Frequency Domain Approach

Spectral decomposition of a PSD into narrow frequency bands has been suggested as a promising way for estimating the fatigue damage of uniaxial wide-band random loadings. The basic idea has been formulated in some recent publications, which also proposed different combination rules to sum up the damage of each narrow frequency band. The purpose of this technical note is to clarify the analogies, relationships and differences among the approaches developed in such publications.

We consider an elastic/viscoelastic problem for the Bresse system with fully Dirichlet or Dirichlet-Neumann-Neumann boundary conditions. The physical model consists of three wave equations coupled in certain pattern. The system is damped directly or indirectly by global or local Kelvin-Voigt damping. Actually, the number of the dampings, their nature of distribution (locally or globally) and the smoothness of the damping coefficient at the interface play a crucial role in the type of the stabilization of the corresponding semigroup. Indeed, using frequency domain approach combined with multiplier techniques and the construction of a new multiplier function, we establish different types of energy decay rate (see the table of stability results at the end). Our results generalize and improve many earlier ones in the literature (see [7]) and in particular some studies done on the Timoshenko system with Kelvin-Voigt damping (see for instance [9], [23] and [25]). Contents 1. Introduction 1 2. Well-posedness of the problem 4 3. Strong stability of the system 6 4. Polynomial stability for non smooth damping coefficients at the interface 10 5. The case of only one local viscoelastic damping with non smooth coefficient at the interface 17 6. Lack of exponential stability 24 7. Additional results and summary 25 Acknowledgments 26 References 26

In this paper, we study the numerical stabilization of a 1D system of two wave equations coupled by velocities with an internal, local control acting on only one equation. In the theoretical part of this study [3], we distinguished two cases. In the first one, the two waves assumed propagate at the same speed. Under appropriate geometric conditions, we had proved that the energy decays exponentially. While in the second case, when the waves propagate at different speeds, under appropriate geometric conditions, we had proved that the energy decays only at a polynomial rate. In this paper, we confirmed these two results in a 1D numerical approximation. However, when the coupling region does not intersect the damping region, the stabilization of the system is still theoretically an open problem. But, here in both cases, we observed an unpredicted behavior : the energy decays at an exponential rate when the propagation speeds are the same or at a polynomial rate when they are different.

We consider an elastic/viscoelastic transmission problem for the Bresse system with fully Dirichlet or Dirichlet-Neumann-Neumann boundary conditions. The physical model consists of three wave equations coupled in certain pattern. The system is damped directly or indirectly by global or local Kelvin-Voigt damping. Actually, the number of the dampings, their nature of distribution (locally or globally) and the smoothness of the damping coefficient at the interface play a crucial role in the type of the stabilization of the corresponding semigroup. Indeed, using frequency domain approach combined with multiplier techniques and the construction of a new multiplier function, we establish different types of energy decay rate (see the table of stability results below). Our results generalize and improve many earlier ones in the literature and in particular some studies done on the Timoshenko system with Kelvin-Voigt damping.

In this paper, the authors want to draw readers' attention to one long-standing question: which approach is preferable for estimation longevities in fatigue problem, the time domain approach (Rainflow) or the frequency domain one (Dirlik and others)? This question is important in engineering problems, particularly in problems of prolongation of the guaranteed service life. The discussion here is restricted by the longevity evaluation only at the post-processing stage of unidirectional loaded machine parts. It means the realizations might be recorded. Some experimental and speculative evidence of preferable use the Rainflow method is shown. Taking into account the huge computer's power nowadays, the question of the irrelevance of appellation to the calculation accelerating using the spectral methods is specially discussed. There are areas, where the spectral methods are really necessary. There is only a need to recommend the restriction of their application scope to these special situations. It seems there is no need in inventing new spectral complicated algorithms only to stress out at the end, that their result coincides with the Rainflow outcome. That might be confusing for the practicing engineers. Currently, the main attention among supporters of spectral methods is focused on non-stationary and non-Gaussian random processes. (to estimate the spectral density is impossible for non-stationary processes, according to definition). These researchers seem to have forgotten, that even for these complicated situations the decision has already existed: that is the Rainflow and its analogues. The paper shows extensive laboratory experiment results of random fatigue testing of aluminum flat specimens under regular (to build the fatigue curve) and irregular (random) loading. The fatigue life curve (Gassner curve) has been built. These results allowed to compare the existing computation methods of longevity estimation. In the particular situation of narrow-band process, the methods seem to provide comparative results. Considered are methodological issues related to the assessment of the necessary and sufficient realization length, the influence of RMS, cycle counting methods and some possibilities of computing resources saving when using the Rainflow method. The stability of the Rainflow estimates is confirmed. Some problems with the choice of parameters during the longevity assessment by the Dirlik method were noted.

