LIBRO - Dhillon - Reliability in the Mechanical Design Process - Chapter 31.pdf

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CHAPTER 31RELIABILITY IN THE MECHANICALDESIGN PROCESS

B. S. DhillonDepartment of Mechanical EngineeringUniversity of OttawaOttawa, Ontario, Canada

1 INTRODUCTION 1000

2 STATISTICAL DISTRIBUTIONSAND HAZARD RATE MODELS 10012.1 Statistical Distributions 10012.2 Hazard Rate Models 1002

3 COMMON RELIABILITYNETWORKS 10033.1 Series Network 10043.2 Parallel Network 10053.3 SeriesParallel Network 10053.4 ParallelSeries Network 10063.5 K-out-of-m-Unit Network 10073.6 Standby System 10083.7 Bridge Network 1009

4 MECHANICAL FAILURE MODESAND CAUSES OF GENERALAND GEAR FAILURES 1010

5 RELIABILITY-BASED DESIGNAND DESIGN-BY-RELIABILITYMETHODOLOGY 1011

6 DESIGN RELIABILITYALLOCATION ANDEVALUATION METHODS 10126.1 Failure Rate Allocation Method 10136.2 Hybrid Reliability Allocation

Method 1013

6.3 Safety Factor and Safety Margin 10146.4 StressStrength Interference

Theory Method 10156.5 Failure Modes and Effect

Analysis (FMEA) 10166.6 Fault Tree Analysis (FTA) 1017

7 HUMAN ERROR ANDRELIABILITY CONSIDERATIONIN MECHANICAL DESIGN 1017

8 FAILURE RATE ESTIMATIONMODELS FOR VARIOUSMECHANICAL ITEMS 10188.1 Brake System Failure Rate

Estimation Model 10188.2 Compressor System Failure Rate

Estimation Model 10198.3 Filter Failure Rate

Estimation Model 10198.4 Pump Failure Rate

Estimation Model 1019

9 FAILURE DATA AND FAILUREDATA COLLECTION SOURCES 1019

REFERENCES 1021

BIBLIOGRAPHY 1022

1 INTRODUCTIONThe history of reliability may be traced back to the early 1930s when probability conceptswere applied to problems related to electric power systems.17 During World War II, Germanresearchers applied the basic reliability concepts to improve reliability of their V1 and V2rockets. During the period 19451950, the U.S. Department of Defense (DOD) conductedvarious studies that revealed a denite need to improve equipment reliability. Consequently,the DOD formed an ad hoc committee on reliability in 1950. In 1952, this committee wastransformed to a permanent body: Advisory Group on the Rer-liability of Electronic Equip-ment (AGREE).8 The group released its report in 1957.

Mechanical Engineers Handbook: Materials and Mechanical Design, Volume 1, Third Edition.Edited by Myer Kutz

Copyright 2006 by John Wiley & Sons, Inc.

2 Statistical Distributions and Hazard Rate Models 1001

In 1951, W. Weibull proposed a function to represent time to failure of various engi-neering items.9 Subsequently, this function became known as the Weibull distribution and isregarded as the starting point of mechanical reliability along with the works of A. M. Freu-denthal.1011

In the early 1960s, the National Aeronautics and Space Administration (NASA) playedan important role in the development of mechanical reliability, basically due to the followingthree factors12:

The loss of Syncom I in space in 1963 due to a bursting high-pressure gas tank The loss of Mariner III in 1964 due to a mechanical failure The frequent failure of components such as valves, regulators, and pyrotechnics in the

Gemini spacecraft systems

Consequently, NASA initiated and completed many projects concerned with mechanical re-liability. A detailed history of mechanical reliability is given in Refs. 1315 along with acomprehensive list of publications on the subject up to 1992.

2 STATISTICAL DISTRIBUTIONS AND HAZARD RATE MODELSVarious types of statistical distributions and hazard rate models are used in mechanicalreliability to represent failure times of mechanical items. This section presents some of thesedistributions and models considered useful to perform various types of mechanical reliabilityanalyses.

2.1 Statistical DistributionsThis section presents three statistical or probability distributions: exponential, Weibull, andnormal.

Exponential DistributionThis is probably the most widely used distribution in reliability work to represent the failurebehavior of various engineering items.16 Moreover, it is relatively easy to handle in perform-ing reliability analysis in the industrial sector. Its probability density function is expressedby14,16

t(t) e for 0 t 0 (1)

where (t) probability density function distribution parameter; in reliability work, it is known as the constant failure

ratet time.

The cumulative distribution function is given by14,16

t t tF(t) (t) dt e dt0 0 (2)t 1 e

where F(t) is the cumulative distribution function.

1002 Reliability in the Mechanical Design Process

Weibull DistributionThis distribution was developed by W. Weibull in the early 1950s and can be used to rep-resent many different physical phenomena.9 The distribution probability density function isexpressed by14,16

1t(t) e(t /) for t 0 0 0 (3)

where and are the distribution scale and shape parameters, respectively. The cumulativedistribution function is given by14,16

t t1t (t / )F(t) (t) dt e dt

0 0

(t / ) 1 e (4)For 1 and 2, the Weibull distribution becomes the exponential and Rayleighdistributions, respectively.

