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Design for reliability in hostile environment
Pergamon Press Printed in U.S.A.
Microelectronics and Reliability Vol. 15 Supplement pp. 75-85, 1976
DESIGN F O R R E L I A B I L I T Y IN H O S T I L E E N V I R O N M E N T BY Leo Fiderer Singer Instrumentation Los Angeles, California USA M e c h a n i s m s and e l e c t r o m e c h a n i c a l d e v i c e s a r e o f t e n s u b j e c t to f a i l u r e m e c h a n i s m s d i f f e r e n t f r o m the ones of c o n v e n t i o n a l e l e c t r o n i c c o m p o n e n t s . In t h i s p a p e r , the t h e o r i e s of r e l i a b i l ity, v i b r a t i o n , s t r e n g t h of m a t e r i a l s and p r o b a b i l i s t i c d e s i g n a r e t r a n s l a t e d i n t o p r a c t i c a l g u i d e l i n e s to p r e d i c t t h e f a i l u r e r a t e s a n d t h e r e l i a b i l i t y of s u c h d e v i c e s when s u b j e c t e d to v a r i ous t y p e s of h o s t i l e e n v i r o n m e n t . Introduction M a n y e l e c t r o n i c s y s t e m s i n t e r f a c e with e l e c t r o - m e c h a n i c a l d e vices or m e c h a n i s m s , such as solenoids or special s w i t c h e s . T h e s e d e v i c e s a r e e x p e c t e d to w i t h s t a n d the h a r m f u l e f f e c t s of c o l d and h o t temperature extremes, shock, vibration, acoustic noise, wear, corros i v e a t m o s p h e r e , and o t h e r h o s t i l e e n v i r o n m e n t s . T h e r e l i a b i l i t y of t h e e n t i r e s y s t e m o f t e n d e p e n d s on t h e p r o p e r m e c h a n i c a l f u n c t i o n of t h e s e devices. C o n v e n t i o n a l h a n d b o o k s t h a t a r e u s e d to c a l c u l a t e f a i l u r e r a t e s and M T B F - f i g u r e s , s u c h as M I L - H D B K - 2 1 7 B , a r e e l e c t r o n i c c i r c u i t o r i e n t e d a n d d e a l p r i m a r i l y with e l e c t r i c a l s t r e s s e s of e l e c t r o n i c c o m ponents, such as r a t e d voltage v e r s u s applied voltage. M e c h a n i c a l s t r e s s e s a r e d e a l t with by u s i n g g e n e r a l i z e d a p p l i c a t i o n f a c t o r s f o r c e r tain environment categories. F i g u r e 1 i l l u s t r a t e s a t y p i c a l t a b l e of s u c h environment application factors, taken from MIL-HDBK-217B. H o w e v e r , t h e s e a p p l i c a t i o n f a c t o r s do n o t a l w a y s r e f l e c t t h e t r u e s t r e s s e s e x p e r i e n c e d by t h e d e v i c e in u s e . A c t u a l m e c h a n i c a l s t r e s s e s a r e a f u n c t i o n of v e h i c l e s i z e and s p e e d , c o m p o n e n t m o u n t i n g and g e o m e t r y , r e s o n a n c e s and m a n y o t h e r p a r a m e t e r s . E a c h one of t h e s e p a r a m e t e r s is a r a n d o m v a r i a b l e . C o n v e n t i o n a l m e t h o d s of s t r e s s c a l c u l a t i o n s a r e i n s u f f i c i e n t to p r o v i d e r e a l i s t i c a n s w e r s a b o u t t h e p r o b a b i l i t y of f a i l u r e . P r o b a b i l i s t i c d e s i g n t e c h n i q u e s w i l l y i e l d b e t t e r r e s u l t s .
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L
rE, INVIRONVIIIITAL FACIDR BASED ON D~VIROIa'~AL SERVICE CONDITION E n v ~ t
'
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Inhabited
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FIGURE 1 E x c e r p t f r o m M I L - H D B K - 217B
Ii
Vol. 15, Supp.
Design for Reliability in Hostile Environment
Probabilistic Design A recently developed design philosophy, named "probabilistic d e s i g n " , m a k e s a d i s t i n c t i o n b e t w e e n d e t e r m i n i s t i c and r a n d o m v a r i a b l e s ( s e e r e f e r e n c e l). A d e t e r m i n i s t i c v a r i a b l e has a definite r e a l value that will always be the s a m e u n d e r a l l c o n d i t i o n s . F o r e x a m p l e , a c e r t a i n m o d e l a u t o m o b i l e h a s f o u r w h e e l s , and t h e n u m b e r " f o u r " is the d e t e r m i n i s t i c d e sign variable. On the o t h e r h a n d , t i r e p r e s s u r e , a v e r a g e s p e e d , o r b a t t e r y s t a t e of c h a n g e a r e a l l c o n s i d e r e d r a n d o m v a r i a b l e s , s i n c e with a l a r g e n u m b e r of a u t o m o b i l e s of the s a m e m o d e l , a c o n s i d e r a b l e v a r i a t i o n in t h e v a l u e of t h e s e v a r i a b l e s w i l l be e n c o u n t e r e d . E a c h o n e of t h e s e r a n d o m v a r i a b l e s is d e f i n e d by two n u m b e r s , n a m e l y the m e a n v a l u e and i t s s t a n d a r d d e v i a t i o n . M o s t p h y s i c a l p a r a m e t e r s e n c o u n t e r e d in m e c h a n i c a l s y s t e m s a r e r a n d o m v a r i a b l e s , s u c h a s t e n s i l e s t r e n g t h , m o d u l u s of e l a s t i c i t y , power supply voltages, p h y s i c a l d i m e n s i o n s and e n v i r o n m e n tal temperatures. In s o m e i n s t a n c e s , with p r e c i s i o n m a c h i n e d p a r t s , t h e t o l e r a n c e s p r e a d is s o n a r r o w t h a t t h e s t a n d a r d d e v i a t i o n is c o n s i d e r e d n e g l i g i b l e c o m p a r e d to t h e m e a n v a l u e of the d i m e n s i o n , and