Miners Fatigue Rainflow RevD2

8/12/2019 Miners Fatigue Rainflow RevD2

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MINER’S CUMULATIVE DAMAGE VIA RAINFLOW CYCLE COUNTINGRevision D

By Tom IrvineEmai! "om#vi$ra"ion%a"a&'om

Fe$r(ary )* +,)+

Vibration fatigue calculations are “ballpark” calculations given uncertainties in S-Ncurves, stress concentration factors and other variables. Perhaps the best that can beexpected is to calculate the accumulated fatigue to the correct “order-of-magnitude.”

In"ro%('"ion

T-is e.am/e is an innova"ion (/on a simiar /ro$em in Re0eren'es ) an% +& I" (ses amore 'onserva"ive me"-o% "-an "-a" in Re0eren'e +&

Consi%er a /o1er s(//y mo(n"e% on a $ra'2e" as s-o1n in Fi3(re )&

Fi3(re )&

)

4o1er

So%er

Termina

A(min(m

Bra'2e"

5&6 in

7&7 in

+&, in

,&+7 in



T-e mo%e /arame"ers are

4o1er S(//y MassM 8 ,&55 $m8 ,&,,))5 $0se'9+:in

Bra'2e" Ma"eria A(min(m aoy ;,;)<T;

Mass Densi"y =8,&) $m:in9>

Eas"i' Mo%((s E8 )&,e?,6 $0:in9+

Vis'o(s Dam/in3Ra"io

,&,7

T-e area momen" o0 iner"ia o0 "-e $eam 'ross<se'"ion I is

3h b12

1I = @)

[ ][ ]3in25.0in0.212

1I = @+

4in0026.0I = @>

T-e s"i00ness EI is

[ ] 4in0026.02^in/lbf 07+1.0eEI =

@5

2inlbf 04e60.2EI +=

@7

T-e mass /er en3"- o0 "-e $eam* e.'(%in3 "-e /o1er s(//y* is

[ ][ ]in25.0in0.2in/lbm1.0 3=ρ @;

in/lbm05.0=ρ @6

+



T-e $eam mass is

[ ][ ]in5.53^in/lbm05.0L =ρ

@

sec^2/inlbf 0.000712lbm275.0L ==ρ @

Mo%e "-e sys"em as a sin3e<%e3ree<o0<0ree%om sys"em s($e'"e% "o $ase in/(" ass-o1n in Fi3(re +&

Fi3(re +&

T-e na"(ra 0re(en'y o0 "-e $eam* 0rom Re0eren'e >* is 3iven $y

( ) 3LmL2235.0

EI3

2

1nf

+ρπ=

@),

( )( ) ( )3in5.5)sec^2/inlbf 0.00114(sec^2/inlbf 0.0007122235.0

2inlbf 04e60.23

2

1nf

+

+

π

=

@))

Hz6.5nf =

@)+

>

m

! '

"

#



0.001

0.01

0.1

10 100 1000 2000

FREQUENCY (Hz)

A C C E L ( G 2 / H z )

POWER SPECTRAL DENSITY 6.1 GRMS OVERALL

Fi3(re >&

Ta$e )& Base In/(" 4SD* ;&) GRMS

Fre(en'y @ A''e @G9+:

+, ,&,,7>

)7, ,&,5

;,, ,&,5

+,,, ,&,,>;

No1 'onsi%er "-a" "-e $ra'2e" assem$y is s($e'"e% "o "-e ran%om vi$ra"ion $ase in/("eve s-o1n in Fi3(re > an% in Ta$e )& T-e %(ra"ion is > min("es&

5



Syn"-esie% Time is"ory

Fi3(re 5&

An a''eera"ion "ime -is"ory is syn"-esie% "o sa"is0y "-e 4SD s/e'i0i'a"ion 0rom Fi3(re>& T-e res("in3 "ime -is"ory is s-o1n in Fi3(re 5& T-e syn"-esis me"-o% is 3iven inRe0eren'e 5&

T-e 'orres/on%in3 -is"o3ram -as a norma %is"ri$("ion* $(" "-e /o" is omi""e% 0or$revi"y&

No"e "-a" "-e syn"-esie% "ime -is"ory is no" (ni(e& For ri3or* "-e anaysis in "-is /a/er'o(% $e re/ea"e% (sin3 a n(m$er o0 s(i"a$e "ime -is"ories&

7



Fi3(re 7&

Veri0i'a"ion "-a" "-e syn"-esie% "ime -is"ory mee"s "-e s/e'i0i'a"ion is 3iven in Fi3(re 7&

;



A''eera"ion Res/onse

Fi3(re ;&

T-e res/onse a''eera"ion in Fi3(re ; 1as 'a'(a"e% via "-e me"-o% in Re0eren'e 7&T-e res/onse is narro1$an%& T-e os'ia"ion 0re(en'y "en%s "o $e near "-e na"(ra0re(en'y o0 7&; & T-e -is"o3ram -as a norma %is"ri$("ion %(e "o "-e ran%omy<varyin3 am/i"(%e mo%(a"ion&

