# Multistate Reliability Analysis of Recessive Fault of Mechanical Meta-Action Unit Based on Markov Process

Authors: Hongyu Ge; Yuan Liu; Chuanwei Zhang; Manzhi Yang
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In order to clearly express the reliability dynamic change process of mechanical meta-action units with recessive fault on continuous time series, this paper proposes a reliability analysis method for multistate systems with recessive failures of mechanical meta-action units based on the fusion of vibration signal analysis and Markov process. By analyzing the vibration signal, five major recessive fault types of mechanical meta-action units are determined. By analyzing the experimental data and based on the average time of the first occurrence of five major recessive faults, a performance level state representation model of mechanical meta-action unit based on fault importance weight is established. Then, based on the repairable characteristics of the mechanical element action unit, the two-way state transition model and state probability differential equation of mechanical meta-action units are established, and the state probability of each state is obtained. Next, under the condition of determining the initial state, the change process curves of the instantaneous availability, instantaneous average performance, and instantaneous average performance deficit of the mechanical meta-action unit are obtained by solving the reliability index calculation formula. Finally, this paper takes the worm rotation meta-action unit as an example to verify the law and probability of the state transition of the mechanical meta-action unit, and the performance level change accompanying the state transition process, by analyzing the failure modes and causes of recessive faults, the corresponding reliability control measures are formulated, and the control effects before and after reliability control are analyzed. The research results show that this method can effectively improve the accuracy of the state probability when calculating the state probability of each state, and compared with the discrete Markov model, when studying the reliability of complex multistate systems, this method can dynamically describe the change process of state probability, instantaneous availability, instantaneous average performance, and instantaneous average performance deficit in real time under the condition of continuous time-dependent variables. It has certain guiding significance for the reliability analysis of mechanical element action units in the long-term range.


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## Body

## 1. Introduction

In recent years, with the development of the manufacturing industry, the domestic demand for high-precision precision machine tools has been increasing year by year. The reliability level of CNC (Computer Numerical Control) machine tools and their key functional components in my country is far from that of foreign countries, and reliability has become one of the important indicators to measure the performance of CNC machine tools and their key functional components. Therefore, it is urgent to study the reliability of CNC machine tools and their key functional components [1–3]. However, as the minimum action unit of a high-precision machine tool, the recessive faults in the machining process and the performance of the mechanical meta-action unit can directly affect the reliability and stability of the whole machine. Therefore, on the premise of pursuing the reliability of the whole machine, the reliability of its mechanical meta-action unit must be considered [4, 5]. In engineering practice, the mechanical meta-action unit often has an intermediate state of failure and integrity. At this time, the mechanical meta-action unit can still continue to work. But if appropriate preventive measures are not taken, the functional failure will occur in advance, from recessive failure to functional failure [6, 7]. The recessive fault state is between the normal state and functional fault and has a certain latency and a variety of states. However, the traditional Markov technology is widely used and effective in the reliability evaluation of multistate systems, and its application consists of two steps: the construction of the overall state space of the system and the system reliability evaluation by solving the corresponding differential equations.At present, many researchers at home and abroad have done a lot of in-depth research on the recessive fault of mechanical equipment, reliability of multistate systems, and performance evaluation. In the aspect of recessive fault detection and status evaluation, Huang [8] analyzed the technical basis of potential hidden fault prediction based on the state inheritance and correlation of electronic equipment in the operation process and puts forward a potential fault state assessment method which is based upon multistate reliability analysis. Kordestani [9] proposed a new fault detection and isolation system, which uses a dynamic neural network to deal with incipient faults at their early stages. The dynamic structure of the neural network designed helps the observer to deal with the nonlinearity of the system and provide fault isolation in the whole operating range. Mousavi [10] introduced a fault detection and isolation strategy for industrial gas turbines based on an ensemble learning method. This method uses the Wiener model and relevant residuals to detect the faults and improves the accuracy and robustness of detection. Kordestani [11] proposed a new hybrid fault detection method, by analyzing vibration signals, extracting time-domain features, adaptively adopting dynamic principal component analysis, identifying fault dynamics by reducing the dimension of time-domain features, and proving that the method is comparable. Compared with the dynamic principal component analysis method of multilayer perceptron neural network, it has more advantages. Chen [12] designed a new type of cable television network with spectrum calculation and fault diagnosis functions. Compared with the traditional diagnosis method, this network model has improved the diagnosis accuracy. Souza [13] proposed using the predictive maintenance model of a convolutional neural network to automatically classify the faults of rotating equipment. This method only uses a set of vibration sensors to classify the faults in rotating machinery and has a high accuracy rate. Tan et al. [14] carried out a comparative study of several state-of-the-art multilabel classification algorithms for simultaneous fault diagnosis of marine machinery based on single fault data and proved the effectiveness of the proposed method.Tosca and Lyonnet [15] proposed a fuzzy classification diagnosis method based on multiple classifiers, and the results show that the performance of the classifier is improved compared with traditional methods.In the field of multistate reliability analysis, the Markov model has been widely used in preventive maintenance, combined forecasting, wind power system reliability research, and steady-state research of multistate complex systems with good predictive accuracy in recent years [16–20]. In order to minimize costs in maintenance of the multistate repairable system, Gu [21] models a preventive maintenance (PM) scheme of the multistate repairable system using the non-Markov process. Li [22] proposed a method of incorporating fuzzy probability and Bayesian networks into multistate systems. This method has been proved to improve the ability of reliability evaluation of complex systems with uncertain issues. Hudson and Kapur [23] mentioned the change process of complex multistate systems from perfect function through various levels of performance degradation to complete failure, providing a new idea for studying multistate. Li et al. [24] proposed a method to analyze the reliability of multistate systems when the available data of components are insufficient. The research results show that this method is effective when the state performance level and/or state probabilities of components are uncertain and/or imprecise. Lisnianski et al. [25] proposed a multistate Markov model of coal power generating units, which proved to be very useful for power system security analysis and short-term operational decisions. Jinsong et al. [26] proposed a novel weighted hidden Markov model (HMM)-based approach for tool wear monitoring and tool life prediction, using the signals provided by TCM techniques. Chan and Yoo [27] proposed a diagnosis procedure to identify the size and location of a defect. The effectiveness of the proposed method is validated using a numerical analysis model of a rotating blade having a crack. Yu et al. [28] suggested a new method based on the combination of the stochastic process method and the universal generating function technique to evaluate the instantaneous availability and the mean instantaneous performance deficit of the proposed repairable MSS. Gregory et al. [29] presented an approach for evaluating the reliability of standby systems composed of multistate elements with constant state transition rates. In performance evaluation, Qiu et al. [30] proposed a new dynamic reliability metric index and an important degree calculation method under performance level constraints. Gu [31] divided the original system with different performance levels, regards the component polymorphism as a discrete stochastic process, establishes its Markov process equation, and solves the state probability and performance expectation of the components at each level during the task time.Based on the above research, this paper proposes a multistate system reliability research method based on the Markov process, and this method considers the recessive fault as the influencing factor affecting the multistate change of complex multistate system for the first time. When studying the reliability change of mechanical element action unit with hidden fault, this method has certain real-time and dynamic observation effect and can effectively analyze the state probability of each state, instantaneous availability, instantaneous average performance, and instantaneous average performance deficit of worm rotation meta-action unit under different time conditions.


