Adaptive Residual Life Prediction for Small Samples of Mechanical Products Based on Feature Matching Preprocessor-LSTM

Abstract

In order to solve the problem of predicting the residual life of mechanical products accurately based on small-sample data, this paper proposes a small-sample adaptive residual life prediction model of mechanical products based on feature matching preprocessor-LSTM. First, aiming at the problem of low accuracy of remaining life prediction for small samples of mechanical products caused by multiple time scales and multiple fault states, the failure time data and performance degradation data are fused, and the failure rate and standard deviation are used as the remaining life prediction criteria to intuitively reflect The possibility of failure of a component or system at a certain point in time. Considering the demand of adaptive small-sample residual life prediction data, this paper establishes the adaptive matching pre-processor model of life characteristics. On this basis, the LSTM neural network is used to establish a small-sample adaptive residual life prediction model. Then, the XJTU-SY bearing life data set and the test data of the small-sample life characteristics measured by the RV reducer are used as the research objects, and a small amount of the data set is randomly selected. The remaining life expectancy is predicted from the sample data and compared with its standard remaining life, respectively. The comparison results show that the overall prediction error is small. This study shows that the remaining life prediction model established can better predict the remaining life of mechanical product sub-sample data and provides a feasible method for predicting the remaining life of mechanical product sub-samples.

1. Introduction

With the development of industrialization, the accuracy of predicting the health status of equipment is becoming more and more demanding. Once an enterprise makes a wrong analysis of its health condition it is likely to cause a lot of property damage and even endanger the lives of employees. Therefore, timely and accurate assessment of equipment health and prediction of its remaining service life has become the object of more and more scholars’ research.

It is difficult to measure the cycle test data of the whole life of mechanical products, and the high cost and long period of obtaining life information through reliability tests lead to difficulties in the application of residual life prediction methods based on big data theory. Life test data generally have small sample sizes, unbalanced sample structure, and low information coverage of the data set. Therefore, in the process of residual life prediction, we can only fully exploit the potential value of these small-sample data to produce more accurate and credible residual life prediction results.

Facing the engineering demand of residual life prediction in the context of small samples, the theory of residual life prediction for small samples has been developed significantly in recent years. With the progress of computing tools, the life prediction methods and accuracy have been continuously updated and improved. From the perspective of data types, the theory of small-sample residual life prediction can be divided into prediction method systems such as the residual life prediction method based on failure time data, the residual life prediction method based on performance degradation data, and the residual life prediction method based on multi-source data fusion.

The method of residual life prediction based on the failure time data is usually on the basis of the failure time of the system. On the basis of assuming the form of the system life distribution, the parameters of the system life distribution are estimated by the method of statistical inference, and then residual life distribution of the system after a period of time is obtained. Whether the form of the system life distribution is suitable or not directly affects the accuracy of the residual life prediction results. Marshall [1] and others summarized the commonly used lifetime distribution functions and discussed the parameter estimation methods for the corresponding distribution functions. In order to broaden the sources of available data and keep the failure mechanisms of the system constant in different environments, we can perform the residual life prediction by accelerated life test and the environmental factor folding method to obtain the failure time data in a short time and then convert it to the life distribution under a normal environment [2,3]. It is too expensive for modern mechanical products through life testing to obtain failure time data, and with the rapid development of condition-monitoring technology, it is increasingly convenient and fast to obtain performance indicators characterizing the health status of the system, so the use of the residual life prediction method based on failure time data is gradually reduced, while residual life prediction methods based on performance degradation data are slowly gaining popularity among experts and scholars.

The residual life prediction method based on performance degradation data can establish the performance degradation trajectory of the system based on the historical operation data of the system and determine the residual life of the system by setting the failure threshold of the system and determining the moment when the amount of performance degradation of the system reaches that value. This kind of method does not need complete monitoring data after complete system failure, but rather need to monitor the system based on actual predicted needs, considering constraints such as time and cost, so as to obtain the corresponding performance degradation data. According to whether the data directly reflect the performance or health status of the system, the method is subdivided into methods based on direct monitoring data and those based on indirect monitoring data. The residual life prediction methods driven by direct monitoring data can be divided into those based on time series modeling and those based on stochastic process modeling. Commonly used time series models include time series analysis techniques [4], gray models [5], artificial neural networks [6], support vector machines [7], and combined prediction models of these methods [8]. However, this method can only obtain the magnitude of the residual life span, not its distribution form, and thus does not reflect the uncertainty of the life span prediction results well.