Normal DistributionThis is one of the most widely known distributions. In mechanical reliability, it is often usedto represent an items stress and strength. The probability density function of the distributionis expressed by

21 (t )(t) exp t (5) 222where and are the distribution parameters (i.e., mean and standard deviation, respec-tively). The cumulative distribution function is given by14,16

t t21 (x )

F(t) (t) dt exp dx (6) 222

2.2 Hazard Rate ModelsIn reliability studies, the term hazard rate is often used. It simply means the constant ornonconstant failure rate of an item. Thus, the hazard rate of an item is expressed by

(t)h(t) (7)1 F(t)where h(t) is the item hazard rate.

This section presents four hazard rate models considered useful to perform various typesof mechanical reliability studies: exponential, Weibull, normal, and general.

Exponential DistributionBy substituting Eqs. (1) and (2) into Eq. (7), we get the following equation for the expo-nential distribution hazard rate function:

3 Common Reliability Networks 1003teh(t)

t1 (1 e ) (8)

As the right-hand side of Eq. (8) is independent of time, is called the failure rate.

Weibull DistributionBy substituting Eqs. (3) and (4) into Eq. (7), we get the following equation for the Weibulldistribution hazard rate function:

1 (t / ) 1[(t / ) e ] th(t) (9)(t / ) 1 [1 e ] For 1 and 2, Eq. (9) becomes the hazard rate function for the exponential andRayleigh distributions, respectively.

Normal DistributionBy substituting Eqs. (5) and (6) into Eq. (7), we get the following equation for the normaldistribution hazard rate function:

2 21/ (2) exp [(t ) /2 ]h(t) (10)t

2 21 1/(2) exp [ (x ) /2 ] dx

General DistributionThe distribution hazard rate function is dened by17

m1 m1 th(t) ct (1 c)mt e for 0 c 1 , , m, 0 (11)where , scale parameters

, m shape parameterst time

The following distribution hazard rate functions are the special cases of Eq. (11): Bathtub; for m 1, 0.5 Makeham; for m 1, 1 Extreme value; for c 0, m 1 Weibull; for c 1 Rayleigh; for c 1, 2 Exponential; for c 1, 1

3 COMMON RELIABILITY NETWORKSComponents of a mechanical system can form congurations such as series, parallel, seriesparallel, parallelseries, k out of m, standby, and bridge. Often, these congurations arereferred to as the standard congurations. Sometime during the design process, it might be

1004 Reliability in the Mechanical Design Process

1 2 3 m

Figure 1 Series system block diagram.

desirable to determine the reliability or the values of other related parameters of systemsforming such congurations. All these congurations or networks are described below.14,16

3.1 Series NetworkThe block diagram of an m-unit series network or conguration is shown in Fig. 1. Eachblock represents a system unit or component. If any one of the components fails, the systemfails; that is, all of the series units must work normally for the system to succeed. Forindependent units, the reliability of the system shown in Fig. 1 is

R R R R R (12)S 1 2 3 mwhere RS series system reliability

m number of unitsRi reliability of unit i for i 1, 2, 3, ... , m

For constant unit failure rates of the units, Eq. (12) becomes14m

t t t t1 2 3 mR (t) e e e e exp t (13) S ii1

where RS(t) series system reliability at time ti constant failure rate of unit i for i 1, 2, 3, ... , m

The system hazard rate is given by14

(t) 1 dR (t)S S (t) S 1 F (t) R (t) dtS Sm

(14) ii1

where S(t) series system hazard rate or total failure rateS(t) series system probability density functionFS(t) series system cumulative distribution function

It is to be noted that the system total failure rate given by Eq. (14) is the sum of the failurerates of all of the units. It means that whenever the failure rates of units are added, it isautomatically assumed that the units are acting in series (i.e., if one unit fails, the systemfails). This is the worst-case assumption often practiced in the design of engineering systems.

The system mean time to failure is given by14

m

MTTF R (t) dt exp t dt S S ii1

0 0

1 (15)

m (1 / )i1 iwhere MTTFS is the series system mean time to failure.

3 Common Reliability Networks 1005

1

2

3

mFigure 2 Parallel system block diagram.