## 2. Vibration Signal Analysis and Performance Analysis of Mechanical Meta-Action Unit

### 2.1. Modeling of the Mechanical Meta-Action Unit Test Bed

The mechanical meta-action unit is the smallest action unit that the CNC machine tool can complete the most basic action (move or rotate) to realize the function of the whole machine. This paper takes the worm rotation meta-action unit as an example, first builds the test bed of the mechanical meta-action unit, and then describes the model of the test bed and the assembly process of the worm rotation meta-action unit. The schematic diagram of the test bench is shown in Figure1.Figure 1
Test bed of worm rotation meta-action unit.In the test bed as shown in Figure1, a reflective strip is attached to the surface of the worm. The speed sensor transmits the collected speed signal to the multifunction data acquisition card. The multifunction data acquisition card transmits the received speed signal to the data acquisition terminal for analysis. At the same time, a magnetic vibration sensor is attached to the surface of the worm base, and the vibration signal captured by the vibration sensor is also transmitted to the data acquisition terminal for analysis through the multifunction data acquisition card. The main research object of the test bed is the worm rotation meta-action unit. In order to intuitively express and further study, the assembly process is disassembled, and the assembly model is shown in Figure 2.Figure 2
Assembly model of worm rotation meta-action unit.The worm rotation meta-action unit is composed of two main components, the servo motor and the worm meta-action. The servo motor and the worm meta-action are connected by a combined coupling. When the coupling is connected with the main parts, the outer surface of the part port is supported by a flat key for physical pressure.


### 2.2. Vibration Signal Analysis of Mechanical Meta-Action Unit

Each state of the multistate repairable system is defined and divided according to the performance level under different time conditions. In the multistate reliability study of the mechanical meta-action unit, taking the worm rotation meta-action unit in the mechanical meta-action unit as the research object, when studying its performance level, it is necessary to divide its performance level first. In the manufacturing process of the worm, due to the differences in technology and materials, there are many performance indexes in evaluating the performance level of the worm rotation meta-action unit. At present, the state corresponding to the performance level in different periods is often divided by testing its transmission performance, speed, and vibration. In the experiment, the vibration signal and speed signal are collected at the same time, and the recessive faults of the worm rotation meta-action unit when different combined signals appear are observed. The recessive fault types are determined by identifying the combined signals. But in the process of the experiment, it is found that when the motor and worm rotate coaxially and the motor speed is maintained at the set fixed value, when the recessive fault occurs, the vibration and speed signals collected by the data acquisition terminal have two kinds of recessive fault signal combinations: (1) the first kind of recessive fault signal combination: the speed of the worm will gradually decrease with the passage of time, but the vibration amplitude is maintained in a small range; (2) the second kind of recessive fault signal combination: the rotational speed of the worm rotating meta-action unit keeps synchronous with that of the motor, but the vibration amplitude changes greatly.In this paper, in order to measure the performance level of each state of the worm rotation meta-action unit, after considering the reference of vibration and speed to evaluate the performance level of the whole worm rotation meta-action unit, the stability before the first failure of the worm rotation meta-action unit is taken as the evaluation basis of the performance level, and stability is defined as the ability to maintain its own state without recessive failure before the first recessive failure of mechanical products. At the same time, the quantitative index of stability, stability degree, is taken as the performance index.In manufacturing systems, common visible problems include product quality defects, lack of precision, equipment failure, and loss of overall operational efficiency. Recessive faults are a prerequisite for such visible problems. Therefore, controlling recessive faults can realize from solving problems to avoiding problems. Because the vibration and speed signals of the worm rotation meta-action unit show certain differences in different states, it is impossible to set the experimental control group under normal operation. Therefore, during the experiment, keep the worm rotation meta-action unit running continuously. When the worm rotation meta-action unit changes in speed and vibration signal, record the recessive fault type of the worm rotation meta-action unit by observing the experimental phenomenon of the worm rotation meta-action unit. Through experiments, it is found that there are five types of recessive faults of worm rotation meta-action unit: coupling slippage, flat key slack, worm axis deviation, axle bearing heating, and accidental worm stuck, and the five recessive faults are independent of each other. According to the test bed shown in Figure1, the speed and vibration signals generated by the worm rotation meta-action unit during operation are collected by the multifunctional data acquisition card through the sensor, and the resulting speed and vibration recessive fault data are uploaded to the data acquisition terminal by the multifunctional data acquisition card, by analyzing the changes of vibration and speed with operation time, to determine the states corresponding to each performance level, as well as the transition law and transition probability between states. Because the single experiment time is too long and the amount of data generated is too large, this paper only presents the vibration and speed signals before and after the stability of the worm rotation meta-action unit changes, so as to show the state of the meta-action unit when the stability of the worm rotation meta-action unit changes. During the experiment, the worm speed is set to 800 r/min, extracting the fault data samples of the worm rotation meta-action unit within 1200 seconds before and after the time when the stability changes, and its sample capacity is all the fault data within 1200 seconds before and after the time when the stability changes. After repeatedly collecting experimental data and extracting fault data samples, the combination forms of recessive fault signals in the occurrence of different recessive faults are obtained. The speed and vibration signals corresponding to each recessive fault are as follows.When there is a recessive fault of the coupling slippage, the recessive fault combination type is the first type of recessive fault signal combination, as shown in Figure3. The vibration amplitude of the worm rotation meta-action unit is maintained within ±700×10−3mm/s2. When the coupling slips, the worm is separated from the motor, and the vibration signal and speed signal are both 0. Throughout the experiment, the vibration amplitude of the worm has always been maintained in a stable range, but the rotation speed of the worm has a tendency to gradually decrease with time. Under the condition that the experimental conditions at the time of separation do not change, reconnect the worm and the motor and run the worm rotation meta-action unit until the entire experimental cycle is completed (see Figure 3).Figure 3
Vibration and speed signal diagram of the recessive fault of coupling slip.When there is a recessive fault of the flat key slack, the recessive fault signal combination type is the second type of recessive fault signal combination, as shown in Figure4. At this time, the vibration amplitude of the worm rotation meta-action unit is maintained within ±1500×10−3mm/s2. Compared with the vibration amplitude of the worm rotation meta-action unit during stable operation, the amplitude is significantly increased at this time. With the increase in time and amplitude, the worm rotation meta-action unit will gradually deviate from the original placement position (see Figure 4).Figure 4
Vibration signal diagram of the recessive fault of flat key slack.When there is a recessive fault of worm axis deviation, the recessive fault signal combination type is the second recessive fault signal combination, as shown in Figure5. When the fault occurs, the vibration amplitude is maintained within ±2300×10−3mm/s2. Compared with the vibration amplitude of the worm rotation meta-action unit when it runs smoothly, the amplitude increases sharply, accompanied by strong noise. By adjusting the horizontal position of the motor and worm and the assembly accuracy of axle bearing and support, the vibration amplitude gradually returns to the level of stable operation (see Figure 5).Figure 5
Vibration signal diagram of the recessive fault of worm axis deviation.When there is a recessive fault of axle bearing heating, the recessive fault signal combination type is the second kind of recessive fault signal combination, as shown in Figure6. The vibration amplitude of the worm rotation meta-action unit is maintained within ±1500×10−3mm/s2. Compared with the vibration amplitude of the worm rotation meta-action unit during stable operation, the amplitude increases significantly at this time, and the amplitude increases continuously with the increase of time and temperature. After the local cooling treatment of the heating part, the time node of the sudden change of the vibration signal in the second experiment was slightly earlier than that in the first experiment (see Figure 6).Figure 6
Vibration signal diagram of the recessive fault of bearing heating.When there is a recessive fault of accidental worm stuck, the recessive fault signal combination type is the second kind of recessive fault signal combination, as shown in Figure7. When the recessive fault occurs, the vibration amplitude of the worm rotation meta-action unit is maintained within +1670×10−3mm/s2, which is significantly higher than the vibration amplitude of the action unit of the worm rotation unit when it runs smoothly. After the recessive fault occurs, the fluctuation range of the vibration signal returns to the level of smooth operation. After many experiments, it is found that a stuck fault is a random event, and there is no obvious rule to follow (see Figure 7).Figure 7
Vibration signal diagram of the recessive fault of accidental worm stuck.