the d i m e n s i o n is t r e a t e d as d e t e r m i n i s t i c v a r i a b l e to s i m p l i f y the c o m p u t a t i o n s . H o w e v e r , in m o s t c a s e s , t o l e r a n c e s in d i m e n s i o n s , r e s i s t o r v a l u e s o r s u p p l y v o l t a g e s h a v e a d e f i n i t e e f f e c t on t h e r e l i a b i l i t y of t h e s y s t e m . H o s t i l e e n v i r o n m e n t , with i t s wide r a n g e of p o s s i b l e v a l u e s , is m o s t c e r t a i n l y a r a n d o m v a r i a b l e , and i t s e f f e c t s c a n b e s t be e v a l u a t e d by p r o b a b i l i s t i c m e t h o d s . R a n d o m v a r i a b l e s can i n t e r a c t with e a c h o t h e r o r with d e t e r m i n i s t i c v a r i a b l e s by a d d i t i o n , s u b t r a c t i o n , m u l t i p l i c a t i o n o r d i v i s i o n . The r e s u l t of t h e s e o p e r a t i o n s , w h e r e at l e a s t one of t h e o p e r a n d s is a r a n d o m v a r i a b l e , is a l s o a r a n d o m v a r i a b l e with i t s own m e a n v a l u e and s t a n d a r d d e v i a t i o n . A d e t a i l e d d i s c u s s i o n of the m a t h e m a t i c s of r a n d o m v a r i a b l e s is f u r n i s h e d in R e f e r e n c e s (1) and (2). T h e b a s i c p r i n c i p l e s and d e f i n i t i o n s , a s t h e y a r e u s e d in t h e p r o b a b i l i s t i c d e s i g n m e t h o d s , a r e summarized below. T h e i l l u s t r a t i o n in F i g u r e 2 s h o w s the p r o b a b i l i t y d i s t r i b u t i o n c u r v e f o r a r a n d o m v a r i a b l e X, a l s o c a l l e d p r o b a b i l i t y d e n s i t y f u n d t i o n . ( P D F ) . T h e m e a n v a l u e , p_, is t h e a v e r a g e of a l l t h e v a l u e s t h a t t h e variable X may assume. ItXis often, but n o t n e c e s s a r i l y , t h e v a l u e of X at t h e m a x i m u m p r o b a b i l i t y d e n s i t y , and is d e f i n e d by t h e e q u a t i o n : co
,--co
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~
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FIGURE 2 Density Function for Random with Normal Distribution
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TABLE I MEAN VALUE
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Basic Operations with Random Variables with Normal Distribution
+
Vol. 15, Supp.
Design for Reliability in Hostile Environment
T h e s t a n d a r d d e v i a t i o n , o - , is a m e a s u r e of t h e d i s p e r s i o n o f a l l t h e p o s s i b l e v a l u e s o f X. A l X r g e v a l u e o f c~ i n d i c a t e s a w i de s p r e a d x b e t w e e n t h e h i g h and l ow v a l u e s of t h e r a n d o m v a r i a b l e . The standard d e v i a t i o n is d e f i n e d by t h e e q u a t i o n ,
~x =~(~- ~x)2f(x)dx "~ - C O
A t h i r d p a r a m e t e r , s o m e t i m e s u s e d in p r o b a b i l i s t i c d e s i g n a s a m e a s u r e o f a r a n d o m v a r i a b l e , is t h e c o e f f i c i e n t of v a r i a t i o n . This is the d i m e n s i o n l e s s r a t i o b e t w e e n t h e s t a n d a r d d e v i a t i o n and t h e m e a n v a l u e . :
provided,
x
,
of c o u r s e ,
that the m e a n value is not z e r o .
T h e i l l u s t r a t i o n in F i g u r e 2 s h o w s a G a u s s i a n o r " n o r m a l " d i s tribution curve, defined by the equation,
i p(x)
=
~ ~2~)
- (X-~x) 2/20 .2 e
A l t h o u g h t h e r e a r e r a n d o m v a r i a b l e s with a v a r i e t y of s h a p e s f o r d i s t r i b u t i o n c u r v e s , m a n y a c t u a l p a r a m e t e r s m a y be a p p r o x i m a t e d b y t h e d i s p e r s i o n of a n o r m a l d i s t r i b u t i o n , s o t h a t in p r a c t i c a l c a l c u l a t i o n s a n o r mal distribution may be assumed without excessive error. R a n d o m v a r i a b l e s w i t h n o r m a l d i s t r i b u t i o n c a n be e a s i l y a l g e b r a ically manipulated. S o m e of t h e b a s i c o p e r a t i o n s a r e s h o w n in T a b l e I, w h e r e X an d Y a r e t h e r a n d o m v a r i a b l e o p e r a n d s , and Z i s t h e r a n d o m variable result. A n o t h e r f e a t u r e of G a u s s i a n d i s t r i b u t i o n c u r v e s is t h e r e a d y availability of probability tables defining the area under the curve bet w e e n g i v e n v a l u e s of s t a n d a r d d e v i a t i o n . T h e s e t a b l e s a r e u s e f u l f o r t h e c a l c u l a t i o n o f r e l i a b i l i t y a s s h o w n in t h e l a t e r s e c t i o n s of t h i s p a p e r
Reliabilit,y versus "Factor of Safety" In c o n v e n t i o n a l s t r e s s c a l c u l a t i o n s , t h e s t r e n g t h of t h e m a t e r i a l , as l i s t e d in h a n d b o o k s , i s c o m p a r e d w i t h t h e w o r k i n g s t r e s s i n d u c e d by t h e e x p e c t e d l o a d . If t h e s t r e n g t h of t h e m a t e r i a l e x c e e d s t h e w o r k i n g s t r e s s b y a s u b s t a n t i a l r a t i o , it i s a s s u m e d t h a t a c o m f o r t a b l e f a c t o r of s a f e t y e x i s t s . It is d e f i n e d a s f o l l o w s : strength F a c t o r of S a f e t y = FS = l o a d s t r e s s
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the goal is a factor of safety of I. 5 or more.
