- Introduction
The reliability of mechanical systems consisting of multiple elements installed in series has been widely studied according to known traditional methods [2], [3] by utilizing the reliability function P(t), with the parameters for failure rate λ i and repair rate µ i collected and statistically analyzed at operational units. Today, the accelerated testing method has been widely applied, allowing for the rapid determination of the failure rate λ i and repair rate µ i parameters of the research object.
To describe the state transition process of the research object, which are mechanical systems with elements installed in series, the Markov model can be fully applied. The use of the Markov model in reliability research allows finding a solution that represents the reliability function P(t) of the object over time, which holds predictive significance regarding its technical condition. From this, recommendations can be made to users regarding the appropriate time to perform technical maintenance and repair, while also helping to predict the remaining technical reserve coefficient at a certain point during the operation of the research object.
2. Research methodology
As presented above, one of the basic parameters that need to be known when studying the reliability of a mechanical system is the failure rate λ i and repair rate µ i . In reality, these parameters need to be collected and statistically analyzed regarding failures and recovery methods at the operational units. However, this requires significant effort; additionally, it is necessary to process abnormal failure data caused by user errors, as these failures do not reflect normal operational rules and must be eliminated. Today, the accelerated testing method has been widely applied. The data on the failure rate λ i and repair rate µ i used in this paper are determined through accelerated testing activities at the Institute of Motive Power Engineering, Le Quy Don Technical University.
Based on the operating states of the clutch, we establish a Markov model corresponding to the respective technical states. Combined with the theoretical basis of reliability applied to technical systems consisting of multiple elements installed in series presented below, we can determine the operational reliability of the system over time.
2.1. Reliability of Mechanical Transmission Systems
a. Probability of failure-free operation of any mechanical transmission system
Let p i (t), (i = 1, 2,..., n) be the probability of failure-free operation or the reliability function of the i-th element of a system consisting of n elements installed in series at a specified time t and P h (t) is the probability of failure-free operation of the system. Then, the probability of failure-free operation of the system consisting of n elements installed in series is determined as follows [2], [3]:
Since p i (t) ≤ 1 then P h (t) ≤ min [p i (t)], i =1, 2,..., n.
The function
b. Failure probability Q h (t) for any mechanical transmission system
The failure probability Q h (t) of the system is determined by [2], [3]:
Where: q i (t) — is the failure probability of the i-th elemen, i = 1, 2,...., n.
If the reliability of the elements at a specified time t is the same, q 1 (t) = q 2 (t) =... = q n (t) = q(t) then the failure probability of the series system of elements is determined by:
c. System failure rate λ h (t):
The failure rate λ
h
(t) of a system consisting of n elements installed in series is determined by the expression [2], [3]:
Where: λ i (t) — s the failure rate of the i-th element, i = 1, 2,.., n.
d. System repair rate of a
series system
[2], [3]:
The above expressions apply to mechanical transmission systems consisting of n elements installed in series with each other.
Thus, we can summarize, for a typical mechanical transmission system consisting of 3 elements installed in series, we have: (6). Therefore, we can evaluate the reliability of any mechanical transmission system through the indices presented in system (6) with 3 series-connected components, for example, the transmission systems of mechanisms such as clutches, gearboxes, etc.
2.2. Markov Model in Studying the Reliability of Technical Systems
In reality, mechanical transmission systems on transport vehicles (such as clutches, gearboxes) are repairable systems. During operation, their technical condition rarely changes abruptly from «perfectly good» to «completely failed», but usually degrades gradually through many intermediate states. At the same time, this degradation process always runs parallel with periodic maintenance and repair activities aimed at restoring the object's working capacity. Therefore, to accurately describe the transition between failure states over time, the application of the Markov model is the optimal approach.
a. Analysis of system states:
For mechanical transmission systems, we divide the technical evolution process into 3 discrete states:
- Good working condition(S0): he system operates normally, without errors.
- Moderate failure (S1): There is a minor error, but the system is still operational.
