PSI - Issue 38

Arvid Trapp et al. / Procedia Structural Integrity 38 (2022) 260–270 A. Trapp / Structural Integrity Procedia 00 (2021) 1–11

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3

frequency domain and signal reconstruction. The main contribution (Sec. 3) is to find meaningful descriptors for the sampling of random vibration and to provide a best-fitting error approximation to relate its frequency domain decomposition to the expected error in a fatigue assessment due to insu ffi cient sampling.

2. Sampled time series of random processes

This section begins by reviewing fundamentals of the statistical characterization of random processes (Sec. 2.1) and ends with a short summary of signal reconstruction (Sec. 2.2).

2.1. Statistical characterization The sampling frequency f s defines the number of samples per time. Its reciprocal is the resolution of a discrete time series ∆ t = 1 f s . According to the Nyquist-Shannon theorem, to uniquely identify a frequency by the Fourier transform, it has to be discretized by at least two samples per period. This implies that for a unique identification the frequency f has to be lower f ≤ f Ny than half of the sampling frequency f s — the Nyquist frequency f Ny = f s 2 (1) Stationary random vibration conforming the central limit theorem follows the Gaussian probability density function (PDF) and is fully defined by its mean µ and the variance σ 2 . Hereby, the PSD G xx ( f ) represents the frequency domain decomposition of the variance σ 2 and consequently fully characterizes stationary random loading. σ 2 = � ∞ 0 G xx ( f ) d f = λ 0 ; for µ = 0 (3) The frequency-domain characterization via the PSD allows for an e ffi cient definition of random vibration as a continuous function. It carries additional information that is accessible via spectral moments λ n , which are defined by [9, 10] λ n = � ∞ 0 (2 π f ) n G xx ( f ) d f (4) Spectral moments λ n relate to essential metrics of the time domain. The zeroth spectral moment λ 0 corre sponds to the integral of the PSD (Eqs. 3, 4) and thus equals the variance λ 0 = σ 2 . The spectral moments of even order n represent the variance of the time derivatives { ˙ x ( t ) , ¨ x ( t ) , ... } of the original series x ( t ). λ 0 = σ 2 x ; λ 2 = σ 2 ˙ x ; λ 4 = σ 2 ¨ x (5) They provide the basis to define statistical time-domain metrics such as the upward zero-crossing- ν 0 and the peak-rate ν p [9] p g ( x ) = 1 √ 2 πσ 2 e    − ( x − µ ) 2 2 σ 2    (2)

1 2 π �

1 2 π �

λ 2 λ 0

λ 4 λ 2

;

(6)

ν 0 =

ν p =

Relating them ( ν 0 ν p

λ 2 √ λ 0 · λ 4

) defines the bandwidth-parameter

= α 2 =

λ n √ λ 0 · λ 2 n

(7)

α n =

Narrowband processes are characterized by α 1 and α 2 → 1. Vice versa broadband processes result in values α 1 and α 2 → 0. For the latter the decomposition of the variance — the PSD — tends to be evenly distributed along the frequency axis.