Spectral decomposition of a PSD into narrow frequency bands has been suggested as a promising way for estimating the fatigue damage of uniaxial wide-band random loadings. The basic idea has been formulated in some recent publications, which also proposed different combination rules to sum up the damage of each narrow frequency band. The purpose of this technical note is to clarify the analogies, relationships and differences among the approaches developed in such publications.

Recent dynamic factor models have been almost exclusively developed under the assumption that the common components span a finite-dimensional vector space. However, this finite-dimension assumption rules out very simple factor-loading patterns and is therefore severely restrictive. The general case has been studied, using a frequency domain approach, in Forni et al. (2000). That paper produces an estimator of the common components that is consistent but is based on filters that are two-sided and therefore unsuitable for prediction. The present paper, assuming a rational spectral density for the common components, obtains a one-sided estimator without the finite-dimension assumption.

The characterization of vibration-fatigue strength is one of the key parts of mechanical design. It is closely related to structural dynamics, which is generally studied in the frequency domain, particularly when working with vibratory loads. A fatigue-life estimation in the frequency domain can therefore prove advantageous with respect to a time-domain estimation, especially when taking into consideration the significant performance gains it offers, regarding numerical computations. Several frequencydomain methods for a vibration-fatigue-life estimation have been developed based on numerically simulated signals. This research focuses on a comparison of different frequency-domain methods with respect to real experiments that are typical in structural dynamics and the automotive industry. The methods researched are: Wirsching-Light, the α0.75 method, Gao-Moan, Dirlik, Zhao-Baker, Tovo-Benasciutti and Petrucci-Zuccarello. The experimental comparison researches the resistance to close-modes, to increased background noise, to the influence of spectral width, and multivibration-mode influences. Additionally, typical vibration profiles in the automotive industry are also researched. For the experiment an electrodynamic shaker with a vibration controller was used. The reference-life estimation is the rainflow-counting method with the Palmgren-Miner summation rule. It was found that the Tovo-Benasciutti method gives the best estimate for the majority of experiments, the only exception being the typical automotive spectra, for which the enhanced Zhao-Baker method is best suited. This research shows that besides the Dirlik approach, the Tovo-Benasciutti and Zhao-Baker methods should be considered as the preferred methods for fatigue analysis in the frequency domain.

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Several popular frequency-based fatigue damage models (Wirsching and Light, Ortiz and Chen, Larsen and Lutes, Benascuitti and Tovo, Benascuitti and Tovo with α.75, Dirlik, Zhao and Baker, and Lalanne) are reviewed and assessed. Seventy power spectrum densities with varied amplitude, shape, and irregularity factors from Dirlik's dissertation are used to study the accuracies of these methods. Recommendations on how to set up the inverse fast Fourier transform to synthesize load data and obtain accurate rainflow cycle counts are given. Since Dirlik's method is the most commonly used one in industry, a comprehensive investigation of parameter setups for Dirlik's method is presented. The mean error and standard deviation of the error between the frequency-based model and the rainflow cycle counting method was computed for fatigue slope exponent m ranging from 3 to 12. The results showed Ortiz and Chen, Benascuitti and Tovo with α .75 , Larsen and Lutes, Dirlik, and Benascuitti and Tovo to be significantly more accurate than Lalanne and Zhao and Baker. These five models have a tendency for the error variation to increase as the fatigue exponent m increases. For this study using Dirlik's seventy spectra, Ortiz and Chen, Benascuitti and Tovo with α .75 , and Larsen and Lutes Single Moment methods had lower mean error than Dirlik's method.

In vibration fatigue, flexible structures operate at or close to their natural frequencies. Therefore, it is common to consider the input excitation as well as the stress/strain response of the structure to be Gaussian and stationary. In reality, a non-Gaussian and non-stationary excitation is frequently observed, resulting in a possibly non-Gaussian and non-stationary response.

The effect of mean stress is a significant factor in design for fatigue, especially under high cycle service conditions. The incorporation of mean stress effect in random loading fatigue problems using the frequency domain method is still a challenge. The problem is due to the fact that all cycle by cycle mean stress effects are aggregated during the Fourier transform process into a single zero frequency content. Artificial neural network (ANN) has great scope for non-linear generalization. This paper presents artificial neural network methods for including the effect of mean stress in the frequency domain approach for predicting fatigue damage. The materials considered in this work are metallic alloys. The results obtained present the ANN method as a viable approach to make fatigue damage predictions including the effect of mean stress. Greater resolution was obtained with the ANN method than with other available methods.

