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Commento
Questa tesi, scritta in inglese, è stata sviluppata per un periodo di circa
7 mesi presso la Fachhocschule di Augsburg in Germania nell’ambito
del progetto Socrates-Erasmus ed ha avuto come oggetto lo studio
della fattibilità di un sistema radar per treni.
Nel 1° capitolo sono state introdotte nozioni teoriche e pratiche
necessarie al miglioramento e allo studio dell’ affidabilità.
In particolare dopo aver introdotto le principali componenti che
caratterizzano l’affidabilità di un sistema , quali : Bathub Hazard
Curve, MTTF,MTBF,disponibilità , si è passati all’ analisi delle
diverse reti di affidabilità :struttura serie , parallelo, standby
ridondante e configurazione K-out-of-n .
Dopo di ciò si è trattato l’uso della tecnica di analisi Fault tree per
sistema con componenti riparabili e utilizzando la tecnica Lambda
Tau.
Si è poi considerata l’affidabilità di un sistema soggetto a causa
comune di fallimento, e si è concluso il capitolo menzionando la
tecnica di ridondanza.
Nel 2° capitolo si è passati al calcolo dell’affidabilità del sistema
composto da tre sensori radar , utilizzando gli strumenti sviluppati nel
primo capitolo;sono stati eseguiti calcoli di affidabilità di R
gt
λ
gt
,MTBF,disponibilità considerado le seguenti situazioni
- Del semplice sistema 2-out-of-3
- Utilizzando la tecnica 1-out-of-2
- Introducendo la condizione di causa comune di fallimento
- Uso di componenti riparabili
- Metodo Lambda Tau
Si è poi introdotta la tecnica FMECA applicata allo studio
dell’affidabilità del singolo sensore radar ,sia dal punto di vista
quantitativo con lo studio della criticità , che dal punto di vista
qualitativo con lo studio della matrice di criticità.
L’ ultimo capitolo è stato dedicato alla progettazione di un sistema d’
antenna per il radar monopulse.
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Nel caso specifico si è progettata e realizzata un’antenna radar Patch
4x4 del tipo monopulse.
Per conferire la caratteristica di monopulse alla patch ci si è serviti di
un ibrido 180°.
La patch è stata progettata (e simulati i relativi diagrammi di
radiazione)utilizzando il software di programmazione MSTRIP40 e
EAGLE, in seguito è stata realizzata in collaborazione con la
SIEMENS.
Sono state poi eseguite le misurazioni dei diagrammi di radiazione
dell’antenna stessa all’ interno di una camera anecoica del laboratorio,
i quali hanno fornito degli ottimi risultati in considerazione del tipo di
antenna.
Alla fine si è introdotta allo scopo di un confronto la realizzazione di
una possibile horn antenna del tipo monopulse.
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CHAPTER 1 - RELIABILITY INTRODUCTION
1.1 Introduction to General Reliability Function
The reliability is defined as the probability that the system will operate
to an agreed level of performance for a specific period , subject to
specified environmental conditions.
-General concepts
Suppose no identical components are under test,after t ,n
f
(t) fail and
n
s
(t) survive .
The reliability function R(t) is defined by
(1.1) R(t) = n
s
(t) / (n
s
(t) + n
f
(t))
Since ns(t) + n
f
(t) =no
The equation becomes
(1.2) R(t) = ns(t) / no
And
(1.3) R(t) + F(t) = 1
Where F(t) is the failure probability at time t.
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By using of few relationship, it is possible to define the istantaneous
failure or hazard rate:
(1.4) λ (t) = -1/R(t) * dR(t)/dt = f(t) / R(t)
Equation (1.4) may be written in the following form
(1.5) -dR(t)/R(t) = λ (t) dt
By integrating both sides of (1.5) over the time range 0 to t, it is
possible to get
t R(t)
Ι λ (t) dt = -Ι 1/R(t) dR(t)
0 t
For the known initial condition that at t=0, R(t)=1 the above integral
expression becomes
t
(1.6) ln R(t) = - Ι λ (t) dt
0
The following general reliability function is obtained from(1.6):
(1.7) R(t) = e
-Ιλ t dt
where λ (t) is th time dependent failure rate, also called the azard rate.
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The above expression can be used to obtain a component reliability
for any known failure time distribution.
1.1.1 Bathtub Hazard Rate Curve
This azard rate curve , shown in fig 1.1 is regarded as a typical hazard
curve, especially when representing the failure behaviour of electronic
componenets.
As shown in figure 1.1 the decreasing hazard rate is sometime s called
the “burn–in period “.
There are also several other names for this period such as debugging
period , infant mortality period , break-in period.
Occurrence of failures during this period is normally attribuited to
design or manufacturing defects .
The constant part of this bathlub hazard rate is called the “useful
period,” which begins just after the infant mortality period and ends
just before the “wear-out period”.
The wear –out period begins when an equipment or component has
aged or bypassed its useful operating life .
Consequently , the number of failures during this time begin to
increase.
Failures that occur randmoly or in another word unpredicadictably.
The hazard rate shown in Figure 1.1 can be represented by the
following function
λ (t) = k λ c t
(c-1)
+(1-k) b t
(b-1)
β e
(β t^b)
for b,c, λ ,β > 0 0<k<1 t>0 and c=0.5 b=1
where b,c = shape parameters
β ,λ = scale parameters
t = time
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λ (t)
Burn –in period Useful life period Wearout period
0 t
Figure1.1 Bathtub curve
1.1.2 Mean Time Between Failures
The most useful measure of performance which does not involve the
period of observations is the mean time between failure (MTBF).