T-e overa res/onse eve is ;&) GRMS& T-is is aso "-e s"an%ar% %evia"ion 3iven "-a"

"-e mean is ero& T-e res/onse an% in/(" eves -ave "-e same overa GRMS va(e*

$(" "-is ony a 'oni'i%en'e&

T-e a$so("e /ea2 is +6& G* 1-i'- res/resen"s a 5&7+<si3ma /ea2&

No"e "-a" some 0a"i3(e me"-o%s ass(me "-a" "-e /ea2 res/onse is ><si3ma an% may

"-(s (n%er/re%i'" 0a"i3(e %ama3e&

S"ress an% Momen" Ca'(a"ion

6



Fi3(re 6&

he follo!ing approach is a simplification. " rigorous method !ould calculate the

stress from the strain at the fixed end.

A 0ree<$o%y %ia3ram o0 "-e $eam is s-o1n in Fi3(re 6&

T-e rea'"ion momen" M$ a" "-e 0i.e%<$o(n%ary is

L%M$ = @)>

T-e 0or'e % is e(a "o "-e e00e'" mass o0 "-e $ra'2e" sys"em m("i/ie% $y "-e

a''eera"ion eve& T-e e00e'"ive mass me is

( )mL2235.0me +ρ=

@)5

( )( ))sec^2/inlbf 0.00114(sec^2/inlbf 0.0007122235.0me +=

@)7

sec^2/inlbf 0013.0me =

@);

T-e $en%in3 momen" M& a" a 3iven %is"an'e L& 0rom "-e 0or'e a//i'a"ion /oin" is

L&'mM& e= (17)

1-ere ' is "-e a''eera"ion a" "-e 0or'e /oin"&

M$

$ %

L"



T-e $en%in3 s"ress b is 3iven $y

I/M&* b = (1+)

T-e varia$e * is "-e s"ress 'on'en"ra"ion 0a'"or& Ass(me "-a" "-e s"ress 'on'en"ra"ion0a'"or is >&, 0or "-e so%er (3 mo(n"in3 -oe&

T-e varia$e is "-e %is"an'e 0rom "-e ne("ra a.is "o "-e o("er 0i$er o0 "-e $eam& T-e'ross<se'"ion is (ni0orm in "-e sam/e /ro$em& T-(s is e(a "o one<-a0 "-e"-i'2ness* or ,&)+7 in&

'(,)I)/L&m* (),( e b = (1)

( ) 4e in0026.0/)in125.0(in5.5sec^2/in)lbf 0013.0)((3.0I)/m* ( =

(20)

sec^2/in^3lbf 02.1I)/m* ( e = (21)

'--l# /ni, cnesin fc,.

( ) ( )( )in/sec^2)/(3+6sec^2/in^3lbf 02.1in/sec^2)/(3+6I)/m* ( e =

(22)

( ) lbf/in^2)/(37in/sec^2)/(3+6I)/m* (e

= (23)

( ) !si/37.0in/sec^2)/(3+6I)/m* ( e = (24)



Fi3(re &

T-e s"an%ar% %evia"ion is +&5 2si& T-e -i3-es" a$so("e /ea2 is ))&, 2si* 1-i'- is 5&7+<si3ma& T-e 5&7+ m("i/ier is aso re0erre% "o as "-e H'res" 0a'"or&

Ne."* a rain0o1 'y'e 'o(n" 1as /er0orme% on "-e s"ress "ime -is"ory (sin3 "-e me"-o%in Re0eren'e ;& T-e $inne% res("s are s-o1n in Ta$e +&

T-e $inne% res("s are s-o1n mainy 0or re0eren'e* 3iven "-a" "-is is a 'ommon/resen"a"ion 0orma" in "-e aeros/a'e in%(s"ry& T-e $inne% res("s 'o(% $e inser"e%in"o a Miner’s '(m(a"ive 0a"i3(e 'a'(a"ion&

T-e me"-o% in "-is anaysis* -o1ever* 1i (se "-e ra1 rain0o1 res("s 'onsis"in3 o0

'y'e<$y<'y'e am/i"(%e eves* in'(%in3 -a0<'y'es& T-is $r("e<0or'e me"-o% is more/re'ise "-an (sin3 $inne% %a"a&

),



Ta$e +& S"ress Res("s 0rom Rain0o1 Cy'e Co(n"in3* Bin Forma"* S"ress Uni"! 2si* Base In/("Overa Leve 8 ;&) GRMS

Ran3e

U//er

Limi"

Lo1er

Limi"

Cy'eCo(n"

s

Avera3eAm/i"(%

e

Ma.

Am/

Min

Am/

Avera3e

Mean

Ma.

Mean

Min

Vaey

Ma.