3.2 Parallel NetworkThis type of conguration can be used to improve a mechanical systems reliability duringthe design phase. The block diagram of an m-unit parallel network is shown in Fig. 2.Each block in the diagram represents a unit. This conguration assumes that all of its unitsare active and at least one unit must work normally for the system to succeed. For indepen-dently failing units, the reliability of the parallel network shown in Fig. 2 is expressed by14

m

R 1 (1 R ) (16)p ii1

where Rp reliability of parallel networkRi reliability of unit i for i 1, 2, 3, ... , m

For constant failure rates of the units, Eq. (16) becomes14m

tiR (t) 1 (1 e ) (17)pi1

where Rp(t) parallel network reliability at time ti constant failure rate of unit i for i 1, 2, 3, ... , m

For identical units, the network mean time to failure is given by14

m1 1MTTF R (t) dt (18)p p ii1

0

where MTTFp parallel network mean time to failure unit constant failure rate

3.3 SeriesParallel NetworkThe network block diagram is shown in Fig. 3. Each block in the diagram represents a unit.This network represents a system having m number of subsystems in series. In turn, eachsubsystem contains n number of active units in parallel. All subsystems must operate nor-mally for the system to succeed. For independent units, the reliability of the seriesparallelnetwork shown in Fig. 3 is given by18,19

1006 Reliability in the Mechanical Design Process

1

1

2

n

1

2

n

1

2

n

2 m

Figure 3 Seriesparallel network block diagram.

m n

R 1 F (19) Sp iji1 j1

where RSp seriesparallel network reliabilitym number of subsystemsn number of units

Fij failure probability of the ith subsystems jth unitFor constant unit failure rates, Eq. (19) becomes18,19

m n

tijR (t) 1 (1 e ) (20) Spi1 j1

where RSp(t) seriesparallel network reliability at time tij constant failure rate of unit ij

For identical units, the network mean time to failure is given by19

MTTF R (t) dtSp Sp0

m in1 1mi1 (1) (21) i ji1 j1where MTTFSp seriesparallel network mean time to failure

unit failure rate

3.4 ParallelSeries NetworkThis network represents a system having m number of subsystems in parallel. In turn, eachsubsystem contains n number of units in series. At least one subsystem must function nor-mally for the system to succeed. The network block diagram is shown in Fig. 4. Each blockin the diagram represents a unit. For independent units, the reliability of the parallelseriesnetwork shown in Fig. 4 is expressed by18,19

m n

R 1 1 R (22) pS iji1 j1

3 Common Reliability Networks 1007

1

1

1

1

2

2

2

2

3

3

3

3

n

n

n

n

1

2

3

m

Figure 4 Parallelseries network block diagram.

where RpS parallelseries network reliabilitym number of subsystemsn number of units

Rij reliability of the jth unit in the ith subsystemFor constant unit failure rates, Eq. (23) becomes18,19

m n

tijR (t) 1 1 e (23) pSi1 j1

where RpS(t) parallelseries network reliability at time tij constant failure rate of unit ij

For identical units, the network mean time to failure is given by19

m1 1MTTF R (t) dt (24)pS pS n ii1

0

where MTTFpS is the parallelseries network mean time to failure.

3.5 K-out-of-m-Unit NetworkThis network is sometimes referred to as a partially redundant network. It is a parallelnetwork with a condition that at least K units out of the total of m units must operate normallyfor the system to succeed.

For independent and identical units, the network reliability is given by14,16m

m i miR R (1 R) (25) K /m iiKwhere

m!m i i!(m i)!

1008 Reliability in the Mechanical Design Process

0

1

2

m

Figure 5 Block diagram of an (m 1)-unit standby system.

RK /m is the K-out-of-m-unit network reliability, and R is the unit reliability. For K 1 andK m, Eq. (25) becomes the reliability expression for parallel and series networks, respec-tively. More specically, parallel and series networks are the special cases of the K-out-of-m-unit network.

For constant failure rates of the units, Eq. (25) becomes14,16m

m it t miR e (1 e ) (26) K /m iiKwhere RK /m K-out-of-m-unit network reliability at time t

unit constant failure rate

The network mean time to failure is given by14,16

m1 1MTTF R (t) dt (27)K / m k /m iik

0

where MTTFK /m is the K-out-of-m-unit network mean time to failure.

3.6 Standby SystemThis is another conguration used to improve reliability. The block diagram of an (m 1)-unit standby system is shown in Fig. 5. Each block in the diagram represents a unit. In thisconguration, one unit operates and m units are kept on standby. As soon as the operatingunit fails, it is replaced by one of the standbys. The system fails when all of its units fail(i.e., operating plus all standbys). For perfect switching, independent and identical units, andas-good-as-new standby units, the standby system reliability is given by14,16

m t i t[ (t) dt] exp( (t) dt)0 0R (t) (28)SS i!i0where RSS(t) standby system reliability at time t

(t) unit hazard rate or time-dependent failure ratem number of standbys

For constant failure rates of the units [i.e., (t) ] Eq. (28) becomesm i t( t) e

R (t) (29)SS i!i0

3 Common Reliability Networks 1009

1 2

3

4 5Figure 6 Block diagram of a ve-unit bridge network.

where is the unit constant failure rate. The system mean time to failure is given by14

MTTF R (t) dtSS SS0

m i t(t) e

dt i!i00

m 1 (30)

where MTTFSS is the standby system mean time to failure.