- Severe failure(S2) : The system stops working completely, requiring major repair or replacement.
b. Establishing the state transition matrix:
Let λ i be the failure rate (transitioning from S i-1 to S i ) and μ i be the repair rate (transitioning from state S i back to S i-1 ). This process is represented by the state transition matrixi Q.
c. Kolmogorov — Markov system of differential equations
Let P0(t), P1(t) and P2(t) — be the probabilities of the system being in states S 0 , S 1 respectively, at time t. The system of differential equations describing the change of these probabilities over time will have the following form:
Let P(t) = [P0(t), P1(t), P2(t)]T be the probability vector representing the likelihood of the system being in state S
i
at time t. The state variation over time is described by a system of differential equations in matrix form:
The general solution of the differential equation (9) has the form: P(t) = e Q.t .P(0) (10)
P(0): is the initial state vector. Assuming that at t = 0 the system is completely new, we have P(0) = [1,0,0] T
e
Q.t
is the matrix exponential of matrix Q multiplied by time t, and can be calculated using the power series formula:
3. Research results and discussion
When conducting accelerated durability testing of the ZF-430 clutch equipped on the Kamaz 43250 vehicle at Le Quy Don Technical University, we determined the values for the failure rate λ i and repair rate µ i respectively of the three elements: friction disk, pressure spring, and release mechanism, which are presented in Table 1 below.
Table 1
Statistical data of failure rate λ and recovery degree values µ of the MFZ-430 Clutch
|
Subject State |
Friction disk |
Pressure spring |
Release mechanism | |||
|
Parameter λ i (h -1 ) |
Parameter μ i (h -1 ) |
Parameter λ i (h -1 ) |
Parameter μ i (h -1 ) |
Parameter λ i (h -1 ) |
Parameter μ i (h -1 ) | |
|
S0 |
6,8.10 –5 |
2,5.10 –1 |
6,6.10 –5 |
2,2.10 –5 |
7,2.10 –5 |
3.10 –5 |
|
S1 |
14.10 –5 |
0,5.10 –1 |
13.10 –5 |
0,4.10 –5 |
15.10 –5 |
0,6.10 –5 |
|
S2 |
--- |
--- |
--- |
--- |
--- |
--- |
Based on the data table above, we establish a system of differential equations for each component. Corresponding to system (8) we have:
— Friction disk P
1
(t):
— Pressure spring P
2
(t):
— Release mechanism P
3
(t):
Solving the equations in system (11) we get:
In system (6), there are two components:
+ The probability function of failure-free operation of the system (system reliability function):
+ The failure probability function (probability function of failure occurrence):
From the system reliability function equations (12) and the system failure probability function (13), we constructed a graph describing the variation law of reliability and failure probability of the MFZ-430 clutch over operating time as shown in Figure 1 below:
Fig. 1. Graph of reliability and failure probability of the MFZ-430 Clutch
Remarks. From the graph above, we see that the reliability P h (t) of the system gradually decreases over operating time, while the failure probability function graph Q h (t) follows an opposite trend, meaning the probability of system failure will gradually increase over time. The characteristics of the curves in this research scope are assumed to follow normal operating rules. In reality, depending on investment costs, the characteristics of the curves can change. For example, when the technical condition of the elements in the system deteriorates and the recovery is carried out by replacement rather than normal repair activities, intervening to change the values of λ and μ the properties of the reliability function and failure probability curves will change; this is also a development direction of the research
Conclusion and recommendations
Studying and evaluating the reliability of technical systems is of great significance, helping to assess the technical condition of objects, serving as a basis for conducting the next research step to provide predictions on the technical reserve coefficient, and thereby proposing more effective exploitation plans for technical equipment in the automotive engineering sector. The report presents the procedure for evaluating the reliability of technical objects with elements installed in series. Within the scope of the report, it is the MFZ-430 clutch system, which consists of 3 basic series-connected elements: Friction disk, pressure spring, and release mechanism, using hypothetical data collected at operational units.
It is necessary to invest effort in statistics to collect data at operational and repair units. Collecting and compiling statistical data must be done meticulously and accurately because the accuracy of the survey results depends on the accuracy of the collected data. For transmission systems such as the powertrains of 4x4 or 6x6 vehicles, bulldozers, and cranes, we can entirely apply this methodology to evaluate their reliability because the transmission systems of these objects have similar structures and operating principles, being mechanical transmission systems with elements installed in series.
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