Xin Q. Diesel Engine System Design

186 Diesel engine system design

temperature and the stress. Very often design modications affect both stress

and strength, for instance, modifying the width of the inter-valve bridge area

in the cylinder head. When the stress and strength are changed, the probability

density function curves are shifted or stretched, and the overlapping area is

changed. Reliability analysis is also helpful for determining a wise target

used in durability validation testing in order to prevent under-design or

over-design.

Reliability simulation for diesel engine design became more active during

the last decade. Gale et al. (1995) introduced a probabilistic reliability

analysis methodology for engine durability. Their analysis had a failure

model, which covered crack, fatigue, wear, or other structural issues.

They believed that in dynamics and fatigue areas analytical prediction of

durability has been well established; however, for thermal and wear-related

mechanisms the models still relied on empirical correlations. Their analysis

output a cumulative distribution function which represents the probability

of failure of the component for a certain lifetime. The analysis predicted the

B10 or B50 durability life of the component. The effects of design change

and noise factor control (e.g., geometry, surface roughness) on the B10 or

B50 life of the component were demonstrated in their analysis. They claimed

that the Monte Carlo simulation could obtain the same result but with 500

times longer computing time than their Fast Probability Integration software

developed at the Southwest Research Institute.

Bignonnet and Thomas (2001) used the statistical stress–strength model to

predict the reliability of automotive components related to fatigue. Niewczas

and Koszalka (2002) presented a method for reliability prediction related

to engine cylinder liner wear and power cylinder durability. They stated

that predicting engine wear reliability with ten-times shortened on-road

accelerated test data was possible. Their method also provided a conversion

factor between the engine test bed time and the vehicle mileage based on

the comparison of wear intensity (e.g., one hour of engine test bed time was

reported to be equivalent to 140 km of vehicle mileage).

Aldridge (2003) presented a methodology to predict engine component/

system reliability and durability by using the Weibull chart. The method was

applied to an electronic diesel fuel injector. The correlations between the

component and system reliability requirements were studied. The method

was able to predict the reliability and durability affected by engine system

performance calibration and manufacturing variations.

Probably the most important engine reliability and durability simulation

work to date, at least from the viewpoint of engine system design, is the

series of work published by Prince et al. (2005), Morin et al. (2005) and

Daubercies et al. (2009). They developed a methodology to estimate and

improve the reliability of a diesel engine cylinder head during the design

stage. Their work focused on the thermo-mechanical damage of the cylinder

Diesel-Xin-02.indd 186 5/5/11 11:44:58 AM



187Durability and reliability in diesel engine system design

head. Their method started with a time history of engine transient speeds

and loads during driving cycles obtained from durability test and in-service

data. A thermal analysis of component temperature history and a thermo-

mechanical stress calculation were conducted. Failure rate was estimated in

their damage–strength reliability model. Design iterations were conducted to

achieve the reliability target. They used lognormal distributions to represent

the stress and strength distributions. They developed simpli ed surrogate

computing models that were derived from the more complex FEA model.

The simpli ed model could simulate the inter-valve bridge stress history of a

10,000–30,000 km vehicle journey in less than one hour of computing time.

Their model was validated with the reliability service data of high-mileage

vehicles. The methodology provided a major advance from empirical design

for reliability to advanced analytical design.

Other engine reliability studies can be found in Pendola et al. (2003),

Rahman et al. (2007), and Kokkolaras et al. (2005).

2.13.6 Reliability allocation and system optimization

System reliability

Reliability can be evaluated for each failure mode or the combination of all the

failure modes. Reliability can also be evaluated for a subsystem/component

or a system. The reliability of each subsystem/component determines the

overall reliability of the entire system. If the subsystem/component malfunction

leads to the entire system malfunction, the system is classi ed as a series

system in terms of its connectivity. The reliability of a series system, R

S

, is

equal to the product of the reliabilities of each independent component R

i

as follows:

Rt

S

Rt

S

Rt

i=

n

i

Rt

i

Rt

()

Rt()Rt

=

()

Rt()Rt

1

P

2.19

where t is the time-to-failure. When a series system comprises a large

number of components, the system reliability may be rather low even if

the individual components have high reliabilities. This fact highlights an

important guideline for system design – design for simplicity (i.e., using

fewer components).