The MTBF of a system may be measured by testing it for a total
period T, during which N fault occur.
Each fault is repaired and the equipment put back on test , the repair
time being excluded from the total test time T .
The observed MTBF is then given by
MTBF = T/N
This observed value is not necessarily the true MTBF since the
equipment is usually observed for only a sample of its total life .
Another way of expressing equipment reliability is the failure rate.
For many electronics system the failure rate is approximately constant
for much of the working life of the equipment .
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Where this is the case :
λ = 1/ MTTF
Howevever , where λ changes with time , more than one parameter
may be required to espress λ as a function of time , and MTBF may
be a more complicated function of λ .
1.1.3 Mean Time To Failure (MTTF)
The MTBF is a measure of reliability for repairable equipment , a
similar measure is useful for components such as thermionic valves ,
resistors , capacitors , transistors , etc…, which are “throw – away “
items that can not be repaired .
The correct for these components is the mean time to failure MTTF.
The expected value E(t), in this case MTTF, of a probability density
function of the continuos random variable time t is given by
E(t)=MTTF=Ι t f(t) dt
Where f(t) is the failure density function.
If a component’s failure times are exponentially distributed , MTTF is
a reciprocal of the constant hazard rate λ :
%
MTTF = Ι t λ e
(-λ t)
dt = 1 / λ
0
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1.1.4 Availability
Availability is the probability that a systemwill operate satisfactorily
at any point in time , where time includes not only operating life but
also active repair time and administrative and logistic time .
The conventional equation for availability is
(1.1.4) A = MTBF / (MTBF) + MTTR )
and also A = 1 - A
Where A is the unavailability and A is the availability , usually
expressed as a percentage .
MTBF is mean time between failures, and MTTR is the mean time to
repair.
MTTR is related to repair hours , while the calculation of MTBF is
related to component operating hours .
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1.2 Reliability Networks
Four kinds of reliabillity networks have been studied :
-Series structure
-Parallel configuraion
-Standby redundancy
-K-out –of-n-configuration
Now will be presented these four structure and later they will be
applicate to the radar system under consideration.
1.2.1 Series Structure
This arrangement reppresents a system whose subsystem or
components form a series network.
If anyone of the subsistem or component fails, the series system
experience an overall system failure.
A typical series system configuration is shown in the following figure
Figure 1.2
R1 R2 R3 Rn
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If the series system component failure are statistically independent,
then the reliability Rs of a series system with nonidentical components
is given by
n
(1.8) Rs = ϑ Ri
i=1
Where n is the number of components or subsystems and Ri is the
reliability of ith component or subsystem.
If the failure times of components are exponentially distributed(i.e.,if
components have constant failure rates), then the ith component
reliability may be written :
(1.9) Ri(t) = e
-λ i t
By substituting (1.9) into (1.8),
n
(1.10) Rs(t) = e
-∑
i=1
λ i t
MTTF is given by
∝
(1.11) MTTF = Ι e
-∑ λ i t
dt = 1 / ∑ λ i
0
The above expression shows that a series system(MTTF) is the
reciprocal of sum of the series network component failure rates.
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1.2.2 Parallel Configuration
This configuration is shown in the following figure:
Figure 1.3 Parallel configuration
This system will fail if and only if all the units in the system
malfunction. The model is based on the assumption that all the system
units are active and load sharing.
In addition it is assumed that the component failures are statistically
independent.
A parallel structure reliability Rp with nonidentical units or component
reliability is given by
n
(1.12) Rp = 1- Π (1- Ri )
i=1
where n is the number of units.
R1
R2
R2
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Ri is the reliability of ith component or subsystem.
If the component failure rates are constant, then by substituing(1.9)
into (1.12),
n
(1.13) Rp = 1- Π (1- e
-λ i t
)
i=1
MTTF is obtained by integrating(1.13) over the interval (0,inf),
∝
(1.14) MTTF = Ι Rp(t) dt
0
For identical components , the above equation reduces to
n
(1.15) MTTF = (1/λ ) Σ 1/i
i=1
1.2.3 Standby Redundancy
This type of redundancy represents a situation with one operating and
n units as standbys.
The standby redundancy arrangement is shown in the following
Figure1.4
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Figure 1.4 Standby Redundancy
Unlike a parallel network where all units in the configuration are
active , the standby units are not active .
The system reliability of the (n+1) unit, which one unit is operating
and n units on the standby mission until the operating unit fails,is
given by
n
(1.16) Rs(t) = Σ ((λ t )
i
e
-λ t
) / i !
i=0
The above equation is true if the following are true:
1.The swithching arrangement is perfect.
2.The units are identical.
3.The units failure rates are constant.
4.The standby units are as good as new.
5.The unit failures are statistically independent.
1
2
n
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In the case of (n+1), nonidentical units whose failure time density
functions are different, the standby redundant system failure density is
given by
t yn yo
(1.17) fst = Ι Ι ....Ι f1(y1)…fn+1(t-yn) dy1..dyn
yn=0 yn-1=0 y1=0
Thus the system reliability can be obtained by integrating fst(t)over the
interval (t,inf) as follows:
∝
(1.18) Rs(t) =Ι fs(t) dt
0
1.2.4 K-out of-N Configuration
This is another form of redundancy.it is used where a specified
number of units must be good for the system success.
The series and parallel configuration in the preceding sections are
special cases of this configuration,that is , k=n and k=1, respectively.
Reliability of this type of configuration is obtained by applying the
binomial distribution.
The system reliabillity for k-out-of-n number of independent and
identical units is given by:
n
(1.19) Rk/n = Σ (n i) R
i
(1-R)
n-i
i=k
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