### 2.3. Performance Index Establishment and Performance Level Analysis of Mechanical Meta-Action Unit

In the process of the experiment, through continuous observation, record the time point and type of recessive fault. When each type of recessive fault appears, record the data of the whole process of one experiment, and carry out the next experiment. Repeat the above experiment 50 times, measure the time before the first failure of the five recessive faults of the worm rotation meta-action unit, and calculate its mean value to get the mean time before the first failure of the five recessive faults separately, as shown in Table1. According to the proportion of the mean time before the first failure of each recessive fault type in the sum of the mean time before the first failure of the five recessive fault types, the index weight of stability degree is obtained (see Table 1).Table 1
Calculation table of recessive fault indexes weight (time unit: hours).
Fault name XmMean time before the first failureIndex weight of performanceCoupling slippage X13.50.071Flat key slack X240.082Worm axis deviation X322.40.457Axle bearing heating X41.50.031Accidental worm stuck X517.60.359According to the above analysis, taking the stability degree of each state as the performance level value, the calculation formula for performance level corresponding to different states is as follows:(1)gi=0.071×ti1+0.082×ti2+0.457×ti3+0.031×ti4+0.359×ti5,where gi is the performance level value when the worm rotation element action unit is in the state i. ti1 is the time before the first failure of the recessive fault of the coupling slippage when in the state i. ti2i is the time before the first failure of the recessive fault of the flat key slack when in the state i. ti3i is the time before the first failure of the recessive fault of the worm axis deviation when in the state i. ti4iis the time before the first failure of the recessive fault of the axle bearing heating when in the state i. ti5iis the time before the first failure of the recessive fault of the accidental worm stuck when in the state i.According to the actual situation, the state of the worm rotation meta-action unit is divided into five categories according to the performance level from high to low: optimal performance, partial fatigue, initial decline, accelerated degradation, and critical failure. Through multiple experiments and data collection, the time before the first failure of various recessive faults of the worm rotation meta-action unit in different states is obtained. The statistical data is shown in Table2 (see Table 2).Table 2
Timetable before the first failure of different recessive failures in different states.
Fault typeximStateiWeightOptimal performancePartial fatigueInitial declineAccelerated degradationCritical failurei=1i=2i=3i=4i=5xi10.071643.532xi20.08265.543.51.5xi30.4572524.72219.515xi40.0312.52.31.710.5xi50.3592018.31813.710.5As shown in Table2, since the state of the worm rotation meta-action unit gradually declines with the passage of time under continuous-time conditions, the first failure time of different recessive faults under different conditions gradually decreases. In different states, the relationship between the time before the first failure of different recessive faults is optimal performance ≥ partial fatigue ≥ initial decline ≥ accelerated degradation ≥ critical failure.According to the performance level calculation formula of formula (1), the performance level value of the worm rotation meta-action unit under each state is obtained, and the rank of the performance level is divided according to the size of the performance level value. The state and state performance levels are shown in Table 3 (see Table 3).Table 3
Table of various states and their performance levels.
Status numberStatus namePerformance levels’ valuegiThe rank of the performance level5Optimal performance19.6U54Partial fatigue18.7U43Initial decline17.1U32Accelerated degradation14.4U21Critical failure≤10.9U1


## 2.1. Modeling of the Mechanical Meta-Action Unit Test Bed

The mechanical meta-action unit is the smallest action unit that the CNC machine tool can complete the most basic action (move or rotate) to realize the function of the whole machine. This paper takes the worm rotation meta-action unit as an example, first builds the test bed of the mechanical meta-action unit, and then describes the model of the test bed and the assembly process of the worm rotation meta-action unit. The schematic diagram of the test bench is shown in Figure1.Figure 1
Test bed of worm rotation meta-action unit.In the test bed as shown in Figure1, a reflective strip is attached to the surface of the worm. The speed sensor transmits the collected speed signal to the multifunction data acquisition card. The multifunction data acquisition card transmits the received speed signal to the data acquisition terminal for analysis. At the same time, a magnetic vibration sensor is attached to the surface of the worm base, and the vibration signal captured by the vibration sensor is also transmitted to the data acquisition terminal for analysis through the multifunction data acquisition card. The main research object of the test bed is the worm rotation meta-action unit. In order to intuitively express and further study, the assembly process is disassembled, and the assembly model is shown in Figure 2.Figure 2
Assembly model of worm rotation meta-action unit.The worm rotation meta-action unit is composed of two main components, the servo motor and the worm meta-action. The servo motor and the worm meta-action are connected by a combined coupling. When the coupling is connected with the main parts, the outer surface of the part port is supported by a flat key for physical pressure.


