Strojniški vestnik - Journal of MechanicalEngineering65(2019)ll-12,631-640 © 2919 JournC of Itehamca^naneermg. AHrightsresvrved. D0I:10.5541/sv-|me.2019.41 sv Original Scientific Paper Recewed for review: 2019-0&10 Receivad rvv^sdformi 2019-1°-14 AccegtaVfvr fublicaticn:2019-10-10 Harmonic Equivalence of the Impulse Loads in Vibration Fatigue Primož Ogrinec - Janko Slavic* - Miha Boltežar University of Ljubljana, Faculty of Mechanical Engineering, Slovenia In vibration fatigue, three unique types of loads are typical: random, harmonic and impulse. In an application any of these loads are possible. A fatigue-life analysis is possible in the time and frequency domains using the frequency-response function of a structure. Recent studies demonstrated that the impulse loads influence the accuracy of a fatigue-life prediction in the frequency domain. The focus of this research is a theoretical study of an equivalent harmonic load to the impulse load on a single-degree-of-freedom system in order to investigate the feasibility of impulse loads in vibration testing. This research shows that there is a relationship between the impulse and harmonic loads that is related to the underlying dynamic properties (e.g., damping, natural frequency). Based on a theoretical analysis an experimental procedure was developed for both cases of excitation, which was able to confirm the theoretical analysis. Using the modal decomposition the single-degree-of-freedom approach can be generalized to multiple-degrees-of-freedom systems. Keywords: vibration fatigue, random loads, spectral methods, fatigue life, stationary and non-stationary loading, non-Gaussian loading, vibration testing Highlights • Non-stationary and non-Gaussian loads lead to significantly shorter fatigue lives of a structure. • Impulses, that can occur in the loading signal, render the loading signal non-Stationary and non-Gaussian. • The equivalence between impulse and harmonic loads, with regard to fatigue testing is presented with an analytical derivation. • Control strategies for impulse and harmonic fatigue tests are presented. • The theoretical procedure was experimentally verified on 18 samples, with impulse and harmonic loading. • Material's fatigue parameters were identified for G-AlSi8Cu3(226). 0 INTRODUCTION As structures are becoming lighter and loads optimised, the effects of structural dynamics [1] and random loads on the fatigue life of flexible structures are becoming more important. This is known as vibration fatigue and has been the subject of various studies in recent years [2] to [7]. Vibration fatigue is focused in loads well below the yield stress (i.e., high-cycle fatigue), which is typically researched in the time domain (e.g., rainflow-counting algorithm [8]) or in the frequency domain (e.g., the narrow band [9], Dirlik [2] or Tovo-Bennasciutti methods [5] and [3]). When studying fatigue life in the frequency domain it is common to assume that the excitation signals and consequently the stress response of a structure are Gaussian and stationary [10] and [11]. In recent years great efforts have been made to develop the frequency-counting methods in the analysis of the fatigue life of structures excited with non-Gaussian and non-stationary excitation signals [12] to [16]. Tovo and Bennasciutti studied non-stationary switching random loads [4]. Song and Wang [17] presented a spectral-moment-equivalence lumped block method that improves the accuracy of the fatigue-life analysis for non-stationary, non-Gaussian loads and incorporates the material parameters into the equivalent spectral moments formula. Bracessi et al. [18] and Niu et al. [19] researched the influence of load Kurtosis and skewness on the damage rate in the case of non-Gaussian signals; Wolfsteiner and Sedlmair [20] and Cianetti et al. [21] presented correction factors based on those two characteristics of the loading signal. In real cases it is common to experience some forms of non-Gaussian loading [22], which can also be the result of the impulses superimposed on the random loading of a structure [23]. These impulses can be the