On the other hand, if the entire system can function properly with at least

one component functioning properly, the system is classi ed as a parallel

system in terms of its connectivity. In other words, a parallel system loses

its function only when all the components fail. The reliability of a parallel

system is given as follows:

Rt

S

Rt

S

Rt

i=

n

i

Rt

i

Rt

()

Rt()Rt

= 1 –

(

1 –

()

Rt()Rt

)

1

P

2.20

Diesel-Xin-02.indd 187 5/5/11 11:44:58 AM



188 Diesel engine system design

Engine components (or subsystems) can be classi ed into two types based

on their different effects on diesel engine system performance:

1. The malfunction of the component results in unrecoverable malfunction

of the entire engine (e.g., piston crack, skirt scuf ng).

2. The malfunction of the component can be compensated by adjusting

its own or other components’ characteristics. The entire engine system

performance is maintained as a result.

The diesel engine is usually considered as a series system for the major

components (or subsystems) when subjected to system design. System reliability

fundamentals were provided by Tillman et al. (1980), Kececioglu (1991),

Leemis (1995), O’Connor (2002) and Rausand and Hoyland (2004).

Reliability allocation and impact on cost

Reliability allocation refers to the optimization process on the reliabilities of

all or some of the components of a given system in order to meet the target

of overall system reliability with minimum cost (Mettas and Savva, 2001).

Reliability allocation needs to occur when the estimated or designed system

reliability is not suf cient, or when the reliabilities of the components are

severely imbalanced, causing a large over-design and waste of lifetime for

some components. In diesel engine system design, reliability allocation and

optimization is essentially allocating the risks of different design constraints.

For example, at rated power there is a trade-off between peak cylinder pressure

and exhaust manifold gas temperature. When fuel injection timing is advanced,

the peak cylinder pressure may increase and the exhaust temperature may

decrease. The pressure and temperature affect the durability of the cylinder

head and exhaust manifold, respectively. Properly assessing the component

reliability and allocating the reliability can achieve an optimum balance at

the system level. Reliability allocation helps to achieve ‘design for reliability’

from the beginning of an engine program in the system design stage.

Reliability importance ranks the impact of a component’s reliability change

on the overall system’s reliability. It is de ned as follows (Leemis, 1995):

I

Rt

Ri

S

i

,

Ri,Ri

=

∂(

Rt∂(Rt

S

∂(

S

Rt

S

Rt∂(Rt

S

Rt

)

∂(

Rt∂(Rt

i

∂(

i

Rt

i

Rt∂(Rt

i

Rt

)

2.21

Cost can be modeled as a function of a component’s reliability. The overall

system cost is the summation of each component’s cost. In general, cost

increases exponentially with reliability. The trade-off between cost and

reliability is managed in reliability allocation and system optimization. There

are two types of guidelines for reliability allocation:

1. The components with high reliability importance can be assigned a high

Diesel-Xin-02.indd 188 5/5/11 11:44:59 AM



189Durability and reliability in diesel engine system design

reliability since a high importance indicates the component has a large

impact on the overall system reliability.

2. The most costly component can be assigned the lowest increase in

reliability.