The residual life prediction methods based on stochastic processes mainly include those based on Gamma process modeling, inverse Gaussian process modeling, Wiener process modeling and Markov chain modeling, etc. The Gamma process is usually only applicable to modeling the degradation trajectory of monotonic data [9,10], such as metal wear and crack growth. However, in the actual working environment the system operation will be disturbed by noise and environment, etc., and its degradation process is not strictly monotonic and will show certain random perturbations. The inverse Gaussian process is also only applicable to describe the monotonic degradation process [11], but compared with the Gamma process, the inverse Gaussian process is mathematically easier to derive and implement and more flexible. The advantages of the Wiener model in mathematical and physical sense make it a widely used degradation modeling and residual life prediction method [12], but due to the introduction of Brownian motion, it will have a direct impact on the system residual life prediction results directly due to the introduction of Brownian motion, in other words, it will increase the uncertainty of the results. Markov chain-based modeling methods often require a large amount of expert knowledge or data for training in order to ensure the accuracy of the prediction results [13]. For the continuous degradation process, a discrete approximation is required, which can also cause certain errors.

Indirect monitoring data-driven methods for residual life prediction include stochastic filtering models [14,15], Hidden Markov Chain Models (HMM) [16,17,18,19], Hidden Semi-Markov Models (HSMM) [20,21,22], and Proportional Hazard Models (PHM) [23]. The random filtering model can absorb the risk of the model in a single step. The stochastic filtering model can absorb the latest monitoring data time by time, thus realizing the real-time prediction of residual life and making the prediction results more and more accurate without setting the failure threshold in advance. However, it is also limited by the amount of operational monitoring data, and the lack of data can lead to inaccurate prediction results. HMM as a probabilistic statistical method with good stochasticity characterization ability and potential structural relationship description ability is widely used in the field of mechanical product modeling. However, to be used directly for the residual life prediction of the system, HSMM is to add the time factor to the HMM structure, which can well describe the intrinsic relationship between the system state and the external observations, but this method can only obtain the expected value of the residual life, which is not convenient for the later health management of the system. It can effectively predict the residual life of the system. However, to ensure the effectiveness of PHM, it must assume that the risk ratio is fixed, i.e., the influence of covariates on the survival probability does not change with time.

The residual life prediction method based on multi-source data fusion mainly considers making full use of failure time data and performance degradation data for residual life prediction. Therefore, the sample size of multi-source data constrains the prediction accuracy of residual life. Generally, the data of accelerated life test is used as a priori information to predict the residual life of the system in real time by fusing field online data with the help of the Bayesian inference method. The residual life prediction method based on multi-source data fusion can improve the low accuracy of prediction results due to the lack of samples for small-sample data.

In summary, the residual life prediction method based on failure time data can only obtain the overall life distribution of the system and does not consider the degradation information during the operation of the system, so it cannot well reflect the residual life distribution after a period of system operation. In order to ensure the accuracy of the prediction results, this method requires a large amount of experimental analysis and evaluation of various small samples of information. The residual life prediction model driven by performance degradation data is constructed from the historical monitoring data of system operation. The advantage is that it does not need the real mechanism expression formula of a complex physical system, and the model is not complicated, which is easier to be applied to practical engineering. However, the disadvantage is that a large amount of monitoring data is required to construct high precision models, which are not representative of the actual system and can be learned by collecting more monitoring data, thus improving the residual life accuracy. The disadvantage of the residual life prediction method based on multi-source data fusion is that the sample size of multi-source data restricts the accuracy of residual life prediction, and the advantage is that it can improve the problem of low accuracy of prediction results caused by the lack of samples for small-sample data.

There are still some shortcomings in addressing the accuracy of residual life prediction of mechanical products in a small-sample environment, which mainly include:

① The problem of low accuracy of residual life prediction under multiple time scales and multiple failure states is that mechanical products undergo different internal structural degradation mechanisms according to different operating states (such as start-up, acceleration, shutdown, load, etc.), and the failure time data monitored by different operating states are fragmented and have the characteristics of multiple time scales “small samples.” Also, the current failure state of mechanical products is not a simple two-state, which undergoes a gradual degradation process during the whole life service cycle and presents multiple failure states before failure, resulting in low accuracy of residual life prediction under multiple time scales and multiple failure states for small samples of mechanical products.

② The adaptive capability of the parameters is an important means to maintain the real-time prediction of the residual life of small-sample mechanical products. However, most of them stay in the experimental stage with a long cycle time and a large sample size, which is not applicable to the conditions of small-sample data.

