Issue 78

F. Veschi et alii, Fracture and Structural Integrity, 78 (2026) 44-73; DOI: 10.3221/IGF-ESIS.78.04

m

  0

¨ ¨ xx QG f f T  0

  

2

m



 

  0

  

m



1    

FDS f

K

(14)

 2 2



3

C

2

 

f 

0

1 K  and

1 C  for comparative purposes between different loadings. A

In comparative studies, it is also possible to set

schematic representation of the FDS evaluation process in the time domain is provided in Fig. 1.

S TATISTICAL CHARACTERISTICS OF S INE - ON -R ANDOM SIGNALS

I

n this section, considerations are made about the statistics of mixed SoR signals and the inapplicability of conventional spectral methods in this context. A Sine-on-Random signal represents a mixed-mode excitation where one or more deterministic sinusoidal components are superimposed on a stochastic Gaussian process. In its simplest form, a single-tone SoR signal   z t can be expressed as the sum of a deterministic sine wave   y t and a random Gaussian process   x t :       z t x t y t   (15) Statistics of the Sine Component This section draws upon the seminal work of S.O. Rice [10], who provided the analytical foundations for the study of random processes and signal theory. A sinusoid is defined by the time-dependent function     sin ω S y t a t    , where a is the amplitude, ω S is the angular frequency, and ψ is the phase. By assuming the argument ω S t     to be a random variable uniformly distributed over   0,2 π , the sine wave   sin y a   represents a transformation of a random variable and, consequently, is a random variable itself. Since every random variable is uniquely characterized by its PDF and an associated characteristic function  1 , the sinusoid can be described analytically through y  [1]:

1

     sin i a

2

  ,

  



i y       

d J a  

 

a E e

e

(16)

y

0



2

0

where 0 J is the Bessel function of order zero and  is the transform variable associated with the characteristic function of the random variable y . Starting from y  , the PDF of the instantaneous values is derived via inverse Fourier transform:

1







1/2

2 a y 

2

y a 

,

  ,



p y a

{

(17)



y

y a 

0,

As shown in Fig. 2, the sine wave spends most of its time near the extremes a  . Even though the PDF of instantaneous values is necessary to describe the process, the statistics of extreme values are very useful in some applications such as fatigue and structural dynamics. The PDF of its maxima (peaks) p y p is represented by a Dirac Delta function centred at the amplitude value a [28,29]:

  x   of a PDF

  x p x is defined as its Fourier transform:

1 The characteristic function



  j x 

    x

 

   



x f x

p x e

dx

 

 

x





  x p x and   x   thus constitute a transform pair .

The functions

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