## 2.2. Vibration Signal Analysis of Mechanical Meta-Action Unit

Each state of the multistate repairable system is defined and divided according to the performance level under different time conditions. In the multistate reliability study of the mechanical meta-action unit, taking the worm rotation meta-action unit in the mechanical meta-action unit as the research object, when studying its performance level, it is necessary to divide its performance level first. In the manufacturing process of the worm, due to the differences in technology and materials, there are many performance indexes in evaluating the performance level of the worm rotation meta-action unit. At present, the state corresponding to the performance level in different periods is often divided by testing its transmission performance, speed, and vibration. In the experiment, the vibration signal and speed signal are collected at the same time, and the recessive faults of the worm rotation meta-action unit when different combined signals appear are observed. The recessive fault types are determined by identifying the combined signals. But in the process of the experiment, it is found that when the motor and worm rotate coaxially and the motor speed is maintained at the set fixed value, when the recessive fault occurs, the vibration and speed signals collected by the data acquisition terminal have two kinds of recessive fault signal combinations: (1) the first kind of recessive fault signal combination: the speed of the worm will gradually decrease with the passage of time, but the vibration amplitude is maintained in a small range; (2) the second kind of recessive fault signal combination: the rotational speed of the worm rotating meta-action unit keeps synchronous with that of the motor, but the vibration amplitude changes greatly.In this paper, in order to measure the performance level of each state of the worm rotation meta-action unit, after considering the reference of vibration and speed to evaluate the performance level of the whole worm rotation meta-action unit, the stability before the first failure of the worm rotation meta-action unit is taken as the evaluation basis of the performance level, and stability is defined as the ability to maintain its own state without recessive failure before the first recessive failure of mechanical products. At the same time, the quantitative index of stability, stability degree, is taken as the performance index.In manufacturing systems, common visible problems include product quality defects, lack of precision, equipment failure, and loss of overall operational efficiency. Recessive faults are a prerequisite for such visible problems. Therefore, controlling recessive faults can realize from solving problems to avoiding problems. Because the vibration and speed signals of the worm rotation meta-action unit show certain differences in different states, it is impossible to set the experimental control group under normal operation. Therefore, during the experiment, keep the worm rotation meta-action unit running continuously. When the worm rotation meta-action unit changes in speed and vibration signal, record the recessive fault type of the worm rotation meta-action unit by observing the experimental phenomenon of the worm rotation meta-action unit. Through experiments, it is found that there are five types of recessive faults of worm rotation meta-action unit: coupling slippage, flat key slack, worm axis deviation, axle bearing heating, and accidental worm stuck, and the five recessive faults are independent of each other. According to the test bed shown in Figure1, the speed and vibration signals generated by the worm rotation meta-action unit during operation are collected by the multifunctional data acquisition card through the sensor, and the resulting speed and vibration recessive fault data are uploaded to the data acquisition terminal by the multifunctional data acquisition card, by analyzing the changes of vibration and speed with operation time, to determine the states corresponding to each performance level, as well as the transition law and transition probability between states. Because the single experiment time is too long and the amount of data generated is too large, this paper only presents the vibration and speed signals before and after the stability of the worm rotation meta-action unit changes, so as to show the state of the meta-action unit when the stability of the worm rotation meta-action unit changes. During the experiment, the worm speed is set to 800 r/min, extracting the fault data samples of the worm rotation meta-action unit within 1200 seconds before and after the time when the stability changes, and its sample capacity is all the fault data within 1200 seconds before and after the time when the stability changes. After repeatedly collecting experimental data and extracting fault data samples, the combination forms of recessive fault signals in the occurrence of different recessive faults are obtained. The speed and vibration signals corresponding to each recessive fault are as follows.When there is a recessive fault of the coupling slippage, the recessive fault combination type is the first type of recessive fault signal combination, as shown in Figure3. The vibration amplitude of the worm rotation meta-action unit is maintained within ±700×10−3mm/s2. When the coupling slips, the worm is separated from the motor, and the vibration signal and speed signal are both 0. Throughout the experiment, the vibration amplitude of the worm has always been maintained in a stable range, but the rotation speed of the worm has a tendency to gradually decrease with time. Under the condition that the experimental conditions at the time of separation do not change, reconnect the worm and the motor and run the worm rotation meta-action unit until the entire experimental cycle is completed (see Figure 3).Figure 3
Vibration and speed signal diagram of the recessive fault of coupling slip.When there is a recessive fault of the flat key slack, the recessive fault signal combination type is the second type of recessive fault signal combination, as shown in Figure4. At this time, the vibration amplitude of the worm rotation meta-action unit is maintained within ±1500×10−3mm/s2. Compared with the vibration amplitude of the worm rotation meta-action unit during stable operation, the amplitude is significantly increased at this time. With the increase in time and amplitude, the worm rotation meta-action unit will gradually deviate from the original placement position (see Figure 4).Figure 4
Vibration signal diagram of the recessive fault of flat key slack.When there is a recessive fault of worm axis deviation, the recessive fault signal combination type is the second recessive fault signal combination, as shown in Figure5. When the fault occurs, the vibration amplitude is maintained within ±2300×10−3mm/s2. Compared with the vibration amplitude of the worm rotation meta-action unit when it runs smoothly, the amplitude increases sharply, accompanied by strong noise. By adjusting the horizontal position of the motor and worm and the assembly accuracy of axle bearing and support, the vibration amplitude gradually returns to the level of stable operation (see Figure 5).Figure 5
Vibration signal diagram of the recessive fault of worm axis deviation.When there is a recessive fault of axle bearing heating, the recessive fault signal combination type is the second kind of recessive fault signal combination, as shown in Figure6. The vibration amplitude of the worm rotation meta-action unit is maintained within ±1500×10−3mm/s2. Compared with the vibration amplitude of the worm rotation meta-action unit during stable operation, the amplitude increases significantly at this time, and the amplitude increases continuously with the increase of time and temperature. After the local cooling treatment of the heating part, the time node of the sudden change of the vibration signal in the second experiment was slightly earlier than that in the first experiment (see Figure 6).Figure 6
Vibration signal diagram of the recessive fault of bearing heating.When there is a recessive fault of accidental worm stuck, the recessive fault signal combination type is the second kind of recessive fault signal combination, as shown in Figure7. When the recessive fault occurs, the vibration amplitude of the worm rotation meta-action unit is maintained within +1670×10−3mm/s2, which is significantly higher than the vibration amplitude of the action unit of the worm rotation unit when it runs smoothly. After the recessive fault occurs, the fluctuation range of the vibration signal returns to the level of smooth operation. After many experiments, it is found that a stuck fault is a random event, and there is no obvious rule to follow (see Figure 7).Figure 7
Vibration signal diagram of the recessive fault of accidental worm stuck.