To address the above problems, this paper proposes a small-sample adaptive residual life prediction model for mechanical products based on feature matching preprocessor-LSTM. First, to address the low accuracy of residual life prediction for small samples of mechanical products caused by multiple time scales and multiple fault states, this paper fuses failure time data and performance degradation data and introduces failure rate and standard deviation of life characteristics as the residual life. In this paper, the failure time data and the performance degradation data are fused, and the failure rate and the standard deviation of life characteristics are introduced as the residual life prediction criteria to reflect the failure of a component or system at a certain time. To meet the current demand of adaptive residual life prediction data with small samples, this paper adopts the statistical matching method and establishes the adaptive matching preprocessor model of life characteristics.

2. Small-Sample Adaptive Residual Life Prediction Model Based on Feature Matching Preprocessor-LSTM

2.1. Overall Framework of the Model

Mechanical products residual life predictions obtained from the test data exist in the time scale, multi-failure state residual life prediction problems in the life test of mechanical products, usually only part of the system in the specified time failure, while another part of the system can still work properly. At this point, the data obtained are not only the failure time data of the system, but also the performance degradation data of the unfailing system. Although the overall life distribution of this type of system can be obtained by using only the failure time data, a more accurate residual life prediction can be obtained if the performance degradation data can also be used. Therefore, to address the problem of low accuracy of residual life prediction for small samples of mechanical products caused by multiple time scales and multiple failure states, the failure time data and performance degradation data are fused, and the failure rate and standard deviation of life characteristics are used as the residual life prediction criteria to visually reflect the probability of failure of a component or system at some point.

The adaptive capability of the parameters is an important tool to maintain the real-time prediction of the residual life of small-sample mechanical products. However, most of them stay in the experimental stage with long cycle time and the required sample size is not applicable to the conditions of small-sample data. So, for the current demand of adaptive small-sample residual life prediction data, a life feature matching preprocessor model is established. On this basis, this paper uses a LSTM neural network to establish a small-sample adaptive residual life prediction model. The overall framework of the model is shown in Figure 1.

2.2. Failure Rate Model of Mechanical Product Life Characteristics

Failure rate is one of the important metrics of mechanical product life. It can visually reflect the possibility of failure of a component or system at a certain moment, and the failure rate is used as an important parameter for product reliability analysis, service life evaluation, risk assessment, maintenance decision making, etc. Therefore, the failure rate of the equipment is used as a matching criterion.

In the framework of using time as a life metric, the failure rate can be described by Equation (1), which is the probability of failure per unit time after the product works up to time t.

$$H(t)=\underset{\Delta t\to 0}{\lim }\frac{P(t<T<t+\Delta t/T>t)}{\Delta t}$$

(1)

in the formula

$$\begin{gathered} P(t<T\le t+\Delta t/T>t)=\frac{P(t<T\le t+\Delta t)}{P(T>t)}=\frac{F(t+\Delta t)-F(t)}{R(t)} \\ H(t)=\underset{\Delta t\to 0}{\lim }\frac{F(t+\Delta t)-F(t)}{\Delta t}·\frac{1}{R(t)}=\frac{F^{\prime }(t)}{R(t)}=\frac{f(t)}{R(t)} \end{gathered}$$

(2)

It is assumed that the mechanical product whose lifetime obeys the Weibull distribution. Its two-parameter Weibull distribution [24] is shown in Equation (3).

$$f(t)=(m/\eta ){(t/\eta )}^{m-1}e-{(t/\eta )}^m>0;t\ge 0$$

(3)

where, $m$ is the shape parameter; $\eta$ is the characteristic life.

The expression for the reliability of its product is

$$R(t)=1-F(t)=e^{-(t/\eta )m};t\ge 0$$

(4)

in the formula

$$\begin{array}{cc}F(t)& ={\displaystyle {\int }_{-\infty }^{t}f(t)dt}\hfill \\ & ={\displaystyle {\int }_{-\infty }^t(m/\eta ){(t/\eta )}^{m-1}{e}^{-(t/\eta )m}dt}\hfill \\ & =1-e^{-(t/\eta )m}\hfill \end{array}$$

(5)

Then the failure rate can be expressed as

$$H(t)=\frac{f(t)}{R(t)}=(m/\eta ){(t/\eta )}^{m-1};t\ge 0$$

(6)

2.3. Standard Deviation Model for Small Sample N-Dimensional Lifetime Characteristics

Since it is known from the reliability life distribution theory that the standard deviation of life characteristics has an important influence on the equipment life prediction, the standard deviation of RV reducer life characteristics is used as another matching criterion.

The change trend of the standard deviation of the general life characteristic curve provided by the enterprise is shown in Figure 2.