The topic of reliability allocation for diesel engines can be formulated as a

constrained nonlinear optimization problem as follows:

Min

imize cost

=

()

s

ubject to

,

=1

,

CC

=CC =

CC

()R()

R

cS

,cS,

CC

cS

CC

=CC =

cS

=CC =

i

n

ci

,ci,

i

()

i

()

S

CCSCC

≥

,

≤

gi

,

= 1

,2

,…

,

,,

RR

, RR,

RR

≤ RR≤

St

,St,

RR

St

RR

ar

RR

ar

RR

StarSt

RR

St

RR

ar

RR

St

RR

ge

RR

ge

RR

argear

RR

ar

RR

ge

RR

ar

RR

t

RR

t

RR

in

= 1in = 1

,2in,2

,…in,…

, in,

ii

≤

ii

≤

,,ii,,

RR

ii

RR

i

iimiii

,,ii,,mi,,ii,,

nm

,,nm,,

nm

,,nm,,

RR

nm

RR

≤ RR≤

nm

≤ RR≤

iinmii

≤

ii

≤

nm

≤

ii

≤

,,ii,,nm,,ii,,

RR

ii

RR

nm

RR

ii

RR

inmi

,,i,,nm,,i,,

ax

ven

vven v

()

=

()

(

)

=1

Rt

()Rt()

Rt

()Rt()

RR

=RR =

(RR (

t

S

Rt

S

Rt

i

n

ii

()

ii

()

ii

Rt

ii

Rt

()Rt()

ii

()Rt()

RR

ii

RR

i

(

i

(

P

Ï

Ì

Ô

Ï

Ô

Ï

Ô

Ì

Ô

Ì

Ô

Ó

Ô

Ì

Ô

Ì

Ô

Ó

Ô

Ó

Ô

2.22

This optimization formulation is to achieve a minimum total system cost C

c,S

under system reliability constraints. When the cost is minimized, the reliability

of each component tends to be reduced to their respective lowest value in order

to avoid over-design for lifetime or a large imbalance of lifetime between the

components. To solve the above optimization problem, one needs to obtain

the time-based probability equation of component reliability as a function

of time-to-failure, and the equation for cost as a function of component

reliability. Note again that the time-based reliability model (e.g., a Weibull

distribution function) usually does not contain engine design parameters.

Once the reliability target is allocated, the method to increase the reliability

of the engine system is usually to re-design the components.

More detailed discussions on reliability allocation and cost–reliability models

are provided by Mettas (2000) and Mettas and Savva (2001). They present

the concept of reliability allocation and optimization, as well as a detailed

cost model as a function of four parameters: (1) feasibility of increasing

a component’s reliability; (2) current reliability; (3) minimum required

reliability; and (4) maximum achievable reliability of the component.

2.13.7 Engine system durability–reliability optimization

model

There are generally two types of calculations for durability and reliability

in engine system design:

1. With a given damage–strength interference model, calculate

reliability.

2. With a given reliability target value (e.g., R = 99%), determine the

corresponding required damage–strength distribution curves. The

Diesel-Xin-02.indd 189 5/5/11 11:44:59 AM



190 Diesel engine system design

system engineer is responsible for optimizing the required system load,

strength, and operation time associated with the reliability. The required

load directly affects both durability and engine performance. The nal

design specication may require several iterations in the design process

by using the damage–strength model.

In engine system design, the second type of calculation is more frequently

encountered. The reliability allocation model in equation 2.22 focuses on the

reliability variation with respect to time and the reliability allocation from

the system level cascaded down to the component level. Equation 2.22 does

not address how to realize the durability design to achieve the component

reliability target because the design parameters such as stress, strength, and

damage are not explicitly involved. Note that the reliability target is related

to two key elements: a failure rate and at a specied lifetime. The reliability

value is dened as ‘one minus the overlapping area under the damage–strength

distribution curves’ (equation 2.16).

The earlier damage–strength model shows that stress, strength, or operation

time (related to damage accumulation) can all be altered in order to reach a

given reliability target. A damage-based durability–reliability optimization

model is developed as follows. The model consists of two types of sub-models:

a deterministic durability damage model and a nondeterministic reliability

damage–strength model. In the model, the system reliability target rst needs

to be allocated to each component to dene the component reliability targets.