consequence of geometric non-linearities, contact conditions, clearances, wear, etc. While the fatigue life under a combination of harmonic and random loads can be studied in the frequency domain [24], [25] and [6], the effects of these impulse loads are not well researched. The presence of impulses renders the signal non-stationary. Capponi et al. [15] and Palmieri et al. [16] noted that in the case of such signals the fatigue-life assessments with spectral methods in the frequency domain return a significantly longer life prediction than were observed in the experimental testing. Hence, the development of new methods that can account for the presence of impulses in the stress response is of great importance. *Corr. Author's Address: University of Ljubljana, Faculty of Mechanical Engineering, Laboratory for Dynamics of Machines and Structures, Aškerčeva 6, 1000 Ljubljana, Slovenia, janko.slavic@fs.uni-lj.si 631 Strojnioki vestnik -Journalof Mechanical Engineering 65(2010119-13,631-640 As was recently shown by Ogrinec et al. [23] the most influential parameters on the accuracy of the frequency-domain methods on the fatigue-life prediction at impulse excitation are: the significant natural frequency, the length of the impulse [26], the coefficient of damping, the modal mass, and the material parameters of the Wohler curve. For that reason, accurate identification of the material's fatigue parameters is crucial. The material parameters, however, can sometimes vary based on the method used to obtain them. It is known, for instance, that the strain rate can significantly influence measurements of the yield stress and the ultimate tensile strength [27]. Similarly, the identification of the material's fatigue parameters can be influenced [28]. For reasons of simplicity, it is common to use a harmonic loading signal for fatigue testing [29], although some other loading signals are possible [30] and [31]. However, since the influence of the material's fatigue parameters on the fatigue-life estimation's accuracy was proven [23], the question of the validity of the parameters of the Wohler curve obtained with a harmonic test remains open. Therefore, an experimental exploration of the effects of impulse loading on the identification of material's fatigue properties is necessary. The main goal of this study is to analyse the possibility to determine the parameters of the Wohler curve, based on a fatigue test, where the sample is excited with impulses. An experimental setup is presented, where only one significant mode shape was excited. The system is free to respond to the half-sine impulse; therefore, the sample experiences a variety of cycles with varying amplitudes. To produce an equivalent load, the hypothesis of a linear accumulation of damage [32] is assumed. The parameters of the Wohler curve are compared to an equivalent test, where the sample is excited harmonically. This manuscript is organised as follows. Sec. 1 presents the theoretical background. The theoretical comparison between the damage accumulation due to harmonic and impulse excitationx is presented in Sec. 2. In Sec. 3 the experimental setup with control algorithms for both load cases is presented. The results are presented in Sec. 4. Sec. 5 draws the conclusions. 1 THEORETICAL BACKGROUND When a dynamic system is excited in the frequency range of its natural frequencies the response of the system is amplified. If the system is excited with a half-sine impulse, the frequency range of the excitation is determined with the length of the impulse. To take advantage of the response amplification, which is particularly useful in fatigue testing, the impulse has to be specified with this range of excitation in mind. However, when the system is excited harmonically, the frequency of excitation has to be close enough to a significant natural frequency to obtain the dynamic amplification. 