Let the lifetime characteristic $r_i(t)$, then when the amount of data is large enough, the standard deviation $S$ of the lifetime characteristic can be calculated by the following equation.

$$S=\sqrt{\frac{{\displaystyle \sum _{i=1}^{n(t)}{(r_i(t)-\overline{r(t)})}^2}}{n(t)-1}}$$

(7)

When the sample data has the characteristics of small samples, the trend of the standard deviation of life characteristics curve can be seen and the standard deviation of life characteristics in the life of incidental failure period with the growth of life is relatively flat in the wear and tear period, with the growth of life changing sharply [25,26]. According to the life characteristic standard deviation curve, the small-sample life characteristic standard deviation model is constructed as:

$$t=S{e}^S+S$$

(8)

where $t$ is the service life

2.4. A Preprocessor Model for Adaptive Matching of Mechanical Product Life Characteristics

Assume that a full life-cycle complete lifetime profile contains data sampled in $Q$ sampling intervals, denoted as total sample $r_1(t),r_2(t),\cdots ,r_Q(t)$, total number of samples $Q$, sampling time $t\in [t_i,t_i+T],i=1,2,\cdots ,Q$, and $T$ is the length of the sampling interval. The life cycle characteristics are absolutized in order to highlight the trend of the overall data. Calculate the failure rate $H(t)$ and the standard deviation $S(t)$ of the lifetime characteristics according to Equations (6)–(8), and find the mean value of $H_i(t),i=1,2,\cdots ,Q$ or $S_i(t),i=1,2,\cdots ,Q$ in each sampling interval $a_1,a_2,a_3,\cdots a_Q$. Randomly extract small samples of total N to match the mean value of the characteristic samples $a_{c+i}$, where $\left\{\begin{array}{l}0\le c+i\le Q\\ 1\le i\le N\end{array}\right.$.

Calculate the mean ${\overline{P}}_c$ of the randomly extracted $N$ small samples $a_{c+i}$ with the following formula.

$${\overline{P}}_c=\frac{1}{N}{\displaystyle \sum _{i=1}^N{a}_{c+i}}$$

(9)

where $c$ is taken as $0,1,2,\cdots ,Q-N$, and ${\overline{P}}_0,{\overline{P}}_1,{\overline{P}}_2,\cdots ,{\overline{P}}_{Q-N}$ is obtained as in Equation (10).

$$\begin{array}{l}{\overline{P}}_0=\frac{1}{N}(a_1+a_2+\cdots +a_N)\\ {\overline{P}}_1=\frac{1}{N}(a_2+a_3+\cdots +a_{N+1})\\ \cdots \\ {\overline{P}}_{Q-N}=\frac{1}{N}(a_{Q-N+1}+a_{Q-N+2}+\cdots +a_Q)\end{array}$$

(10)

From the failure rate model and standard deviation model and the trend of the standard deviation of mechanical product life characteristics, it can be seen that ${\overline{P}}_0,{\overline{P}}_1,{\overline{P}}_2,\cdots ,{\overline{P}}_{Q-N}$ has a monotonically increasing tendency, that is

$${\overline{P}}_0\le {\overline{P}}_1\le {\overline{P}}_2,\cdots ,\le {\overline{P}}_{Q-N}$$

(11)

With the small-sample life characteristics ${\widehat{r}}_i(t),i=1,2,\cdots ,N$ to be predicted, ${\widehat{H}}_i(t),i=1,2,\cdots ,N$, ${\widehat{S}}_i(t),i=1,2,\cdots ,N$ and ${\widehat{a}}_i,i=1,2,\cdots ,N$ are calculated in turn, then $\widehat{P}=\frac{1}{N}({\displaystyle \sum _{i=1}^N{\widehat{a}}_i})$ is to be matched with the characteristics, and the adaptive life matching model is established as follows using the obtained series of RV reducer instantaneous failure rate and standard deviation matching criteria ${\overline{P}}_0,{\overline{P}}_1,{\overline{P}}_2,\cdots ,{\overline{P}}_{Q-N}$.