4ea2

)&;; +)&5 >&7 ),&5> ),&+ <,&+ ,&,6> ,&75 <))&,+ ),&+

)6&56 )&;; +)&, &)) &, <,&>7 ,&)7+ ,&7 <&+ ),&))

)7&+ )6&56 ),&, &,6 &6, <)&>; ,&,,+ ,&;6 <&7> &,

)>&), )7&+ >6+&, ;& 6&;> <)&,6 <,&,+; ,&6) <&7) &>5

),&+ )>&), 5>&, 7&5 ;&77 <)&,+ ,&,,; )&,, <6&); 6&+,

&65 ),&++,76&

75&; 7&5; <)&+> <,&,), ,& <;&75 ;&)7

;&77 &65>;76&

,

>&6 5&>6 <)&) <,&,,+ )&)7 <7&>, 7&+,

5&>6 ;&775,&

7+&6+ >&+ <)&,+ ,&,,+ )&,; <5&++ 5&)>

>&+ 5&>6++6>&

7)&+ +&) <,&> ,&,,7 ,&5 <>&,; +&5

+&) >&+)65)&

7)&> )&;5 <,& ,&,,+ ,&+ <+&>; +&7;

)&, +&)))5,&

,,&> )&, <)&,5 ,&,+, )&+5 <+&,> )&

,&77 )&, ;6,&, ,&5, ,&77 <)&;> <,&,,> )&; <)&+ +&5,

,&,, ,&7765>&

,

,&,5 ,&+6 <;&,, <,&,+5 7&> <;&,) 7&5

No"e "-a"! Am/i"(%e 8 @/ea2<vaey:+

))



Miner’s C(m(a"ive Fa"i3(e

Le" n $e "-e n(m$er o0 s"ress 'y'es a''(m(a"e% %(rin3 "-e vi$ra"ion "es"in3 a" a 3iveneve s"ress eve re/resen"e% $y in%e. i&

Le" $e "-e n(m$er o0 'y'es "o /ro%('e a 0a"i3(e 0ai(re a" "-e s"ress eve imi" 0or "-e'orres/on%in3 in%e.&

Miner’s '(m(a"ive %ama3e in%e. $ is 3iven $y

∑=

=m

1i i

i

n$

@+7

1-ere m is "-e "o"a n(m$er o0 'y'es or $ins %e/en%in3 on "-e anaysis "y/e&

In "-eory* "-e /ar" s-o(% 0ai 1-en

Rn @"-eory 8 )&, @+;

For aeros/a'e ee'"roni' s"r('"(res* -o1ever* a more 'onserva"ive imi" is (se%

Rn@aero 8 ,&6 @+6

T-e n(m$er o0 ao1a$e 'y'es 0or a 3iven s"ress eve is %e"ermine% 0rom an S<N0a"i3(e '(rve in A//en%i. A* Fi3(re A<5 0or "-e ;,;)<T; a(min(m $ra'2e" in "-e sam/e

/ro$em&

In/(" Leve S"(%y

)+



Ta$e 5& SDOF Sys"em* So%er Termina Lo'a"ion* Fa"i3(eDama3e Res("s 0or Vario(s In/(" Leves* ), se'on%D(ra"ion* Cres" Fa'"or 8 5&7+

In/(" OveraLeve@GRMS

In/(" Mar3in@%B

Res/onseS"ress S"%Dev @2si

R

;&) , +&5 6&5e<,

&6 > >&5 +&,e<,;

)+&> ; 5& 7&>e<,7

)6&> ;& ,&,,)5+

+5&7 )+ &6 ,&,>

+6&5 )> ),&

U"ima"e

Fai(re

T-e a''(m(a"e% 0a"i3(e %ama3e 1as 'a'(a"e% 0or a 0amiy o0 'ases as s-o1n inTa$e 5& Ea'- 'ase (se% "-e $ase in/(" 4SD 0rom Fi3(re 5 1i"- "-e in%i'a"e% a%%e%mar3in& F(r"-ermore* ea'- (se% a s'ae% version o0 "-e same syn"-esie% "ime -is"ory&

Ea'- 0( an% -a0<'y'e 0rom "-e rain0o1 res("s 1as a''o(n"e% 0or& An ao1a$e Nva(e 1as 'a'(a"e% 0or ea'- s"ress am/i"(%e S (sin3 e(a"ion @+ 0or ea'- 'y'e or-a0<'y'e&

A r(nnin3 s(mma"ion 1as ma%e (sin3 e(a"ion @+7&

A3ain* "-e s(''ess 'ri"erion 1as R J ,&6& T-e 0a"i3(e 0ai(re "-res-o% is some1-ere$e"1een "-e )+ an% )> %B mar3in&

T-e %a"a s-o1s "-a" "-e 0a"i3(e %ama3e is -i3-y sensi"ive "o "-e $ase in/(" an%res("in3 s"ress eves&

D(ra"ion S"(%y

)>



A ne1* 6+,<se'on% si3na 1as syn"-esie% 0or "-e ; %B mar3in 'ase& T-e "ime -is"ory/o" is omi""e% 0or $revi"y&

