determination coefficient sets is calculated to evaluate the derived degradation model in describing the performance degradation of ball screw. The average determination coefficient set also contains 27 average determination coefficients because every group of degradation data collected in one working condition by one accelerometer could generate one corresponding average determination coefficient. The values of average determination coefficient set are shown in Figure 11. The averages of average determination coefficients collected in one working condition are also calculated and shown in Figure 11.

It can be seen from the results that average determination coefficients change in [0.6633, 0.8594], and the average of all average determination coefficients is calculated as 0.7848. The results mean that the derived degradation model fits well with the real degradation path of ball screw. The high fitting degree indicates the correctness of degradation analysis about ball screw in this study. The results also illustrate that the derived exponential degradation model performs well in describing the degradation process of ball screw.

##### 6.4. Ball Screw RUL Prediction Using the Proposed Degradation Model
###### 6.4.1. RUL Prediction Method of Ball Screw

To further demonstrate the correctness and capability of the proposed exponential degradation model for RUL prediction, the vibration signals collected in working condition 1 by accelerometer 3 are selected to estimate the RUL of ball screw based on the particle filtering (PF) algorithm proposed in [45]. The PF algorithm is utilized to update the model parameters and predict the RUL value by incorporating the measured data, the developed degradation model, and failure threshold. In this paper, RUL is considered as the interval between current time and the first time to reach the predefined failure threshold of monitored degradation data. The flowchart of the RUL prediction method based on PF algorithm is shown in Figure 12.

The initial prediction point of ball screw is defined as the 120th point (120000 strokes), and the failure threshold of ball screw is set as when the amplitude of the acceleration signal overpassed 20g [15]. The PF algorithm is utilized to integrate the degradation model with measured dataset for parameters updating and RUL prediction after the degradation starts. At each inspection interval after initial prediction point, a corresponding RUL distribution can be obtained at this time based on failure threshold and newly collected degradation data. The 50% percentile of the RUL distribution is decided as the predicted remaining useful life of current time.

###### 6.4.2. RUL Prediction Results

The RUL prediction results of ball screw based on the derived exponential degradation model and PF algorithm are shown in Figure 13. It can be observed that the predicted RUL converges to the real RUL as time goes, and the RUL can be estimated in a high and stable accuracy after 168000 strokes. At the beginning of predicting process, the predicted RULs deviate a lot from the real values because of the existence of multiple uncertainties due to lack of measurements. The uncertainties reduce as more measured degradation data become available, which enables estimating the RUL in a higher accuracy.

The prediction error of RUL is calculated to evaluate the performance of the prediction method and corresponding degradation model, which is defined as formwhere is the RUL prediction error of the ith point and and are the actual RUL and the predicted RUL of the ith point, respectively.

The RUL prediction errors based on the proposed degradation model are shown in Figure 14. It can be learned from Figure 14 that the prediction error becomes smaller as the amount of measured degradation data increase with time. The prediction error reduced to a very low value after 168000 strokes.

For comparison, the linear model is also applied as degradation model to predict the RUL of ball screw. The RUL prediction results and the prediction errors of ball screw based on the linear model-based prediction method are shown in Figures 13 and 14, respectively. As observed from the figures, there exist large gaps between the estimated RULs and the actual RULs of ball screw for a long time at the beginning by using the linear model. The predicted RUL by linear degradation model deviates to some extent until 195000 strokes, which is much larger than 169000 strokes of the proposed model. The linear model-based RUL estimation converges much slower than the proposed exponential degradation model.

In general, the proposed exponential model-based prediction method converges quickly and has small value of prediction error after reaching 169000 strokes, which indicates that the prediction method based on the proposed degradation model can accurately predict the RUL of the ball screw, thus proving the effectiveness and correctness of the proposed exponential degradation model.

#### 7. Conclusions

In the future, intensive study on the RUL prediction of ball screw based on the proposed degradation model will be the focus of our work. Multiple signals like current signal, acoustic emission signal, and temperature will be collected to combine with vibration signal to further research the degradation model of ball screw.

#### Data Availability

The data used to support the findings of this study have not been made available because the authors can use but do not have the right to share the data.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This research was supported by the National Natural Science Foundation of China (No. 51775452 and No. 51805457), Open Foundation of the State Key Laboratory of Fluid Power and Mechatronic Systems (No. GZKF-201709), and the China Scholarship Council (CSC).

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