PSI - Issue 17

Arvid Trapp et al. / Procedia Structural Integrity 17 (2019) 379–386 Arvid Trapp/ Structural Integrity Procedia 00 (2019) 000 – 000 = 60 Hz. The left-hand side of Fig. 3 assembles the information of the excitation. It shows a Gaussian realization ( ) ∼ , ( ) that is derived from the PSD. The Gaussian realization is modulated by the synthetically generated modulating signal representing the non-stationary character. Its effects can be summarized by the irregularity factors 2 , 4 that preserve the variance but scale the kurtosis. The representation of the latter over frequency 4 ( ) suggests a uniform modulation of the frequency content. Applied to measured data, this would mark the starting point of the method. The bottom shows the load spectra (inverted) that are derived from the Gaussian and the AM excitation. Their ratio defines the transformation (⋅) due to the non-stationary behavior which is to be retrieved in a response analysis. 385 7

Ͳ ͳͲ ʹͲ ͵Ͳ ͶͲ ͷͲ ͸Ͳ – ȏ•Ȑ Ǧͳ ǦͲǤͷ Ͳ ͲǤͷ ͳ Gaussian response (displacement) ʹ α ͲǤͲͶ͹Ǣ α ͵ǤͲͳ Ͳ ͳͲ ʹͲ ͵Ͳ ͶͲ ͷͲ ͸Ͳ – ȏ•Ȑ Ͳ ͲǤͷ ͳ ͳǤͷ ʹ modulating signal ʹ α ͳǢ Ͷ α ͵Ǥ͵Ͳ Ͳ ͳͲ ʹͲ ͵Ͳ ͶͲ ͷͲ ͸Ͳ – ȏ•Ȑ ǦͲ Ǧ Ǥ ͳ Ͳ ͷ ͲǤͷ ͳ modulated Gaussian response ʹ α ͲǤͲͶ͹Ǣ α ͻǤ͸ͷ Ͳ ͳͲ ʹͲ ͵Ͳ ͶͲ ͷͲ ͸Ͳ – ȏ•Ȑ ǦͲ Ǧ Ǥͷ ͳ ͲǤͷ Ͳ ͳ non-Gaussian response (displacement) ʹ α ͲǤͲͶ͹Ǣ α ͻǤͷ͸ response PSD

Gaussian excitation (acceleration) ʹ α ʹͻǤͳ͸Ǣ α ͵ǤͲͳ

ǦʹͲ Ͳ ʹͲ

y (t) g

x (t) g

d c

š ‰ ȏȀ• ʹ Ȑ

m

› ‰ ȏ Ȑ

Ͳ ͳͲ ʹͲ ͵Ͳ ͶͲ ͷͲ ͸Ͳ – ȏ•Ȑ Ͳ ͳͲ ʹͲ ͵Ͳ ͶͲ ͷͲ ͸Ͳ – ȏ•Ȑ Ͳ ͲǤͷ ͳ ͳǤͷ ʹ modulating signal ʹ α ͳǢ Ͷ α ͵Ǥ͵Ͳ modulated excitation (acceleration) ʹ α ʹͻǤʹͲǢ α ͳͲǤͲͲ

 ȏǦȐ š ƒ ȏȀ• ʹ Ȑ Ͷ ȋˆ Ȍ ȏǦȐ

 ȏǦȐ

ǦʹͲ Ͳ ʹͲ

m  ȏȐ › ‰

d c

x (t) am

ͳͲ ʹͲ ͵Ͳ ͶͲ ͷͲ ͸Ͳ – ȏ•Ȑ 4th-order irregularity

Ͳ

y (t) am

Ͳ Ͳ ͳ ʹ ͵ Ͷ

‡ƒ ‘ˆ Ͷ ȋˆ Ȍ α ͵Ǥ͵ͳʹ Ͷ α ͵Ǥ͵Ͳ

› ƒ ȏȐ

ʹͷͲ ˆ ȏ œȐ

ͷͲ

ͳͲͲ

ͳͷͲ

ʹͲͲ

excitation PSD

SDOF transfer function ˆ

››ǡ‰ Ǣ ››ǡƒ

ššǡ‰ Ǣ šš ǡƒ

α ͸Ͳ œ

ͳͲ ǦͶ ȁ ȋˆ Ȍȁ ȏȋȀȋȀ• ʹ ȌȐ

ͳͲ Ǧͳ

šš ȋˆ Ȍ ȏȋȀ• ʹ Ȍ ʹ Ȁ œȐ

›› ȋˆȌ ȏȋȌ ʹ Ȁ œȐ

Ͳ ͳͲͲ ʹͲͲ ͵ͲͲ ˆ ȏ œȐ ͳͲ Ǧͷ freq.-domain fatigue assessment

Ͳ ͳͲͲ ʹͲͲ ͵ͲͲ ˆ ȏ œȐ ͳͲ Ǧ͸ time-domain fatigue assessment

——Žƒ–‡† •–”‡•• …›…Ž‡•  ‹ ȏǦȐ Ͳ ʹͲ ͶͲ ͸Ͳ ͺͲ excitation load spectra Ͳ ͳͲͲ ʹͲͲ ͵ͲͲ ˆ ȏ œȐ ͳͲ Ǧʹ ͳͲ Ͳ ͳͲ ʹ ͳͲ Ͷ ‘Ǧ ƒ—••‹ƒ ‡š…‹–ƒ–‹‘ ȋš ƒ Ȍ ƒ—••‹ƒ š…‹–ƒ–‹‘ ȋš ‰ Ȍ –”ƒ•ˆ‘”ƒ–‹‘ ȋ‰Ȍ ‰ ȏǦȐ

Ͳ ͲǤͷ ͳ ͳǤͷ ʹ ʹǤͷ ͵ Ͳ ͳͲ ʹͲ ͵Ͳ ͶͲ ͷͲ ͸Ͳ •–”‡•• ƒ’Ž‹–—†‡ • ȏ ƒȐ ͳͲ Ͳ

ͳͲ Ͳ Ͳ ͳͲ ʹͲ ͵Ͳ ͶͲ ͷͲ ͸Ͳ •–”‡•• ƒ’Ž‹–—†‡ • ȏ ƒȐ

ƒ—••‹ƒ ”‡•’‘•‡ ȋ ››ǡ‰ ǡ Ȍ ‘””‡…–‡† ƒ—••‹ƒ ”‡•’‘•‡ ȋ ››ǡ‰ ǡ‰ǡ Ȍ ‘Ǧ ƒ—••‹ƒ ”‡•’‘•‡ ȋ› ƒ ǡ Ȍ

ƒ—••‹ƒ ”‡•’‘•‡ ȋ› ‰ ǡ Ȍ ‘””‡…–‡† ƒ—••‹ƒ ”‡•’‘•‡ ȋ› ‰ ǡ‰ǡ Ȍ ‘Ǧ ƒ—••‹ƒ ”‡•’‘•‡ ȋ› ƒ ǡ Ȍ ‘†—Žƒ–‡† ƒ—••‹ƒ ”‡•’‘•‡ ȋ› ‰ ǡǡ Ȍ

ȏȀ• ʹ Ȑ

Ž‘ƒ† ƒ’Ž‹–—†‡ •

ͳͲ ʹ ——Žƒ–‡† •–”‡•• …›…Ž‡•  ‹ ȏǦȐ ͳͲ Ͷ

ͳͲ ʹ ——Žƒ–‡† •–”‡•• …›…Ž‡•  ‹ ȏǦȐ ͳͲ Ͷ

Figure 3. Simulation of SDOF system subjected to AM non-stationary loading.