Fourth International Conference on Material and Component Performance under Variable Amplitude Loading
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Vibration Fatigue

Session chair: Carpinteri, Andrea, Professor (Germany)
Shortcut: A
Date: Monday, 30. March 2020, 11:00
Room: Hall A
Session type: Oral


11:00 A-01

Vibration fatigue at impact excitation: time vs frequency domain (#19)

P. Ogrinec1, J. Slavič1, M. Česnik1, M. Boltežar1

1 University of Ljubljana, Faculty of Mechanical Engineering, Ljubljana, Slovenia

In vibration fatigue three types of loads are typical; random, harmonic and impact. When dealing with real loads, any combination of these three types is possible. While the analysis of fatigue life in the cases of harmonic and random loads is well researched, the impact loads, or the combination of impact and any of the other two load types, were not yet investigated in detail in the frequency domain.

The main focus of this research is a theoretical study of the fatigue damage analysis in time- and frequency-domain of a single-degree-of-freedom dynamic system under well-separated half-sine impulse excitation. Based on this theoretical study a relation between the time- and frequency-domain damage is introduced, which is based on the dynamic properties of a flexible structure (natural frequency, damping), material parameters, and the loading characteristics. It was found that the frequency domain damage estimation differs significantly from the time domain damage; however, based on the introduced theoretical relation between the time- and frequency-domain damage estimation a frequency-domain correction can be applied. Besides theoretical, also experimental validation is presented.

This research provides the fundamental theoretical background for further research of impact loads, for instance for cases where the impacts are not well separated, or combination of different load types.


[1] D. E. Newland, An introduction to random vibrations, spectral & wavelet analysis, Courier Corporation, 2012. 215
[2] F. Cianetti, A. Alvino, A. Bolognini, M. Palmieri, C. Braccesi, The design of durability tests by fatigue damage spectrum approach, Fatigue & Fracture of Engineering Materials & Structures 41 (4) (2018) 787–796.

[3] D. Benasciutti, R. Tovo, Frequency-based analysis of random fatigue loads: Models, hypotheses, reality: Frequenzbasierte analyse zufälliger ermüdungsbelastungen: Modelle, hypothesen, praxis, Materialwissenschaft und Werkstofftechnik 49 (3) (2018) 345–367.

[4] M. Mršnik, J. Slavič, M. Boltežar, Frequency-domain methods for a vibration-fatigue-life estimation–application to real data, International
Journal of Fatigue 47 (2013) 8–17.

[5] M. Mršnik, J. Slavič, M. Boltežar, Vibration fatigue using modal decomposition, Mechanical Systems and Signal Processing 98 (2018) 548–556.

[6] P. Ogrinec, J. Slavič and M. Boltežar. Harmonic Equivalence of the Impulse Loads in Vibration Fatigue, Journal of Mechanical Engineering 65(2019)11-12, 631-640

[7] P. Ogrinec, J. Slavič, M. Česnik and M. Boltežar. Vibration fatigue at half-sine impulse excitation in the time and frequency domains, International Journal of Fatigue, Volume 123, June 2019, Pages 308-317

Response of a single degree of freedom (SDOF) system at impact excitation
The SDOF system response to impact excitation result in several load cycles with decreasing amplitude loads. The damage can be estimated in the time- and frequency-domain.
Ratio between the time- and frequency-domain methods (narrow-band), with fixed material parameters
The ratio between the time- and frequency-domain damage estimation differs with the natural frequency and coefficient of damping of the system.
Keywords: vibration fatigue, impact excitation, narrow-band
11:20 A-02

Using The Experimental Data In Form Of Gasser Curve To Compare The Methods For Longevity Estimation (#30)

I. Gadolina1, A. V. Erpalov1

1 IMASH RAS, Moscow, Russian Federation

Having the good results in some scientific problems (vibration of bridges, high-rise buildings in the wind, earthquake impacts) spectral approach, namely employing stress power spectral density Sσ(f) for predicting fatigue at post-processing stage for fatigue estimation, seem to be overestimated and over-promoted. The rain-flow estimation in the time domain, having good experimental approbation and being the result of the clear algorithm, has the physical sense (closed loops explanation) and is applicable at post-processing stage in a natural way even to non-stationary and non-gaussian signals. For some reasons since the early stages of random loading investigating in fatigue, scientists are trying to employ the methods, which deal with Sσ(f) in predicting fatigue at the post-processing stage. The most likely reason for this substitution lays in the fact, that long time ago there was no proper equipment for registration of long enough loading processes causing the fatigue. Many experiments used to be conducted on vibrating machines and researchers possessed only information about spectrum power distribution among the frequencies. Nowadays the means for registration are abundant, as well as the methods for the compact loading history presentation. It is very difficult to understand why some scientists develop the complicated formulae for estimating the rain-flow distribution through Sσ (f) just to finally report, that their results are in the good agreement with the traditional rain-flow method being applied in the time domain.

