Frequency-domain methods for a vibration-fatigue-life ...

Frequency-domain methods for a vibration-fatigue-lifeestimation - application to real data

J. Slavic, M. Mrsnik, M. BoltezarUniversity of Ljubljana, Faculty of Mechanical Engineering,Askerceva 6, 1000 Ljubljana, Sloveniae-mail: [email protected]

AbstractIn automotive industry the accelerated vibration tests are used as the product’s final experimental verificationagainst vibratory loads. Based on the time-domain load measurements the fatigue due to vibration excitationis in general relatively easy to estimate; however, as the time-domain analysis requires explicit numericalapproach it proves disadvantageous compared to the frequency-domain methods for vibration fatigue es-timation. In the frequency domain the vibration fatigue is frequently estimated with the empirical Dirlikmethod, which was found as one of the most accurate compared to the well established Rainflow count-ing time-domain method. In this research the Dirlik vibration fatigue estimation method is theoreticallyand experimentally compared to competitive frequency domain methods: Benasciutti and Tovo, Wirsching-Light, Zhao-Baker, Petrucci-Zuccarello, empirical α0.75, Tovo-Benasciutti, and the Gao and Moan method.The definitions of the discussed methods are given and discussed. Previously, some of the methods havebeen compared versus each other, while this research compares all methods (including very recent ones)side by side. Further, if most of previous comparison research was made on simulated numerical signals,this research focuses on the real experimental data obtained with an electro-dynamical shaker experiment.The vibration profiles typical for automotive industry accelerated test are analyzed. Compared to the Dirlikmethod, the Tovo-Banascutti and the improved Zhao-Baker methods were found more accurate. In detail,the Zhao-Baker method was found most accurate and the only method giving conservative fatigue damageestimation for the typical automotive industry accelerated test profiles. For the selected automotive excita-tion profiles, this research shows that the Tovo-Banascutti and the improved Zhao-Baker methods should beconsidered as more accurate in vibration fatigue damage estimation.

1 Introduction

Due to the fatigue, dynamically loaded mechanical structures are inclined to failure [1]. It is thus a commonengineering practice to test such structures and evaluate their fatigue strength. Tools for estimating fatiguestrength must be available during the design stage, to enable efficient product development. One such tool isa well established [1, 2, 3, 4, 5] time-domain approach to fatigue-life estimation [6, 7, 8].

The approach in the time domain is a combination of the rainflow counting method (first developed byMatsuishi and Endo [6]), that is a set of rules on how to identify the damaging cycles inside the loadinghistory, and the linear damage accumulation method (devised separately by Palmgren [7] and Miner [8]),that dictates the rules on how to add the collected damaging cycles. This procedure can turn out to be quitetedious and slow, especially when working with structural dynamics, where calculations are usually made inthe frequency domain [2].

It is customary to provide a PSD, describing the distribution of power over a frequency interval [9], when

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working with dynamic loads (i.e. those present in a moving vehicle), that are caused by random processes.The immediate benefit is that with finite-element-models, the frequency response calculations are muchfaster than the transient dynamic analysis in the time domain [2]. Frequency-domain methods for a vibration-fatigue-life estimation can then be used to estimate fatigue life based on the properties of PSD, where thetrue value of fatigue life is one that would be calculated by the combination of rainflow and Palmgren-Minermethods [3, 10, 11].

All frequency-domain methods, listed and evaluated in this research are derived for Gaussian process. Someare only appropriate for narrow-band processes (such as the Narrow-band approximation by Miles [12])and others can be considered for the mathematically more complex wide-band processes (such as Tovo-Benasciutti [11]). In general, most of frequency-domain methods are empirical solutions (e.g., Dirlik [10]and Zhao-Baker [3]) with few exceptions (e.g., Bishop [13]) that are theoretical, although computationallyintensive; Bishop’s method shows little improvement in accuracy over Dirlik and the latter is usually thepreferred choice [2].

Several comparison studies exist, that were made since the Dirlik method was first introduced [14, 15, 11,16, 17]. The results of those studies show that Dirlik performs better than most of other methods.

A study by Benasciutti and Tovo [18] in 2004 compared the following frequency-domain methods: Wirsching-Light [19], Zhao-Baker [3], Dirlik [10], the empirical α0.75 [18], and Tovo-Benasciutti [4, 11]. It was con-cluded, based on numerically simulated power spectral densities, that the Tovo-Benasciutti method is almostas accurate as the Dirlik method. A new method has been presented recently by Gao and Moan [17].