1.1 Structural Dynamics Any real structure can be regarded as a continuum and can therefore be modelled as a multi-degrees-of-freedom (MDOF) system. The equation of motion for a general MDOF structure is [33], [34]: M x + Cx + Kx = f, (1) where M, C and K are the mass, viscous damping and stiffness matrices of the structure, respectively. f represents the vector of the excitation forces and x is the displacement vector of the stucture's degrees of freedom. After introducing the proportional viscous damping ^ and modal coordinates q, the equations of motion become uncoupled [35]. iq+ 2|m o x q+ m 0 x q & f, where: x = & q. (2) (3) & is the mass-normalized modal matrix [35]. I represents the identity matrix, [\0|mo\] is the diagonal damping matrix and [\ m0\ is the diagonal matrix of the natural frequencies. For the i-th modal coordinate qi the uncoupled equation of motion is [35]: q + 2^imo,i qi + m0 i qi = &f, (4) where is i-th mass-normalized eigenvector. It is worth noting that the transformation to the modal coordinates transforms a MDOF system with N degrees of freedom, to N decoupled single-degree-of-freedom (SDOF) systems. 1.2 Impulse Response When a SDOF system [35] is excited with an ideal unit impulse, which can be defined with the Dirac delta function: 5 (t) t = 0 0, t = 0 (5) 632 Ogrinec, P. - Slavic, J. - Boltezar, M. Stroj ntsklvestnlk- Journalnf MechanlcalEnginnnring the system responds with the impulse response function g(t) [33] and [35]: g(t) e-8«ot sin (yf—S2 Vl - 82«0 (6) When the damping coefficient is small (8 << 1, Vl - 82 « 1) then Eq. (6) simplifies to: g(t ) e-« sin («0t ) «o (7) In order to obtain the response for a general force excitation f (t), the convolution integral of the impulse f(t) and the impulse-response function g(t) are required [36]. x(t )= f(t ) * g(t)= Í f(T) g(t - T)dT, J —œ (8) where t is the time variable and x(t) is the time response of the SDOF system. The half sine impulse excitation and the impulse load of a SDOF structure are presented in Fig. 1. 1.3 Fatigue-life Estimation In order to obtain the fatigue damage in the case of an impulse excitation a counting method has to be employed. In this paper the rainflow counting method [8] will be used in combination with the hypothesis of linear damage accumulation [32] and [37]. Here, the basics (used later in this paper) of the time-domain high-cycle approach to the fatigue-life estimation are presented (for details see, e.g. [38]). The rainflow algorithm [8] transforms the stress-time history into a series of stress reversals. After the stress-time history is determined, the Miners rule of damage accumulation is applied and the damage is calculated as: D RF k — k N ' (9) where P is the number of ranges in the stress-reversal series that are identified by the rainflow algorithm and ni is the number of stress reversals at each individual stress level. Together with the fatigue parameters, which determine the expected number of cycles Ni at a stress level ai, Eq. (9) leads to the accumulated damage in the structure. The number of expected cycles Ni that a structure survives at the stress level oi is theoretically described with the Wohler curve [39]: C, (10) where k and C are the material's fatigue parameters. It is important to be aware that this simple S-N relationship does not account for the material's endurance limit and can only be used when dealing with high-cycle fatigue [40] or with materials that do not exhibit a fatigue limit. 2 HARMONIC EQUIVALENCE OF THE IMPULSE LOADS 2.1 Damage at impulse excitation In order to compare the harmonic and impulse excitations, the accumulated fatigue damage, which is the consequence of the individual loads, has to be determined analytically. The modal decomposition, as presented in Sec. 1, makes it possible to apply the methods developed for the single-degree-of-freedom (SDOF) system, to multiple-degree-of-freedom (MDOF) systems [35]. As was shown in [23] the dynamic stress response to an impulse excitation can be simplified to a scaled impulse-response function Eq. (7) for the analysis of the fatigue damage in the time domain, assuming the impulses are well separated, meaning that the response of the impulse is completely damped before the next impulse occurs. Therefore, the relatively complex solution of the convolution integral Eq. (8) is not necessary. The