CHINESE JOURNAL OF MECHANICAL
A Comprehensive Reliability Allocation Method for CNC Lathes Based on Cubic Transformed Functions of FMEA Yang Zhou1, Zhu Yunpeng1, Ren Hongrui1, Zhang Yimim1* 1. School of Mechanical Engineering & Automation, Northeastern University, Shenyang 110819, PR China Abstract: Computerized numerical controlled (CNC) lathes are very important in manufacturing industry and can be regard as a series system. Aiming at solving the problem of its reliability allocation, a comprehensive reliability allocation method based on cubic transformed functions of failure mode and effects analysis (FMEA) is introduced. Firstly, conventional reliability allocation methods are reviewed. Then the limitations of directly combination of comprehensive allocation method and the exponential transformed FMEA method are investigated. Subsequently, a cubic transformed function is established in order to overcome this problem. The properties of the new transformed functions are discussed by dividing the failure severity and the failure occurrence into three intervals. Designers can choose appropriate transform amplitudes according to their requirements. Finally, a CNC lathe and a spindle system are used as examples. The allocation results of the new transformed method and the conventional methods are compared to emphasis the characteristics of the present approach.
Key words: CNC lathes; Reliability; FMEA; Cubic transformed function; Comprehensive allocation;
1
Introduction
1
A system is generally designed as an assembly of subsystems. Each subsystem has its own reliability attributes. Reliability allocation is a process to allocate the system reliability to subsystems according to a determined principle or method [1]. The reliability of a mechanical system depends on the rationality of the pre-allocation process of subsystems to some extent. Computerized numerical controlled (CNC) lathes are widely used in the manufacturing industry, thus the investigation of CNC lathes’ reliability is very important [2-4]. However, because of the complexity of CNC lathes’ structure and the enormous failure modes, the pre-allocations of the system are often irrational and lack of credibility. These may affect the efficiency and the service life of lathes [5]. Therefore, it is necessary to discuss the allocation and re-allocation method of CNC lathes. *Corresponding author. E-mail:
[email protected] This project is supported by Chinese National Natural Science Foundation (51135003, 51205050), Major State Basic Research Development Program of China (973 Program) (2014CB046303), Key National Science & Technology Special Project on “High-Grade CNC Machine Tools and Basic Manufacturing Equipments” (2013ZX04011011). Fundamental Research Funds for the Central Universities (N130503002), State Key Laboratory of Mechanical System and Vibration Fund Project (MSV201402), Liaoning province Doctor Startup Foundation (20121005). © Chinese Mechanical Engineering Society and Springer-Verlag Berlin Heidelberg 2012
A CNC lathe can be regard as a series system combined by multiple subsystems [6]. Many results have been presented by researchers: Wang et al. [7] presented a comprehensive allocation method. Seven reliability criterions were considered in determining allocation weights. However, the criticality number is not related with failure severity but only with the failure probability. Itabashi et al. [8] introduced an allocation method considering the criticality and frequency of failure by using the FMEA. In their study, the failure severity and the failure occurrence were associated in allocating the system reliability. Kim et al. [9] reviewed the traditional allocation methods and pointed the limitations of the conventional FMEA method. An exponential transformed failure severity was presented in order to reduce the occurrence of the unacceptable failure effects in a newly designed system. Subsequently, this method was developed in the study of Yadav et al. [10]. In their study, a three-dimensional method was introduced to analysis the reliability allocation of mechanical systems. A new concept known as the improvement effort was introduced to analyze the reliability allocation. In addition, Gu et al. [11] and Sriramdas et al. [12] employed an approach of fuzzy mathematics in the reliability allocation of a CNC lathe. The fuzzy set theory and analytic hierarchy process are used to decrease the influence of uncertain factors during the decision process. In conclusion, there are various allocation methods for CNC lathes. However, most of these methods have
limitations. It is a natural thought to combine the transformed method presented by Kim et al. [9] and the comprehensive allocation method [1]. Nevertheless, a directly combination is not always rational, and the reason is investigated in the present study. In order to overcome these limitations, a new cubic transformed function was designed based on the FMEA. The properties of this new transformation of failure severity are discussed, as well as the failure occurrence. The characteristics and the applicability of this new allocation method are investigated through two examples. In what follows, the comprehensive allocation method and the exponential transformed functions are reviewed in section 2. In section 3, the limitation of the conventional method is analyzed and a new cubic transformed function is designed based on FMEA. Two examples are employed in section 4, the one is to compare the conventional comprehensive allocation method and the present method, the other is an example of a CNC lathe’s spindle system. Finally, the properties of the present method are summarized in section 5.