## 2.3. Performance Index Establishment and Performance Level Analysis of Mechanical Meta-Action Unit

In the process of the experiment, through continuous observation, record the time point and type of recessive fault. When each type of recessive fault appears, record the data of the whole process of one experiment, and carry out the next experiment. Repeat the above experiment 50 times, measure the time before the first failure of the five recessive faults of the worm rotation meta-action unit, and calculate its mean value to get the mean time before the first failure of the five recessive faults separately, as shown in Table1. According to the proportion of the mean time before the first failure of each recessive fault type in the sum of the mean time before the first failure of the five recessive fault types, the index weight of stability degree is obtained (see Table 1).Table 1
Calculation table of recessive fault indexes weight (time unit: hours).
Fault name XmMean time before the first failureIndex weight of performanceCoupling slippage X13.50.071Flat key slack X240.082Worm axis deviation X322.40.457Axle bearing heating X41.50.031Accidental worm stuck X517.60.359According to the above analysis, taking the stability degree of each state as the performance level value, the calculation formula for performance level corresponding to different states is as follows:(1)gi=0.071×ti1+0.082×ti2+0.457×ti3+0.031×ti4+0.359×ti5,where gi is the performance level value when the worm rotation element action unit is in the state i. ti1 is the time before the first failure of the recessive fault of the coupling slippage when in the state i. ti2i is the time before the first failure of the recessive fault of the flat key slack when in the state i. ti3i is the time before the first failure of the recessive fault of the worm axis deviation when in the state i. ti4iis the time before the first failure of the recessive fault of the axle bearing heating when in the state i. ti5iis the time before the first failure of the recessive fault of the accidental worm stuck when in the state i.According to the actual situation, the state of the worm rotation meta-action unit is divided into five categories according to the performance level from high to low: optimal performance, partial fatigue, initial decline, accelerated degradation, and critical failure. Through multiple experiments and data collection, the time before the first failure of various recessive faults of the worm rotation meta-action unit in different states is obtained. The statistical data is shown in Table2 (see Table 2).Table 2
Timetable before the first failure of different recessive failures in different states.
Fault typeximStateiWeightOptimal performancePartial fatigueInitial declineAccelerated degradationCritical failurei=1i=2i=3i=4i=5xi10.071643.532xi20.08265.543.51.5xi30.4572524.72219.515xi40.0312.52.31.710.5xi50.3592018.31813.710.5As shown in Table2, since the state of the worm rotation meta-action unit gradually declines with the passage of time under continuous-time conditions, the first failure time of different recessive faults under different conditions gradually decreases. In different states, the relationship between the time before the first failure of different recessive faults is optimal performance ≥ partial fatigue ≥ initial decline ≥ accelerated degradation ≥ critical failure.According to the performance level calculation formula of formula (1), the performance level value of the worm rotation meta-action unit under each state is obtained, and the rank of the performance level is divided according to the size of the performance level value. The state and state performance levels are shown in Table 3 (see Table 3).Table 3
Table of various states and their performance levels.
Status numberStatus namePerformance levels’ valuegiThe rank of the performance level5Optimal performance19.6U54Partial fatigue18.7U43Initial decline17.1U32Accelerated degradation14.4U21Critical failure≤10.9U1