Several frequency-domain methods were compared theoretically and experimentally. The stress distributionon a real structure relates significantly to the structural dynamics [20] which can influence the accuracy of thefatigue life estimate. The main focus of this research is thus the performance of frequency-domain methodson typical vibration profiles that are used in accelerated test in the automotive industry.

Section 2 of this research gives concise theoretical background, fundamental to many of the frequency-domain methods. These methods are presented in Section 3. What follows is the experimental set-up de-scription and a discussion of the results in Section 5. The Conclusions are given in Section 6.

2 Theoretical background

To facilitate an understanding of frequency-domain methods and their definitions, a brief introduction to thestochastic process theory will be presented first. For a deeper understanding and an elaborate derivation thereader is referred to the reference works of Shin and Hammond [9] and Newland [21]. The basics of themodal analysis that can be used to determine the stress function at different material points of the vibratingstructure are covered next. Finally, a general approach to specifying the fatigue strength in the frequencydomain is presented.

2.1 Properties of random processes

In the frequency domain the random loading of a random process X is defined by the power spectral densitySXX(f), with f denoting the frequency. It is customary to use a one-sided power spectral density GXX(f),defined on the positive half-axis only. The statistical properties of a stationary process can be described bythe moments of the power spectral density. The general form for the i-th spectral moment mi is given by:

mi =∫ ∞

0f iGXX(f) df. (1)

For a fatigue analysis the moments up to m4 are normally used. The even moments represent the varianceσ2X of the random process X and its derivatives:

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σ2X = m0 σ2

X= m2 (2)

The spread of the process or the spectral width is estimated using the parameter αi, which has the generalform:

αi =mi√m0m2 i

(3)

The most commonly used α2 is the negative of the correlation between the process and its second derivative,as described by Tovo [4]. It takes values from 0 to 1. The higher the value, the narrower is the process in thefrequency domain and vice-versa.

Another frequently used spectral parameter is Vanmarcke’s parameter δ [22]:

δ =√

1− α21 (4)

The expected peak occurrence frequency νp and the expected positive zero-crossing rate ν0 are defined as:

νp =√m4

m2ν0 =

√m2

m0(5)

They are imperative for determining the fatigue-life intensity. The analytical derivation of both was describedby Newland [21].

The foundations for a frequency-domain approximation of a cycle distribution were set by Rice [23], whomanaged to analytically define the probability density function of the peaks pp(a) based on the power spectraldensity:

pp(a) =

√1− α2

2√2π σX

e− a2

2σ2X

(1−α22) +

α2 a

σ2X

e− a2

2σ2X Φ

(α2 a

σX√

1− α22

)(6)

where a is the peak amplitude and σX and α2 are defined with Equations (2) and (3), respectively. Φ (z) isthe standard normal cumulative distribution function:

Φ (z) =1√2π

∫ z

−∞e−

t2

2 dt (7)

2.2 Structural dynamics and modal analysis

The objective of the modal analysis is to establish a mathematical model of the structure. The structuraldynamics and the response can already be characterized during the design stage, usually with the help ofFEM methods [24, 20, 25]. The linear structural dynamics is described in the frequency domain, whichis why the frequency-domain methods for fatigue analysis are of great interest. Recent research [26] evensuggests it is possible to evaluate more complicated multi-axial loadings in the same manner. So as not tolose focus, here only the basics are given and the interested reader is advised to read references [27, 25].

Complex structures can be viewed as linear, multi-degree-of-freedom systems that are described by a systemof second-order differential equations [27]:

[M] {x}+ [C] {x}+ [K] {x} = {f} (8)

DYNAMIC TESTING: METHODS AND INSTRUMENTATION 1507

where [M] is the mass matrix, [C] is the damping matrix, [K] is the stiffness matrix, {x} is the vector ofdegrees of freedom and {f} is the excitation force vector. Furthermore, through modal analysis the transferfunctions from one point on the structure to the other can be deduced, mathematically or by experiment.For a selected geometrical location on the structure the displacement response spectra SXX(ω) can be ob-tained by knowing the frequency-response function H(ω) (phase information is truncated with PSD, thus|H(ω)|2) and the force excitation spectra SFF (ω) [27]:

SXX(ω) = |H(ω)|2 SFF (ω) (9)

The response characteristics other than the displacements (e.g., stress, acceleration) can also be expressed.As mechanical structures are usually exposed to a known acceleration profile, for a fatigue analysis thefrequency-transfer function Has(ω) from the acceleration a to the material stress s is of special interest. Formulti-degree-of-freedom systems the response is dependent on multiple inputs and Equation (9) is rewrittenin the matrix form as:

s = Has a (10)

where s is the stress tensor in the frequency domain, a is the forced vibration acceleration profile and Has isthe frequency-response function matrix.