3.7 Bridge NetworkThe block diagram of a bridge network is shown in Fig. 6. Each block in the diagramrepresents a unit. Mechanical components sometimes can form this type of conguration.For independent units, the reliability of the bridge network shown in Fig. 6 is20

5 4

R 2 R R R R R R R R R bn i i 1 3 5 1 4 2 5i1 i1

5 4 3 5

R R R R R R R R R R (31) i i 5 i 1 i 1 2 4 5i2 i1 i1 i3

where Rbn bridge network reliabilityRi unit i reliability for i 1, 2, 3, 4, 5

For identical units and constant failure rates of the units, Eq. (31) becomes145t 4t 3t 2tR (t) 2e 5e 2e 2e (32)bn

where is the unit constant failure rate. The network mean time to failure is given by

49MTTF R (t) dt (33)bn bn 600

where MTTFbn is the bridge network mean time to failure.

1010 Reliability in the Mechanical Design Process

4 MECHANICAL FAILURE MODES AND CAUSES OF GENERALAND GEAR FAILURES

A mechanical failure may be dened as any change in the shape, size, or material propertiesof a structure, piece of equipment, or equipment part that renders it unt to carry out itsspecied mission adequately.13 Thus, there are many different types of failure modes asso-ciated with mechanical items. Good design practices can reduce or eliminate altogether theoccurrence of these failure modes. Some of these failure modes are as follows2123:

Fatigue failure Material aw failure Bending failure Metallurgical failure Bearing failure Instability failure Shear loading failure Compressive failure Creep/rupture failure Tensile yield strength failure Ultimate tensile strength failure Stress concentration failure

There are many causes of product failures. Some of these are as follows24:

Defective design Wear-out Defective manufacturing Wrong application Incorrect installation Failure of other parts

A study performed over a period of 35 years reported a total of 931 gear failures.24They were classied under four categories: breakage (61.2%), surface fatigue (20.3%), wear(3.2%), and plastic ow (5.3%). The causes of these failures were grouped into the vecategories shown in Fig. 7.

These ve categories were further divided into various elements. The elements of theservice-related classication were continual overloading (25%), improper assembly (21.2%),impact loading (13.9%), incorrect lubrication (11%), foreign material (1.4%), abusive han-dling (1.2%), bearing failure (0.7%), and operator errors (0.3%). The elements of the heat-treatment-related classication were incorrect hardening (5.9%), inadequate case depth(4.8%), inadequate core hardness (2%), excessive case depth (1.8%), improper tempering(1%), excessive core hardness (0.5%), and distortion (0.2%). The elements of the design-related classication were wrong design (2.8%), specication of suitable heat treatment(2.5%), and incorrect material selection (1.6%). The elements of the manufacturing-relatedclassication were grinding burns (0.7%) and tool marks or notches (0.7%). Finally, the threeelements of the material-related classication were steel defects (0.5%), mixed steel or in-correct composition (0.2%), and forging defects (0.1%).

5 Reliability-Based Design and Design-by-Reliability Methodology 1011

Heattreatment related (16.2%)

Design related (6.9%)

Materialrelated (0.8%)

Manufacturingrelated (1.4%)

Service related (74.7%)

Classifications

Figure 7 Classications of gear failure causes.

5 RELIABILITY-BASED DESIGN AND DESIGN-BY-RELIABILITY METHODOLOGYIt would be unwise to expect a system to perform to a desired level of reliability unless itis specically designed for that level. Desired system/equipment /part reliability design spec-ications due to factors such as well-publicized failures (e.g., the space shuttle Challengerdisaster and the Chernobyl nuclear accident) have increased the importance of reliability-based design. The starting point for reliability-based design is during the writing of the designspecication. In this phase, all reliability needs and specications are entrenched into thedesign specication. Examples of these requirements might include item mean time to failure(MTTF), mean time to repair (MTTR), test or demonstration procedures to be used, andapplicable documents.

Over the years the DOD has developed various reliability documents for use during thedesign and development of an engineering item. Many times, such documents are entrenchedinto the item design specication document. Table 1 presents some of these documents.Many professional bodies and other organizations have also developed documents on variousaspects of reliability.15,22,25 References 22 and 26 provide descriptions of documents devel-oped by the DOD.

Reliability is an important consideration during the design phase. According to Ref. 27,as many as 60% of failures can be eliminated through design changes. There are manystrategies the designer could follow to improve design:

Eliminate failure modes. Focus design for fault tolerance. Focus design for fail safe. Focus design to include mechanism for early warnings of failure through fault diag-

nosis.

1012 Reliability in the Mechanical Design Process

Table 1 Selected Design-Related Documents Developed by the U.S. Department of Defense

Document No. Document Title

MIL-STD-721 Denitions of Terms for Reliability and MaintainabilityMIL-STD-217 Reliability Prediction of Electronic EquipmentMIL-STD-781 Reliability Design, Qualication, and Production Acceptance Tests: Exponential DistributionMIL-STD-756 Reliability Modeling and PredictionMIL-STD-785 Reliability Program for Systems and EquipmentMIL-HDBK-251 Reliability /Design Thermal ApplicationsMIL-STD-1629 Procedures for Performing a Failure Mode, Effects, and Criticality AnalysisRADC-TR-75-22 Non-Electronic Reliability NotebookMIL-STD-965 Parts Control ProgramMIL-STD-2074 Failure Classication for Reliability Testing

During the design phase of a product, various types of reliability and maintainability analysescan be performed, including reliability evaluation and modeling, reliability allocation, main-tainability evaluation, human factors / reliability evaluation, reliability testing, reliabilitygrowth modeling, and life-cycle cost. In addition, some of the design improvement strategiesare zero-failure design, fault-tolerant design, built-in testing, derating, design for damagedetection, modular design, design for fault isolation, and maintenance-free design. Duringdesign reviews, reliability and maintainability-related actions recommended/ taken are to bethoroughly reviewed from desirable aspects.