When ${\overline{P}}_c\le \widehat{P}\le {\overline{P}}_{c+1}$, matching yields a range of lifetime $\widehat{b}$ (becomes a linear model)

$$\begin{array}{l}\frac{a_c-a_{\min }}{a_{\max }-a_{\min }}\le \widehat{b}\le \frac{a_{c+1}-a_{\min }}{a_{\max }-a_{\min }},a_c<a_{c+1}\\ \frac{a_{c+1}-a_{\min }}{a_{\max }-a_{\min }}\le \widehat{b}\le \frac{a_c-a_{\min }}{a_{\max }-a_{\min }},a_{c+1}<a_c\end{array}$$

(12)

and standard remaining life

$$\begin{array}{l}l=\frac{(\widehat{a}-a_c)l_{c+1}-(a_{c+1}-\widehat{a})l_c}{(2\widehat{a}-a_c-a_{c+1})}\\ l_c=\frac{a_{\max }-a_c}{a_{\max }-a_{\min }}\\ l_{c+1}=\frac{a_{\max }-a_{c+1}}{a_{\max }-a_{\min }}\end{array}$$

(13)

Rectify to get

$$l=\frac{2\widehat{a}{a}_{\max }-(a_c+a_{c+1})(\widehat{a}+a_{\max })+2a_c{a}_{c+1}}{(2\widehat{a}-a_c-a_{c+1})(a_{\max }-a_{\min })}$$

(14)

After completing the feature matching preprocessing, the input feature instantaneous failure rate ${\widehat{H}}_c(t)$ and the standard deviation ${\widehat{S}}_c(t)$ of the life feature and the standard remaining life $l(t)$ of the output feature are obtained for the small-sample adaptive remaining life prediction model [27,28,29].

2.5. Small-Sample Adaptive Residual Life Prediction Model Based on Feature Matching Preprocessor-LSTM

The lifetime features of the small-sample data are brought into Equations (9)–(11) for preprocessing and into Equations (12) and (14) for lifetime feature matching, and based on the advantages of LSTM in processing and predicting long- and short-term information in time series, the instantaneous lapse rate ${\widehat{H}}_c(t)$ and standard deviation ${\widehat{S}}_c(t)$ of the lifetime features after completing feature matching preprocessing are used as LSTM inputs, and the standard remaining lifetime $l_c(t)$ after completing feature matching preprocessing is used as LSTM output to establish a small-sample adaptive remaining lifetime prediction model based on feature matching preprocessor-LSTM neural network [30].

The prediction model is shown in Figure 3.

Step 1 Based on the data provided by the enterprise from the reliability life provided by the manual, the reliability function $R(t)$ is calculated by Equation (4), and the undefined parameter value $\eta$ of the failure rate function is derived, and the failure rate function $H(t)$ is calculated by Equation (6).

Step 2 After deriving the complete lapse rate function, a series of instantaneous lapse rates $H(t)$ are derived based on the mechanical product lapse rate model.

Step 3 According to the general life characteristic curve, construct the life characteristic standard deviation model Equation (8), or use the measured life characteristic data by Equation (5) to find the standard deviation $S(t)$ of mechanical product life characteristics in each time period.

Step 4 takes the obtained instantaneous failure rate $H(t)$ and standard deviation $S(t)$ of a series of RV reducers as matching thresholds, substitutes the measured life feature data $r(t)$ into the adaptive life matching model Equations (12) and (14) for feature matching, and obtains the input feature instantaneous failure rate ${\widehat{H}}_c(t)$ and life feature standard deviation ${\widehat{S}}_c(t)$ of the small-sample adaptive remaining life prediction model and the output feature standard remaining life $l_c(t)$.

Step 5 The instantaneous failure rate ${\widehat{H}}_c(t)$ and the standard deviation ${\widehat{S}}_c(t)$ of the lifetime features after feature matching preprocessing are used as LSTM inputs, and the standard remaining lifetime $l(t)$ after completing feature matching preprocessing is used as LSTM output, which is imported into the LSTM neural network for fitting to obtain the predicted remaining lifetime ${\widehat{l}}_c(t)$.

3. Adaptive Remaining Life Prediction for Small Samples of Bearings Based on Feature Matching Preprocessor-LSTM

In order to verify the validity of the proposed model, the bearing life dataset from Xi’an Jiaotong University [31] was used as the research object, and its dataset acquisition process is shown in Figure 4. Two acceleration sensors were used for testing the vibration data in horizontal and vertical directions, respectively. The sampling parameters were set as shown in

Table 1. This is a table. RV reducer parameter.

Sampling Frequency	Sampling Interval	Sampling Time
25.6 kHz	1 min	1.28 s

, and a total of 123 sets of experimental data were obtained.

The bearing life dataset from Xi’an Jiaotong University was brought into the constructed model for remaining life prediction according to steps 1 to 5. Following the flow of the above model, the instantaneous failure rate and standard deviation of the partial set were calculated and substituted as the main features into the small-sample adaptive remaining life prediction model based on LSTM neural network [32,33].

Ten random groups of data are set for sampling simulation, and among the ten randomly selected groups, interpolation is used, i.e., every other group of data is used as the experimental set, and a total of five groups are obtained for network training and five groups are used for model testing, where the number of neuron nodes is set to 180.