2.3 Fatigue-life estimation

A possible scenario in fatigue analysis is to numerically simulate the operating time until failure for a struc-tural member with a given critical stress function in the frequency domain at selected node(s) (10). Instead ofthe time-until-failure criteria, this research focuses on the damage intensityD, see Benasciutti and Tovo [11].The damage intensity estimates the damage per unit of time. It is defined with:

D = νpC−1

∫ ∞0

sk pa(s) ds, (11)

where νp is the expected peak occurrence frequency, given by Equation (5), C and k are material parametersand pa(s) is the cycle amplitude probability density function. The function pa(s) is the unknown and a keyfactor considered by frequency-domain methods for fatigue analyses.

The evaluation of frequency-domain methods is carried out by a comparison of the fatigue-life estimate T inseconds, which is obtained from the damage intensity D:

T =1D

(12)

3 Frequency-domain methods

Typical methods for fatigue analyses in the frequency domain designed to deal with Gaussian randomloadings are discussed in this section. While the narrow-band (NB) approximation is only appropriatefor narrow-band processes, others are also appropriate for wide-band loading processes. These are theWirsching-Light method (WL) [19], the α0.75 method (AL) [18], the Gao-Moan method (GM) [17], theDirlik method (DK) [10], both Zhao-Baker methods (ZB1 and ZB2) [3], the Tovo-Benasciutti methods (TB1and TB2) [4, 11] and the Petrucci-Zuccarello method (PZ) [5]. The α0.75 method and the Wirsching-Lightmethod differ from the others in the way that they provide the correction factor to correct the narrow-bandapproximation for wide-band processes.


3.1 Narrow-band approximation

The narrow-band approximation (presented by Miles in 1956 [12]) is the most basic approximation of dam-age intensity and should only be used with narrow-band processes. It is calculated with:

DNB = ν0C

−1(√

2m0

)k Γ(

1 +k

2

)(13)

where ν0 is the expected positive zero-crossings intensity, C and k are material fatigue parameters, m0 is the0-th spectral moment, α2 is determined with (3) and Γ(·) is the Euler gamma function, which is defined as:

Γ(z) =∫ ∞

0tz−1 e−t dt (14)

3.2 Wirsching-Light method

Wirsching and Light [19] in 1980 used an additional spectral width parameter α2 to correct the narrow-bandapproximation with the empirical factor ρWL:

DWL = ρWLD

NB (15)

where DNB is the narrow-band approximation obtained with Equation (13) and ρWL is defined as:

ρWL = a(k) + [1− a(k)] (1− ε)b(k) (16)

with the spectral width parameter ε being:

ε =√

1− α22 (17)

and the best fitting parameters a(·) and b(·) dependent on the S-N slope k:

a(k) = 0.926− 0.033 k b(k) = 1.587 k − 2.323 (18)

3.3 α0.75 method

The second correction method, named α0.75, was proposed by Benasciutti and Tovo in 2004 [18]. It is basedon the spectral parameter α0.75 and is given with:

DAL = α2

0.75DNB (19)

where DNB is the narrow-band approximation (13) and α0.75 is defined with (3).


3.4 Gao-Moan method

Recently, a trimodal method was presented by Gao and Moan [17] along with a generalized variant that canbe applied to other (wide-band) types of processes. The fatigue-damage intensity DGM is the sum of thehigh-frequency process RH(t), the intermediate-frequency process RM (t) and the low-frequency processRL(t) contributions, each are being narrow-banded and having a Rayleigh-distributed amplitude with therespective variances equal to σH , σM , σL:

DGM = DP +DQ +DH (20)

where RH is treated as a narrow-band process and DP and DQ are damage intensities due to the processesRP (t) and RQ(t), respectively:

RP (t) = RH(t) +RM (t)RQ(t) = RH(t) +RM (t) +RL(t)

(21)

The damage component DH is determined using (13), where ν0H is given by (5). Equation (11) is used forDP and DQ, where the expected zero-crossing frequencies are used instead of the peak frequencies and arecalculated with:

ν0P =√m2H δ2

H +m2MσM(

σ2H + σ2

M

) (22)