Many spectral approach methods suppose utilization of the so-called spectral moments of Sσ (f) for the estimation of the distribution of the rain-flow cycles. Fig. 1 gives the example, how differently the estimated spectral density for the same signal might be [1]. Looking at the Fig.1 it became clear, that the moments for those two spectra options would be completely different. A big question arises about the representativeness of the estimations based on so unstable characteristic.

The main arguments explaining the possible benefits of spectral approach from the adepts of spectral methods are: 1) accelerating of computer productivity; 2) Better consideration of the random nature of the events. Answering to argument 1), we dare to contradict, that there is no problem with computer productivity as well as with computer memory limitation these days. Best rain-flow algorithms in time domain allow treating long enough realization in the real-time mode. As for the argument 2), it is possible to speculate, that indeed, there might be a problem with the statistical representation of the situation in reality, and it will not disappear if the problem is being represented in the frequency domain instead of the time domain.

To demonstrate the fact, that the spectral density does not influence much upon the longevity, a model for the approximation of the peak sequence by the continuous random process was developed. The model could revile some contradictions in the spectral approach. The model substitutes the sequence of turning points by the continuous process, consisting of the half-cosines, utilizing the special rules for amplitudes, periods and phases. The known stress loading realization in the part of a cantilever machine (realization I) was substituted by the simulated one following the proposed model (realization II) with different frequency content. The spectral densities, therefore, are different for two processes (Fig.2). On the other hand, the rain-flow distributions, as well as estimated longevities, are the same for both processes.


1. S.L Marpl. Digital spectral analysis with applications. 1986. Prentice-Yall, New Jersey. 584 p.

Fig.1 Two results of spectral density estimation for the same process [1]
Fig.2. Comparing the spectral densities of the two similar by longevity estimation processes
Keywords: metal fatigue, rain-flow, spectral methods
11:40 A-03

The role of uncertainty of power spectral density data in estimating the fatigue damage of random uniaxial loadings through frequency-domain methods (#38)

D. Benasciutti1

1 University of Ferrara, Department of Engineering, Ferrara, Italy

Two approaches are commonly followed for estimating the fatigue damage of a random stress time-history. One of them (time-domain approach) is based on a direct analysis of stress signals though the rainflow counting and the Palmgren-Miner rule. The other one (frequency-domain approach) is based on the so-called spectral methods and exploits a frequency-domain analysis in which a random loading or stress is characterized through its frequency spectrum – called Power Spectral Density (PSD). Through the theory of random processes, analytical expressions can be derived by which the fatigue damage and service life can be estimated directly from the PSD of the random stress (see Figure 1). In the literature, analytical solutions were made available for stationary random loadings with normally distributed values [1,2], although corrections for non-Gaussian case were also proposed [3,4].

In spectral methods, the analytical expressions of damage are functions of several quantities (spectral moments, bandwidth parameters) that are computed directly from the stress PSD (see Figure 1). Such expressions have usually been calibrated on results of numerical simulations, in which the stress PSD – and so the spectral parameters – were known quantities (i.e. in simulations, the stress PSD was a known input).

In engineering applications, however, the stress PSD is not known in advance and needs to be estimated from measurements. For example, in the Welch’s method a measured time-history is divided into “n” overlapped sub-segments from which the PSD is derived. It can be demonstrated that the estimated spectrum follows a Chi-square distribution with “n” degrees of freedom. A confidence interval can also be constructed to enclose the “true” (but unknown) spectrum [5]. The confidence interval depends on the signal time length and on the number of sub-segments.

Sampling variability effects are present in the estimated PSD computed from stress time-histories, and in turn in the spectral parameters derived from it and thus in the corresponding frequency-domain damage derived from the estimated spectrum (see Figure 2). Understanding the sampling variability of frequency-domain fatigue damage is of paramount importance to increase the confidence in the use of spectral methods in practical applications.

Starting from the formula of the confidence interval for an estimated spectrum, this work aims to investigate how sampling variability is transferred from an estimated PSD down to the fatigue damage assessed through a frequency-domain approach.


1. Benasciutti D, Tovo R. Spectral methods for lifetime prediction under wideband stationary random processes. Int J Fatigue 2005;27(8):867-77.