A 0a"i3(e anaysis 1as "-en /er0orme% (sin3 "-e SDOF sys"em in Fi3(re )& T-eanaysis 1as "-en re/ea"e% (sin3 "-e , "o >;, se' an% , "o ), se' se3men"s o0 "-ene1 syn"-esie% "ime -is"ory& T-e R va(es 0or "-ese "-ree 'ases are s-o1n in Ta$e 7&

Ta$e 7& SDOF Sys"em* So%er Termina Lo'a"ion* Fa"i3(e Dama3eRes("s 0or Vario(s D(ra"ions* )+&+ GRMS In/("

D(ra"ion @se' S"ress RMS @2si Cres" Fa'"or R

), 5&+ 5&;7 7&>6e<,7

>;, 5& 5&) ,&,,,)+>

6+, 5& 5&6 ,&,,,+>5

T-e R va(e is a//ro.ima"ey %ire'"y /ro/or"iona "o "-e %(ra"ion* s('- "-a" a %o($in3o0 %(ra"ion neary yie%s a %o($in3 o0 R&

)5



Time is"ory Syn"-esis Varia"ion S"(%y

A se" o0 "ime -is"ories 1as syn"-esie% "o mee" "-e $ase in/(" 4SD ? ; %B& T-e

res/onse o0 "-e SDOF sys"em in Fi3(re ) 1as 'a'(a"e%& T-e res("s are 3iven inTa$es ; an% 6&

Ta$e ;& SDOF Sys"em* So%er Termina Lo'a"ion* Fa"i3(e Dama3eRes("s 0or Vario(s Time is"ory Cases* ),<se'on% D(ra"ion* )+&+ GRMSIn/("

S"ress RMS @2si Cres" Fa'"or K(r"osis R

5&; 7&55 >&) ;&E<,7

5& 5&5, >&, 7&)E<,7

5&, 5&5> >&, 5&>E<,7

5&, 5&5; >&) 6&,;E<,7

5& 7&6 >&, 6&;,E<,7

5& 5&7 >&, ;&,+E<,7

5&5 5&;5 >&, 5&6;E<,7

5&+ 5&;7 >&) 7&>6E<,7

5&) 5&>6 >&, 5&>E<,7

5&; 5&76 >&, 7&>,E<,7

5&5 5&;, >&, 7&))E<,7

5&; 5&+6 >&, 5&;6E<,7

Ta$e 6& Limi"s 0or S"ress Res/onse4arame"ers

4arame"er Min Ma.

S"ress @2si 5&, 5&,

Cres" Fa'"or 5&+6 7&6

K(r"osis +& >&)+

R5&>E<

,76&;,E<,7

)7



No"e "-a" "-e 'res" 0a'"or is "-e ra"io o0 "-e /ea2<"o<s"an%ar% %evia"ion* or /ea2<"o<rmsass(min3 ero mean&

No"e "-a" Rayei3- %is"ri$("ion /re%i'"s a 5&77 'res" 0a'"or 0or a 7&; os'ia"or over a),<se'on% %(ra"ion& T-e 0orm(a is 3iven in A//en%i. B&

T-e 'res" va(e varies s('- "-a" "-e ma.im(m va(e is >; -i3-er "-an "-e minim(m&

T-e overa s"ress eve an% 2(r"osis remain neary 'ons"an" a'ross "-e se" o0 "ime-is"ories& K(r"osis is %e0ine% in A//en%i. C&

T-e R va(e varies 1i"- "-e ma.im(m $ein3 6> -i3-er "-an "-e minim(m&

T-e R va(e is sensi"ive "o "-e overa s"ress RMS eve* 'res" 0a'"or an% 2(r"osis*

"en%in3 "o in'rease 1i"- even sma /osi"ive '-an3es in ea'- o0 "-ese /arame"ers& SeeFi3(res "-ro(3- ))* 1-i'- in'(%e a inear '(rve<0i"&

!10"#

#!10"#

6!10"#

$!10"#

%!10"#

.$% .$& .%0 .%1 .%2 .%' .% .%# .%6 .%$ .%% .%& .&0 .&1

*2.1'E"!1 "&.$6E"+ ,-! /1.'2E"#+

20.2#

STRESS RMS (345)

R

FATIGUE DAMAGE R+ 1%0 SECOND DURATION+ 12.2 GRMS INPUT

Fi3(re &

);



!10"#

#!10"#

6!10"#

$!10"#

%!10"#

.0 .2 . .6 .% #.0 #.2 #. #.6 #.% 6.0

*1.$&E"#!1 "2.%E"#+ ,-! /1.&1E"#+

20.616

CREST FACTOR

R


Fi3(re ),&

!10"#

#!10"#

6!10"#

$!10"#

%!10"#

2.&% '.00 '.02 '.0 '.06 '.0% '.10 '.12

*1.$E"!1 ".$'E"+ ,-! /2.1$E"#+

20.0%

URTOSIS

R


Fi3(re ))&

)6



T-e %a"a in Fi3(res "-ro(3- )) are a sna/s-o" 0or a /ar"i'(ar SDOF sys"em an% $asein/(" 4SD&