2
B(k )
11( k ) (k ) 21 (k ) n1
Assuming a series system U can be divided in to n subsystems, which can be represented by U U1 , U 2 , , U n . V is the allocation criterion, and assuming there are m reliability allocation criterions: V V1 , V2 , , Vm . The relationship of U and V is shown in Fig.1 U1
U2
V1
Fig. 1.
V2
...
...
Un-1
Un
n( 2k )
ki
1 n (k ) ij , k 1, 2, n j 1
Vm
Reliability allocation criterions and a series system
11 12 22 Γ 21 m1 m 2
A a1 , a2 , where
(3)
1n 2 n
mn mn
(4)
, an W Γ
(5)
W — the weighting vector
W w1 , w2 , , wm
wk (1 k m) — the weight of the k th allocation criterion. By substituting Eq. (4) into Eq. (5), the allocation vector A can be obtained. To each subsystem, Ui (i 1,2, , n) , the allocation of failure rates satisfies the relationship:
: n a1 : a2 :
: an
(6)
Therefore, the coefficients of each subsystem’s failure rate are:
i
ratio for the failure rate allocation between the i th subsystem and the j th subsystem is denoted as
ij( k ) 1 (jik ) (i, j 1,2, , n; k 1,2, , m)
n
Defining ai represents the comprehensive failure rate allocation coefficients, the system’s allocation vector, A , is defined as:
Considering the Vk (1 k m) criterion, the relative
ij( k ) (0 ij( k ) 1) , and [7]
, m; i 1, 2,
Assuming that there are m allocation criterions in a system, according to Eq. (3), a new allocation matrix Γ can be obtained as:
1 : 2 : Vm-1
1(nk ) 2(nk ) , k 1, 2, , m (2) nn( k ) nn
Each row of the matrix B ( k ) denotes the failure rate allocation ratio between a specific subsystem and every other subsystem. Solving the average value of the i th subsystem that represents the relative allocation coefficient of the i th subsystem under the k th criterion:
Review of allocation methods
2.1 Comprehensive allocation method
12( k ) 22( k )
ai
s*
n
a j 1
(7)
j
* where s —the desired failure rate.
(1)
where if ij( k ) 0.5 , it means that the i th subsystem should have higher failure rate than the j th subsystem. (k ) Obviously, ii 0.5 . The failure rate allocation ratio matrix B ( k ) of the k th allocation criterion can be represented as:
Further, the reliability allocation of subsystems with respect to time t is [13]:
Ri e i t e
ai
n
j 1
a j s*t
(8)
2.2 The exponential transformed functions of failure severity According to the comprehensive allocation method, the key point of using this method is to calculate the relative
CHINESE JOURNAL OF MECHANICAL ratio ij( k ) (0 ij( k ) 1) . In order to contribute the example 1 shown in section 4, seven criterions are chosen according to the reference [7], they are: the failure occurrence ( k 1 ), the failure severity ( k 2 ), the maintainability ( k 3 ), the complexity ( k 4 ), the manufacturing technology ( k 5 ), the working condition ( k 6 ) and the cost ( k 7 ). FMEA is a method used to analysis all possible failure modes of a system. In FMEA, failure modes are classified by considering their severity and occurrence [14]. The conventional allocation method based on FMEA can be expressed as [15-16]:
Ci where
1 Ri
Ri
S
Oif , i 1, 2,
if
f 1
,n
(9)
Ci — the criticality of subsystem i Ri — the number of subsystems’ failure modes
Sif and Oif — the severity and the occurrence of the
ith subsystem with respect to the f failure mode. The reliability allocation weight wi is:
wi
i
, i 1
n
i 1
i
Ci n
C i 1
(10)
i