A systematic series of steps are taken to design a reliable mechanical item. The designby methodology is composed of such steps14,28,29:

Dene the design problem under consideration. Identify and list all associated design variables and parameters. Perform failure mode, effect, and criticality analyses according to MIL-STD-1629.30

Verify critical design parameter selection. Establish appropriate relationships between the failure-governing criteria and the crit-

ical parameters. Determine the failure-governing stress and strength functions and then the most ap-

propriate failure-governing stress and strength distributions. Estimate the reliability utilizing the failure-governing stress and strength distributions

for all critical failure modes. Iterate the design until reliability goals are achieved. Optimize design with respect to factors such as safety, reliability, cost, performance,

maintainability, weight, and volume. Repeat the design optimization process for all critical components. Estimate item reliability. Iterate the design until item reliability goals are fully satised.

6 DESIGN RELIABILITY ALLOCATION AND EVALUATION METHODSOver the years, many reliability allocation and evaluation methods have been developed foruse during the design phase.14,29 This section presents some that are considered useful, par-ticularly in designing mechanical items.

6 Design Reliability Allocation and Evaluation Methods 1013

6.1 Failure Rate Allocation MethodThis method is used to allocate failure rates to system components when the overall systemrequired failure rate is given. The method is based on the following three assumptions14:

All system components fail independently. Component failure rates are constant. System components form a series network.

Thus, the system failure rate using Eq. (14) ism

(34)S ii1

where S system failure ratem total number of system componentsi failure rate of component i for i 1, 2, 3, ... , m

If the specied failure rate of the system is Sp, then the component failure rate is allocatedsuch that

m

* (35) i Spi1

where is the failure rate allocated to component i for i 1, 2, 3, ... , m. The following*ithree steps are associated with this approach:

Estimate failure rates of the system components (i.e., i for i 1, 2, 3, ... , m) usingthe eld data.

Calculate the relative weight i of component i using the preceding step failure ratedata and the expression

i for i 1, 2, 3, ... , m (36)i m i1 iIt is to be noted that i represents the relative failure vulnerability of component i and

m

1 (37) ii1

Allocate the failure rate to part or component i by using the equation

* for i 1, 2, 3, ... , m (38)i i SpA solved example in Ref. 14 demonstrates the application of this method.

6.2 Hybrid Reliability Allocation MethodThis method combines two reliability allocation methods: similar familiar systems and factorsof inuence. The method is more attractive because it incorporates the benets of bothsystems.14,29

The basis for the similar-familiar-systems approach is the familiarity of the designerwith similar systems as well as the utilization of failure data collected on similar systemsfrom various sources during the allocation process. The principal disadvantage of the similar-

1014 Reliability in the Mechanical Design Process

familiar-systems method is the assumption that the reliability and life-cycle cost of similarsystems are satisfactory or adequate.

The factors-of-inuence method is based upon four factors that are considered to affectitem reliability: failure criticality, environment, complexity / time, and state of the art.

The failure criticality factor is concerned with how critical is the failure of an item (e.g.,the failure of some auxiliary equipment in an aircraft may not be as critical as the failureof an engine). The environmental factor takes into account the susceptibility of items toconditions such as vibration, humidity, and temperature.

The complexity / time factor relates to the number of item/subsystem components /partsand the operating time of the item under consideration during the total system operatingperiod. Finally, the state-of-the-art factor relates to the advancement in the state of the artfor an item under consideration.

During the reliability allocation process, each item is rated with respect to each of thesefour factors by assigning a number from 1 to 10. The assignment of 1 means the item isleast affected by the factor under consideration and 10 means the item is most affected bythe same factor. Subsequently, reliability is determined by weighing these numbers for allfour factors.

6.3 Safety Factor and Safety MarginThe safety factor and safety margin are arbitrary multipliers used to ensure the reliability ofmechanical items during the design phase. These indexes can provide satisfactory design ifthey are established using considerable past experiences and data.

Safety FactorA safety factor can be dened in many different ways.13,3135 Two commonly used denitionsfollow.