## 3. The Availability Analysis of Mechanical Element Action Unit Based on the Markov Model

As far as the mechanical meta-action unit is concerned, the parts assembled by it are repairable and replaceable. Therefore, the research object of this paper, the worm rotation meta-action unit, is also a repairable multistate assembly structure. The state space diagram of the worm rotation meta-action unit is shown in Figure8.Figure 8
State transfer diagram of worm rotating meta-action unit.In the process of multistate reliability analysis based on the Markov chain, the differential equations of the reliability function of the worm rotation meta-action unit are as follows:(2)dp5tdt=∑e=14μe,5pet−p5t∑e=14λ5,e,dpitdt=∑e=i+15λe,ipet+∑e=1i−1μe,ipet−pit∑e=1i−1λi,e+∑e=i+15μi,e,i<j<5,⋮dp1tdt=∑e=25λe,1pet−p1t∑e=25μ1,e.The initial condition is(3)p50=1,p40=p30=p20=p10=0.The reliability function is(4)Rit=1−p1t=∑j=i+15pjt,1≤i≤4.Whent⟶∞, the worm rotation meta-action unit always enters the absorption state, so the final state probability of equation group (2) is(5)p5=p4=p3=p2=0,p1=1.The repair of a multistate repairable mechanical meta-action unit can be divided into major repair and minor repair. Minor repair makes the mechanical meta-action unit from statei to state i+1, while major repair makes it from state j to state i, where i>j+1. According to the statistics of historical fault data, the failure rate and maintenance rate of the worm rotation meta-action unit are shown in Table 4 (see Table 4).Table 4
Statistics of failure and maintenance rate of the meta-action unit.
Minor faultMajor faultFailure rateλijMaintenance rateμjiFailure rateλijMaintenance rateμjiλ5,415 times/quarterμ4,540 times/quarterλ5,36 times/quarterμ3,515 times/quarterλ4,310 times/quarterμ3,435 times/quarterλ5,22.5 times/quarterμ2,510 times/quarterλ3,29 times/quarterμ2,330 times/quarterλ4,23 times/quarterμ2,415 times/quarterλ2,17 times/quarterμ1,220 times/quarterλ4,11 times/quarterμ1,43 times/quarterλ3,12 times/quarterμ1,35 times/quarterAccording to the above statistical table of failure rate and maintenance rate, the state probability of the worm rotating meta-action unit is obtained by solving the following differential equations:(6)dp5tdt=−λ5,4+λ5,3+λ5,2p5t+μ4,5p4t+μ3,5p3t+μ2,5p2t,dp4tdt=λ5,4p5t−λ4,3+λ4,2+λ4,1+μ4,5p4t+μ3,4p3t+μ2,4p2t+μ1,4p1t,dp3tdt=λ5,3p5t+λ4,3p4t−λ3,2+λ3,1+λ3,4+μ3,5p3t+μ2,3p3t+μ1,3p1t,dp2tdt=λ5,2p5t+λ4,2p4t+λ3,2p3t−λ2,1+λ2,3+λ2,4+μ2,5p2t+μ1,2p1t,dp1tdt=λ4,1p4t+λ3,1p3t+λ2,1p2t−μ1,2+μ1,3+μ1,4p1t.The solution is obtained by the Laplace-Stieltjes transformation and inverse transformation. The equations of equation (6) after the Laplace-Stieltjes transformation are as follows:(7)p1˜s=λ4,1p4˜s+λ3,1p3˜s+λ2,1p2˜ss+μ1,4+μ2,4+μ3,4,p2˜s=λ5,2p5˜s+λ4,2p4˜s+λ3,2p3˜s+μ1,4p1˜ssλ2,1+μ2,3+μ2,4+μ2,5,p3˜s=λ5,3p5˜s+λ4,3p4˜s+λ2,3p2˜s+μ1,3p1˜ss+λ3,2+λ3,1+μ3,4+μ3,5,p4˜s=λ5,4p5˜s+λ3,4p3˜s+λ2,4p2˜s+μ1,4p1˜ss+λ4,3+λ4,2+λ4.1+μ4,5,p5˜s=λ4,5p4˜s+λ3,5p3˜s+λ2,5p2˜ss+λ5,4+λ5,3+λ5,2,where pk˜s=Lpkt=∫0∞e−stpkt is the Laplace-Stieltjes transformation of the function pktpkt, and according to formula (7), carry out the Laplace-Stieltjes inverse transformation and obtain the state probability equations under the condition that the initial state is formula (3), and the equations are accurately and approximately solved as(8)p1t=0.00051×e−81.93t−0.00019×e−68.62t−0.06120×e−24.42t+0.02738×e−53.53t+0.03351,p2t=0.00950×e−68.62t−0.00484×e−81.93t−0.02884×e−24.42t−0.03831×e−53.53t+0.06230,p3t=0.01313×e−81.93t−0.00117×e−68.62t−0.01385×e−24.42t−0.12436×e−53.53t+0.12625,p4t=0.01026×e−24.42t−0.01984×e−81.93t−0.05655×e−68.62t−0.18202×e−53.53t+0.24816,p5t=0.04841×e−81.93t+0.01104×e−81.93t+0.09364×e−24.42t+0.31732×e−53.53t+0.52958.Through solving, the state probability curve of each state corresponding to the different performance levels of the worm rotation meta-action unit is obtained, and the state curve is shown in Figure9.Figure 9
State probability curve.As shown in Figure9,(1)
When the worm rotation meta-action unit is in the initial state, it is in the state of optimal performance, that is, the probability of occurrence of the fifth state with a performance level of U5 is 1, and the probability of occurrence of the other four states is 0 at this time. Because at this time the worm rotation element action unit has not yet been worn, with the passage of time, the wear of the worm rotation element action unit gradually increases, and the probability of occurrence of each state gradually changes.(2)
The state probability of each state at a certain moment can also be understood as the sum of the probabilities of transitioning from the other four states to its own state at a certain moment. That is,p5t is the sum of the probabilities of the fifth state transitioning from the other four states to its own state. Similarly, p4t，p3t，p2t，andp1t can be known.In the actual situation, the mechanical meta-action unit has different performance requirements for the mechanical meta-action unit under different processing requirements and processing environments. It is assumed that the performance requirement iswnn=1,2,3,4 . Then, the instantaneous availability equations of the worm rotation meta-action unit under different performance requirements are(9)A4t=p5t=1−p1t−p2t−p3t−p4t,g4<w4<g5,A3t=p4t+p5t=1−p1t−p2t−p3t,g3<w3<g4,A2t=p3t+p4t+p5t=1−p1t−p2t,g2<w2<g3,A1t=p2t+p3t+p4t+p5t=1−p1t,g1<w1<g2.Through calculation, the instantaneous availability of the worm rotation meta-action unit under different performance requirements is obtained, and the instantaneous availability is shown in Figure10.Figure 10
Instantaneous availability curve.From Figure10 and equation (9), the following can be seen:(1)
A1t，A2t，A3t，andA4t are the instantaneous availability curves when the performance requirement conditions w1，w2，w3，andw4 are met, respectively. The lower the performance level required by the performance requirement conditions, the higher the instantaneous availability of the worm rotation meta-action unit within a quarter.(2)
In the same way, when the performance requirement conditions increase, the instantaneous availability of the worm rotation meta-action unit decreases; that is, the probability of normal use under the requirement conditions decreases. The availability of the worm rotation meta-action unit that meets the established performance requirements before reaching the failure state is 96.6%.At the same time, according to the state probability of each state of the worm rotation meta-action unit and the performance level value corresponding to each state, the instantaneous average performance of the worm rotation meta-action unit within a quarter can be obtained. The calculation formula of the instantaneous average performance is(10)Et=∑i=15gipit=19.6×p5t+18.7×p4t+17.1×p3t+14.4×p2t+10.9×p1t.Through calculation, the instantaneous average performance of the worm rotation element action unit is obtained, and its instantaneous average performance is shown in Figure11.Figure 11
Instantaneous average performance curve.As shown in Figure11,(1)
Within a quarter, with the change of time, the average instantaneous performance of the worm rotation meta-action unit gradually stabilizes at a certain performance level value, and its performance level value is 18.44.(2)
During the whole quarter and normal use under general conditions, the probability of initial decline, accelerated degradation, and critical failure of the worm rotation element action unit is 0, and the probability of operating in the optimal state is 5.1%. The probability of partial fatigue operation is 94.9%.In addition, according to the instantaneous average performance of the worm rotation meta-action unit, the instantaneous average performance deficit can be obtained. For different performance requirementswn, the calculation formula of the instantaneous average performance deficit is(11)Dtn=∑i=14pitmaxwn−gi,0n=1,2,3,4,5.Then, the instantaneous average performance deficit equations under different performance requirements are(12)Dt1=0,w1=g1=10.9,Dt2=3.5×p1t,w2=g2=14.4,Dt3=2.7×p2t+6.2×p1t,w3=g3=17.1,Dt4=1.6×p3t+4.3×p2t+7.8×p1t,w4=g4=18.7,Dt5=0.9×p4t+2.5×p3t+5.2×p2t+8.7×p1t,w5=g5=19.6.Through calculation, the instantaneous average performance deficit of the worm rotation meta-action unit is obtained, and the instantaneous average performance deficit is shown in Figure12.Figure 12
Instantaneous average performance deficit curve.As shown in Figure12,(1)
With the improvement of performance requirements, the instantaneous average performance deficit of the worm rotation meta-action unit increases. When the performance level of the worm rotation meta-action unit meets the minimum performance requirements under the critical failure condition and the worm rotation element action unit reaches the steady state, the deficit value of its instantaneous average performance deficit is 0, and the deficit rate is 0%.(2)
When the performance level of the worm rotation meta-action unit meets the highest performance requirements under the optimal performance condition, the instantaneous average performance deficit value is 1.15, and the deficit rate is 5.9%. That is, before the failure of the worm rotation meta-action unit, the average instantaneous performance deficit rate during the entire quarter of operation is between 0% and 5.9%, and the probability of meeting the performance requirements before the failure event is greater than or equal to 94.1%.