T-is HTime is"ory Varia"ion s"(%y s-o(% $e re/ea"e% 1i"- varyin3 na"(ra 0re(en'ies*%am/in3 ra"ios* in/(" eves* %(ra"ions* e"'&

Con'(sion

T-is is a H1or2<in</ro3ress&

Fa"i3(e anaysis area%y 'arries (n'er"ain"y 3iven "-a" S<N '(rves an% s"ress'on'en"ra"ion 0a'"ors are $o"- "en(o(s&

T-e 'res" 0a'"or an% 2(r"osis are very im/or"an"& Res/onse /ea2s a$ove ><si3ma ma2ea si3ni0i'an" 'on"ri$("ion "o 0a"i3(e %ama3e& Even minor '-an3es in "-e res/onse

s"ress 'an -ave si3ni0i'an" e00e'" on "-e 0a"i3(e %ama3e R&

T-e res("s in Fi3(re ; s-o1 varia"ion in "-e 0a"i3(e %ama3e R 1i"- a se" o0 in%e/en%en"$ase in/(" "ime -is"ories ea'- o0 1-i'- sa"is0ies "-e same $ase in/(" 4SD s/e'i0i'a"ion&T-e %i00eren'e is %(e varia"ion in (n%eryin3 res/onse s"a"is"i'a /arame"ers& A'onserva"ive a//roa'- 0or a 3iven %esi3n /ro$em 1o(% "-(s $e "o "a2e "-e ma.im(mR 0or "en or more in%e/en%en" syn"-esis 'ases&

No"e "-a" "-e s"ress res/onse varia"ion 1o(% aso o''(r 0or a s-a2er "a$e "es"* even i0"-e $ase in/(" is /ea2<imi"e%&

T-ere are a%%i"ion 'on'erns arisin3 0rom mo%ein3 sim/i0i'a"ions&

T-e anaysis (se% /(re $en%in3 s"ress& A $e""er a//roa'- 1o(% -ave $een "o (se "-ema.im(m /rin'i/a s"ress or von Mises s"ress 1-i'- 1o(% -ave in'(%e% "-e s-ears"ress& T-e s"ress 'on"ri$("ions o0 -i3-er mo%es s-o(% aso $e 'onsi%ere%&

T-ere are a%%i"iona ma"eria 'on'erns as 3iven in A//en%i. D&

I%eaiin3 a sys"em as a sin3e<%e3ree<o0<0ree%om sys"em may yie% "o an (n%er</re%i'"ion o0 "-e 0a"i3(e %ama3e as s-o1n in A//en%i. E& T-e in'(sion o0 -i3-ermo%es may no" in'rease "-e s"ress eve m('- $(" 1i in'rease "-e 0a"i3(e %ama3e*$e'a(se -i3-er mo%es a%% rea"ivey -i3-<0re(en'y s"ress reversa 'y'es&

A%%i"iona A//i'a"ions

T-e 0a"i3(e me"-o% is a//ie% "o a 'on"in(o(s $eam mo%e in A//en%i. E&

Re0eren'es

)



)& Dave S"ein$er3* Vi$ra"ion Anaysis 0or Ee'"roni' E(i/men"* Se'on% E%i"ion*Wiey* Ne1 Yor2* )&

+& T& Irvine* Ran%om Vi$ra"ion Fa"i3(e* Revision B* Vi$ra"ion%a"a* +,,>&

>& T& Irvine* Ben%in3 Fre(en'ies o0 Beams* Ro%s* an% 4i/es* Revision S*Vi$ra"ion%a"a* +,)+&

5& T& Irvine* A Me"-o% 0or 4o1er S/e'"ra Densi"y Syn"-esis* Rev B* Vi$ra"ion%a"a*+,,,&

7& Davi% O& Sma1oo%* An Im/rove% Re'(rsive Form(a 0or Ca'(a"in3 S-o'2Res/onse S/e'"ra* S-o'2 an% Vi$ra"ion B(e"in* No& 7)* May ))&

;& ASTM E ),5<7 @+,,7 Rain0o1 Co(n"in3 Me"-o%* )6&

6& -""/!::en&1i2i/e%ia&or3:1i2i:Fa"i3(e@ma"eria

& MIL<DDK<7* De/ar"men" o0 De0ense an%$oo2! Me"ai' Ma"erias an%Eemen"s 0or Aeros/a'e Ve-i'e S"r('"(res* >) an +,,>&

& Ka"-een S& Dra3oi'-* Ni22i D& DiMa""eo* Fa"i3(e Da"a Boo2! Li3-" S"r('"(raAoys* ASM In"erna"iona&

),& -""/!::s'-oar&i$&v"&e%(:"-eses:avaia$e:e"%<,6)+,,)<)+5;+5:(nres"ri'"e%:Comea(C-a/"er>%o'&/%0