However, the conventional FMEA allocation method only focuses on two factors ( Si and Oi ). Therefore, discussing the combination of the FMEA allocation and the comprehensive allocation method introduced in section 2.1 is necessary. In order to obtain the relative ratio of the failure severity, the severities Si , S j of the i th and the j th subsystems are required. Obviously, when R 1 , there is Si Sif . When R 1 , the maximum method [7] is employed to calculate the failure severity of subsystems: (12) Si max Si1 , Si 2 , , SiR Considering the case that the failure severities of a subsystem are evenly distributed in the range of 1 to 10, which means Si [1,10] . The advantage of the severity’s exponential transformed function is shown as follow: If the conventional FMEA allocation method and the comprehensive allocation method are directly combined, considering the severity of the i1 th and the j1 th subsystems: Si1 1 and S j1 2 . Similarly, the severity of the i2 th and the
j2 th subsystems are chosen as
Si2 5 and S j2 10 . Then the severity relative ratios of the two cases are the same:
In Eq. (9)-(10), the failure severity Sif is a linear function, which means the failure severity is divided into 10 levels, and the rate of each level increase in a linear trend. However, this method has limitations in allocating the reliability of series systems [17]. For example, the difference between Sif 1 and Sif 2 is the same as
of the subsystem i1 and j1 , Si1 S j1 , is far less than
the difference between Sif 9 and Sif 10 . In order to
that of the subsystem i2 and
overcome this problem, Kim et al. [9] presented an exponential transformed function of failure severity: S (11) Sif e if
allocation ratios of these two cases are the same, which is not rational. By introducing the exponential transformed function of failure severity (Eq. (11)), Eq. (13) can be more rational with 0.8 :
where Sif — the transformed failure severity;
— the severity modified coefficient. The transformed method can well overcome the problem discussed above, and Fig.2 depicts the curves of transformed failure severity Sif under different and the failure severity Sif .
2 3
(2) i(2)j i(2)j ; (2) ji j i 1 1
2 2
11
22
1 3
(13)
According to Eq. (13), although the severity’s difference
j2 ,
Si2 S j2 , the
(2) i(2)j 0.69 i(2)j 0.982; (2) j i 0.31 j i 0.018 (14) 1 1
2 2
11
22
In Eq. (14), it can be seen that the exponential transformed function can highly emphasis the effect of a high failure severity. However, there are still limitations in this method, which will be discussed in the following section.
3
Cubic transformed functions
3.1 Limitations In the case that the failure severities are evenly distributed in the range of 1 to 10, considering Si 9 and
S j 10 , when 0.8 , the relative ratio is:
Fig. 2.
The transformed severity Sif and the severity Sif
ij(2) 0.69; (2) ji 0.31
(15)
It is noticeable that although the exponential transformed
function distinguish the relative ratio between Si1 1, S j1 2 and Si2 5, S j2 10 , the difference of high failure severities, such as Si 9 and S j 10 , are the same as the difference of low failure severities, such as Si 1 and S j 2 . In order to explain the limitation of this phenomenon, assuming Si S0 and S j S0 1 , the severity relative ratio is:
ij(2)
e
S0 S0 1
1 e 1
Fig. 3.