This paper treats the theoretical fatigue analysis of non-Gaussian random loadings. It first reviews a method recently developed by the Authors, which is capable to account for the effect of load bandwidth and non-normality. Fatigue lives from simulated Gaussian and non-Gaussian loadings, as well as data from experiments taken from the literature, are compared with theoretical estimations. The non-Gaussian loading was modelled as a transformed Gaussian process. The results show how the load non-normality has a great effect on both the cycle distribution and the fatigue damage; the proposed method seems able to account for this effect correctly. Furthermore, the theoretical predictions are shown to be substantially comparable (with some exceptions) with the experimental data.

The rainflow fatigue damage in bimodal Gaussian processes is discussed. Some methods specifically developed for bimodal processes (e.g. single-moment, Jiao-Moan, Sakai-Okamura and Fu-Cebon method) are compared with the Tovo-Benasciutti method, which is a new approach valid for the general case of wide-band processes. Further, an improvement of the Fu-Cebon method is also proposed. Numerically simulated bimodal random processes are used to compare the accuracies of all methods and to find the optimal range for their applicability. The single-moment, the modified Fu-Cebon and the TB methods are seen as the most accurate methods over a wide combination of bimodal spectra.

The dependence of fatigue damage accumulation on power spectral density (psd) is investigated for several types of random processes relevant to the stresses in offshore structures. Representative data have been obtained from rainflow analysis of time histories simulated from both unimodal and bimodal psd functions. These data are used to test the accuracy of spectral approximation methods in which the damage predictions are made from fairly simple calculational operations on the psd function. Primary emphasis is placed on the commonly used Rayleigh approximation method and a new method that involves calculation of only one spectral moment (i.e., one integral of the psd). The new method is shown to be substantially more accurate than the Rayleigh approximation for some situations.

This research activity refers to the problem of fatigue damage evaluation of mechanical components subjected to random loads. In detail, the present paper describes a procedure, developed by the author, that, starting from component modal modelling, can very quickly gives an answer to the request not only of a qualitative evaluation of its stress state but also of a quantitative and very reliable estimation of the component damage. This estimation is available (both in time and in frequency domain), regardless of the stress state recovery, only by an appropriate elaboration of lagrangian coordinates and elements stress mode shapes. This allows for a quick assessment of the fatigue behaviour upon the whole model or just on an elements subset, prior to the exact evaluation of the damage that always requires a very high computational burden.

The aim of the present paper is to show and validate an innovative method, developed by authors to evaluate , in frequency domain, the fatigue damage of mechanical components modeled by modal approach and subjected to random dynamic loads. The authors, in particular, have theoretically demonstrated that the exact statistical properties (spectral moments) of the PSD functions matrix of stress tensor of the model are obtainable only from PSD functions matrix of its modal coordinates and from PSD functions matrix of inputs. To show the capabilities of this new approach and to verify the obtainable speeding up of the evaluation process two test cases are analyzed and discussed.

The evaluation of the fatigue behavior of mechanical systems and components subjected to random loads is increasingly being addressed to the frequency domain environment both in terms of dynamic analysis and in terms of results post-processing. The procedure set out in the paper (bands method) is formulated without assumptions about the shape of the Power Spectral Density (PSD) function to be analyzed, in order to obtain a criterion that can be applied to any typology of the related random process. The basic idea of the proposed approach is that each PSD function can be divided into frequency bands, for each of them a Rayleigh distribution could be associated and that the aforementioned distributions could be opportunely combined to obtain the actual damage. The method is theoretically developed and numerically validated on a particular subset of PSD functions by comparing the results obtained with those carried out by the time domain damage evaluation reference method and a frequency domain direct approach (Dirlik's method). With regard to the time domain reference criterion the stress time histories are generated by the indirect method (a previously defined authors' method), the cycles spectra are obtained by Rain Flow Counting, (RFC) and the damage is evaluated by Palmgren–Miner's rule.