Denition I. The safety factor is dened by36

mShSF 1 (39)mSS

where SF safety factormSh mean failure governing strengthmSS mean failure governing stress

This index is a good measure of safety when both stress and strength are normally distributed.However, when the spread of both strength and/or stress is large, the index becomes mean-ingless because of positive failure rate.14

Denition II. The safety factor is dened by3237

USSF (40)WSwhere SF safety factor

WS working stress expressed in pounds per square inch (psi)US ultimate strength expressed in psi

6 Design Reliability Allocation and Evaluation Methods 1015

Safety MarginThe safety margin (SM) is dened as14,31

SM SF 1 (41)The negative value of this measure means that the item under consideration will fail. Thus,its value must always be greater than zero.

The safety margin for normally distributed stress and strength is expressed by14,31

as msSM (42)Sh

where as average strengthms maximum stressSh strength standard deviation

In turn the maximum stress ms is expressed by

C (43)ms SS SSwhere SS mean stress

SS stress standard deviationC factor between 3 and 6

6.4 StressStrength Interference Theory MethodThis method is used to determine the reliability of a mechanical item when its associatedstress and strength probability density functions are known. The item reliability is denedby13,14,29

R P(y x) P(x y) (44)where R item reliability

P probabilityx strength random variabley stress random variable

Equation (44) is rewritten in the form13,14,29

R (y) (x) dx dy (45) y

where (x) strength probability density function(y) stress probability density function

Special Case Model: Exponentially Distributed Stress and StrengthIn this case, an items stress and strength are dened by

y(y) e 0 y (46)and

1016 Reliability in the Mechanical Design Processx(x) e 0 x (47)

where and are the reciprocals of the mean values of stress and strength, respectively.Using Eqs. (46) and (47) in Eq. (45) yields

y xR e e dx dy 0 y (48)

For and Eq. (48) becomes1/y, 1 /x,x

R (49)x y

where x mean strengthy mean stress

Similarly, models for other stress and strength probability distributions can be developed. Anumber of such models are presented in Ref. 13.

6.5 Failure Modes and Effect Analysis (FMEA)FMEA is a vital tool for evaluating system design from the point of view of reliability. Itwas developed in the early 1950s to evaluate the design of various ight control systems.38

The difference between the FMEA and the failure mode, effect, and criticality analysis(FMECA) is that FMEA is a qualitative technique used to evaluate a design, whereasFMECA is composed of FMEA and criticality analysis (CA). Criticality analysis is a quan-titative method used to rank critical failure mode effects by taking into account their occur-rence probabilities.

As FMEA is a widely used method in industry, there are many standards/documentswritten about it. In Ref. 30 45 such publications were collected and evaluated, prepared byorganizations such as the DOD, NASA, and the Institute of Electrical and Electronic Engi-neers (IEEE). These documents include39:

DOD: MIL-STD-785A (1969), MIL-STD-1629 (draft) (1980), MIL-STD-2070 (AS)(1977), MIL-STD-1543 (1974), AMCP-706-196 (1976)

NASA: NHB 5300.4 (1A) (1970), ARAC Proj. 79-7 (1976) IEEE: ANSI N 41.4 (1976)

Details of the above documents as well as a list of publications on FMEA are given in Ref.24.

The main steps involved in performing FMEA are as follows29:

Dene carefully all system boundaries and detailed requirements. List all parts /subsystems in the system under consideration. Identify and describe each part and list all its associated failure modes. Assign failure rate /probability to each failure mode. List effects of each failure mode on subsystem/system/plant.

7 Human Error and Reliability Consideration in Mechanical Design 1017

Enter remarks for each failure mode. Review each critical failure mode and take appropriate measures.

This method is described in detail in Ref. 14.

6.6 Fault Tree Analysis (FTA)This method, so called because it arranges fault events in a tree-shaped diagram, is one ofthe most widely used techniques for performing system reliability analysis. In particular, itis probably the most widely used method in the nuclear power industry. The technique iswell suited for determining the combined effects of multiple failures.

The fault tree technique is more costly to use than the FMEA approach. It was developedin the early 1960s at Bell Telephone Laboratories to evaluate the reliability of the MinutemanLaunch Control System. Since that time, hundreds of publications on the method have ap-peared.15

The FTA begins by identifying an undesirable event, called the top event, associatedwith a system. Fault events that could cause the occurrence of the top event are generatedand connected by AND and OR logic gates. The construction of a fault tree proceeds bygeneration of fault events (by asking the question How could this event occur?) in asuccessive manner until the fault events need not be developed further. These events areknown as primary or elementary events. In simple terms, the fault tree may be described asthe logic structure relating the top event to the primary events. This method is described indetail in Ref. 14.