F o r e x a m p l e , if a m i s s i l e c o m p o n e n t , m a d e of 7075 a l u m i n u m a l l o y , w i t h a l i s t e d t e n s i l e s t r e n g t h of 70, 000 p s i i s s u b j e c t e d to an a n t i c i p a t e d l o a d s t r e s s o f 4 2,000 p s i d u r i n g l a u n c h , c o n v e n t i o n a l a n a l y s i s w o u l d a s s u m e t h a t t h e f a c t o r of s a f e t y is 70, 000 p s i 1. 67 FS = 42, 000 p s i = P r o b a b i l i s t i c d e s i g n , h o w e v e r , w o u l d c o n s i d e r 70, 000 p s i and 42, 000 p s i as t h e m e a n v a l u e s of t h e s t r e n g t h and t h e l o a d , a n d w oul d a s k t h e q u e s t i o n : " H o w s a f e a r e we r e a l l y ? " T h e a n s w e r w oul d p r i m a r i l y d e p e n d on t h e v a l u e of t h e s t a n d a r d d e v i a t i o n f o r e a c h o f t h e s e r a n d o m variables. F o r m a n y m e t a l s , t h e c o e f f i c i e n t of v a r i a t i o n of s t r e n g t h is 0 . 0 8 , s o t h a t t h e r a n d o m v a r i a b l e of s t r e n g t h i s g i v e n b y Ps : 70, 000 ps i ,
o-s = 0 . 0 8 Ps = 5, 600 p s i
The load stress is a function of the acceleration during launch, the mass supported by the component, its geometry and dimensional tolerances. Every one of these parameters is a random variable, so that a considerable spread between high and low values can be exPected. Let us assume that in this example, the calculation of the load stress from these random variables resulted in a coefficient of variation of 0.2, and P L : 42, 000 p s i ,
crL = 0 . 2 P L = 8, 400 p s i .
If we p l o t t h e d i s t r i b u t i o n c u r v e s of b o t h r a n d o m v a r i a b l e s , s t r e n g t h an d l o a d , we find a r e g i o n of o v e r l a p in w h i c h t h e l o a d s t r e s s would exceed the strength. T h e a r e a of t h i s o v e r l a p is t h e p r o b a b i l i t y of f a i l u r e . In o r d e r t o c a l c u l a t e t h e r e l i a b i l i t y o f t h i s l o a d / s t r e n g t h c o m b i n a t i o n , we d e f i n e t h e d i f f e r e n c e b e t w e e n s t r e n g t h and l o a d as a n e w r a n d o m v a r i a b l e D: D:S-L
We e s t a b l i s h t h e m e a n v a l u e and s t a n d a r d d e v i a t i o n of t h i s n e w v a r i a b l e a c c o r d i n g to t h e o p e r a t i o n s i n d i c a t e d in T a b l e h PD = ~ s -
PL = 28,000 psi;
CrD= ~
+ 2=
10,100 p s i
N e x t , a p l o t of t h e n o r m a l d i s t r i b u t i o n c u r v e f o r t h i s r a n d o m v a r i a b l e is e s t a b l i s h e d a s s h o w n in F i g u r e 4. T h e a r e a u n d e r t h e n e g a t i v e p o r t i o n of t h i s c u r v e i s t h e p r o b a b i l i t y of f a i l u r e , and t h e p o s i t i v e p o r t i o n is t h e p r o b a b i l i t y of s u c c e s s , which is a l s o the r e l i a b i l i t y . T o find t h e n o r m a l i z e d v a l u e of t h e c o o r d i n a t e at t h i s c r o s s - o v e r l i n e b e t w e e n s u c c e s s and f a i l u r e , we e s t a b l i s h t h e r a t i o :
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~L /cL)
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/Is
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look
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PDI~D
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: 28, 0 0 0 / 1 0 , 1 0 0 : 2 . 7 7
From a table of normal distribution, If, we find the values of "Probability "Probability of Failure".
similar to the one shown in Table of Success" P(s) and P(f), the
F r o m t h e t a b l e , we find t h a t t h e v a l u e s f o r P ( s ) and P(f) c o r r e s p o n d i n g to 2.77cr a r e 0. 9972 and 0. 0028 r e s p e c t i v e l y .
Thus, with the probabilistic design concept, instead of a questionable factor of safety, a reliability of 99.72% for the part during the launch was established. Random
Vibration
Environment
By definition, random vibration is a random variable. In many equipment specifications for airborne devices, this type of vibration is now indicated instead of sinusoidal vibration environment. The environment is usually specified in the form of an "acceleration spectral density curve", sometimes also called "power spectral density curve", such as the one shown in Figure 5. In this case, the mean value is zero and the standard deviation is represented by the square root of the area under the curve. The standard deviation is in this case called RMS - (Root Mean Square) acceleration, expressed in gravity units as multiples of g. T w o a s p e c t s of t h e e f f e c t of t h e r a n d o m v i b r a t i o n e n v i r o n m e n t h a v e to b e c o n s i d e r e d f o r t h e r e l i a b i l i t y of e l e c t r o - m e c h a n i c a l d e v i c e s . O n e p r o b l e m is t h e o n e of f a t i g u e , w h e r e b y a s t r u c t u r a l m e m b e r m a y f a i l as a r e s u l t of r e p e a t e d s t r e s s r e v e r s a l s i n d u c e d by v i b r a t i o n . T h i s p r o b l e m h a s b e e n d i s c u s s e d by t h e a u t h o r in a p a p e r p r e s e n t e d at t h e 1975 C a n a d i a n SR E S y m p o s i u m and is e x t e n s i v e l y c o v e r e d in R e f e r e n c e s 3, 4 an d 5.
