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Available online at www.sciencedirect.com Structural Integrity Procedia 00 (2021) 1–11

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ScienceDirect Structural Integrity Procedia 00 (2021) 1–11 Structural Integrity Procedia 00 (2021) 1–11 Structural Integrity Procedia 00 (2021) 1–11 Structural Integrity Procedia 00 (2021) 1–11 Structural I tegrity Procedia 00 (2021) 1–11 Structural Integri y Procedia 0 (2021) 1–11 Procedia Structural Integrity 38 (2022) 260–270

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Fatigue Design 2021, 9th Edition of the International Conference on Fatigue Design

© 2021 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the Fatigue Design 2021 Organizers This paper investigates and quantifies the e ff ects of insu ffi cient sampling on counting algorithms in vibration fatigue. Even though a stress series upholds the Nyquist-Shannon theorem, insu ffi ciently sampled series lead to discrepancies in the fatigue damage estimation, if their peaks are not well resolved in the time domain. Therefore, these discrepancies are investigated by relating the fatigue damage potential of stress series to its up-sampled series using popular signal reconstruction algorithms. In order to find suitable parameters for describing and specifying the error due to insu ffi cient sampling (up to 35%) time series are considered that represent synthetic and realistic PSDs. A best-fitting error formula is obtained that estimates from a PSD whether and to what extent a stress series is subjected to insu ffi cient sampling. This provides more accurate results for lifetime estimation without changing the computational of cycle counting algorithms. Keywords: rainflow counting; insu ffi cient sampling; signal reconstruction; random vibration fatigue; pseudo damage; zero-padding; spectral moments This paper investigates and quantifies the e ff ects of insu ffi cient sampling on counting algorithms in vibration fatigue. Even though a stress series upholds the Nyquist-Shannon theorem, insu ffi ciently sampled series lead to discrepancies in the fatigue damage estimation, if their peaks are not well resolved in the time domain. Therefore, these discrepancies are investigated by relating the fatigue damage potential of stress series to its up-sampled series using popular signal reconstruction algorithms. In order to find suitable parameters for describing and specifying the error due to insu ffi cient sampling (up to 35%) time series are considered that represent synthetic and realistic PSDs. A best-fitting error formula is obtained that estimates from a PSD whether and to what extent a stress series is subjected to insu ffi cient sampling. This provides more accurate results for lifetime estimation without changing the co putational of cycle counting algorithms. Keywords: rainflow counting; insu ffi cient sampling; signal reconstruction; random vibration fatigue; pseudo damage; zero-padding; spectral moments This paper investigates and quantifies the e ff ects of insu ffi cient sampling on counting algorithms in vibration fatigue. Even though a stress series upholds the Nyquist-Shannon theorem, insu ffi ciently sampled series lead to discrepancies in the fatigue damage estimation, if their peaks are not well resolved in the time domain. Therefore, these discrepancies are investigated by relating the fatigue damage potential of stress series to its up-sampled series using popular signal reconstruction algorithms. In order to find suitable parameters for describing and specifying the error due to insu ffi cient sampling (up to 35%) time series are considered that represent synthetic and realistic PSDs. A best-fitting error formula is obtained that estimates from a PSD whether and to what extent a stress series is subjected to insu ffi cient sampling. This provides more accurate results for lifetime estimation without changing the computational of cycle counting algorithms. Keywords: rainflow counting; insu ffi cient sampling; signal reconstruction; random vibration fatigue; pseudo damage; zero-padding; spectral moments This paper investigates and quantifies the e ff ects of insu ffi cient sampling on counting algorithms in vibration fatigue. Even though a stress series upholds the Nyquist-Shannon theorem, insu ffi ciently sampled series lead to discrepancies in the fatigue damage estimation, if th ir peaks are not well resolved in the time domain. Therefore, these discrepancies are investi ated by relati g the fatigu damage p tential of stress series t its up-sampled s ries using popular si nal reconstr ction algorithms. In order to find s itable parameters for describing an specifying the error du to insu ffi cient sampling (up to 35%) ti e series are considered that represent synth tic and realistic PSDs. A b st-fitting error formula is obtained that estimates from a PSD whether and to what extent a stress series is subjected to insu ffi cie t sampling. This provides more ccurate results for lifetime estimation without changing the computational of cycle counting algorithms. Keywords: rainflow counting; insu ffi cient sampling; signal reconstruction; random vibration fatigue; pseudo damage; zero-padding; spectr l moments Nomenclature α 2 bandwidth parameter ( ν ∗ p ) ν p (standardized) peak rate x zp ( t ) zero-padded time series bstract