2. Benasciutti D. Fatigue analysis of random loadings. A frequency-domain approach. LAP Lambert Academic Publishing; 2012.

3. Benasciutti D, Tovo R. Cycle distribution and fatigue damage assessment in broad-band non-Gaussian random processes. Probab Eng Mech 2005;20(2):115-127.

4. Benasciutti D, Tovo. Frequency-based analysis of random fatigue loads: Models, hypotheses, reality. Mater und Werkstoff 2018;49(3):345-367.

5. Bendat, J.S., Piersol, A.G. (2010). Random data. Analysis and measurements procedures, 4th ed., Wiley, USA.

Figure 1. A stress Power Spectral Density (PSD) and related spectral parameters. An example of analy

The figure displays a stress Power Spectral Density (PSD) with related spectral moments and bandwidth parameters. An example of analytical expression for estimating the fatigue damage from PSD parameters is also shown.

Measured time-history, estimated PSD and fatigue damage (confidence intervals).

The figure shows how a measured time-history is used to estimate a PSD with confidence interval. A proposal of confidence intervals for fatigue damage is also shown.

Keywords: Random loading, Power Psectral Density, statistical variability
12:00 A-04

Fatigue damage assessment model in frequency domain for non-Gaussian strain energy density parameter (#37)

M. J. Böhm1, D. Benasciutti2

1 Opole University of Technology, Department of Mechanics and Machine design, Faculty of Mechanical Engineering, Opole, Poland
2 University of Ferrara, Department of Engineering, Ferrara, Italy

The strain energy density model often referred as the energy parameter model describes the energy of the combined stress-strain state of the material. This parameter allows to take into account either the elastic or plastic state of the material in terms of calculations or a combination of these states [1]. As it was noticed by many scientists like Kujawski the plastic strain amplitude alone has a major influence on the fatigue damage, but it is insufficient in many cases especially if we are analyzing the multiaxial state of material [2]. That’s why we should take into account information of both strain and stress. Other important fact that tips the scale on the side of this method is that for low cycle and high cycle fatigue life regions the strain energy density is a constant damage parameter, which is not the case with either standing alone strain or stress models. That’s why the energy of the hysteresis loop seems to be a good factor to describe fatigue damage. The time domain is using cycle counting algorithms to assess the damage degree of the material. Beside this it has many advantages towards to the frequency domain which uses statistical information obtained from the power spectral density (PSD), but one huge disadvantage which is the computation time. There are still unsolved issues like the proper use of the strain energy density directly with the proper domain description. If we take a look at a simple case of strain energy density described with the use of the stress then we can notice, that even though the stress loading time history is Gaussian the energy history will be non-Gaussian. For those occupied with frequency domain methods it is well known that non-Gaussianity is a major issue in damage assessment as described by Benasciutti and Tovo [3]. A simple Fourier Transformation (FFT) in the case of a non-Gaussian signal will create a PSD that is Gaussian and therefore it will give inaccurate results in the next steps of the damage assessment. Therefore we need to compensate the information about the non-Gaussianity in another form by either using transformations or like in the case of this paper to propose a direct damage intensity model that takes into account the kurtosis and skewness values of the base signal. This paper presents a first attempt to describe the damage intensity of strain energy parameter directly with the use of a narrowband stress distributed random signal. For comparison the calculations are performed for the time domain damage assessment with the use of the rainflow and Palmgren-Miner hypothesis for stress as well as strain energy density as presented in Figure 1 for an idealized material state with slope value k=3 and C=1. The frequency domain damage calculations are performed with the classic formula for narrow band damage and the use of a new model, that is using spectral moment information of the narrow band power spectral density of stress, that is used in the strain energy density description process. The formulation of the model is explained stepwise. The important fact of the model is that it takes into account the non-Gaussian characteristic of the strain energy signal. The obtained results show good compatibility between the damage models.


1. Böhm M., Łagoda T., Fatigue life assessment with the use of the spectral method for non-gaussian loading histories with the use of the energy parameter, Journal of Machine Construction and Maintenance, vol. 1 (2018) 27–31.

2. Ellyin F., Kujawski D., Plastic strain energy in fatigue failure, Journal of Engineering Materials and Technology, vol 106 (1984) 342-347

3. Benasciutti D., Tovo R., Cycle distribution and fatigue damage assessment in broad-band non-Gaussian random processes, Probabilistic Engineering Mechanics, vol. 20 (2005) 115-127

Figure 1
Damage intensity comparison between the stress and strain energy density time histories in both the time and frequency domain
Keywords: fatigue, frequency domain, damage model, strain energy density