Another problem, not less significant, is the ability of a spring loaded electrical contact to maintain electrical continuity during vibration. The same applies to solenoids, relays and magnetically energized actuators. Probabilistic techniques of analysis are helpful in obtaining the reliability of these devices, too. M a n y of t h e s e d e v i c e s c a n be r e p r e s e n t e d by a s i n g l e - d e g r e e - o f f r e e d o m s y s t e m f o r t h e p u r p o s e s of v i b r a t i o n a n a l y s i s , s u c h as t h e o n e in t h e d i a g r a m in F i g u r e 6. In t h i s c a s e , t h e s p r i n g i s m a i n t a i n e d in a c o m p r e s s e d c o n d i t i o n and f o r c e s t he c o n t a c t , r e p r e s e n t e d b y t h e m a s s M, a g a i n s t t h e f r a m e , w h i c h in t h i s c a s e is p a r t of t h e m o v i n g f o u n d a t i o n . As l o n g as t h e p r o d u c t of t h e m a s s t i m e s t h e a c c e l e r a t i o n d o e s n o t e x c e e d t h e f o r c e in t h e c o m p r e s s e d s p r i n g , c o n t a c t w i l l b e m a i n t a i n e d . T o d e t e r m i n e t h e p r o b a b i l i t y of c o n t a c t b e i n g m a i n t a i n e d , we f i r s t e s t a b l i s h t h e v a l u e f o r RMS a c c e l e r a t i o n b y t a k i n g t h e s q u a r e r o o t of t h e area under the acceleration spectral density curve. This random variable
Vol. 15, Supp.
Design for Reliability in Hostile Environment
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Analysis.
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is multiplied by the mass of the contact (another random variable) by the formula from Table I. Note that the mean value of the force random variable is zero in this case. Next we compare this variable with the force of the compressed spring by the method shown in the preceding section of strength versus load stress. The area of overlap is the probability of loss of contact and the reliability is again obtained by the equation
R = P(s) = 1 - P(f) F o r e x a m p l e , i f w e o b t a i n a v a l u e f o r P ( f ) = 0. 000015, t h e n t h e r e l i a b i l i t y w o u l d b e 0. 9 9 9 9 8 5 , m e a n i n g t h a t d u r i n g a n y t i m e i n t e r v a l o f s u f f i c i e n t l e n g t h , c o n t a c t w o u l d b e m a i n t a i n e d 99. 9 9 8 5 % of t h e t i m e . Tolerances
and
Simplifications
In many instances, the value of the standard deviation in a parameter is not specified. A maximum and minimum value is mentioned instead. Without loss of too much accuracy, the following simplifications can be made for use in the random variable calculations.
~ x =' ( X m a x + X m i n ) / 2 crx = ( X m a x - X m i n ) / 6 In the normal distribution curve, 99.74% of the area under the curve lies between the values of (/~ - 3(y) and (g + 3o'), so that the error would not be significant. The same procedure can also be applied to the use of dimensional tolerances with fabricated parts. The calculations for the values of standard deviations of products and quotient of random variables according to Table I can also be simplified. If the coefficient of variation C V is less than 0. 2, a fair approximation can be obtained as follows:
Multiplication:
%y
+.y%
Divis ion: '22
2 2 + #y~x
Vol. 15, Supp.
Design for Reliability in Hostile Environment
REFERENCES
New
i. York:
E.B. Haugen, Probabilistic Approaches John Wiley (1968)
to Desi6n.
2. E.B. Haugen and P. H. Wirsching, Probabilistic Design, A Realistic Look at Risk and Reliability in Engineering, Machine Design, Series of 5 Articles April 1975 through June 1975, Vol. 47, No. 9 through 14
3. L. F i d e r e r , Dynamic Environment F a c t o r s in Determinin~ Electronic Assefnbl~ Reliability, P r o c e e d i n g s of the 1975 Canadian SRE Reliability Symposium, Ottawa, 10 May 1975 4. A. Sorensen, J r . , Product Reliability and Random F a t i ~ e . IEEE Transactions on Reliability, Vol. R-20, No. 4, pp. 244-248, November 1971 5. E . B . Haugen and P. H. Wirsching, Probabilistic Design Alternatives to M i n e r ' s Cumulative Damage Rule, Proceedings of the Annual Reliability and Maintainability Symposium, Philadelphia, Pennsylvania, 1973, Index Serial No. 1102 6. MIL-HDBK-217B, U. S. Department of Defense Reliability P r e d i c t i o n of Electronic Equipment. 20 September 1974
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Microelectronics and Reliability Vol. 15 Supplement pp. 75-85, 1976