This paper investigates a d quantifies the e ff ects f insu ffi cient sa pling counting algorith s in vibration fatigue. Eve hough a stress series upholds the yq ist-Shannon theore , insu ffi cie tly sa pled seri s lead to discrepancies in the fatigue da age estimation, if heir peaks are not e l resolved in the ti e do in. Therefore, these discrepancies are investigated by relating the fatigue da age potenti l of tress series to its up-sa pled seri s using p pular signal reconstruction lgo ith s. In order to fi d suitable para et rs for describing and specifying the error due to insu ffi cient sa pli g (up to 35 ) time series are considered that represent synthetic and realistic PS s. best-fitting error for ula is obtained that estimates from a PS hether and to hat extent a stress series is subjected to insu ffi cient sa pling. This provides more accurate results for lifetime estimation without changing the computational of cycle counting algorith s. ey ords: rainflo counting; insu ffi cient sa pling; signal reconstruction; rando vibration fatigue; pseudo damage; zero-padding; spectral mom nts Abstract This paper investiga es and quantifi s the e ff ects of insu ffi cient sampling o counting algorithms in vibration fatigue. Even thou h a stress eries upholds the Nyquist-Shannon th orem, insu ffi ciently sampled eri s lead o discrepancies in th fat g e dama e es imation, if their p aks re n t well resolved in the t me omain. Therefore, these discrepancie a e i vestigat d by relating he fat gue damage potential f stress s ries to its up-sampled seri using popular signal reconstruction algorithms. In order to fi d suitable param ters for describing and specifying the error due to insu ffi cient sampling (up to 35%) time series are considered that represent sy thetic and realistic PSDs. A best-fitting error formula is btained that estimates from a PSD whether and to what extent a stress series is subjected to insu ffi ient sampling. This provides more accurate results for lifetime estimation without hanging the computation l of cycle counting algorithms. K ywor s: rainflow counting; insu ffi cient sampling; signal reconstruction; random vibration fatigue; pseudo damage; zero-padding; spectral moments Ab tract This paper investigates and quantifi s the e ff ects of insu ffi cient sampling on counting algorithms in vibration fatig e. Even thou h a stress ser es upholds th Nyquist-Shannon theorem, insu ffi ciently sampled seri s lead to discrep ncies in the fatigue damage estimation, if their peaks are n t ell resolve i the time omain. Therefore, these discrepancies are investigated by relating the fatigue damage potential of stress series to its up-sampled series using popular signal reconstruction algorithm . In order to find suitable param ters for describing and specifying the error due to insu ffi cient sampling (up to 35%) time s ries are considered that represent sy thetic and realis ic PSDs. A best-fitting error formula is obtained that estimates from a PSD whe her and to what ext nt a t ess series is subjected t insu ffi cient sampling. This provides mo e accu te results for lifetime estimation without hanging the computational of cycle counting algorithms. Keywords: rainflow counting; insu ffi cient sampling; signal reconstruction; random vibration fatigue; pseudo damage; zero-padding; spectral moments Abstract Abstract Abstract ∆ t time-domain resolution M , N number of samples ζ up-sampling factor Fatigue Design 2021, 9th Edition of the International Conference on Fatigue Design Fatigue Design 2021, 9th Edition of the International Conference on Fatigue Design Fatigue Design 2021, 9th Edition of the International Conference on Fatigue Design E ff ects of insu ffi cient sampling on counting algorithms in vibration fatigue atigue esign 2021, 9th dition of the International onference on Fatigue esign Fatigue Design 2021, 9th Edition of the International Conference on Fatigue Design E ff ects of insu ffi cient sampli g on counting algorithms in vibration fatigue Fatigue Design 2021, 9th Edition of the International Conference on Fatigue Design E ff ects of insu ffi cient sampling on counting algorithms in vibration fatigue E ff ects of insu ffi cient sampling on counting algorithms in vibration fatigue E ff ects of insu ffi cient sampling on counting algorithms in vibration fatigue E ff ects of insu ffi cient sa pling on counting algorith s in vibration fatigue Arvid Trapp ∗ , Quirin Hoesch, Peter Wolfsteiner ts f i s i t s li ti l rit s i i ration fatigue Arvid Trapp ∗ , Quirin Hoesch, Peter Wolfsteiner Arvid Trapp ∗ , Quirin Hoesch, Peter Wolfsteiner Arvid Trapp ∗ , Quirin Hoesch, Peter Wolfsteiner Munich University of Applied Sciences, Dachauer Str. 98b, Munich 80335, Germany rvid rapp ∗ , uirin oesch, Peter olfsteiner Arvid Trapp ∗ , Quirin Hoesch, Peter Wolfsteiner Arvid Trapp ∗ , Quirin Hoesch, Peter Wolfsteiner Munich University of Applied Sciences, Dachauer Str. 98b, Munich 80335, Germany Munich University of Applied Sciences, Dachauer Str. 98b, Munich 80335, Germany Munich University of Applied Sciences, Dachauer Str. 98b, Munich 80335, Germany unich University of Applied Sciences, Dachauer Str. 98b, Munich 80335, Germany Munich University of Applied Sciences, Dachauer Str. 98b, Munich 80335, Germany Munich University of Applied Sciences, Dachauer Str. 98b, Munich 80335, Germany Abstract