stress response of the structure oj can be written as a scaled impulse-response function: oi (t ) = Op e-8«ot sin(«ot ) «0 (11) for small coefficients of damping 8 << 1. In Eq. (11) the ®o stands for the natural frequency of a SDOF system and oP the peak stress of the response (envelope). As the rainflow algorithm [8] considers only the peaks and valleys of the stress response, Eq. (11) can be further simplified for the peaks: Gl (i) = e-1 n8 (2í'+1)(-1)í' (12) where i is an integer. In this manner the stress response can be discretized to a series of local maximum and minimum values of the stress response, also known as a turning-point sequence. Since the rainflow algorithm requires only the amplitudes of the extreme stress values, the individual stress cycles %i can be defined as the summation of the absolute values of the successive extreme points: Xi = 1 G(i) + G(i + 1)). After introducing Eq. (12), Eq. (13) becomes: Xi = 1 (en8 + 1) e-1 n8 (2i+V (13) (14) p Harmonic Equivalence of the Impulse Loads in Vibration Fatigue 633 Strojnioki vestnik -Journalof Mechanical Engineering 65(2010119-13,631-640 Excitation impulse Time Time b) Fig. 1. a) Normalised half-sine impulse excitation, andb) the impulse response of a SDOF structure The Palmgren-Miner rule of linear damage accumulation Eq. (9) can now be employed to determine the damage D as: D 1 Nj Y -^'Ni i=0 (15) where Ni is the number of cycles until failure at the level Xi and Ni is the number of impulse excitations (resulting in the impulse-response cycle load xO. By using the definition of the cycles Eq. (14) and the Wohler curve Eq. (10), the respective number of cycles Ni that the structure would endure for any given stress-cycle amplitude can be determined as: D, = Ni £ C i=0 (16) The Cauchys criterion for convergence [41] states that for each e > 0 a pair k > n exists for which: £ Xj j=n+1 \Xn+1 + Xn+2 + ■■■+ Xr\ < £. (17) Since the stress cycles of a damped SDOF structure limit towards 0, lim xn = 0, the Cauchys criterion is satisfied and the sum in Eq. (16) is final: Ni2-k (e-2(3n8) (e%s + 1) OpY Di =--V /, 8)k x-- • (18) c( (e-n8) k - 1 2.2 Damage at Harmonic Excitation If the load on the structure is harmonic with a constant stress amplitude oeq, using the Palmgren-Miner rule for linear damage accumulation [32], the cumulative damage is: Dh ak Nh a? ■ (19) where NH is the number of stress cycles at aeq. 2.3 The Equivalent Impulse Load to Harmonic Excitation Here, the damage Dj Eq. (18) of the Nj repetitions of impulse excitation, where the response load is a damped impulse response as shown in Fig. 1, will be related to the damage DH Eq. (19) due to the harmonic excitation with NH harmonic cycles at amplitude aeq: Di = Dh . (20) Using Eqs. (18) and (19) and the assumption that the fatigue parameters are the same for impulse and harmonic excitations: Nj2-k (e-2(3%s) (e%s + 1) a, NhC , (21) C((e-n*)k - 1) which can be simplified to: aeq = cosh ( ^ 1 a, Nj 1/k 2 J p\Nh (en5k - 1) . (22) For small coefficients of damping 5 << 1, which is often the case in real structures, this can be further simplified to: Nj 1/k eq Nh (en5k -1) (23) As can be seen from this derivation, the influencing factors on the relation between the impulse and harmonic loads are the coefficient of damping 8, the amplitudes oeq for harmonic and oP impulse loads, the number of harmonic cycles NH and impulses Ni and the slope of the Wohler curve k. This analytical derivation proves that for any impulse-response load there exists an equivalent harmonic load that yields the k p 634 Ogrinec, P. - Slavic, J. - Boltezar, M. Strojniskl vestnik- Journalof Mechanical Engineering 65(2019)11-12,631-640 same damage. Eq. (22) can be rearranged into a form that resembles the notation for the Wohler curve: '•'eq Gp 1 ii 5 k 1 1/k f N¡ \ 1/k (24) 3 EXPERIMENTAL SETUP For the experimental validation an electrodynamic shaker capable of impulse and harmonic excitations with closed-loop control was used. In order to obtain the parameters of the Wohler curve at impulse excitation, the sample's mode shape of interest must be excited with a translational