DESIGN F O R R E L I A B I L I T Y IN H O S T I L E E N V I R O N M E N T BY Leo Fiderer Singer Instrumentation Los Angeles, California USA M e c h a n i s m s and e l e c t r o m e c h a n i c a l d e v i c e s a r e o f t e n s u b j e c t to f a i l u r e m e c h a n i s m s d i f f e r e n t f r o m the ones of c o n v e n t i o n a l e l e c t r o n i c c o m p o n e n t s . In t h i s p a p e r , the t h e o r i e s of r e l i a b i l ity, v i b r a t i o n , s t r e n g t h of m a t e r i a l s and p r o b a b i l i s t i c d e s i g n a r e t r a n s l a t e d i n t o p r a c t i c a l g u i d e l i n e s to p r e d i c t t h e f a i l u r e r a t e s a n d t h e r e l i a b i l i t y of s u c h d e v i c e s when s u b j e c t e d to v a r i ous t y p e s of h o s t i l e e n v i r o n m e n t . Introduction M a n y e l e c t r o n i c s y s t e m s i n t e r f a c e with e l e c t r o - m e c h a n i c a l d e vices or m e c h a n i s m s , such as solenoids or special s w i t c h e s . T h e s e d e v i c e s a r e e x p e c t e d to w i t h s t a n d the h a r m f u l e f f e c t s of c o l d and h o t temperature extremes, shock, vibration, acoustic noise, wear, corros i v e a t m o s p h e r e , and o t h e r h o s t i l e e n v i r o n m e n t s . T h e r e l i a b i l i t y of t h e e n t i r e s y s t e m o f t e n d e p e n d s on t h e p r o p e r m e c h a n i c a l f u n c t i o n of t h e s e devices. C o n v e n t i o n a l h a n d b o o k s t h a t a r e u s e d to c a l c u l a t e f a i l u r e r a t e s and M T B F - f i g u r e s , s u c h as M I L - H D B K - 2 1 7 B , a r e e l e c t r o n i c c i r c u i t o r i e n t e d a n d d e a l p r i m a r i l y with e l e c t r i c a l s t r e s s e s of e l e c t r o n i c c o m ponents, such as r a t e d voltage v e r s u s applied voltage. M e c h a n i c a l s t r e s s e s a r e d e a l t with by u s i n g g e n e r a l i z e d a p p l i c a t i o n f a c t o r s f o r c e r tain environment categories. F i g u r e 1 i l l u s t r a t e s a t y p i c a l t a b l e of s u c h environment application factors, taken from MIL-HDBK-217B. H o w e v e r , t h e s e a p p l i c a t i o n f a c t o r s do n o t a l w a y s r e f l e c t t h e t r u e s t r e s s e s e x p e r i e n c e d by t h e d e v i c e in u s e . A c t u a l m e c h a n i c a l s t r e s s e s a r e a f u n c t i o n of v e h i c l e s i z e and s p e e d , c o m p o n e n t m o u n t i n g and g e o m e t r y , r e s o n a n c e s and m a n y o t h e r p a r a m e t e r s . E a c h one of t h e s e p a r a m e t e r s is a r a n d o m v a r i a b l e . C o n v e n t i o n a l m e t h o d s of s t r e s s c a l c u l a t i o n s a r e i n s u f f i c i e n t to p r o v i d e r e a l i s t i c a n s w e r s a b o u t t h e p r o b a b i l i t y of f a i l u r e . P r o b a b i l i s t i c d e s i g n t e c h n i q u e s w i l l y i e l d b e t t e r r e s u l t s .
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L
rE, INVIRONVIIIITAL FACIDR BASED ON D~VIROIa'~AL SERVICE CONDITION E n v ~ t
'
S~t.~z
rE
G' B
0.2
sF
0.2
%
Z.o
AI
4.0
Naval, S h e l t e r e d
NS
~.0
Gbsm/nd, Mobile
GM
~.0
Naval, Unsheltered
NU
5.0
~me,
AU
6,0
ML
i0.0
Gax~, Benign s~ace
nx~t Fixed
~ e ,
Inhabited
t~ite~
Missile, Imam~.h
FIGURE 1 E x c e r p t f r o m M I L - H D B K - 217B
Ii
Vol. 15, Supp.
Design for Reliability in Hostile Environment
Probabilistic Design A recently developed design philosophy, named "probabilistic d e s i g n " , m a k e s a d i s t i n c t i o n b e t w e e n d e t e r m i n i s t i c and r a n d o m v a r i a b l e s ( s e e r e f e r e n c e l). A d e t e r m i n i s t i c v a r i a b l e has a definite r e a l value that will always be the s a m e u n d e r a l l c o n d i t i o n s . F o r e x a m p l e , a c e r t a i n m o d e l a u t o m o b i l e h a s f o u r w h e e l s , and t h e n u m b e r " f o u r " is the d e t e r m i n i s t i c d e sign variable. On the o t h e r h a n d , t i r e p r e s s u r e , a v e r a g e s p e e d , o r b a t t e r y s t a t e of c h a n g e a r e a l l c o n s i d e r e d r a n d o m v a r i a b l e s , s i n c e with a l a r g e n u m b e r of a u t o m o b i l e s of the s a m e m o d e l , a c o n s i d e r a b l e v a r i a t i o n in t h e v a l u e of t h e s e v a r i a b l e s w i l l be e n c o u n t e r e d . E a c h o n e of t h e s e r a n d o m v a r i a b l e s is d e f i n e d by two n u m b e r s , n a m e l y the m e a n v a l u e and i t s s t a n d a r d d e v i a t i o n . M o s t p h y s i c a l p a r a m e t e r s e n c o u n t e r e d in m e c h a n i c a l s y s t e m s a r e r a n d o m v a r i a b l e s , s u c h a s t e n s i l e s t r e n g t h , m o d u l u s of e l a s t i c i t y , power supply voltages, p h y s i c a l d i m e n s i o n s and e n v i r o n m e n tal temperatures. In s o m e i n s t a n c e s , with p r e c i s i o n m a c h i n e d p a r t s , t h e t o l e r a n c e s p r e a d is s o n a r r o w t h a t t h e s t a n d a r d d e v i a t i o n is c o n s i d e r e d n e g l i g i b l e c o m p a r e d to t h e m e a n v a l u e of the d i m e n s i o n , and the d i m e n s i o n is t r e a t e d as d e t e r m i n i s t i c v a r i a b l e to s i m p l i f y the c o m p u t a t i o n s . H o w e v e r , in m o s t c a s e s , t o l e r a n c e s in d i m e n s i o n s , r e s i s t o r v a l u e s o r s u p p l y v o l t a g e s h a v e a d e f i n i t e e f f e c t on t h e r e l i a b i l i t y of t h e s y s t e m . H o s t i l e e