e 0 e Eq. (16) indicates that the adjacent failure severity always has the same relative ratio. On the other hand, failure severities can be divided into three intervals: the low interval ( Si [1,3] ), the moderate interval ( Si [4,7] ) and the high interval ( Si [8,10] ). Considering the case that all failure severities concentrate on the low interval and the moderate interval, the transformed function is not suitable because it cannot reveal the difference of low severities’ effects. This is because the derivative of the exponential transformed function is: S
dSi dSi S E e i e dE dSi
Curves of Eq. (15) and Eq. (17)
(16)
(17)
where E — the level of the failure severity in FMEA； Si E . In Eq. (17), the value of Si increase fast as the level of failure severity increase. In fact, in the evenly distributed case, the increasing speed of the high severity should decrease. It is noticeable that the continuous increasing trend is only acceptable when the severities are concentrated on the high interval. 3.2 Determination of cubic transformed functions According to the discussion above, a new transformed function is required in order to reveal the effects of FMEA factors. Consider a quadratic equation： 2 2 dSˆi a E 1 E 2 1 (18) 0 dE 0 E 2 1 where a0 — a constant — the level where the failure severity of a subsystem concentrate on 1 n Si n i 1 Sˆ — a transformed function needs to be
i
determined. Consider R 1 , Fig.3 shows the curves of Eq. (17) and Eq. (18)
It can be seen from Fig.3 that the derivative of the exponential transformed function increases sharply as the level of failure severity increase. By contrast, the quadratic equation (Eq. (18)) reaches the maximum value and then decreases. The maximum amplitude can be designed by designers. According to Eq. (18), a cubic transformed function can be obtained as:
E3 E 2 2 1 E c0 a0 ˆ Si ( E ) 3 c1
E 2 1
E 2 1 (19a) where a0 , c0 and c1 — undetermined coefficients. Determining constrained functions
Sˆi (10) S 2 1 10 ˆ (19b) Si (2 1) S 2 1 10 ˆ Si (1) 1 and a0 , c0 and c1 can be obtained by substituting Eq. (19b) into Eq. (19a). In Eq. (19), S is determined by designers according to requirements. Fig.4 depicts the new cubic transformed functions with S 10 and the exponential transformed function with 0.23 .
Fig. 4.
Curves of the cubic transformed functions and the exponential transformed function
In Fig.4, it can be seen that when is in the high interval, the cubic transformed function has the same
CHINESE JOURNAL OF MECHANICAL properties as the exponential transformed function. Therefore, if the average severity falls into the high interval, it is acceptable to use either the cubic transformed function or the exponential transformed function. However, When is in the low interval, the transformed severity becomes unchanged in the high interval, this is acceptable because if the average severity falls into the low interval, it means that there are few severities in the high interval and their differences are insignificant which can be regard as the same level. In general, this case is not common in practical situations. Therefore, the relative ratio of the failure severity can be rewritten as:
ij(2)
Sˆ j Sˆi Sˆ j
, (i, j 1, 2,
n)
(20)
In FMEA, the failure occurrence of the subsystem i under the f th failure mode, Oif , and its exponential transformed function, Oif , also have the same limitations as the failure severity we discussed above. Therefore, the relative ratio of a subsystem’s failure occurrence is [7]
Oi , (i, j 1, 2, Oi O j
n)
(21)
E3 E 2 2 1 E c0 a0 ˆ Oi ( E ) 3 c 1 ˆ Oi (10) O 2 1 10 ˆ s.t. Oi (2 1) O 2 1 10 ˆ Oi (1) 1
E 2 1 E 2 1
, (i, j 1, 2,
n)
(24)
Examples
Based on the reliability allocation method presented above, two examples are employed in order to explain the application of this method, as well as to investigate the proprieties of this method. The first example is the same with that in reference [7], which is used to verify and analyze the present method. The second example is to allocate reliability to subsystems of a CNC lathe’s spindle system by using the present allocation approach.
can be obtained by Eq. (19). According to the relative failure severity Si shown in Tab.1, the transformed failure severity Sˆi is obtained as:
E3 2 0.604 E 2 1 E 3.577 E 2 1 Sˆi ( E ) 3 50 E 2 1 and
where
a0 , c0 and c1 — undetermined coefficients Oˆ — the cubic transformed failure occurrence i
O — determined by designers according to requirements without confusion
E — the level of the failure occurrence in FMEA；
E Oi . Similarly, the failure occurrence of a subsystem can be defined as
, OiR
In general, a CNC lathe can be divided into 15 subsystems. They are the turret, the clamping accessory, the electric and electronic system, the main transmission, the X feed system, the Z feed system, the CNC system, the power supply, the hydraulic system, the servo system, the cooling system, the swarf conveyors, the lubricant system, the spindle assembly and the guard. Assuming S 50 , the transformed failure severity Sˆi
(25)
(22)
(23)
The transformed relative ratio of failure occurrence
ij(1) is defined as
Oˆ i Oˆ j
failure severity Si (Provided by [7]) and the FMEA
According to the analysis in section 3.2, the cubic transformed failure occurrence is:
Oi max Oi1 , Oi 2 ,
4
Oˆi
4.1 Example 1
3.3 Cubic transformed functions of failure occurrence
ij(1)
ij(1)
15
1 Si 4.933 . 15 i 1
Similarly, let O 50 , the relative failure occurrence
Oi and the FMEA failure occurrence Oi are shown in Tab.2. According to Eq. (22), the transformed failure occurrence Oˆ i is:
E3 2 0.431 E 2 1 E 3.042 E 2 1 Oˆi ( E ) 3 50 E 2 1 (26) and
1 15 Oi 5.4 . 15 i 1
The values of the cubic transformed severity and occurrence are listed in the last column of Tab.1 and Tab.2. Fig.5 depicts the transformed severity and occurrence of Eq. (25) and Eq. (26).