## 4. Reliability Control Measures and Control Effect Analysis of Mechanical Meta-Action Unit

For various types of recessive faults, we analyze and get the reasons for the faults, find out the corresponding reliability control points, and give corresponding reliability control measures for each reliability control point. Its reliability control measures table is shown in Table5 (see Table 5).Table 5
Reliability control measures table.
Recessive fault typesCause of failureReliability control pointsReliability control measuresCoupling slippageExcessive torque during operationDeformation of rubber materialReplace the rubber coupling with a light metal one-piece double-port couplingFlat key slackLong-term wearGenerate assembly gapWorm axis deviationStrong vibration occurs during operationThe snap ring between the worm, the support, and the axle bearing is looseInstall the base plate of the meta-action unit, drill holes on the base plate according to the size of the base, and assemble and strengthen the base and the base plateAxle bearing heatingToo long running time results in friction heat generationThe ball inside the axle bearing is not lubricated enoughPhysical cooling with coolant and regular lubrication and maintenanceAccidental worm stuckRoutine maintenance is less, the worm port is corroded and rustedThere is too much water vapor in the assembly gapAccording to the established reliability control measures, after the reliability control, the overall performance of the worm rotation meta-action unit has been improved to a certain extent. Due to the replacement of the coupling and the installation of the base plate in the worm rotation meta-action unit, the two recessive faults of coupling slippage and worm axis deviation are overcome, and the base plate effectively fixes the worm rotation meta-action unit, so the whole machine vibration of the worm rotation meta-action unit is effectively controlled. The time before the first failure of various recessive faults of worm rotation meta-action units under different states is effectively prolonged. Then, the operation data of the worm rotation meta-action unit are collected and recorded again and analyzed. Among them, the weight calculation table of each recessive fault index after reliability control is shown in Table6.Table 6
Calculation table of recessive fault indexes weight after reliability control (Time unit: hour).
Fault nameXm'Mean time before the first failureIndex weight of performanceFlat key slackX1'9.60.207Axle bearing heatingX2'4.40.095Accidental worm stuckX3'32.40.698According to the above analysis and calculation, the calculation formula of performance level after reliability control is obtained as follows:(13)gi′=0.207ti1′+0.095ti2′+0.698ti3′,where gi′ is the performance level value of worm rotation meta-action unit when in the state i after reliability control. ti1′ is the time before the first failure of the flat key slack recessive fault when it is in the state i after reliability control. ti2′ is the time before the first failure of the axle bearing heating recessive fault when it is in the state i after reliability control. ti3′ is the time before the first failure of the accidental worm stuck recessive fault when it is in the state i after reliability control.After the reliability control of the worm rotation meta-action unit, the experiment was carried out again, and the time before the first failure of the three recessive faults was collected. Through recording, the time before the first failure of different recessive faults in different states is obtained as Table7 shows.Table 7
Timetable before the first failure of different recessive failures in different states after reliability control.
Statei′ FaultType xim'WeightOptimal performancePartial fatigueInitial declineAccelerated degradationCritical failurei′=1i′=2i′=3i′=4i′=5xi1'0.20716.412.29.55.53.7xi2'0.0956.85.64.22.51.3xi3'0.69856.44831.819.512.6According to formula (13), the performance level value of the worm rotation meta-action unit in each state after reliability control is obtained by calculation, and the rank of performance level is divided according to the size of the performance level value. Each state and its state performance levels are shown in Table 8.Table 8
Table of various states and their performance levels after reliability control.
Status numberStatus namePerformance levels’ valuegi′The rank of the performance level5′Optimal performance43.4U5′4′Partial fatigue36.6U4′3′Initial decline24.6U3′2′Accelerated degradation15U2′1′Critical failure≤9.7U1′After reliability control, we remake the statistics of the historical failure times and maintenance times of the worm rotation meta-action unit in four quarters and record the type of recessive fault each time the recessive fault occurs. Through calculation, the average number of historical failures and maintenance times of the worm rotation meta-action unit in each quarter is obtained. The failure rate and maintenance rate are shown in Table9.Table 9
Statistics of failure and maintenance rate of the meta-action unit after reliability control.
Minor faultMajor faultFailure rateλij'Maintenance rateμji'Failure rateλij'Maintenance rateμji'λ5,4'11 times/quarterμ4,5'28 times/quarterλ5,3'4 times/quarterμ3,5'8 times/quarterλ4,3'7 times/quarterμ3,4'25 times/quarterλ5,2'2 times/quarterμ2,5'6 times/quarterλ3,2'7 times/quarterμ2,3'18 times/quarterλ4,2'2 times/quarterμ2,4'10 times/quarterλ2,1'4 times/quarterμ1,2'12 times/quarterλ4,1'1 times/quarterμ1,4'2 times/quarterλ3,1'1 times/quarterμ1,3'2 times/quarterAccording to formulas (6) and (7), the state probability curve of each state corresponding to different performance levels of the worm rotation meta-action unit after reliability control is obtained by calculation. The state probability curve is shown in Figure 13.Figure 13
State probability curve after reliability control.It can be seen from Figure13 that after reliability control, the state probability of each state when the worm rotation meta-action unit reaches the steady state within a quarter: the state probability of “optimal performance” state when reaching the steady state is 0.50, the state probability of “partial fatigue” state when reaching the steady state is 0.25, and the state probability of “initial decline” state when reaching the steady state is 0.13. The state probability of the “accelerated degradation” state when reaching the steady state is 0.08, and the state probability of the “critical failure” state when reaching the steady state is 0.04.According to formula (9), the instantaneous availability of the worm rotation meta-action unit after reliability control under different performance requirements is obtained by calculation. The instantaneous availability curve is shown in Figure 14.Figure 14
Instantaneous availability curve after reliability control.It can be seen from Figure14 that when the stable state is reached, when the performance demand is 9.7≤w1,≤43.4, the instantaneous availability of the worm rotation meta-action unit is 95.7%; When the performance requirement is 15≤w2,≤43.4, the instantaneous availability of the worm rotation meta-action unit is 88.1%. When the performance requirement is 24.6≤w3,≤43.4, the instantaneous availability of the worm rotation meta-action unit is 75.3%. When the performance requirement is 36.6≤w4,≤43.4, the instantaneous availability of the worm rotation meta-action unit is 50.1%.According to formula (10), the instantaneous average performance of the worm rotation meta-action unit after reliability control within one-quarter can be obtained by calculation. The change curve of the instantaneous average performance is shown in Figure 15.Figure 15
Instantaneous average performance curve after reliability control.It can be seen from Figure15 that when the worm rotation meta-action unit reaches the stable state after reliability control, its instantaneous performance level value is 35.67. In the normal use of the whole quarter and under general conditions, the probability of initial decline, accelerated degradation, and critical failure of the worm rotation meta-action unit is 0, and the probability of maintaining the operation in the optimal performance state is 9.5%.The probability of keeping running in partial fatigue state is 90.5%.According to formula (11), the instantaneous average performance deficit of the worm rotation meta-action unit after reliability control under different performance requirements is obtained by calculation, and the change curve of the instantaneous average performance deficit is shown in Figure 16.Figure 16
Instantaneous average performance deficit after reliability control.It can be seen from Figure16 that after reliability control, the deficit value of the worm rotation meta-action unit meeting the performance requirements of a critical failure state is 0, the deficit value meeting the performance requirements of an accelerated degradation state is 0.22, the deficit value meeting the performance requirements of initial decline state is 1.22, the deficit value meeting the performance requirements of partial fatigue state is 4.34, and the deficit value meeting the performance requirements of optimal performance state is 7.74.We compare the state probability, instantaneous availability, and instantaneous average performance deficit before and after reliability control, to analyze the improvement of the performance level of the worm rotation meta-action unit and the effect of reliability control after the implementation of reliability control measures. By comparing the state probabilities before and after the reliability control, the state probability changes of each state are obtained, and the state probability changes are shown in Figure17.Figure 17
Change of state probability before and after reliability control.It can be seen from Figure17 that after reliability control, the change value of the state probability of the “optimal performance” state is −0.03, the state probability of the “partial fatigue” and “initial decline” states does not change, the change value of the state probability of the “accelerated degradation” state is +0.02, and the change value of the state probability of the “critical failure” state is +0.01. The overall performance level of the worm rotation meta-action unit after reliability control has been improved. Because the state running at a higher performance level is more vulnerable to damage, the ability of the worm rotation meta-action unit running at a higher performance level to maintain its best state is relatively weak. At the same time, the sum of the state probability of the worm rotation meta-action unit is 1, so when the state probability of the “optimal performance” state decreases, the state probability of other states increases. In addition, in the whole quarter, the instantaneous average performance level value when reaching the stable state remains close to 35.67, which is between the two states of “partial fatigue” and “initial decline”, so the ability of “partial fatigue” and “initial decline” to maintain their own state is strong, and there is no obvious change in state probability. Therefore, the state probability of “accelerated degradation” and “critical failure” states increases accordingly, but the performance level of the worm rotation meta-action unit will change sharply before failure, so the state probability of the “accelerated degradation” state is greater than that of “critical failure” state.By comparing the instantaneous availability before and after reliability control, the instantaneous availability change of each state is obtained. The instantaneous availability change is shown in Figure18.Figure 18
Change of Instantaneous availability before and after reliability control.It can be seen from Figure18 that the instantaneous availability of the worm rotation meta-action unit after reliability control under different performance requirements is less than that before reliability control. According to formula (9), it can be inferred that the instantaneous average performance of the worm rotation meta-action unit after reliability control is improved, and its ability to maintain its best performance state is weak. The state probability of the “optimal performance” state decreases, and the state probability of “accelerated degradation” and “critical failure” state increases. Therefore, it can be concluded that the instantaneous availability of worm rotation meta-action unit with an overall improved performance level is lower than that before the overall improvement of performance level under different performance demand conditions.By comparing the instantaneous average performance deficit before and after reliability control, the instantaneous average performance deficit change of each state is obtained. The instantaneous average performance deficit change is shown in Figure19.Figure 19
Change of Instantaneous average performance deficit before and after reliability control.It can be seen from Figure19 that the instantaneous average performance deficit of the worm rotation meta-action unit after reliability control is greater than that before reliability control. In the study, the performance level values corresponding to each state of the worm rotation meta-action unit before and after reliability control are set as the performance demand conditions of the worm rotation meta-action unit before and after reliability control. The instantaneous average performance deficit of the worm rotation meta-action unit before and after reliability control is calculated under different performance requirements. According to the analysis, the performance level of each state of the worm rotation meta-action unit after reliability control has been improved, but the D-value between the performance levels of each state has increased, which shows that the higher the overall performance level of the mechanical meta-action unit, the greater the difference between the performance levels of each state.