7 HUMAN ERROR AND RELIABILITY CONSIDERATIONIN MECHANICAL DESIGN

As in the reliability of any other system, human reliability and error play an important rolein the reliability of mechanical systems. Over the years many times mechanical systems/equipment have failed due to human error rather than hardware failure. Careful considerationof human error and reliability during the design of mechanical systems can help to eliminateor reduce the occurrence of non-hardware-related failures during the operation of such sys-tems. Human errors may be classied under the following seven distinct categories4042:

Design errors Operator errors Assembly errors Inspection errors Maintenance errors Installation errors Handling errors

Each of these categories is described in detail in Ref. 14.There are numerous causes for the occurrence of human error including poor equipment

design, complex tasks, poor work layout, poorly written operating and maintenance proce-dures, poor job environment (e.g., poor lighting, high/ low temperature, crowded work space,high noise level), inadequate work tools, poor skill of involved personnel, and poor moti-vation of involved personnel.4042

1018 Reliability in the Mechanical Design Process

Human reliability of time-continuous tasks such as aircraft maneuvering, scope moni-toring, and missile countdown can be calculated by using the equation42,43

t

R (t) exp (t) dt (50)h 0

where Rh(t) human reliability at time t(t) time-dependent human error rate

For constant human error rate [i.e., (t) ], Eq. (50) becomest

tR (t) exp dt e (51)h 0

where is the constant human error rate. The subject of human reliability and error isdiscussed in detail in Ref. 42.

Example. A person is performing a certain time-continuous task. Assume his or her errorrate is 0.004 per hour. Calculate the persons probability of performing the task correctlyduring a 5-h period.

By using the values in Eq. (51), we get(0.004)5R (5) e 0.98h

So, there is approximately a 98% chance that the person will perform the task correctlyduring the specied period.

8 FAILURE RATE ESTIMATION MODELS FOR VARIOUS MECHANICAL ITEMSMany mathematical models available in the literature can be used to estimate failure ratesof items such as bearings, pumps, brakes, lters, compressors, and seals.4446 This sectionpresents some of these models.

8.1 Brake System Failure Rate Estimation ModelThe brake system failure rate is expressed by44

6

(52)brs ii1

where brs brake system failure rate, expressed in failures /106 h1 brake housing failure rate2 total failure rate of actuators3 total failure rate of seals4 total failure rate of bearings5 total failure rate of springs6 total failure rate of brake friction materials

The values of i for i 1, 2, ... , 6 are obtained through various means.44,47

9 Failure Data and Failure Data Collection Sources 1019

8.2 Compressor System Failure Rate Estimation ModelThe compressor system failure rate is expressed by45

6

(53)comp ii1

where comp compressor system failure rate, expressed in failures /106 h1 failure rate of all compressor bearings2 compressor casing failure rate3 failure rate due to design conguration4 failure rate of valve assay (if any)5 failure rate of all compressor seals6 failure rate of all compressor shafts

Procedures for calculating 1, 2, 3, 4, 5, and 6 are presented in Ref. 45.

8.3 Filter Failure Rate Estimation ModelThe lter failure rate is expressed by46

6

(54)ft b ii1

where ft lter failure rate, expressed in failures /106 hb lter base failure ratei ith modifying factor; i 1 is for temperature effects, i 2 for water contam-

ination effects, i 3 for cyclic ow effects, i 4 for differential pressureeffects, i 5 for cold-start effects, and i 6 for vibration effects

Procedures for estimating b, 1, 2, 3, 4, 5, and 6 are given in Ref. 46.

8.4 Pump Failure Rate Estimation ModelThe pump failure rate is expressed by46

5

(55)pm ii1

where pm pump failure rate, expressed in failures /106 cycles1 pump uid driver failure rate2 pump casing failure rate3 pump shaft failure rate4 failure rate of all pump seals5 failure rate of all pump bearings

Procedures for calculating 1, 2, 3, 4, and 5 are presented in Ref. 46.

9 FAILURE DATA AND FAILURE DATA COLLECTION SOURCESFailure data provide invaluable information to reliability engineers, design engineers, man-agement, and so on, concerning the product performance. These data are the nal proof of

1020 Reliability in the Mechanical Design Process

Table 2 Failure Rates for Selected Mechanical Items

Item Description Failure Rate 106 h

Roller bearing 8.323Bellows (general) 13.317Filter (liquid) 6.00Compressor (general) 33.624Pipe 0.2Hair spring 1.0Pump (vacuum) 10.610Gear (spur) 3.152Seal (O-ring) 0.2Nut or bolt 0.02Brake (electromechanical) 16.00Knob (general) 2.081Washer (lock) 0.586Washer (at) 0.614Duct (general) 2.902Note: Use environment: ground xed or general.

Table 3 Selected Failure Data Sources for Mechanical Items

Source Developed By

Ref. 48 Reliability Analysis Center, Rome Air DevelopmentCenter, Grifs Air Force Base, Rome, New York

Component Reliability Data for Use inProbabilistic Safety Assessment (1998)

International Atomic Energy Agency, Vienna,Austria

R. G. Arno, Non-Electronic Parts ReliabilityData (Rept. No. NPRD-2, 1981)

Reliability Analysis Center, Rome Air DevelopmentCenter, Grifs Air Force Base, Rome, New York

Government Industry Data ExchangeProgram (GIDEP)

GIDEP Operations Center, U.S. Dept. of Navy,Seal Beach, Corona, California.