E D F W K L S2 Q
Power supply Hydraulic system Servo system Cooling system Swarf conveyors Lubricant system Spindle assembly Guard
0.049 0.041 0.038 0.035 0.030 0.029 0.013 0.008
5 4 4 4 3 3 2 1
22.168 14.201 14.201 14.201 7.4424 7.4424 2.7545 1.0001
In Fig.4, it can be seen that the difference between
Si 5 and Si 10 , ΔSi , is larger than the difference between Oi 5 and Oi 10 , ΔOi . ΔSi ΔOi
Fig. 5.
Curves of the cubic transformed severity and occurrence Tab 1.
The failure severities of subsystems
Symbol
Subsystems
Si / 104 [7]
Si ( E )
Sˆi
M J
Turret Clamping accessory Electric and electronic system Main transmission X feed system Z feed system CNC system Power supply Hydraulic system Servo system Cooling system Swarf conveyors Lubricant system Spindle assembly Guard
3.177 2.813
10 9
49.999 49.999
2.092
8
48.346
1.693 1.402 1.427 1.575 1.418 0.834 1.368 1.219 0.7905 0.48 0.4308 0.3148
7 5 5 6 5 3 4 4 3 2 2 1
43.032 26.122 26.122 35.221 26.122 8.8910 16.943 16.943 8.8910 3.1741 3.1741 1.0001
V S1 X Z NC E D F W K L S2 Q
Tab 2. Symbol M J V S1 X Z NC
Oi [7]
Symbol M J
0.269 0.099
49.998 48.857
0.096
9
48.857
0.084 0.075 0.065 0.063
8 7 6 6
44.687 38.273 30.479 30.479
Subsystems Turret Clamping accessory Electric and electronic system Main transmission X feed system Z feed system CNC system Power supply Hydraulic system Servo system Cooling system Swarf conveyors Lubricant system
V S1 X Z NC E D F W K L S2 Q
Oˆ i
Oi ( E ) 10 9
The mean down time and the mean repair cost of subsystems
Spindle assembly Guard
Tab 4.
The complexities C of subsystems
TD
TH
2.72 1.99
1.97 1.16
0.75
0.50
2.64 2.38 2.00 1.03 0.76 1.82 0.89 3.05 4.04 2.33 2.89 4.79
1.03 1.35 1.92 1.01 0.55 0.98 2.42 2.76 1.71 1.17 1.37 2.36
M
J
V
S1
X
Z
NC
E
D
F
W
K
L
S2
Q
300
150
250
80
100
100
30
50
80
50
40
50
30
60
10
The manufacturing technology M , the working condition E and the cost C0
Tab 5. Subsystems Manufacturing technology
Tab 3.
The failure occurrences of subsystems
Subsystems Turret Clamping accessory Electric and electronic system Main transmission X feed system Z feed system CNC system
Subsystems Number of components
indicates that the high occurrence subsystem can obtain higher reliability than the high severity system in the same level. That means the method, in this case, emphasis the effect of high occurrence. In addition, the mean down time TD and the mean repair cost TH , the complexity C , the manufacturing technology M , the working condition E and the cost C0 are listed in Tab.3-5. The cost C0 is defined as the ratio C R of the cost increment C and the reliability increment R , and the rating scores of M , E and C0 should be in the range of 0 to 1 [7].