movement in the axis of the shaker. Additionally, the sample's natural frequency of interest must be distinct, well separated from the other natural frequencies and be within the shaker's frequency range [42]. The area where the maximum stresses in the sample occur, must be on a surface where strain-gauge measurements are possible. It is also preferable to correlate strain with the relative displacements, since they can be measured throughout the test more reliably [40],[42]. The Y-shaped sample was chosen as it meets all of these criteria. Its natural frequencies are well separated; therefore, only one mode can be excited. Subsequently, the sample can be regarded as a SDOF system. The specimen consists of three rectangular cross-sections, 10 mm x 10 mm in size, that are arranged at 120° around the main axis. A circular hole in the centre ensures the fatigue zone is concentrated in the centre between the arms of the specimen. The samples are cast from the aluminium alloy G-AlSi8Cu3(226). The thickness is determined by milling and the overall shape by wire EDM (electrical discharge machining). The fatigue zones, which can be clearly seen in Fig. 2, are additionally ground and polished in order to remove any initial surface defects, which could start the crack growth prematurely. On the arms of the specimen two steel dead-weights can be positioned; using different dead-weights the natural frequency and the damping of the system can be changed. The design idea of the specimen is explained in detail in [40]. A total of 18 samples were experimentally tested: 9 of them at impulse excitation and the other 9 samples at harmonic excitation, similar to the description in [23] and [40]. During preparation of the Y-sample, extreme care was taken while machining, as well as during the preparation of the fixture of the sample in order to excite only one significant mode. The material G-AlSi8Cu3(226) was chosen because of its good casting properties that reduce the porosity of the cast sample. The material is, however, notorious for relatively large silicon inclusions, which affect the variance in the fatigue lives between samples. The natural frequency of the system was varied with the selection of different weights; three different system setups were researched (each setup was researched with 3 samples with impulse and 3 with harmonic excitation), see Tables 1 and 2 for the natural frequency of the samples, while other parameters given in the table will be discussed later. 3.1 Impulse Test During the fatigue testing the sample was excited with well-separated half-sine impulses. In the first step an experimental modal analysis was performed to obtain the natural frequencies of the samples. The duration of the impulse was determined with: 2n (25) where o0 is the focused natural frequency that was excited and a is a constant a = 2, as proposed in [23]. Due to the fatigue damage the natural frequency will decrease. In vibration fatigue a change in the natural frequency of 2 % is considered as fatigue failure [43]; with the small change it is reasonable to assume that the mechanical system is still linear. The control loop for impulse testing is presented in Fig. 3. If xS is the amplitude of displacement of the shaker base and xA is the amplitude of displacement of one of the Y-sample arms, then the relative displacement amplitude Ax can be related to the stress a in the fatigue zone via the coefficient ka [23]: ka = T- = E em 2 Ax xs — xA'" (26) where e is the strain, measured in the fatigue zone, rn is the frequency of excitation, XA is the acceleration of the arm of the sample, XS is the acceleration of the armature of the shaker, and E is the elastic modulus. The relation between the accelerations and deflections is simple since the response of the structure is harmonic in both excitation cases. This calibration was performed on a separate sample, so as not to induce any initial damage to the samples that were used for the identification of the material's fatigue properties. As the natural frequency was well excited in the impulse testing, it was possible to identify and track the natural frequency for every impulse-excitation response. Thus it was possible to identify the critical t Harmonic Equivalence of the Impulse Loads in Vibration Fatigue 635 Strojniskivestnik-Journalof Mechanical Engneering65(2019)ll-12,631-640 Fig. 2. Experimental setup Table 1. Samples for the impulse test Sample nr. Nntnrnl frequency [Hz] öp [MPn] N [/] 5^ [/ Dtl [/ 5fl M Dfl M ).) 