In vibration fatigue,
exible structures operate at or close to their natural frequencies. Therefore, it is common to consider the input excitation as well as the stress/strain response of the structure to be Gaussian and stationary.
In reality, a non-Gaussian and non-stationary excitation is frequently observed, resulting in a possibly non-Gaussian and non-stationary  response.
The importance of this non-Gaussianity (typically observed via the kurtosis) has resulted in signicant research on the relevance of the Gaussian assumption in fatigue life. For dynamic structures the prior research was mainly theoretically and numerically focused. This work researches the importance of non-Gaussianity and non-stationarity theoretically, numerically and experimentally.
Y-shaped specimens were used in this research. The excitation close to the natural frequency is random and in all the researched cases with the same power spectral density (PSD). While the PSD was kept the same, the rate of non-Gaussianity and the non-stationarity were changed.
The results show that when the excitation is stationary and non-gaussian, the fatigue life is not signicantly impacted, and that standard frequency counting methods are applicable. However, for the case of a non-stationary, non-Gaussian excitation, the fatigue life was found to be signicantly impacted and the Gaussian theoretical approach is questionable.

In frequency domain evaluation of fatigue damage, (Vibration Fatigue) most of the evaluation criteria are based on spectral moments of the power spectral density function (PSD) of stress signal. In numerical simulation this means to perform a dynamic analysis in frequency domain of the whole model (i.e. Finite Element Model) and, consequently, to evaluate the stress PSD function of each element or node. In this paper the authors introduce a method to fast evaluate the spectral moments of the stress power spectral density functions of a finite element model analyzed by modal approach. The authors theoretically demonstrate and validate that the statistical properties (spectral moments) of the power spectral density functions matrix of the stress tensor of each element of a generic numerical modal model are obtainable only by the evaluation of spectral moments of the power spectral density functions matrix of the model modal coordinates and by a simple linear combination with their modal stress shapes. The proposed method obtains the same results than the standard approach, resulting much faster. Only the evaluation, one time, of the spectral moments of the modal coordinates PSD functions matrix is needed. By a simple linear combination of these with the stress mode shapes is then possible to directly obtain, for each element, stress spectral moments, avoiding a lot of integral calculations, proportional to the elements numerousness.

The well-known problem with the fatigue lifetime assessment of non-Gaussian loading signals with the use of spectral method has been presented in the paper. A correction factors that transform the non-Gaussian signal into an equivalent Gaussian signal proposed by Bracessi et al. (2009) has been used for the purpose of lifetime calculations together with Palmgren-Miner Hypothesis. The calculations have been performed for the 10HNAP steel under random non-Gaussian load with four dominating frequencies. The signal has been generated on the test stand SHM250 for random tension-compression tests. The results with zero and non-zero mean stresses have been used to calculate the fatigue life with the frequency domain method based on Dirlik's model and with a time domain method with the use of the rainflow cycle counting algorithm. The obtained calculation results have been compared with experimental results.

A review of the available methods for estimating fatigue damage from variable amplitude loading is presented. The dependence of fatigue damage accumulation on power spectral density (psd) is investigated for stationary Gaussian random processes. Improvements to the Rayleigh approximation are examined by comparison to rainflow analysis of time histories simulated from psd functions representative of simple theoretical and real world applications. Spectral methods investigated include those by Wirsching and Light, Ortiz and Chen, the Dirlik formula, and the Single-Moment method, among other recent methods. Guidelines and limitations are given for using spectral methods to increase confidence in the damage estimate.

Fatigue failures of components subject to random stress histories with bi-modal spectral densities are investigated. Such stress spectral densities are often experienced by chassis and suspension components of road vehicles. Some of the methods for predicting fatigue damage using frequency-domain statistics of random stress processes are reviewed. A new 'bi-modal' model is devised for calculating the probability density function of stress ranges in the case of a spectral density with two peaks. The method has the attraction of converging to the same results as the well known Rayleigh distribution for narrow band processes, if either of the spectral peaks is small. A component in a heavy road vehicle suspension is used as a case study. Sensitivity analyses are conducted to compare the new method with various other frequency-domain approaches, and with the widely accepted rain-flow method of cycle counting in the time domain.