M
J
V
S1
X
Z
NC
E
D
F
W
K
L
S2
Q
0.98
0.70
0.26
0.34
0.40
0.40
0.14
0.20
0.40
0.20
0.30
0.44
0.34
0.38
0.10
0.96
0.76
0.20
0.44
0.40
0.40
0.10
0.24
0.34
0.14
0.30
0.56
0.26
0.44
0.14
0.98
0.70
0.30
0.58
0.50
0.50
0.14
0.26
0.40
0.20
0.36
0.46
0.34
0.54
0.10
M (k 5) Working condition
E (k 6) Cost
C0 (k 7)
CHINESE JOURNAL OF MECHANICAL According to Tab.1-5, an allocation matrix Γ can be obtained through Eq. (1)-(4), where the failure severity 0.703 0.290 0.425 Γ 0.791 0.741 0.736 0.714
and the failure occurrence of subsystems are replaced by Sˆi and Oˆ i :
0.699 0.699 0.682 0.653 0.608 0.608 0.543 0.453 0.453 0.453 0.332 0.332 0.188 0.094 0.290 0.296 0.317 0.414 0.414 0.355 0.414 0.623 0.500 0.500 0.623 0.783 0.783 0.899 0.518 0.721 0.482 0.478 0.467 0.619 0.712 0.546 0.507 0.376 0.378 0.494 0.447 0.331 0.671 0.763 0.541 0.589 0.589 0.334 0.439 0.541 0.439 0.392 0.439 0.334 0.479 0.159 (27) 0.675 0.452 0.514 0.552 0.552 0.316 0.392 0.552 0.392 0.485 0.574 0.514 0.540 0.252 0.692 0.396 0.575 0.553 0.553 0.257 0.437 0.516 0.321 0.487 0.628 0.455 0.575 0.321 0.646 0.453 0.604 0.571 0.571 0.291 0.421 0.520 0.363 0.495 0.552 0.482 0.588 0.230
0.99. Substituting s* into Eq. (7), yields:
Assuming the weight vector W is:
W 0.20 0.10 0.08 0.14 0.17 0.15 0.16
(28)
i 1.773
1.469 1.045 1.248 1.404 1.108 1.228 1.335 1.236 1.309 0.760 10
According to Eq. (5), the allocation vector A is:
A 0.665 0.626 0.542 0.553 0.561 0.551 0.392 0.468 0.527 0.415
(29)
0.460 0.500 0.464 0.491 0.285 Considering the desired mean time between failure (MTBF) of the system is 500 h, which means s* 0.002 , the desired reliability of the system is over Tab 6. Symbol
Subsystems
M J
Turret Clamping accessory Electric and electronic system Main transmission X feed system Z feed system CNC system Power supply Hydraulic system Servo system Cooling system Swarf conveyors Lubricant system Spindle assembly Guard
V S1 X Z NC E D F W K L S2 Q
1.669 1.446 1.474 1.495 (30) 4
In Eq. (25), the reciprocal of each element is the MTBF. Tab.6 shows the results of subsystems’ MTBF allocation [7], the allocated results of the conventional FMEA ( Oi , Si ) and the allocated results of the present method ( Oˆ i , Sˆi ).
The comparision of the reliability allocation methods Observed MTBF /h
Allocated MTBF /h [7]
Conventional FMEA Oi , Si /h
Transformed FMEA Oˆ i , Sˆi /h
752.14 2044.06
5523.56 6140.60
5680.28 6049.73
5640.30 5990.53
2118.84
7067.28
6967.94
6913.39
2413.13 2714.77 3102.59 3217.50 4136.79 4964.14 5265.00 5791.50 6682.50 6949.80 15795.00 24820.71
6890.52 6318.62 6398.46 9679.27 8154.97 7230.30 9222.45 8305.45 7412.56 8346.89 7645.18 13099.09
6848.95 6776.49 6875.05 9657.02 8029.57 7151.62 8953.74 8086.74 7362.13 8060.15 7437.08 12741.44
6783.67 6687.41 6808.22 9567.79 8015.78 7120.02 9026.82 8146.56 7491.85 8087.66 7640.61 13150.53
It can be seen from Tab.6 that the allocation result is similar to the reference, which indicates the applicability of the present method. According to the fifth and the sixth columns in Tab.6, the subsystems with severity and occurrence that fall into the high interval (M, J, V, S1, X) have higher failure rates than those before the transformation. By comparing the failure severity and occurrence of these five subsystems, it can be seen that the cubic transformed method emphasizes the effect of failure occurrence Oi , which reveals the conclusion as Fig.5 indicates.
