PSI - Issue 31

Damjan Čakmak et al. / Procedia Structural Integrity 31 (2021) 98– 104

101

4

Damjan Č akmak et al. / Structural Integrity Procedia 00 (2021) 000–000

( ) ( 1 j −

j

2 1 j +

x

x

2

2

S

m γ

∞



( ) x

( ) x

(6a-c)

0 

2

e d t −

erf

0 erf ≤

1.

x

t

=

=

≡

≤





r

)

! 2 1 j +

8

π

π

λ

0

j

=

0

As outlined by curly brackets in Eq. (5), weighted sum of the normal/Gaussian and Rayleigh distributions is utilized in RL function, where term [1 + erf( x )] serves as transcendent weighting coefficient. Weight is relative to frequency spectrum width governed by γ and λ from Eqs. (2g,h). Eq. (5) is investigated in numerous referent publications, e.g. Lalanne (2014), Lü and Jiao (2000), however its exact solution I RL ( S r ) with respect to Eq. (1b) is currently not known. This is due to the transcendental Gauss error function erf( x ), which is mathematically challenging to integrate. To the best of authors’ knowledge, Lü and Jiao (2000) were the last investigators trying to solve this integral analytically. One may note that for irregularity factor γ = 0 from Eq. (2g), PDF from Eq. (5) algebraically morphs into pure Gaussian/BB distribution. However, for γ → 1, original Rayleigh/NB distribution is only asymptotically obtained by performing corresponding limit analysis on Eq. (5). These relations write as

2

2

S

S

−

−

1

S

r

r

( ) (

)

( )

( )

( ) ( r

)

(7a,b)

8

8

m

m

0

e

,

lim

e

1

.

RL r p S

BB r p S

RL r p S

NB p S

γ

=

=

≡

=

≡

γ

=

r

0

0

4

m

0 8 π m

1

γ

→

0

Therefore, RL distribution cannot explicitly reproduce pure NB distribution due to observed singularity for γ = 1. Integration of Rice distribution from Eq. (5) with pivotal constraining assumption that 0 ≤ γ < 1 and 0 < λ ≤ 1, and with respect to Eq. (1b), yields final simplified analytical expression ( ) ( ) ( ) 2 2 3 2 1 2 RL r 0 1 3 3 , ; ; 1 8 2 2 2 1+ 1 , 2 2 2 π m m F m m m I m S m γ γ λ λ γ λ +     + − + +             Γ + Γ +               = (8) where term 2 F 1 (·) is known as the Gaussian or ordinary HyperGeometric (HG) function, see Bouyssy et al. (1993). Mathematical details for obtaining this result are omitted due to substantial length. Eq. (8) is considered to be a major contribution of this work and can be used as presented, without additional mathematical manipulation or iterative procedures. This special HG function is well documented in the literature and can be easily invoked in standard mathematically oriented software, e.g. WolframMathematica, MathWorks MATLab etc. Moreover, Eq. (8) represents an analytical solution for peak distribution of given process and only an approximate solution for RFC ranges. It also provides more conservative results compared to already conservative NB Eq. (3b), Tovo (2002). Hence, damage intensity for RL distribution, according to Eq. (1) and (5), can be written as

[ ] P

[ ] ( ) RL r 0 I S

E

E

D

( ) ( )

( )

( )

(9a,b)

,

RL r p S

RL r p S

I S

γ γ =

=

γ

≡



RL

RL r

'

'

t

B

B

max

rf

rf

where regularity factor γ is introduced as an additional correction factor to account for over conservative PA prediction, Tovo (2002). This γ -concept was already implied and utilized by Lalanne (2014). However, it must be emphasized that only γ -governed correction may provide inconsistent, i.e. sometimes too liberal and sometimes too conservative

fatigue results for general broad-band stress PSD. 3. Bishop and Sherratt spectral benchmark

In order to test the validity and accuracy of derived RL-based integral from Eq. (8), statistical vibration fatigue benchmark example by Bishop and Sherratt (2000) is employed. All described spectral theories from previous chapter are benchmarked against RFC in the time domain. Stress PSD is defined as an ideal rectangular bi-modal spectrum shown in Fig. 1a. Initially imposed stress PSD ( S PSD,init ) is shown with solid line. Two dominant frequencies, f 1 = 1 Hz