After establishing the small-sample adaptive remaining life prediction model based on LSTM neural network, the standard deviation and instantaneous failure rate are extracted from the experimental data as the main features according to the above formula, and some other features in the time domain and frequency domain are taken as auxiliary features, and the output is in the form of percentage of remaining life to train the network.

Model simulation was performed on the sample data with the established prediction model: since the LSTM neural network fitting is stochastic and the results vary slightly from run to run, the data were recorded when the training error [34] was minimal. The network model eventually converges.

To test the accuracy of the random matching model, 13~22, 26~35, 53~62 and 113~122 groups of experimental data were taken as the test data, and the results are shown in Figure 5 below.

When 13 to 22 groups of experiments were selected, the results of the comparison between the predicted and real values, the relative error values of prediction, and the training process are shown in Figure 5a,b. LSTM network prediction absolute mean error MAE = 2.7668 × 10⁻⁴, LSTM network prediction mean absolute error percentage MAPE = 2.8233 × 10⁻⁴, LSTM network prediction root mean square error RMSE = 3.8662 × 10⁻⁴.

When 26~35 groups of experiments are selected, the results are shown in Figure 5c,d. LSTM network prediction absolute mean error MAE = 0.0167, LSTM network prediction mean absolute error percentage MAPE = 0.0193, LSTM network prediction root mean square error RMSE = 0.0217.

When 53~62 groups of experiments are selected, the LSTM network prediction results are shown in Figure 5e,f, with absolute mean error MAE = 2.8484 × 10⁻⁴, LSTM network prediction mean absolute error percentage MAPE = 2.9381 × 10⁻⁴, and LSTM network prediction root mean square error RMSE = 3.2739 × 10⁻⁴.

When 113 to 122 sets of experiments were selected, the LSTM network prediction results are shown in Figure 5g,h, with absolute mean error MAE = 0.0110, LSTM network prediction mean absolute error percentage MAPE = 0.0203, and LSTM network prediction root mean square error RMSE = 0.0140.

In summary, the proposed residual life prediction model has less fitting error and accurate prediction results in the case of less data, which proves the effectiveness of the small sample adaptive residual life prediction model based on LSTM neural network.

4. Adaptive Residual Life Prediction for Small Samples of RV Reducers Based on Feature Matching Preprocessor-LSTM

Using the RV reducer test rig [35] shown in Figure 6, the RV reducer life characteristics data were collected.

The power of the servo motor used in the experiment is 5 kw, the range of the torque sensor is 15 N·m, the range of the torque sensor is 200 N·m, and the specification of the magnetic powder brake is 1.5 kw. The speed of the spindle is set to 151 r/min, the input and output shafts of the RV reducer are measured by the torque sensor, and the model of the RV reducer is RV-20E. The data are sampled every 0.5 s. The experiment was conducted at room temperature, and the model parameters of the main equipment of the experimental platform are shown in

Table 2. RV reducer platform model parameters.

Equipment	Model
Servo Motor	Panasonic MSME502GCG
Torque Sensor	NCNJ-101(0 ± 15 N·m)
RV reducer	RV-20E
Torque Sensor	NCNJ-101(0 ± 200 N·m)
Magnetic Powder Brake	FZ400A-1

.

The collected experimental data are shown in

Table 3. Experimental part of the collected data.