According to Eq. (8), the original reliability of the allocated system ( t 1 ) is 15
15
i 1
i 1
Rs Ri e i t 0.998 0.99
(31)
Eq. (31) indicates that the present method is acceptable. 4.2 Example 2 The second example is focused on a spindle system of a CNC lathe. In this study, six criterions are
under consideration: the failure severity ( k 1 ), the failure occurrence ( k 2 ), the complexity ( k 3 ), the manufacturing technology ( k 4 ), the working condition ( k 5 ) and the cost ( k 6 ). The failure severity and occurrence of each failure modes are listed Tab.7 Subsystems
The FMEA of the spindle system
Subsystem’s failure occurrence
Si
Transformed severity Sˆi ( S 50 )
22.014
3 2
3
5.868
7
37.919
8
8
38.344
6
30.237
6
24.603
6
30.237
6
24.603
Failure modes
Occurrence
4 5
5
Oif
in Tab.7, the transformed systems’ severity and occurrence are calculated according to Eq. (19) and (22). The values of the other four criterions are shown in Tab.8, which are obtained by expert rating.
Oi
Transformed occurrence Oˆ i ( O 80 )
Severity
Sif
Subsystem’s failure severity
1
Support
FM11 FM12
2
Measuring assembly
FM21
7
3
Front support
4
Back support
FM31 FM32 FM33 FM41 FM42 FM43
2 6 2 2 6 2
FM51
3
3
7.405
6
6
24.603
FM61
3
3
7.405
6
6
24.603
FM71 FM72 FM81 FM82
8 4 5 2
8
44.198
5
17.562
5
22.014
9
43.952
5 6 7 8
Front Seal Back seal Driving assembly Braking assembly
Tab.8 Subsystems Number of components
Nˆ (k 3)
Manufacturing technology
Mˆ (k 4)
Working condition
Eˆ (k 5) Cost
C0 (k 6)
4 2 6 4 2 6
3 5 9 7
Criterion values of subsystems from k 3 to k 6
1
2
3
4
5
6
7
8
1
4
8
5
5
3
4
3
0.45
0.54
0.54
0.73
0.39
0.31
0.67
0.48
0.44
0.58
0.55
0.52
0.32
0.26
0.69
0.49
0.46
0.61
0.58
0.53
0.37
0.31
0.65
0.50
According to the process of example 1, the allocation matrix Γ can be obtained as: 0.505 0.779 0.231 Γ 0.476 0.488 0.485
0.627 0.577 0.577 0.276 0.246 0.659 0.505 0.381 0.482 0.482 0.482 0.482 0.561 0.351 0.521 0.676 0.572 0.572 0.454 0.521 0.454 0.520 0.520 0.593 0.441 0.386 0.573 0.491 0.555 0.543 0.529 0.411 0.363 0.597 0.514 0.555 0.542 0.520 0.432 0.390 0.570 0.506
(32)
Considering the desired MTBF is 1500 h, which means s* 0.00067 , the desired reliability of the system is over 0.99. Substituting s* into Eq. (7), yields:
i 0.404
0.452 0.469 0.461
0.354 0.318 0.491 0.402 104
(35)
The MTBF of the subsystems are:
The weight vector W is:
W 0.22 0.11 0.15 0.18 0.16 0.18 (33)
0.422 0.379 0.586 0.480
22105.32 21344.86
21676.30 28269.72 31469.12
According to Eq. (6), the allocation vector A is:
A 0.482 0.540 0.559 0.551
MTBFi 24756.05
(36)
20375.79 24902.72 / h (34) According to Eq. (8), the original reliability of the
CHINESE JOURNAL OF MECHANICAL allocated system ( t 1 ) is 8
8
i 1
i 1
Rs Ri e
References [1] Hudoklin A, Rozman V. Reliability allocation [J]. Elektrotehniski i t
0.9997 0.99
(37)
Eq. (37) indicates that the allocated results are acceptable.