n v i r o n m e n t , with i t s wide r a n g e of p o s s i b l e v a l u e s , is m o s t c e r t a i n l y a r a n d o m v a r i a b l e , and i t s e f f e c t s c a n b e s t be e v a l u a t e d by p r o b a b i l i s t i c m e t h o d s . R a n d o m v a r i a b l e s can i n t e r a c t with e a c h o t h e r o r with d e t e r m i n i s t i c v a r i a b l e s by a d d i t i o n , s u b t r a c t i o n , m u l t i p l i c a t i o n o r d i v i s i o n . The r e s u l t of t h e s e o p e r a t i o n s , w h e r e at l e a s t one of t h e o p e r a n d s is a r a n d o m v a r i a b l e , is a l s o a r a n d o m v a r i a b l e with i t s own m e a n v a l u e and s t a n d a r d d e v i a t i o n . A d e t a i l e d d i s c u s s i o n of the m a t h e m a t i c s of r a n d o m v a r i a b l e s is f u r n i s h e d in R e f e r e n c e s (1) and (2). T h e b a s i c p r i n c i p l e s and d e f i n i t i o n s , a s t h e y a r e u s e d in t h e p r o b a b i l i s t i c d e s i g n m e t h o d s , a r e summarized below. T h e i l l u s t r a t i o n in F i g u r e 2 s h o w s the p r o b a b i l i t y d i s t r i b u t i o n c u r v e f o r a r a n d o m v a r i a b l e X, a l s o c a l l e d p r o b a b i l i t y d e n s i t y f u n d t i o n . ( P D F ) . T h e m e a n v a l u e , p_, is t h e a v e r a g e of a l l t h e v a l u e s t h a t t h e variable X may assume. ItXis often, but n o t n e c e s s a r i l y , t h e v a l u e of X at t h e m a x i m u m p r o b a b i l i t y d e n s i t y , and is d e f i n e d by t h e e q u a t i o n : co
,--co
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~
X
0,~"
/ \
0.40,3
/
O.Z 0.1
X
J Probability
FIGURE 2 Density Function for Random with Normal Distribution
Variable
TABLE I MEAN VALUE
ALGEBRAIC FUNCT I ON
STANDARD DEVIATION
~dition: Z=X+Y
Oz=V4÷
~z=~x+~
Subtraction: Z = X - Y
~z
= Bx
- ~y
Multiplication: Z = X Y
Bz = Bx By
'
22
22
22
Division:
~z = )'t/Fy
~z - ~ y
Basic Operations with Random Variables with Normal Distribution
+
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Design for Reliability in Hostile Environment
T h e s t a n d a r d d e v i a t i o n , o - , is a m e a s u r e of t h e d i s p e r s i o n o f a l l t h e p o s s i b l e v a l u e s o f X. A l X r g e v a l u e o f c~ i n d i c a t e s a w i de s p r e a d x b e t w e e n t h e h i g h and l ow v a l u e s of t h e r a n d o m v a r i a b l e . The standard d e v i a t i o n is d e f i n e d by t h e e q u a t i o n ,
~x =~(~- ~x)2f(x)dx "~ - C O
A t h i r d p a r a m e t e r , s o m e t i m e s u s e d in p r o b a b i l i s t i c d e s i g n a s a m e a s u r e o f a r a n d o m v a r i a b l e , is t h e c o e f f i c i e n t of v a r i a t i o n . This is the d i m e n s i o n l e s s r a t i o b e t w e e n t h e s t a n d a r d d e v i a t i o n and t h e m e a n v a l u e . :
provided,
x
,
of c o u r s e ,
that the m e a n value is not z e r o .
T h e i l l u s t r a t i o n in F i g u r e 2 s h o w s a G a u s s i a n o r " n o r m a l " d i s tribution curve, defined by the equation,
i p(x)
=
~ ~2~)
- (X-~x) 2/20 .2 e
A l t h o u g h t h e r e a r e r a n d o m v a r i a b l e s with a v a r i e t y of s h a p e s f o r d i s t r i b u t i o n c u r v e s , m a n y a c t u a l p a r a m e t e r s m a y be a p p r o x i m a t e d b y t h e d i s p e r s i o n of a n o r m a l d i s t r i b u t i o n , s o t h a t in p r a c t i c a l c a l c u l a t i o n s a n o r mal distribution may be assumed without excessive error. R a n d o m v a r i a b l e s w i t h n o r m a l d i s t r i b u t i o n c a n be e a s i l y a l g e b r a ically manipulated. S o m e of t h e b a s i c o p e r a t i o n s a r e s h o w n in T a b l e I, w h e r e X an d Y a r e t h e r a n d o m v a r i a b l e o p e r a n d s , and Z i s t h e r a n d o m variable result. A n o t h e r f e a t u r e of G a u s s i a n d i s t r i b u t i o n c u r v e s is t h e r e a d y availability of probability tables defining the area under the curve bet w e e n g i v e n v a l u e s of s t a n d a r d d e v i a t i o n . T h e s e t a b l e s a r e u s e f u l f o r t h e c a l c u l a t i o n o f r e l i a b i l i t y a s s h o w n in t h e l a t e r s e c t i o n s of t h i s p a p e r
Reliabilit,y versus "Factor of Safety" In c o n v e n t i o n a l s t r e s s c a l c u l a t i o n s , t h e s t r e n g t h of t h e m a t e r i a l , as l i s t e d in h a n d b o o k s , i s c o m p a r e d w i t h t h e w o r k i n g s t r e s s i n d u c e d by t h e e x p e c t e d l o a d . If t h e s t r e n g t h of t h e m a t e r i a l e x c e e d s t h e w o r k i n g s t r e s s b y a s u b s t a n t i a l r a t i o , it i s a s s u m e d t h a t a c o m f o r t a b l e f a c t o r of s a f e t y e x i s t s . It is d e f i n e d a s f o l l o w s : strength F a c t o r of S a f e t y = FS = l o a d s t r e s s
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the goal is a factor of safety of I. 5 or more.