322.2 022.) 7322 2.2240 0.24 2.2320 0.09 0.2 322.2 88.34 055222 2.2208 0.29 2.2232 0.02 0.3 322.2 025.9 2422 2.2242 0.38 2.2253 0.33 2.) 332.2 88.4 025522 2.2078 0.02 2.2088 0.28 2.2 334.2 84.2 222 082 2.2089 0.42 2.2323 2.55 2.3 328.2 83.2 233282 2.2025 0.57 2.2082 0.34 3.) 422.2 74.2 329222 2.2045 2.82 2.2053 2.72 3.2 398.5 78.3 022422 2.2022 2.83 2.2029 2.44 3.3 422.2 82.7 222 042 2.2072 0.27 2.2083 0.05 Table 2. Samples for the harmonic test Sample nr. Nntnrnl frequency [Hz] 5 [f\ oh [MPn] NH [/] Dh [/] 4.0 298.2 2.2272 55.3 03 022 482 m 4.2 322.2 2.2284 25.2 2 229 897 0.22 4.3 322.2 2.2232 52.2 37 224 975 0.09 5.0 330.2 2.2029 22.3 4 423 822 2.83 5.2 333.2 2.2070 42.0 528 457 272 0.94 5.3 332.2 2.2080 49.9 22 829 448 2.85 2.0 422.5 2.2038 44.2 28 827 222 2.28 2.2 399.2 2.2045 82.2 284 922 2.85 2.3 422.2 2.2020 42.0 27 229 488 0.22 damage of the sample when the natural frequency fell by 2 %. During the fatigue test, as the damage increases, the damping can change [29]. For an accurate damage estimation it is therefore important to identify the damping coefficient throughout the test. Table 1 shows the initial 5init and the final 5fm values of the coefficient of damping. Due to the open-loop strategy, also the stress load, estimated from the two accelerometers, can slightly change. During the fatigue test, a peak amplitude deviated less than to 2 % for all samples, but the sample 1.1, a 6 % decrease was observed. natural frequency. A control strategy was used, where the phase shift O between the accelerometer mounted on the shaker's armature and the arm of the sample (see Fig. 2) was monitored to adjust the excitation frequency and to control the stress amplitude [40], see Fig. 4. During the harmonic test, the amplitude and natural frequency were updated every 500 ms. The damping coefficient does not influence the fatigue-life prediction in the case of harmonic excitation. The excitation amplitudes can, however, still vary. Although a closed-loop control algorithm was used, the excitation amplitudes could vary by up to 0.5 %. 3.2 Harmonic Test Under the harmonic test, the samples were excited harmonically with a frequency close to the significant 3.3 Damage Criteria As the structural dynamics of the sample changes long before its complete failure, it is practical to identify 636 Ogrinec, P. - Slavic, J. - Boltezar, M. Strojniski vestnik - Journal of Mechanical Engineering 65(2019)11-12, 631-640 Fig. 3. Control loop for Impulse excitation Fig. 4. Control loop for harmonic excitation a failure when the natural frequency starts to change rapidly [40] and [42]. Therefore, in this research a drop of 2 % in the natural frequency was considered as a failure. For details about the failure on damage identification from the natural frequency, the interested reader is referred to [44]. During the impulse test the identification of the natural frequency is fairly straight forward, as it can be identified as the frequency at the peak value in the Fourier transform of the stress signal, see Fig. 5. During the harmonic test, the natural frequency had to be identified indirectly, from the Harmonin Equiualeencofthe Impulse LaaSsin ViaratlonFatlgue 637 Strojnioki vestnik -Journalof Mechanical Engineering 65(2010119-13,631-640 phase difference between the signals of the top and bottom accelerometers. As the frequency falls, the excitation frequency required to keep the same dynamic gain and the phase shift between the signals falls likewise. Therefore, the fall of the significant natural frequency can be determined by recording the excitation frequency. The tracked excitation frequency is shown in Fig. 6. 