))&K& A-in* Com/arison o0 Tes" S/e'i0i'a"ions an% Meas(re% Fie% Da"a* So(n% Vi$ra"ion* Se/"em$er +,,;&

)+& V& A%ams an% A& As2enai* B(i%in3 Be""er 4ro%('"s 1i"- Fini"e Eemen"Anaysis* OnWor% 4ress* San"a Fe* N&M&* )&

)>&T& Irvine* Mo%a Transien" Vi$ra"ion Res/onse o0 a Can"iever Beam S($e'"e% "oBase E.'i"a"ion* Vi$ra"ion%a"a* +,)>&

)



A44ENDIP A

Fa"i3(e S<N C(rves

#0

100

1#0

200

2#0

'00

'#0

100

102

10

106

101

10'

10#

10$

L57/ (889/4)

S 4 4 ( M P - )

S"N CURVE FOR :RITTLE ALUMINUM WITH A UTS OF '20 MP-

Fi3(re A<)&

T-e '(rve in Fi3(re A<) is "a2en 0rom Re0eren'e 6&

I" may $e (ns(i"a$e 0or en3ineerin3 'a'(a"ion $e'a(se "-e aoy an% o"-er %e"ais are

(ni%en"i0ie%& B(" i" is very (se0( as a (ai"a"ive e.am/e o0 a(min(m 0a"i3(e&

T-e '(rve 'an $e ro(3-y %ivi%e% in"o "1o se3men"s& T-e 0irs" is "-e o1<'y'e 0a"i3(e

/or"ion 0rom ) "o ),,, 'y'es& T-is '(rve is 'on'ave as vie1e% 0rom "-e ori3in& T-e

se'on% /or"ion is "-e -i3-<'y'e '(rve $e3innin3 a" ),,,* 1-i'- is 'onve. as vie1 0rom

"-e ori3in&

+,



F(r"-ermore* "-e s"ress eve 0or one 'y'e is "-e ("ima"e s"ress imi"&

T-ere is no en%(ran'e imi"&

Fi3(re A<+&

T-e 0a"i3(e '(rve 0or a(min(m ;,;)<T; is s-o1n in Fi3(re A<+* 0rom Re0eren'e *Fi3(re >&;&+&+&&

T-e s"ress ra"io in "-e e3en% is "-e @/ea2:vaey 0or a 3iven 'y'e& T-e KT va(ein%i'a"es "-e s"ress 'on'en"ra"ion 0a'"or o0 )&,&

+)



T-e ("ima"e s"ress 0or A ;,;)<T; is 57 2si /er Re0eren'e & T-e yie% s"ress is 5, 2si&

T-ese s"ress imi"s are 0or "-e "es"e% sam/es in Fi3(re A<+& T-e a'"(a va(es aso%e/en% on "-e 0orm& S-ee"* /a"e* $ars* ro%s* e."r(sions* 0or3in3s an% 'as"in3s a -ave"-eir o1n se" o0 imi"s&

T-e a%van"a3e o0 "-e '(rve in Fi3(re A<+ is "-a" i" is a("-ori"a"ive an% 1e<%o'(men"e%&T-e %isa%van"a3e is "-a" is %oes no" 'over o1<'y'e 0a"i3(e $eo1 ),,, 'y'es&

Fi3(re A<>&

T-e %a"a in Fi3(re A<> is "a2en 0rom Re0eren'e &

T-e "es" %a"a 0or a(min(m ;,;)<T; a//ears "o $e3in a" +@),5 'y'es& A /a"ea( a" "-e("ima"e s"ress imi" o0 57 2si is ass(me% 0rom ), "o @),5 'y'es&

T-e /a"ea( -as a /-ysi'a e./ana"ion& ys"eresis oo/s 0orm in "-e s"ress<s"rain'(rves 0or ma"erias (n%er3oin3 'y'i'a oa%in3 1-ere "-e ma.im(m s"ress e."en%s in"o"-e /as"i' %e0orma"ion re3ime& In /ar"i'(ar* a(min(m (n%er3oes s"rain<-ar%enin3&I"s -ys"eresis oo/s s"a$iie a0"er re/ea"e% 'y'in3 so "-a" "-e s"ress am/i"(%e remainsrea"ivey 'ons"an" over a ar3e /or"ion o0 0a"i3(e i0e& F(r"-er in0orma"ion is 3iven inRe0eren'e ),&

++



0

#

10

1#

20

2#

'0

'#

0

#

#0

100

102

106

10%

101

10'

10#

10#

10$

CYCLES

M A ; S T R E S S ( 6 S I )

S"N CURVE ALUMINUM 6061"T6 T1 STRESS RATIO "1FOR REFERENCE ONLY

Fi3(re A<5&

T-e ma.im(m s"ress is ero<"o</ea2 0or "-e s"ress ra"io 8 <)&

T-e /(r/ose o0 "-is se'"ion is no" "o 3ive "-e %e0ini"ive S<N '(rve 0or a(min(m ;,;)<T;*