F o r e x a m p l e , if a m i s s i l e c o m p o n e n t , m a d e of 7075 a l u m i n u m a l l o y , w i t h a l i s t e d t e n s i l e s t r e n g t h of 70, 000 p s i i s s u b j e c t e d to an a n t i c i p a t e d l o a d s t r e s s o f 4 2,000 p s i d u r i n g l a u n c h , c o n v e n t i o n a l a n a l y s i s w o u l d a s s u m e t h a t t h e f a c t o r of s a f e t y is 70, 000 p s i 1. 67 FS = 42, 000 p s i = P r o b a b i l i s t i c d e s i g n , h o w e v e r , w o u l d c o n s i d e r 70, 000 p s i and 42, 000 p s i as t h e m e a n v a l u e s of t h e s t r e n g t h and t h e l o a d , a n d w oul d a s k t h e q u e s t i o n : " H o w s a f e a r e we r e a l l y ? " T h e a n s w e r w oul d p r i m a r i l y d e p e n d on t h e v a l u e of t h e s t a n d a r d d e v i a t i o n f o r e a c h o f t h e s e r a n d o m variables. F o r m a n y m e t a l s , t h e c o e f f i c i e n t of v a r i a t i o n of s t r e n g t h is 0 . 0 8 , s o t h a t t h e r a n d o m v a r i a b l e of s t r e n g t h i s g i v e n b y Ps : 70, 000 ps i ,
o-s = 0 . 0 8 Ps = 5, 600 p s i
The load stress is a function of the acceleration during launch, the mass supported by the component, its geometry and dimensional tolerances. Every one of these parameters is a random variable, so that a considerable spread between high and low values can be exPected. Let us assume that in this example, the calculation of the load stress from these random variables resulted in a coefficient of variation of 0.2, and P L : 42, 000 p s i ,
crL = 0 . 2 P L = 8, 400 p s i .
If we p l o t t h e d i s t r i b u t i o n c u r v e s of b o t h r a n d o m v a r i a b l e s , s t r e n g t h an d l o a d , we find a r e g i o n of o v e r l a p in w h i c h t h e l o a d s t r e s s would exceed the strength. T h e a r e a of t h i s o v e r l a p is t h e p r o b a b i l i t y of f a i l u r e . In o r d e r t o c a l c u l a t e t h e r e l i a b i l i t y o f t h i s l o a d / s t r e n g t h c o m b i n a t i o n , we d e f i n e t h e d i f f e r e n c e b e t w e e n s t r e n g t h and l o a d as a n e w r a n d o m v a r i a b l e D: D:S-L
We e s t a b l i s h t h e m e a n v a l u e and s t a n d a r d d e v i a t i o n of t h i s n e w v a r i a b l e a c c o r d i n g to t h e o p e r a t i o n s i n d i c a t e d in T a b l e h PD = ~ s -
PL = 28,000 psi;
CrD= ~
+ 2=
10,100 p s i
N e x t , a p l o t of t h e n o r m a l d i s t r i b u t i o n c u r v e f o r t h i s r a n d o m v a r i a b l e is e s t a b l i s h e d a s s h o w n in F i g u r e 4. T h e a r e a u n d e r t h e n e g a t i v e p o r t i o n of t h i s c u r v e i s t h e p r o b a b i l i t y of f a i l u r e , and t h e p o s i t i v e p o r t i o n is t h e p r o b a b i l i t y of s u c c e s s , which is a l s o the r e l i a b i l i t y . T o find t h e n o r m a l i z e d v a l u e of t h e c o o r d i n a t e at t h i s c r o s s - o v e r l i n e b e t w e e n s u c c e s s and f a i l u r e , we e s t a b l i s h t h e r a t i o :
Vol. 15, Supp.
~L /cL)
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/Is
/
LOAb
STRENGTH
i
42,ooo
0
70,000 P$~
psi FIGURE 3
D i s t r i b u t i o n C u r v e s f o r S t r e n g t h and L o a d S t r e s s
/a-L) P(~:)
2
/
D-S-I_
JzD
z8,ooO = 2 . 7 T
0
FIGURE Distribution
Curve
for Excess
o"D
4 Strength
over
Load
I
look
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PDI~D
Vol. 15, Supp
: 28, 0 0 0 / 1 0 , 1 0 0 : 2 . 7 7
From a table of normal distribution, If, we find the values of "Probability "Probability of Failure".
similar to the one shown in Table of Success" P(s) and P(f), the
F r o m t h e t a b l e , we find t h a t t h e v a l u e s f o r P ( s ) and P(f) c o r r e s p o n d i n g to 2.77cr a r e 0. 9972 and 0. 0028 r e s p e c t i v e l y .
Thus, with the probabilistic design concept, instead of a questionable factor of safety, a reliability of 99.72% for the part during the launch was established. Random
Vibration
Environment
By definition, random vibration is a random variable. In many equipment specifications for airborne devices, this type of vibration is now indicated instead of sinusoidal vibration environment. The environment is usually specified in the form of an "acceleration spectral density curve", sometimes also called "power spectral density curve", such as the one shown in Figure 5. In this case, the mean value is zero and the standard deviation is represented by the square root of the area under the curve. The standard deviation is in this case called RMS - (Root Mean Square) acceleration, expressed in gravity units as multiples of g. T w o a s p e c t s of t h e e f f e c t of t h e r a n d o m v i b r a t i o n e n v i r o n m e n t h a v e to b e c o n s i d e r e d f o r t h e r e l i a b i l i t y of e l e c t r o - m e c h a n i c a l d e v i c e s . O n e p r o b l e m is t h e o n e of f a t i g u e , w h e r e b y a s t r u c t u r a l m e m b e r m a y f a i l as a r e s u l t of r e p e a t e d s t r e s s r e v e r s a l s i n d u c e d by v i b r a t i o n . T h i s p r o b l e m h a s b e e n d i s c u s s e d by t h e a u t h o r in a p a p e r p r e s e n t e d at t h e 1975 C a n a d i a n SR E S y m p o s i u m and is e x t e n s i v e l y c o v e r e d in R e f e r e n c e s 3, 4 an d 5.