Input Torque (N·m)	Input Speed (r/min)	Input Power (kw)	Output Torque (N·m)	Output Speed (r/min)	Output Power (kw)	Efficiency	Speed Ratio
0.796	151.169	0.013	83.42	1.251	0.011	0.867	120.824
0.796	151.161	0.013	83.463	1.25	0.011	0.867	120.9
0.796	151.147	0.013	83.515	1.25	0.011	0.868	120.939
0.795	151.133	0.013	83.561	1.25	0.011	0.87	120.927
0.793	151.116	0.013	83.598	1.25	0.011	0.872	120.914
0.792	151.112	0.013	83.63	1.25	0.011	0.873	120.91
0.792	151.128	0.013	83.66	1.25	0.011	0.874	120.927
0.792	151.134	0.013	83.671	1.25	0.011	0.874	120.94
0.792	151.137	0.013	83.674	1.25	0.011	0.874	120.953
0.793	151.135	0.013	83.672	1.25	0.011	0.873	120.958
0.793	151.127	0.013	83.657	1.25	0.011	0.873	120.952
0.793	151.119	0.013	83.637	1.25	0.011	0.872	120.945
0.795	151.109	0.013	83.605	1.25	0.011	0.87	120.937
0.797	151.11	0.013	83.574	1.25	0.011	0.867	120.938
0.8	151.125	0.013	83.53	1.25	0.011	0.864	120.927
0.804	151.14	0.013	83.483	1.25	0.011	0.859	120.911
0.807	151.142	0.013	83.426	1.25	0.011	0.855	120.885
0.811	151.133	0.013	83.366	1.25	0.011	0.851	120.878
0.812	151.118	0.013	83.299	1.25	0.011	0.849	120.866
0.813	151.112	0.013	83.232	1.25	0.011	0.847	120.861
0.812	151.109	0.013	83.172	1.25	0.011	0.847	120.858
0.812	151.106	0.013	83.107	1.25	0.011	0.847	120.845
0.812	151.104	0.013	83.045	1.251	0.011	0.847	120.823
0.811	151.103	0.013	82.998	1.251	0.011	0.847	120.805
0.811	151.108	0.013	82.96	1.251	0.011	0.847	120.801
0.81	151.121	0.013	82.926	1.251	0.011	0.847	120.811
0.81	151.132	0.013	82.906	1.251	0.011	0.848	120.819
0.808	151.16	0.013	82.908	1.251	0.011	0.849	120.842
0.807	151.195	0.013	82.928	1.251	0.011	0.851	120.854
0.806	151.222	0.013	82.964	1.252	0.011	0.852	120.821
0.805	151.248	0.013	83.012	1.252	0.011	0.853	120.794

.

According to step 1 to step 5, firstly, according to the definition of B10 life [36] in the manual of mechanical products, when the unreliability is 0.1, its reliable life is

$$L_h=K\times \frac{N_0}{N_m}\times {(\frac{T_0}{T_m})}^{\frac{10}{3}}$$

(15)

$K$: life factor; $N_0$: rated output speed r/min; $N_m$: average output speed; $T_0$: rated output torque; $T_m$: average output torque.

$$T_m=\sqrt[\frac{10}{3}]{\frac{t_1·N_1·T_1{}^{\frac{10}{3}}+t_2·N_2·T_2{}^{\frac{10}{3}}+\cdots +t_n·N_n·{T_n}^{\frac{10}{3}}}{t_1·N_1+t_2·N_2+\cdots +t_n·N_n}}$$

(16)

$$N_m=\frac{t_1·N_1+t_2·N_2+\cdots +t_n·N_n}{t_1+t_2+\cdots +t_n}$$

(17)

From Equation (15)

$$t_0={{\displaystyle \sum _{i=1}^n\left[\frac{T_i}{T_0}\right]}}^p\frac{n_i}{n_0}{t}_i$$

(18)

get

$$t_i=t_0\times \frac{n_0}{n_i}\times {\left[\frac{T_0}{T_i}\right]}^p(i=1,2,3,\cdots ,n)$$

(19)

In this case, P is taken as 10/3.

Where according to the data provided by the enterprises, $T_1=266$ N·m; $T_2=53$ N·m; $T_3=160$ N·m; $T_{em}=745$ N·m; $N_1=N_3=10$ r/min; $N_2=20$ r/min; $N_{em}=20$ r/min; $t_1=0.2$ s; $t_2=0.5$ s; $t_3=0.2$ s; $t_{em}=0.05$ s, substituting the data into Equations (16) and (17), we get

$$\begin{array}{cc}{T}_m& =\sqrt[\frac{10}{3}]{\frac{t_1·N_1·{T_1}^{\frac{10}{3}}+t_2·N_2·{T_2}^{\frac{10}{3}}+\cdots +t_n·N_n·{T_n}^{\frac{10}{3}}}{t_1·N_1+t_2·N_2+\cdots +t_n·N_n}}\hfill \\ & =158.8611\hfill \end{array}$$

(20)

$$\begin{array}{cc}{N}_m& =\frac{t_1·N_1+t_2·N_2+\cdots +t_n·N_n}{t_1+t_2+\cdots +t_n}\hfill \\ & =17.8947\hfill \end{array}$$

(21)

Also check the manual of the RV reducer and get $N_0=20$ r/min; $T_0=167$ N·m Substitute the data into Equation (15) to get

$$L_h=K\times \frac{N_0}{N_m}\times {(\frac{T_0}{T_m})}^{\frac{10}{3}}=7921.1$$

(22)

Then from Equation (5) we get

$$F(t)=1-e^{-{(L_h/\eta )}^m}=0.1$$

(23)

where m is taken as the empirical constant 2.