ν0Q =√m2H δ2

H +m2M δ2M +m2L

×

2σL√σ2H + σ2

M + σ2L − π σH σM

2(√

σ2H + σ2

M + σ2L

)3

+2σH σM arctan σH σM

σL√σ2H+σ2

M+σ2L

2(√

σ2H + σ2

M + σ2L

)3

(23)

where δH and δM are Vanmarcke’s parameters from Equation (4), calculated for the processes RH(t) andRM (t), respectively. The probability density functions for the sum of two or more random variables, alsoneeded for calculating (11), are determined by means of the convolution integral:

pa,HM =∫ ∞

0pa,H(s) pa,M (t− s) ds (24)

pa,HML =∫ ∞

0pa,HM (s) pa,L(t− s) ds (25)

where pa,H , pa,M and pa,L are the cycle-amplitude probability densities of the respective narrow-band pro-cesses. Hermite integration was used by Gao and Moan to calculate the distribution of the sum of the multipleRayleigh random variables [17].

The trimodal approach is generalized to a general wide-band process by splitting the power spectral densityinto three parts, according to the chosen criteria. The authors recommend splitting into three parts with equalvariance, with each part then being treated as one of the three modes. This same approach is used in thispaper, referenced with GM.


3.5 Dirlik method

The Dirlik method [10] was devised in 1985. It has long been considered to be one of the best and has alreadybeen subject to modifications, e.g., for the inclusion of the temperature effect, by Zalaznik and Nagode [28].The rainflow-cycle amplitude probability density estimate is given by:

pa(s) =1√m0

[G1

Qe−ZQ +

G2 Z

R2e−Z2

2R2 +G3 Z e−Z2

2

](26)

whereZ is the normalized amplitude and xm is the mean frequency, as defined by the author of the method [10]:

Z =s√m0

xm =m1

m0

(m2

m4

) 12

(27)

and the parameters G1 to G3, R and Q are defined as:

G1 =2(xm − α2

2

)1 + α2

2

G2 =1− α2 −G1 +G2

1

1−R (28)

G3 = 1−G1 −G2 R =α2 − xm −G2

1

1− α2 −G1 +G21

(29)

Q =1, 25 (α2 −G3 −G2R)

G1(30)

while α2 has already been defined with Equation (3). The closed-form expression for the fatigue-life intensityhas been derived in the form:

DDK = C−1 νpm

k20

[G1Q

k Γ(1 + k)

+(√

2)k

Γ(1 +k

2)(G2 |R|k +G3

)] (31)

3.6 Zhao-Baker method

This method was developed by Zhao and Baker in 1992 [3]. It gives the following expression for the cycleamplitude probability density function:

pa(Z) =

Weibull︷︸︸︷w αβ Zβ−1 e−αZ

β+

Rayleigh︷︸︸︷(1− w)Z e−

Z2

2 (32)

where Z is the normalized amplitude from Equations (27) and w is the weighting factor defined by:

w =1− α2

1−√

2π Γ(

1 + 1β

)α−1/β

(33)

and α and β are the Weibull parameters:

α = 8− 7α2 (34)


β =

{1.1; α2 < 0, 9

1.1 + 9 (α2 − 0, 9); α2 ≥ 0.9(35)

For small values of the S-N slope k (k = 3) the rainflow damage is more closely related to α0.75 than α2.Zhao and Baker offered an improved method of their calculation (ZB2) where α is calculated as:

α = d−β (36)

with d calculated as a root of:

Γ(

1 +3β

)(1− α2) d3 + 3 Γ

(1 +

1β

)(ρZB α2 − 1) d+

+3√π

2α2 (1− ρZB) = 0

(37)

and the correction factor ρZB at k = 3 is determined by:

ρZB|b=3 =

{−0.4154 + 1.392α0.75; α0.75 ≥ 0.5

0.28; α0.75 < 0.5(38)

A closed-form expression for directly calculating the fatigue-damage intensity is defined with [18]:

DZB1/ZB2 =

νpCmk/20

[wα

− kβ Γ(

1 +k

β

)+ (1− w) 2k/2 Γ

(1 +

k

2

)](39)

where the use of the ordinary approach is denoted with ZB1 and the use of the improved method with ZB2.