Nomenclature α 2 Nomenclature α 2 Nomenclature α 2 ε error f Nomenclature α 2 Nomenclature α 2 Nomenclature α 2 ∆ t ∆ t ∆ t f Ny f s ∆ t ε ∆ t

( ν ∗ p ) ν p ( ν ∗ p ) ν p ( ν ∗ p ) ν p p ( x ) s eq ( ν ∗ p ) ν p M , N ( ν ∗ p ) ν p ( ν ∗ p ) ν p M , N M , N M , N M , N M , N p ( x ) p ( x ) p ( x ) σ / σ p ( x ) p ( x ) s eq s eq s eq T x ( t ) s eq s eq p ( x ) σ / σ σ / σ x zp ( t ) zero-padded time series x zp ( t ) zero-padded time series x zp ( t ) zero-padded time series CPD conformity of pseudo damage FDS Fatigue Damage Spectrum x zp ( t ) zero-padded time series up-sampling factor CPD conformity of pseudo damage x zp ( t ) zero-padded time series ζ up-sampling factor CPD conformity of pseudo damage x zp ( t ) zero-padded time series ζ up-sampling factor CPD c nformity of pseudo da age ζ ζ ζ ζ up-sampling factor up-sampling factor up-sampling factor MSE mean squared error CPD conformity of pseudo damage CPD conformity of pseudo damage CPD conformity of pseudo damage probability density function DK Dirlik probability density function DK Dirlik probability density function DK Dirlik 2 standard deviation / variance PDF probability density function PSD power spectr l density irlik FDS Fatigue a age Spectru probability density function DK Dirlik FDS Fatigue Damage Spectrum FDS Fatigue Damage Spectrum FDS Fatigue Damage Spectrum RFC rainflow counting MSE mean squared error F S Fatigue Damage Spectrum F S Fatigue Damage Spectrum MSE mean squared error MSE mean squared error MSE mean squared error MSE mean squared error MSE mean squared error σ / σ 2 standard deviation / variance PDF probability density function 2 standard deviation / variance PDF probability density function 2 standard deviation / variance PDF probability density function σ / σ 2 standard dev ation / variance DF probability density function 2 standar dev ation / variance PDF probability density function probability density function DK Dirlik s eq 2 standard deviation / variance PDF probability density function PSD power spectral density RFC rainflow counting PSD power spectral density RFC rainflow counting PSD power spectral density RFC rainflow counting PSD power spectral density RFC rainflow counting PSD power spectral density RFC rainflow counting PSD power spectral density RFC rainflow counting x ( t ) x ( t ) x ( t ) x ( t ) σ / σ σ / σ n probability density function DK Dirlik x ( t ) x ( t ) n n n s n n n s s s s s s T T T T T T (standardized) peak rate (standardized) peak rate (standardized) peak rate order; number of cycles pseudo damage (standardized) eak rate nu ber of sa ples o der; number of cycles robability density function (standardized) peak rate number of samples o der; number of cycles (standardized) eak rate number of s mples order; number of cycles number of samples number of samples number of samples response amplitude order; number of cycles order; number of cycles order; number of cycles pseudo damage pseudo damage pseudo damage period; uration (origi al) time seri s pseudo da age response amplitude pseudo damage response amplitude pseudo amage resp nse amplitude response amplitude response amplitude response amplitude period; duration period; duration period; duration period; duration pe od; duration period; duration (original) time series (original) time series (original) time series (original) time series (original) time series (original) time series

bandwidth parameter time-domain resolution bandwidth parameter time-domain resolution bandwidth parameter time-domain resolution signal frequency Nyquist frequency sampling frequency band idth para eter ti e-do ain resolution bandwidth pa ameter t me-domain resolution bandwidth parameter time-domain resolution signal frequency Nyquist frequency sampling frequency signal frequency Nyquist frequency sampling frequency signal frequency Nyquist frequency sampling frequency Miner exponent n -th order spectral moment (standardized) zero-crossing rate signal frequency yquist frequency sampling frequency signal frequency Nyquist frequency sampling frequency signal frequency Nyquist frequency n -th order spectral moment n -th order spectral moment n -th order spectral moment n -th order spectral moment (standardized) zero-crossing rate n - h order spectral moment (standardized) zero-crossing rate n -th order spectral moment (standardized) zero-c ossing r te (standardized) zero-crossing rate (standardized) zero-crossing rate (standardized) zero-crossing rate error error error error error error