330 325 320 315 310 305 300 -Failure criteria "1. -V i 0 25000 50000 75000 100000 125000 150000 Number of impacts [/] Fig. 5. Decreasing of natural frequency during the impulse fatigue test 305 K ^ 300 Ç e 295 u fT e 290 c c 285 at it 280 li 275 Failure criteria 10000 20000 30000 40000 Time [s] Fig. 6. Fall of excitation frequency during harmonic excitation. 4 RESULTS 4.1 Identification of the Wohler Curve Since the goal of this paper is to prove the equivalence of the impulse and harmonic loads, the Wohler curve was identified on the harmonic test results, only. The fatigue parameters for the equiprobability curve identified on the harmonic test results were: Ch ku - - 7.2450 x 10 9.6347. ,23 (27) (28) The parameters were identified with the cost function: e (k, C) = min EDi(k, C) - 1 i=i (29) where S is the number of samples, k and C are the material parameters used in each individual iteration of the optimisation, and e (C, k) is the error between the optimal value of the damage and the damage in current iteration. 4.2 Comparing the Two Loading Strategies With the fatigue parameters obtained from the harmonic test Eq. (27), Eq. (19) is used to theoretically estimate the damage for the harmonic test. As the fatigue parameters were fitted to the experimental results of the harmonic test it is expected that DH would be close to 1. From Table 2 we can see that this is the case for most of the samples; the deviation from 1 is reasonable and expected in the fatigue test. It is reasonable to assume that the fatigue parameters between the harmonic and the impulse test are the same. Therefore, the fatigue parameters obtained from the harmonic test Eq. (27) can be used with the damage estimation at impulse excitation Eq. (18). The resulting damage D'"it and D{in based on the damping identified at the beginning and the end of the test are given in Table 1. We can see that the damage is reasonably close to 1. This experimental result validates the theoretical relation between the impulse and the harmonic load introduced in Eq. 24. 5 CONCLUSIONS The harmonic loads for high-cycle fatigue are well understood and can be theoretically and experimentally researched relatively easily. The contrary is true for the impulse loads; the impulse excitation is frequently exciting mechanical systems, leading to a damped free response related to the structural dynamic's of the system. Based on the structural dynamics theoretical background and classic fatigue theory in the time domain, a theoretical relation between the vibration-fatigue damage in the case of the impulse load and the harmonic load is introduced. The theoretical model was experimentally evaluated using the Y-sample. A total of 18 samples were experimentally tested; 9 for harmonic loads and 9 for impulse excitation. The samples were grouped in sets of 3. Using different mass loadings to the Y-sample, three different dynamic systems were researched (different natural frequencies and damping). Based on the time-domain experimental data from the harmonic test, the fatigue parameters were identified. The identified material parameters were then used to recalculate the damage at failure for every individual sample, for both excitation strategies. The identified damage was reasonably close to 1 confirming the introduced theoretical relation between the impulse and harmonic excitations. The introduced relation between the impulse and harmonic excitations can be used for the 638 Ogrinec, P. - Slavic, J. - Boltezar, M. Stroj ntsklvestnlk- Journalnf Pf^chan^calE^r^^irneering further development of frequency-counting methods, specifically with impulse loads as a source of non-stationarity. 6 ACKNOWLEDGEMENTS The authors acknowledge the partial financial support from the Slovenian Research Agency (research core funding No. P2-0263). 