When $m=2$, the solution is $\eta =24403$, then

$$H(t)=\frac{f(t)}{R(t)}=\frac{2}{24403}·\frac{t}{24403};t\ge 0$$

(24)

According to the failure rate model Formula (6), the standard deviation model Formula (8), and the life characteristic matching preprocessor model Formulas (12) and(14), a small-sample adaptive life curve matching model of the RV reducer is established, as shown in

Table 4. Remaining life table corresponding to failure rate and standard deviation.

Instantaneous Failure Rate (h−1)	Standard Deviation (h)	Remaining Life (h)
1.93 × 10−6	0.175675	7721.1
5.11 × 10−6	0.35135	7521.1
8.29 × 10−6	0.527025	7321.1
1.15 × 10−5	0.7027	7121.1
1.47 × 10−5	0.878375	6921.1
1.78 × 10−5	1.05405	6721.1
2.10 × 10−5	1.229725	6521.1
2.42 × 10−5	1.4054	6321.1
2.74 × 10−5	1.581075	6121.1
3.06 × 10−5	1.75675	5921.1
3.37 × 10−5	1.932425	5721.1
3.69 × 10−5	2.1081	5521.1
4.01 × 10−5	2.283775	5321.1
4.33 × 10−5	2.45945	5121.1
4.65 × 10−5	2.635125	4921.1
4.96 × 10−5	2.8108	4721.1
5.28 × 10−5	2.986475	4521.1
5.60 × 10−5	3.16215	4321.1
5.92 × 10−5	3.337825	4121.1
6.24 × 10−5	3.5135	3921.1
6.56 × 10−5	3.689175	3721.1
6.87 × 10−5	3.86485	3521.1
7.19 × 10−5	4.040525	3321.1
7.51 × 10−5	4.2162	3121.1
7.83 × 10−5	4.391875	2921.1
8.15 × 10−5	4.56755	2721.1
8.46 × 10−5	4.743225	2521.1
8.78 × 10−5	4.9189	2321.1
9.10 × 10−5	5.094575	2121.1
9.42 × 10−5	5.27025	1921.1
9.74 × 10−5	5.445925	1721.1
0.000100549	5.6216	1521.1
0.00010373	5.797275	1321.1
0.000106912	5.97295	1121.1
0.000110093	6.148625	921.1
0.000113274	6.3243	721.1
0.000116456	6.499975	521.1
0.000119637	6.67565	321.1
0.000122819	6.851325	121.1
0.000126	7.027	0

.

After establishing the threshold models for the instantaneous failure rate as well as the standard deviation, the experimentally measured 200-h actual data of the RV reducer were preprocessed by Equations (9)–(11) and imported into Equations (12) and (14) adaptive life matching models, and the instantaneous failure rate and standard deviation after completing the matching were imported into the LSTM neural network as the input, and the remaining life was used as the output, and the interpolation method was used to construct a training set and a test set for model training [37] to determine the remaining life of the RV reducer, and the obtained results are shown in Figure 7 [38].

It can be seen from Figure 7 that the predicted data has a general effect at the beginning, but the fitting error gradually decreases from the 8th sample, because the experimental data is only 200 h of sample data, which is far less than the full cycle life, which can be regarded as a small sample data, which verifies the feasibility and effectiveness of the model based on it.

5. Conclusions

In view of the problem of low prediction accuracy of the remaining life of small samples of mechanical products caused by multiple time scales and multiple fault states, the failure time data and performance degradation data are fused, and the failure rate and standard deviation are introduced as the remaining life prediction criteria, intuitively reflecting the possible failure of a component or system at a certain moment. Aiming at the current demand for self-adaptation of residual life prediction data for small samples, a statistical matching method is used to establish a preprocessor model of life feature self-adaptive matching. On this basis, the LSTM neural network is used to establish a small-sample adaptive residual life prediction model.

Using the bearing life dataset of Xi’an Jiaotong University as the research object, a small amount of sample data in the dataset is randomly selected for remaining life prediction, substituted into the constructed small-sample adaptive remaining life prediction model for prediction, and compared with the standard remaining life of bearings. The prediction results show that the remaining life of the RV reducer is about 6121.1~5921.1 h, and compared with the standard remaining life of the RV reducer, the results show that the overall prediction error is small, which verifies the accuracy of the proposed model and feasibility of the proposed model.

The proposed model can match the life characteristics of mechanical products based on a small number of life characteristics samples and give accurate remaining life prediction curves, which overcomes the shortcomings of traditional life prediction methods with large feature requirements and provides a feasible method for the remaining life prediction of mechanical products with small samples, and this method can also provide reference significance for other equipment with similar life prediction, which can effectively reduce the economic loss of enterprises and help improve the efficiency of enterprises.