The use of the frequency domain approach in the virtual estimation of mechanical component fatigue life under random loads is related to two conditions regarding the dynamic behaviour of components and the state of stress. The mechanical system must have linear behaviour and the probability density function of stress must be Gaussian, respectively. Obviously, these conditions are not independent, because there is a close tie between the transformations induced by the system to the random inputs and stress distribution. The rigorous procedure for the extension of these hypotheses is not available and only approximated approaches can be used: normally these are based on a corrective coefficient to the narrow-band formula. The main goal of this report is to suggest a separation of the effects on the corrective coefficient. In this manner, the global coefficient can be seen as the product between a partial coefficient related only to the wide-band effects of stress power spectral density function and another one dependent on non-normality indices of stress probability density function. A meaningful application has been investigated to validate the practical employment of this approach. By this example the authors also defined an original analytical expression of a corrective coefficient for Gaussian damage; however, the formulation has to be improved by other applications, because its validity is tested only on a too much limited domain of Kurtosis values. Moreover, the authors suggest that a modal approach to the stress recovery procedure of a flexible body might be an interesting way to the rapid identification of non-Gaussianity indices in the analysis of frequency and time domain dynamics. For this reason, they believe that the investigation of tying the stress non-Gaussianity to the non-Gaussianity of the component modal coordinates to be useful.

The aim of this paper was the identification of the load conditions to consider during the design and assessment of military vehicles. This meant identifying test tracks and their corresponding ground speeds and track lengths, and to define their combinations that are useful to simulate the load spectrum representing the system/component life in an accelerated way. The result of this activity was the development of a formulation for the synthesis of a single power spectrum density function (PSD) representing multiple load conditions expressed by a set of PSD functions. This function is equivalent in terms of damage to the given PSD functions set. Another important result was the definition of a virtual test ring, equivalent to those defined in standards for equipment and supplies, but useful for the virtual check of the military vehicle integrity. This new formulation and the proposed test ring intend to supply tools that would be very useful in the early phases of vehicle design and assessment. In order to verify these results, numerical and experimental analyses of a generic military vehicle were undertaken. Results demonstrated the validity of the load synthesis formulation and the load combination that are useful for the assessment of military vehicle durability.

The use of the frequency domain approach in the virtual estimation of mechanical component fatigue life under random loads is related to two conditions regarding the dynamic behaviour of components and the state of stress. The mechanical system must have linear behaviour and the probability density function of stress must be Gaussian, respectively. Obviously, these conditions are not independent, because there is a close tie between the transformations induced by the system to the random inputs and stress distribution. The rigorous procedure for the extension of these hypotheses is not available and only approximated approaches can be used: normally these are based on a corrective coefficient to the narrow-band formula. The main goal of this report is to suggest a separation of the effects on the corrective coefficient. In this manner, the global coefficient can be seen as the product between a partial coefficient related only to the wide-band effects of stress power spectral density function and another one dependent on non-normality indices of stress probability density function. A meaningful application has been investigated to validate the practical employment of this approach. By this example the authors also defined an original analytical expression of a corrective coefficient for Gaussian damage; however, the formulation has to be improved by other applications, because its validity is tested only on a too much limited domain of Kurtosis values. Moreover, the authors suggest that a modal approach to the stress recovery procedure of a flexible body might be an interesting way to the rapid identification of non-Gaussianity indices in the analysis of frequency and time domain dynamics. For this reason, they believe that the investigation of tying the stress non-Gaussianity to the non-Gaussianity of the component modal coordinates to be useful.

The use of the frequency domain approach in the virtual estimation of mechanical component fatigue life under random loads is related to two conditions regarding the dynamic behaviour of components and the state of stress. The mechanical system must have linear behaviour and the probability density function of stress must be Gaussian, respectively. Obviously, these conditions are not independent, because there is a close tie between the transformations induced by the system to the random inputs and stress distribution. The rigorous procedure for the extension of these hypotheses is not available and only approximated approaches can be used: normally these are based on a corrective coefficient to the narrow-band formula. The main goal of this report is to suggest a separation of the effects on the corrective coefficient. In this manner, the global coefficient can be seen as the product between a partial coefficient related only to the wide-band effects of stress power spectral density function and another one dependent on non-normality indices of stress probability density function. A meaningful application has been investigated to validate the practical employment of this approach. By this example the authors also defined an original analytical expression of a corrective coefficient for Gaussian damage; however, the formulation has to be improved by other applications, because its validity is tested only on a too much limited domain of Kurtosis values. Moreover, the authors suggest that a modal approach to the stress recovery procedure of a flexible body might be an interesting way to the rapid identification of non-Gaussianity indices in the analysis of frequency and time domain dynamics. For this reason, they believe that the investigation of tying the stress non-Gaussianity to the non-Gaussianity of the component modal coordinates to be useful.

Abstract: This research activity refers to the problem of fatigue damage evaluation of mechanical components subjected to random loads. In detail, the present paper describes a procedure, developed by the author, that, starting from component modal modelling, can very quickly gives an answer to the request not only of a qualitative evaluation of its stress state but also of a quantitative and very reliable estimation of the component damage.