Another problem, not less significant, is the ability of a spring loaded electrical contact to maintain electrical continuity during vibration. The same applies to solenoids, relays and magnetically energized actuators. Probabilistic techniques of analysis are helpful in obtaining the reliability of these devices, too. M a n y of t h e s e d e v i c e s c a n be r e p r e s e n t e d by a s i n g l e - d e g r e e - o f f r e e d o m s y s t e m f o r t h e p u r p o s e s of v i b r a t i o n a n a l y s i s , s u c h as t h e o n e in t h e d i a g r a m in F i g u r e 6. In t h i s c a s e , t h e s p r i n g i s m a i n t a i n e d in a c o m p r e s s e d c o n d i t i o n and f o r c e s t he c o n t a c t , r e p r e s e n t e d b y t h e m a s s M, a g a i n s t t h e f r a m e , w h i c h in t h i s c a s e is p a r t of t h e m o v i n g f o u n d a t i o n . As l o n g as t h e p r o d u c t of t h e m a s s t i m e s t h e a c c e l e r a t i o n d o e s n o t e x c e e d t h e f o r c e in t h e c o m p r e s s e d s p r i n g , c o n t a c t w i l l b e m a i n t a i n e d . T o d e t e r m i n e t h e p r o b a b i l i t y of c o n t a c t b e i n g m a i n t a i n e d , we f i r s t e s t a b l i s h t h e v a l u e f o r RMS a c c e l e r a t i o n b y t a k i n g t h e s q u a r e r o o t of t h e area under the acceleration spectral density curve. This random variable
Vol. 15, Supp.
Design for Reliability in Hostile Environment
tlJ>.
z~N (3-1uP, ,j~J tajuJ o0. ol/)
CURVE SHAPE AE THROUGH AP
'•REQUIRED CURVE
.f 20
I00 IOOO 2 0 0 0 FREQUENCY ( H z )
RANDOM
VIBRATION
Random
ACCELERATION TEST CURVE
POWER SPECTRAL, DENSITY Wo ( G ~ H , , |
,~E AF AG AH
0.02 0.04 0.06 0.1 0
C O M P O SI TE G-RMS MINIMUM
AJ AK
0,2 0 0.30
5.4 7.6 9.3 12.0 16.9 20.7
AL AM AN . . . .
0.40 0.60 I .00 I ,50
23.9 29.3 37.9. 46.4
[
"ENVELOPE
Vibration
83
FIGURE 5 Specification from
MIL-STD-810C.
I .~--C
FRAME FIGURE Schematic
Representation
--I
--]ct F
.6
of Switch Contact for Vibration
Analysis.
84
L. Fiderer
Vol. 15, Supp
is multiplied by the mass of the contact (another random variable) by the formula from Table I. Note that the mean value of the force random variable is zero in this case. Next we compare this variable with the force of the compressed spring by the method shown in the preceding section of strength versus load stress. The area of overlap is the probability of loss of contact and the reliability is again obtained by the equation
R = P(s) = 1 - P(f) F o r e x a m p l e , i f w e o b t a i n a v a l u e f o r P ( f ) = 0. 000015, t h e n t h e r e l i a b i l i t y w o u l d b e 0. 9 9 9 9 8 5 , m e a n i n g t h a t d u r i n g a n y t i m e i n t e r v a l o f s u f f i c i e n t l e n g t h , c o n t a c t w o u l d b e m a i n t a i n e d 99. 9 9 8 5 % of t h e t i m e . Tolerances
and
Simplifications
In many instances, the value of the standard deviation in a parameter is not specified. A maximum and minimum value is mentioned instead. Without loss of too much accuracy, the following simplifications can be made for use in the random variable calculations.
~ x =' ( X m a x + X m i n ) / 2 crx = ( X m a x - X m i n ) / 6 In the normal distribution curve, 99.74% of the area under the curve lies between the values of (/~ - 3(y) and (g + 3o'), so that the error would not be significant. The same procedure can also be applied to the use of dimensional tolerances with fabricated parts. The calculations for the values of standard deviations of products and quotient of random variables according to Table I can also be simplified. If the coefficient of variation C V is less than 0. 2, a fair approximation can be obtained as follows:
Multiplication:
%y
+.y%
Divis ion: '22
2 2 + #y~x
Vol. 15, Supp.
Design for Reliability in Hostile Environment
REFERENCES
New
i. York:
E.B. Haugen, Probabilistic Approaches John Wiley (1968)
to Desi6n.
2. E.B. Haugen and P. H. Wirsching, Probabilistic Design, A Realistic Look at Risk and Reliability in Engineering, Machine Design, Series of 5 Articles April 1975 through June 1975, Vol. 47, No. 9 through 14
3. L. F i d e r e r , Dynamic Environment F a c t o r s in Determinin~ Electronic Assefnbl~ Reliability, P r o c e e d i n g s of the 1975 Canadian SRE Reliability Symposium, Ottawa, 10 May 1975 4. A. Sorensen, J r . , Product Reliability and Random F a t i ~ e . IEEE Transactions on Reliability, Vol. R-20, No. 4, pp. 244-248, November 1971 5. E . B . Haugen and P. H. Wirsching, Probabilistic Design Alternatives to M i n e r ' s Cumulative Damage Rule, Proceedings of the Annual Reliability and Maintainability Symposium, Philadelphia, Pennsylvania, 1973, Index Serial No. 1102 6. MIL-HDBK-217B, U. S. Department of Defense Reliability P r e d i c t i o n of Electronic Equipment. 20 September 1974
85