Ref. 49 Reliability Analysis Center, Rome Air DevelopmentCenter, Grifs Air Force Base, Rome, New York

the success or failure of the effort expended during the design and manufacture of a productused under designed conditions. During the design phase of a product, past informationconcerning its failures plays a critical role in the reliability analysis of that product. Failuredata can be used to estimate item failure rate, perform effective design reviews, predictreliability and maintainability of redundant systems, conduct trade-off and life-cycle coststudies, and perform preventive maintenance and replacement studies. Table 2 presents failurerates for selected mechanical items.13,48,49

There are many different ways to collect failure data. For example, during the equipmentlife cycle, there are eight identiable data sources: repair facility reports, development testingof the item, previous experience with similar or identical items, customers failure-reportingsystems, inspection records generated by quality control and manufacturing groups, testsconducted during eld demonstration, environmental qualication approval, and eld instal-lation, acceptance testing, and warranty claims.50 Table 3 presents some sources for collectingfailure data for use during the design phase.13

References 1021

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trol, 7, 2134 (1957).21. J. A. Collins, Failure of Materials in Mechanical Design, Wiley, New York, 1981.22. W. Grant Ireson, C. F. Coombs, and R. Y. Moss (eds.), Handbook of Reliability Engineering and

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1022 Reliability in the Mechanical Design Process

30. Procedures for Performing Failure Mode, Effects, and Criticality Analysis, MIL-STD-1629, Depart-ment of Defense, Washington, DC, 1980.

31. D. Kececioglu, and E. B. Haugen, A Unied Look at Design Safety Factors, Safety Margin, andMeasures of Reliability, Proceedings of the Annual Reliability and Maintainability Conference,1968, pp. 522530.

32. G. M. Howell, Factors of Safety, Machine Design, July 12, 1956, pp. 7681.33. R. B. McCalley, Nomogram for Selection of Safety Factors, Design News, Sept. 1957, pp. 138

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10, 104109 (1961).42. B. S. Dhillon, Human Reliability: With Human Factors, Pergamon, New York, 1986.43. T. L. Regulinski, and W. B. Askern, Mathematical Modeling of Human Performance Reliability,

in Proceedings of the Annual Symposium on Reliability, 1969, pp. 511.44. S. Rhodes, J. J. Nelson, J. D. Raze, and M. Bradley, Reliability Models for Mechanical Equipment,

Proceedings of the Annual Reliability and Maintainability Symposium, 1988, pp. 127131.45. J. D. Raze, J. J. Nelson, D. J. Simard, and M. Bradley, Reliability Models for Mechanical Equip-

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Mechanical Equipment, Proceedings of the Annual Reliability and Maintainability Symposium,1989, pp. 146153.

47. T. D. Boone, Reliability Prediction Analysis for Mechanical Brake Systems, NAVAIR/SYSCOMReport, Department of Navy, Department of Defense, Washington, DC, Aug. 1981.

48. M. J. Rossi, Non-Electronic Parts Reliability Data, Report No. NRPD-3, Reliability Analysis Center,Rome Air Development Center, Grifss Air Force Base, NY, 1985.

49. R. E. Schafer, J. E. Angus, J. M. Finkelstein, M. Yerasi, and D. W. Fulton, RADC Non-electronicReliability Notebook, Report No. RADC-TR-85-194, Reliability Analysis Center, Rome Air Devel-opment Center, Grifss Air Force Base, NY, 1985.

50. B. S. Dhillon, and H. C. Viswanath, Bibliography of Literature on Failure Data, Microelectron.Reliabil., 30, 723750 (1990).

BIBLIOGRAPHYBompas-Smith, J. H., Mechanical Survival, McGraw-Hill, London, 1973.Carter, A. D. S., Mechanical Reliability, Macmillan Education, London, 1986.Carter, A. D. S., Mechanical Reliability and Design, Wiley, New York, 1997.Dhillon, B. S., Robot Reliability and Safety, Springer-Verlag, New York, 1991.Frankel, E. G., Systems Reliability and Risk Analysis, Martinus Nijhoff, The Hague, 1984.Haugen, E. B., Probabilistic Mechanical Design, Wiley, New York, 1980.Kapur, K. C., and L. R. Lamberson, Reliability in Engineering Design, Wiley, New York, 1977.Kivenson, G., Durability and Reliability in Engineering Design, Hayden, New York, 1971.Little, A., Reliability of Shell Buckling Predictions, MIT Press, Cambridge, MA, 1964.

Bibliography 1023

Little, R. E., Mechanical Reliability Improvement: Probability and Statistics for Experimental Testing,Marcel Dekker, New York, 2003.

Mechanical Reliability Concepts, American Society of Mechanical Engineers, New York, 1965.Middendorf, W. H., Design of Devices and Systems, Marcel Dekker, New York, 1990.Milestone, W. D. (ed.), Reliability, Stress Analysis and Failure Prevention Methods in Mechanical Design,

American Society of Mechanical Engineers, New York, 1980.Shooman, M. L., Probabilistic Reliability: An Engineering Approach, Krieger, Melbourne, FL, 1990.Siddell, J. N., Probabilistic Engineering Design, Marcel Dekker, New York, 1983.

Additional publications on mechanical design reliability may be found in Refs. 13 and 15.