$(" is ra"-er "o 3ive a 1or2in3 '(rve 0or "-e e.am/es in "-is /a/er&

T-e <) '(rve 1i $e (se% 0or "-e e.am/es& No"e "-a" "-e <) ra"io ro(3-y mee"s "-e

e./e'"e% rain0o1 'y'e $e-avior&

An e."ra/oa"ion is ma%e 0or "-e o1<'y'e 0a"i3(e re3ion 0rom /oin" @) 'y'e* 57 2si "o

@)7>5 'y'es* >&6> 2si (sin3 a sim/e inear '(rve& T-is is a %o'(men"e%ass(m/"ion& A "-oro(3- 'onsi%era"ion o0 o1<'y'e 0a"i3(e is $eyon% "-e s'o/e o0 "-is

/a/er&

T-e '(rve<0i" e(a"ions are 3iven as 0oo1s&

Le" N $e "-e n(m$er o0 'y'es& Le" S $e "-e 'orres/on%in3 ma.im(m s"ress am/i"(%e&

+>



A44ENDIP B

Rayei3- Dis"ri$("ion Cres" Fa'"or 0or an SDOF Sys"em Res/onse

T-e 0orm(a is 0or "-e ma.im(m /re%i'a"e% 'res" 0a'"or is

( )( )fnln2

5772.0fnln2 += @B<)

1-ere

fn is "-e na"(ra 0re(en'y

is "-e %(ra"ion

ln is "-e na"(ra o3ari"-m 0(n'"ion

E(a"ion @B<) is "a2en 0rom Re0eren'e ))&

+7



A44ENDIP C

K(r"osis

K(r"osis is a /arame"er "-a" %es'ri$es "-e s-a/e o0 a ran%om varia$e’s -is"o3ram or i"se(ivaen" /ro$a$ii"y %ensi"y 0(n'"ion @4DF&

T-e 2(r"osis 0or a "ime series i is

*/,sis

[ ]

4

n

1i

4i

n

σ

µ−

∑= @C<)

1-ere

µ Mean

σ s"an%ar% %evia"ion

n n(m$er o0 sam/es

T-e "erm in "-e n(mera"or is "-e H0o(r"- momen" a$o(" "-e mean&

A /(re sine "ime -is"ory -as a 2(r"osis o0 )&7&

A "ime -is"ory 1i"- a norma %is"ri$("ion -as a 2(r"osis o0 >&

Some a"erna"e %e0ini"ions o0 2(r"osis s($"ra'" a va(e o0 > so "-a" a norma %is"ri$("ion 1i -ave a 2(r"osis o0 ero&

A 2(r"osis ar3er "-an > in%i'a"es "-a" "-e %is"ri$("ion is more /ea2e% an% -as -eavier

"ais "-an a norma %is"ri$("ion 1i"- "-e same s"an%ar% %evia"ion&

+;



A44ENDIP D

Fa"i3(e Cra'2s

A %('"ie ma"eria s($e'"e% "o 0a"i3(e oa%in3 e./erien'es $asi' s"r('"(ra '-an3es&T-e'-an3es o''(r in "-e 0oo1in3 or%er!

)& #rack $nitiation. A 'ra'2 $e3ins "o 0orm 1i"-in "-e ma"eria&

+& %ocali&ed crack gro!th& Lo'a e."r(sions an% in"r(sions o''(r a" "-es(r0a'e o0 "-e /ar" $e'a(se /as"i' %e0orma"ions are no" 'om/e"eyreversi$e&

>& #rack gro!th on planes of high tensile stress. T-e 'ra'2 /ro/a3a"esa'ross "-e se'"ion a" "-ose /oin"s o0 3rea"es" "ensie s"ress&

5& 'ltimate ductile failure. T-e sam/e r(/"(res $y %('"ie 0ai(re 1-en "-e'ra'2 re%('es "-e e00e'"ive 'ross se'"ion "o a sie "-a" 'anno" s(s"ain "-ea//ie% oa%s&

Desi3n an% Environmen"a Varia$es a00e'"in3 Fa"i3(e Li0e

T-e 0oo1in3 0a'"ors %e'rease 0a"i3(e i0e&

)& Stress concentrators. oes* no"'-es* 0ie"s* s"e/s* 3rooves* an% o"-erirre3(ar0ea"(res 1i 'a(se -i3-y o'aie% re3ions o0 'on'en"ra"e% s"ress* an%"-(s re%('e 0a"i3(e i0e&

+& Surface roughness. Smoo"- s(r0a'es are more 'ra'2 resis"an" $e'a(sero(3-ness 'rea"es s"ress 'on'en"ra"ors&

>& Surface conditioning. ar%enin3 /ro'esses "en% "o in'rease 0a"i3(es"ren3"-* 1-ie /a"in3 an% 'orrosion /ro"e'"ion "en% "o %iminis- 0a"i3(es"ren3"-&

5& (nvironment. A 'orrosive environmen" 3rea"y re%('es 0a"i3(e s"ren3"-& A'om$ina"ion o0 'orrosion an% 'y'i'a s"resses is 'ae% corrosion fatigue.