The damage of a given failure mode for a given component is calculated with

a deterministic model by using an initial estimated operation time based on a

given strength and stress (load). Such a calculation is repeated with the Monte

Carlo simulation to cover all the statistical variations in order to generate

the probability density function curve of the damage. Then, the component

reliability can be calculated with the damage–strength interference model. If

the calculated reliability value does not match the goal value, the estimated

operation time needs to be iterated until they match. It is possible that the

operation time obtained from this damage–strength model is different from

the lifetime calculated by using the time-based reliability model in the form

of equation 2.14 due to different assumptions used by these models. In order

to make the operation time match with the goal of a desirable reliability/

durability lifetime of the engine, optimization is required to change the stress

or strength for a given component. The durability–reliability optimization

model structure is:

Diesel-Xin-02.indd 190 5/5/11 11:44:59 AM



191Durability and reliability in diesel engine system design

DD

iH

DD

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DD

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ox

i

,

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,,

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,

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DD= DD

+

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ft (ft

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iL

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,

iL,iL

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Dt

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ii

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+

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ii

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ii

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+ …

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=

== =

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Ì

Ô

Ï

Ô

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Ô

Ì

Ô

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Ó

Ô

Ì

Ô

Ì

Ô

Ó

Ô

Ó

Ô

2.23

where t is the operation time, and i represents a component or subsystem such

as cylinder head, piston, exhaust manifold, valvetrain, piston ring, cylinder

liner, crankshaft, bearing, connecting rod, compressor, turbine, DPF, etc. The

unknowns in equation 2.23 are t

i,HCF

, t

i,LCF

, t

i,wear

, …, t

i,other

. Each component

may be subject to multiple failure modes, for example, HCF crack at one

location and LCF crack at another location. For each failure mode, there may

be multiple failure mechanisms to accumulate the damage in the durability

model in 2.23, for example, fatigue, creep, and oxidation. The probability

density functions used in the reliability model in 2.23 can be evaluated by

the Monte Carlo simulation to account for statistical probabilistic variations.

Equation 2.23 can be formulated for each component to cover all the failure

modes and failure mechanisms. The operation time of each component for

the reliability goal is determined as the minimum of all failure modes:

t

i

= min(t

i,HCF

, t

i,LCF

, t

i,wear

, …, t

i,other

) 2.24

The engine system’s operation time for the reliability goal is given by:

t

S

= min (t

1

, t

2

, t

3

, …, t

n

) 2.25

The operation time may not meet the desirable reliability lifetime or durability

life, t

S, goal

. The durability–reliability design optimization is formulated as:

Min

imize component lif

e di

imize component life diimize component lif

ffe

re

nce

(

=1

S

i

n

i

tt

– tt –

i

tt

i

ttttt

tt

S

goal

SS

tt

SS

tt

goal

iS

tt

iS

tt

goal

i

,

2

,,

goal,,goal

iS,,iS

)

s

ubject to

tt =tt

tt

SS

tt =tt

SS

tt

,

SS

,

SS

tt

SS

tt ,tt

SS

tt

goal

,

goal

,,

,

,,

goal,,goal

,

goal,,goal

≥

tt ≥tt

tt

iS

tt ≥tt

iS

tt

= 1,

2,

3, …

,

by changing s

tr

e

tretr

ss

or

s

or s or

tr

ength i

trength itr

n

n de

nn den

si

gn

Ï

Ì

Ô

Ï

Ô

Ï

Ô

Ì

Ô

Ì

Ô

Ó

Ô

Ì

Ô

Ì

Ô

Ó

Ô

Ó

Ô

2.26

Diesel-Xin-02.indd 191 5/5/11 11:45:01 AM



192 Diesel engine system design

By changing system loading or component strength, the system operation

time can be designed on the target requirement. Meanwhile, the lifetime of

the over-designed components is reduced by changing the stress or strength

(if possible) so that all the components may ideally have the same or a

similar lifetime at their own component reliability target with minimum

system cost.

In summary, the durability–reliability optimization model covers the

following key elements for engine system design: balance between the

system and components; balance among operation time, stress, and strength;

failure modes; failure mechanisms; damage accumulation; and probabilistic

distribution.

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