7 REFERENCES [1 ] Luznar.J..Slavic, J., Boltezar, M. (2(^1.S^).Stru cturehtome noise atoWM Excitation nslng so exOessed field recannrruction mt^nhoSatOmoSal Oecomnossian. SOrnnniski nestpO-Peurnsr ofMechanical Engineering, sol. 65, no. 9, p. AH-Am, D 0l:10.5545/sv -jme.2019.6115. [2] Diriik.T. (198f^)./h5gli^^Cieo .eComp-tsriin F^gg-e Analysis. P hthDesin.r^n iversityefWorwicEiWsnt ic0. [3] Benaidst^D^004.. ^^sAngis^ erf .andom pc^z^rliseai Po^grgra.ferrtrc. [4] Benaaduttj H, Tovo, R. (2007). Frequency-based fatigue analysiso.nonrstatisnary switching random Fractore oOEngi-eeriog Materials & Stcuc-urss, vol. 3., no. 1 f t. e016rE026, e001g.0gl6/j.9450-2696.200^1671 [5] BenainiuSi,D., Soza, R. (hOBP.. Freqi^^r^r^s^te^^^cnnn^^r^EW^f randotzS^l^^j^ue Isaes: MooCr hypothonesi r^^lWy^./t^^neea/ Sc/esce and En4/neqo/em Tesheoiegy, vsi.4W,no. 3,p. 30M-3etm DOlllm.a0023me&eMml700190 [6] Mtssiki M., V^^r/i6, g.EhsS-c^r.M. (2gC3). ^6ggsec;o-65,rpig meOhodsforaoibration-fatigue-lite estimation-application to retl datM Internatioval Jouma[ of Fai&ne, vol. 47( p. e-17, d 4cnq^r^esswjnjfm¡sr^e .oon2.pn.woPi [7] Mi-snmitPSIstit. h. ^i^ite^or, M. (2.18)° VibrcEi-nfatigue unisg marlc/l dogomf/63i.tiot. Meteiastca/ Csstnms ore S/Mna/ Prorevslcg. ooi. 98m 0( f5-^i-^mi56, DskOO.COieAe ^ies^srmoh7.ns.D5e. [8] MttSMShiiM,EcSn, T. (S960). sae^ue of meteis subjected to oarBrnigr^rresw, Jeean Sonimtynf Mech6dica1EnVi65em,vol. 64,no.6,p. 3^-4g. [9] Miles, t.Wi(69a4), On Etmctoroi Ostigue snoor randae! lonempmoeonel o8theCm4onautfc5iSci6S5es,ic\. 26,00.65, p. 753-762, D0I:10.2514/8.3199. [ M/ Newland^.E. ;&062). An(gt3345henn to gc3ecm Vibrmtiens, Spec^E.1. eam/et6ngCs/y.hoverPuSiication1Mineoia. [ 1C] Bvc6fS J., P.l^rnol, A. poll). RanOoc jfo-r<3: Assiysis mn3 Mcoaurgment Frete.^., stL 7no, J..0 W^^^^s &Sons, H^tro ken. [12] Bmosess o.. Ciaoetti, f^omassist L. (W0nhii sast ovomu^ina of state nsectral memente. leiamationff Cour6a/ ofMeehani.a/ Sciences, vol. 127, p. 4-9, D0l:10.1016/j. ij mec2ci.s0is.sn.h07 [13] Bnhrn.M., f^^c^na,M[^Slmi(^sPnupim^^^os^mr^^rv^i^ilfre useoftSn s|tectralmefnon fornon-MauwsiEt ioanMg histories wso vhe use oM.s ncer1 para^ietem coufna| ofMb^iny Cmsfruefosanc/M5/6tencn5/\ vol. 1, p. 27-31. [ 14] Fan, Z.Jiang, Y, zfong, S., Taz.O, Chen, X. (2007). Research onv^t^roti^a fatigus ofci^i^lDon f^tfreo^^r^fhrced composited unfernon-Vaussianvandoro ioaZ.PrzgnostZzhond Syofom Health Management Conference, p. 1-8, D0l:10.1109/ p Hroininsoieisp [15] Capponi, L, Cesnik, M., Slavic, J., Cianetti, F., Boltezar, M. (20S7). Nenzhtatiooarin-z infeoin vihroVionJatieu^: ahez-oOinvl andeose-ie^^n^ireeeN^c^lo./^^^^fia^^í^t^í^Z^^ismc^l nfFitiguo, vol. íou, p zp0--270, e4l:]Ld.a01Z/j.eaie(ZUe.S0 [ 14] Palmieri, de Cesoik, n, J, ziaeettt F, ZalteOar: OS (20S7| Non-Gaussianity anOnon-oJa7ioaoaty i-L aO|Lptioa fatieuz. tntepz^Onvsl Journilof Uotigug, vul. 97, p. 9-11, D 0l:10.1416/|.ijfet igue.2016.12.017. [ 17] Songs of Wang, S. (z009). A novel speetrzl m^omeata eqmrnatenve vauza lum.N^ bleok me^c-ao tor el'fie0,^"-es- mmtiznoJ o ffe h ooeafrortur a^iti^e (J a m age g tesz. t/zov. Journal of Foi/Jt^, voi. 1P0, |1. :L64-175, DOaiO.lOlL/g ij PaCiglle(00t8.09.01U [18] BoacgesJO,, d^.atetC( i, Lorl, Go n,:dli, CO, (0009! The fre-zency cComs^néipproF-^1 it virtoil Udlguo eotimpoot 1f non-llneennystems/Tho problem of non-Gaussian states of stressi.mtematigzhiJouro7l oFFatigue, voi. ge,noi4| p.7o\-7 7VIDOl:N0tlO00/j.onaCimueiegnVi03.070. [19] Nia, Qiz Oncg, s, a. Xo (20oOF. -Lo empirisoi rnoVe dpcomfusition-bpiocl frequ^ncy-Fon^^ii t0prDair foi the fafigue ai^lysiz vf r^c^6tiat^^6ary processes. Fatigue & Fracturo P^oOinezring Ma tenate SSfructucss, aol. 4Zinei(31 p. Z980-C9eF. DOI:V0.1g01Cffe.yd83Ui [20] WaT7te¡venP,1Sedlmag0.(;t0y5).De^l¡ng iai^^ilmnj^l^^í^L^e tetr UL^zafr^ fram meavvred non (3t^l^c^niiar^erGcefe^-;ra¡ Praoed^a Engneeringl vol^oo, e. C8: l^^l-V0.Pz^OlC p rmz^i^^^C^V 5.(^0.16^ [21] CiaioO^l fo f^^í^i^rfo Mr 1rircysi,C,Mor8it:nh a, (2018). Cerrete/n no^mu^^vn^rmaco toa^vel^^zm rafo^etnunaou indidd by g-^nn1^^,1^ Proo5S/a Structural Inamgrity l/o-^, j 3,0 398, DO.:la^-^r^^e/j.o^i^7tr0S [22] Rycu|iz1 i.i1 C3rl(^)iCat:puocilra atocmNncbave^. Szao0/nhWac JopmvrofSthtist:ce,e. 37omz4. [2/] Ogrrnea, 1, ^^^vii:, ro(^^í5;iik, M, BdneLsfi0.^n^01 faogu^ v0 faz-almo imfLuize- oxcNr^ü^c m thz eme z^cS free^u^v-tpemarnaVnteroationaLOmernai zdFo0igueiNolilmm p. 3(^8^31V,00::10.10^(^^j80F^í0e.20Cg.ne^031l [24] Ang¡^ij^i|C^ryelT,í^.,Thos^(6>stM.-2 018). Synthesis of sine-on-^