5
Conclusions
The present study establishes a comprehensive allocation method for series systems based on the cubic transformed functions of FMEA. The properties of this new transformed function are discussed. Two examples, known as the CNC lathe system and the spindle system, are employed to verify the applicability of the present method. Specific conclusions are listed as follows: (1) Cubic transformed functions of FMEA factors are established by introducing the low, moderate and high intervals, shown in Eq. (19) and Eq. (22). (2) The cubic transformed function can be designed to emphasis the failure severity or the failure occurrence of subsystems according to designers’ requirements. It is proved that the cubic transformed functions and the exponential transformed functions can be exchanged when the average subsystem’s severity or occurrence concentrate on the high interval. (3) The cubic transformed functions are employed in the comprehensive allocation method. The new relative ratios are defined and the process of using the present method is discussed. (4) Two examples are employed at the end of the study. Example 1 is used to verify the applicability and the properties of the present method. Example 2 is employed to show the process of applying the present method. The allocation results can be adjusted by changing the coefficients O and S according to the requirement of designers.
Conflicts of interests The researchers claim no conflicts of interests
Vestnik, 1980, 47(1): 10-14.
[2] Keller A Z, Kamath A R R, Perera U D. Reliability analysis of CNC machine tools [J]. Reliability Engineering, 1982, 3(6): 449-473.
[3] Wu J, Deng C, Shao X Y, Xie S Q. A reliability assessment method based on support vector machines for CNC equipment [J]. Science in China Series E: Technological Sciences, 2009, 52(7): 1849-1857. [4] Kuo W, Wan R. Recent Advances in Optimal Reliability Allocation [J]. IEEE Transactions on Systems, Man, and Cybernetics—Part A: Systems And Humans, 2007, 37(2): 143-156. [5] Wang Y, Jia Y, Yu J, et al. Field failure database of CNC lathes [J]. International Journal of Quality & Reliability Management, 1999, 16(4): 330-343. [6] Wang Y, Jia Y, Yu J, et al. Failure probabilistic model of CNC lathes [J]. Reliability Engineering & System Safety, 1999, 65(3): 307-314. [7] Wang Y, Yam R, Zuo M J, Tse P. A comprehensive reliability allocation method for design of CNC lathes [J]. Reliability engineering & system safety, 2001, 72(3): 247-252. [8] Itabashi C R R, Yadav O P. System Reliability Allocation based on FMEA Criticality [R]. SAE Technical Paper, 2009. [9] Kim K O, Yang Y, Zuo M J. A new reliability allocation weight for reducing the occurrence of severe failure effects [J]. Reliability Engineering & System Safety, 2013, 117: 81-88. [10] Yadav O P, Zhuang X. A practical reliability allocation method considering modified criticality factors [J]. Reliability Engineering & System Safety, 2014, 129: 57-65. [11] Yadav O P, Singh N, Goel P S. Reliability demonstration test planning: A three dimensional consideration [J]. Reliability Engineering & System Safety, 2006, 91(8): 882-893. [12] Gu Y K, Tang S Y. A fuzzy reliability allocation method for the product based on the knowledge [C]// Measuring Technology and Mechatronics Automation (ICMTMA), 2010 International Conference on. IEEE, 2010, 1: 101-104. [13] Sriramdas V, Chaturvedi S K, Gargama H. Fuzzy arithmetic based reliability allocation approach during early design and development [J]. Expert Systems with Applications, 2014, 41(7): 3444- 3449. [14] Keskin G A, Ozkan C. An Alternative Evaluation of FMEA: Fuzzy ART Algorithm [J]. Quality and Reliability Engineering International, 2009, 25: 647-661. [15] Hu S, Fang Q , Xia H, Xi Y. Formal safety based on relative risks model in ship navigation [J]. Reliability Engineering and System Safety, 2007, 92: 369-77. [16] Saadi S, Djebabra M, Boubaker L. Proposal for a new allocation method of environmental goals applied to an Algerian cement factory [J]. Management of Environmental Quality: An International Journal, 2011, 22: 581-94. [17] Zhuang X, Limon S, Yadav O P. Considering Modified Criticality Factor and Functional Dependency for Reliability Allocation Purposes [C]// Proceedings of the 2014 Industrial and Systems Engineering Research Conference. 2014.