## 5. Conclusion

(1)
In the process of multistate reliability analysis of recessive faults for the worm rotating meta-action unit, the test bed model of the worm rotation meta-action unit is established, the assembly process model diagram of the worm rotation meta-action unit is drawn, then the speed signal and vibration signal are collected when the worm rotation element action unit failed, and five types of recessive faults are summarized according to the characteristics of speed signal and vibration signal and experimental observation.(2)
The performance level evaluation index and state space of the worm rotation meta-action unit are established. By calculating the data collected during the experiment, the performance level value of each state is obtained. According to the state space of the worm rotating meta-action unit, the state transition diagram is drawn, the state transition paths between the states are clarified, and the Markov model is established to determine the mathematical relationship between the states. Through accurate calculation and visual processing, the state probability curve, instantaneous availability curve, instantaneous average performance curve, and instantaneous average performance deficit curve of the worm rotation meta-action unit are obtained and analyzed.(3)
Finally, by analyzing the failure mode and cause, the reliability control point was obtained, and the corresponding reliability control measures were formulated. After reliability control, the state probability, instantaneous average performance, instantaneous availability, and instantaneous average performance deficit of the worm rotation meta-action unit after reliability control are reanalyzed. By comparing and analyzing the three indicators of state probability, instantaneous availability and instantaneous average performance deficit before and after reliability control, the changes of worm rotation element action unit in overall and each state performance level, the ability to maintain its own states, and the difference of performance level between states after reliability control are summarized.



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*Source: 3587380-2022-04-28.xml*