ε ε ε ε ε f f f f f f f Ny f Ny f Ny k λ n f Ny f s f Ny ∆ t f Ny f s f s f s f s f s λ n λ n λ n λ n λ n λ n k k k k k k

G xx ( f ) power spectral density

G xx ( f ) power spectral density G xx ( f ) power spectral density G xx ( f ) power spectral density ( ν ∗ 0 ) ν 0 G xx ( f ) pow r spectral density G xx ( f ) pow r spectral density sampling frequency G xx ( f ) pow r spectral density Miner exponent Miner exponent Miner exponent iner exponent Miner expon n Miner expon nt 1. I troduction

( ν ∗ 0 ) ν 0 ( ν ∗ 0 ) ν 0 ( ν ∗ 0 ) ν 0 ( ν ∗ 0 ) ν 0 ( ν ∗ 0 ) ν 0 ( ν ∗ 0 ) ν 0

2452-3216 © 2021 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the Fatigue Design 2021 Organizers 10.1016/j.prostr.2022.03.027 ∗ Corresponding author. Tel.: + 49-89-1265-3345; email: atrapp@hm.edu ∗ Corresponding author. Tel.: + 49-89-1265-3345; email: atrapp@hm.edu ∗ Corresponding author. Tel.: + 49-89-1265-3345; email: atrapp@hm.edu ∗ Corresponding author. Tel.: + 49-89-1265-3345; email: atrapp@hm.edu ∗ Corresponding author. Tel.: + 49-89-1265-3345; email: atrapp@hm.edu ∗ Corresponding author. Tel.: + 49-89-1265-3345; email: atrapp@hm.edu 1. Introduction 1. Introduction 1. Introduction Perf rming a fatigue strength assessment for variable amplitude loading generally consists of deriving classified load spectra from local stress states in order to compare them with structural strength parameters (e.g. stress-l fe- / Wo¨hler-cu ves). Commonly, stress st tes are available s time-discrete series obtained from strain measurements or nu rical simula ions (e.g. fini e-ele ent metho , mul i-body syst ms). Processing 1. Introduction Perfor ing a fatigue strength assess ent for variable a litude loading generally consists of deriving classified load spectra from local stress states in order to co are the ith structural strength parameters (e.g. stress-life- / Wo¨hler-curves). Commonly, stress states are available as ti e-discrete series obtained fro strain easure ents or nu erical si ulations (e.g. finite-ele ent ethod, ulti-body syste s). Processing 1. Introduction Performing a fatigue strength assessment for variable a litude loading generally consis s of deriving classified load spectra from local stress states in order to compare them with structural strength parameters (e.g. stress-life- / Wo¨hler-curves). Commonly, stress states are available as ti e-discrete series obtained from strain measurements or numerical simulations (e.g. finite-element method, multi-body systems). Processing 1. Intr duction Performing a fatigue strength assessment for variable amplitude loading generally consists of deriving classified load spectra from local stress states in order to compare them with structural strength parameters (e.g. stress-life- / Wo¨hler-curves). Co monly, stress states are available as ti e-discrete series obtained from strain measurements or numerical simulations (e.g. finite-element method, multi-body systems). Processing Performing a fatigue strength assessment for variable amplitude loading generally consists of deriving classified load spectra from local stress states in order to compare them with structural strength parameters (e.g. stress-life- / Wo¨hler-curves). Commonly, stress states are available as time-discrete series obtained from strain measurements or numerical simulations (e.g. finite-element method, multi-body systems). Processing Performing a fatigue strength assessment for variable amplitude loading generally consists of deriving classified load spectra from local stress states in order to compare them with structural strength parameters (e.g. stress-life- / Wo¨hler-curves). Commonly, stress states are available as time-discrete series obtained from strain measurements or numerical simulations (e.g. finite-element method, multi-body systems). Processing Performing a fatigue strength asse sment for variable a plitude loading generally consists of deriving classified load spectra from local stress states in order to compare them with stru tural st ength parameters (e.g. stress-lif - / Wo¨hler-curves). Co monly, stress stat s are available as ti e-discrete series obtained from strain measurements or numerical simulations (e.g. finite-element method, multi-body systems). Processing ∗ Corresponding author. Tel.: + 49-89-1265-3345; email: atrapp@hm.edu