Tem/era"(re may aso $e a 0a'"or&

+6



Mass /er Len3"- ρ 8 ,&,7 $m:in

Vis'o(s Dam/in3 Ra"io 8 ,&,7 0or a mo%es

Ca'(a"e "-e $en%in3 s"ress a" "-e 0i.e% $o(n%ary& Omi" "-e s"ress 'on'en"ra"ion0a'"or& T-e $ase in/(" "ime -is"ory is "-e same as "-a" in Fi3(re 5 1i"- +) %B mar3in&

Fi3(re E<+&

T-e res/onse anaysis is /er0orme% (sin3 Ma"a$ s'ri/"! 'on"in(o(s$eam$asea''e&m&

T-e norma mo%es res("s are 3iven in Ta$e E<)& A "y/i'a res/onse is s-o1n in

Fi3(re E<+&

Ta$e E<)& Na"(ra Fre(en'y Res("s* Fi.e%<FreeBeam

+



Mo%e 0n @4ar"i'i/a"ion

Fa'"orE00e'"ive Mo%a

Mass @$m

) 77 ,&,>) ,&>;

+ >57 ,&,)6 ,&))>

> ;6 ,&,), ,&,>

5 )7 ,&,,6 ,&,+,

7 >)>+ ,&,,; ,&,)+

A3ain* $o"- "-e mo%e s-a/e an% /ar"i'i/a"ion 0a'"or are 'onsi%ere% as %imensioness*

$(" "-ey m(s" $e 'onsis"en" 1i"- res/e'" "o one ano"-er&

Ta$e E<+& Con"in(o(s Beam* S"ress a" Fi.e% Bo(n%ary* Fa"i3(e Dama3eRes("s* ),<se'on% D(ra"ion* ;& GRMS In/("

Mo%esIn'(%e%

S"ress RMS@2si

Cres" Fa'"or K(r"osis R

) ;&,6 7&6 >&, ,&,,,55+;

+ ;&5) 7&>> >&, ,&,,))>

> ;&5+ 7&>6 >&, ,&,,)+5>

5 ;&5+ 7&5, >&, ,&,,)+;,

T-e 0a"i3(e %ama3e res("s in Ta$e E<+ 1o(% -ave $een -i3-er i0 a s"ress

'on'en"ra"ion 0a'"or 1as in'(%e%& T-e /(r/ose o0 "-is inves"i3a"ion 1as ra"-er

"o %e"ermine "-e e00e'" o0 in'(%in3 -i3-er mo%es 0or a sam/e 'on"in(o(s

sys"em&

T-e res("s s-o1 "-a" "-e s"ress RMS 'an $e a''(ra"ey 'a'(a"e% (sin3 ony"1o mo%es&

T-e 0a"i3(e %ama3e R rea'-es a" /a"ea( a" "-ree mo%es&

No"e "-a" 0i0"- mo%a 0re(en'y is 1e a$ove "-e ma.im(m 0re(en'y o0 "-e$ase in/("* so i" 1as ne3e'"e%&

>,



A more "-oro(3- inves"i3a"ion 1o(% invove re/ea"in3 "-is anaysis 0or a 0amiyo0 "ime -is"ory in/("s* ei"-er 1i"- "-e same or varyin3 overa eves&

A44ENDIP F

Time S'ain3 0or E(ivaen" Tes"in3

T-e 0oo1in3 a//ies "o s"r('"(res 'onsis"in3 ony o0 a(min(m ;,;)<T; ma"eria&

I" is $ase% on "-e se3men" o0 "-e S<N '(rve in Fi3(re A<5 0or s"ress eves $eo1 >&6>2si&

Re1ri"e e(a"ion @A<> as

()l8100.10+9()l810 ∝ (%91)

)(l810()l810 0.109∝ (%92)

0.109∝ (%93)

1

1/.26

∝ (%94)

1

.26

∝ (%95)

( .26 cns,n, (%96)

No1 'onsi%er a re0eren'e "es" (sin3 in%e. ) an% an e(ivaen" "es" (sin3 in%e. +&

( 4( 4.26

22.2611 = (%97)

(

(

4

4.26

2

1

1

2

= (%9+)

>)



Ass(me inear $e-avior& A %o($in3 o0 "-e s"ress va(e re(ires ):;)> "imes "-en(m$er o0 re0eren'e 'y'es& T-(s* i0 "-e a''eera"ion GRMS eve is %o($e%* "-en ane(ivaen" "es" 'an $e /er0orme% in ):;)> "- o0 "-e re0eren'e %(ra"ion* in "erms o0/o"en"ia 0a"i3(e %ama3e& T-is is aso s-o1n $y 'onsi%erin3 "-e n(meri'a e./erimen"

res("s o0 Ta$es 5 an% 7&

Miners Fatigue Rainflow RevD2

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