3.7 Tovo-Benasciutti method

In 2002, Benasciutti and Tovo [4, 11] first proposed an approach where the fatigue life is calculated as a linearcombination of the upper and lower fatigue-damage intensity limits. The final expression for the estimatedfatigue-damage intensity is:

DTB =

[b+ (1− b)αk−1

2

]α2D

NB (40)

where b is unknown and is approximated with numerical simulation data. The narrow-band damage intensityD

NB is defined with Equation (13), where the zero-crossing intensity ν0 is replaced with the peak intensityνp for wide-band processes.

Two different equations are suggested for determining the factor b in order to substitute into Equation (40),both were proposed by Benasciutti and Tovo [4, 11]:

bTB1app = min

{α1 − α2

1− α1, 1}

(41)

bTB2app =

(α1 − α2)[1.112 (1 + α1 α2 − (α1 + α2)) e2.11α2 + (α1 − α2)

](α2 − 1)2

(42)

where using the bTB1app from Equation (41) will be referenced with TB1 and using bTB2

app from Equation (42)with TB2.


3.8 Petrucci-Zuccarello method

This method was proposed in 2004 by Petrucci and Zuccarello [5] who sought a connection between themoments of the probability density of the equivalent Goodman stress and the parameters α1, α2, k and γ,the last of these being the ratio:

γ =xmax

Su(43)

where xmax is the absolute maximum value of the stress process and Su is the material tensile strength.

As already mentioned, the Petrucci-Zuccarello method implicitly predicts equivalent stress ranges by calcu-lating the Goodman equivalent stress range [29]:

re =r

1− mSu

(44)

where r is the original stress range, m is the range mean and Su is the tensile strength. The expression forcalculating the damage-intensity estimate is:

DPZ = C−1 νp

√mk

0 eΨ(α1,α2,b,γ) (45)

where νp is defined with (5) and the spectral moment m0 with (1). The approximation function Ψ(·) [5]:

Ψ(α1, α2, k, γ) =(Ψ2 −Ψ1)

6(b− 3) + Ψ1 +

[29

(Ψ4 −Ψ3

− Ψ2 + Ψ1) (k − 3) +43

(Ψ3 −Ψ1)]

(γ − 0, 15)(46)

depends on the material parameter k, the ratio γ and the coefficients Ψ1 to Ψ4 given as polynomials:

Ψ1 = −1, 994− 9, 381α2 + 18, 349α1

+ 15, 261α1 α2 − 1, 483α22 − 15, 402α2

1

Ψ2 = 8, 229− 26, 510α2 + 21, 522α1

+ 27, 748α1 α2 + 4, 338α22 − 20, 026α2

1

Ψ3 = −0, 946− 8, 025α2 + 15, 692α1

+ 11, 867α1 α2 + 0, 382α22 − 13, 198α2

1

Ψ4 = 8, 780− 26, 058α2 + 21, 628α1

+ 26, 487α1 α2 + 5, 379α22 − 19, 967α2

1

(47)

with α1 and α2 taken from Equation (3).

4 Experiment

In real life dynamical structures exposed to vibrations respond according to their natural dynamics, and toavoid real-life (in application) experiments, accelerated tests are used. The load during accelerated vibra-tion tests is based on real-life vibration loads, but these tests are usually accelerated with higher vibration


amplitudes. The accelerated vibration tests are usually made by using electro-dynamical shakers, where acontrolled excitation profile in the frequency domain is used. According to Equation (10) the excitationprofile is controlled at a selected point of the structure and this results in a stress tensor frequency profile atall the other points. The transfer function from the excitation to the response differs for each material point;so as not to focus on the structure, the excitation acceleration profile is considered as the stress profile andanalyzed from the point of the damage estimation (1 m/s2 is considered 1 MPa)1. The response profile wassimulated on a real electro-dynamical shaker and controlled with an industry-standard vibration controller.

4.1 Experimental set-up

The test site consisted of a personal computer for control and data acquisition, an acceleration sensor andelectrodynamic shaker equipment, see Figures 1 and 2.

Figure 1: Experiment schematics

Professional hardware and software with a control loop were used to generate the excitation according to agiven power spectral density in the range from 10 to 1000 Hz. At the same time, the measured signal wasalso discretized with a DAQ module and saved for later analysis.

The duration of one measurement was 5 minutes. The sampling frequency was 10000 Hz. The root-mean-square (RMS) value of every random process run on the shaker was 20 MPa.

4.2 Simulated spectra

In accelerated vibration tests, the structure is fixed to the electro-dynamical shaker and excited with a givenprofile. Because the structural response natural modes discussed above are developed for the frequencyregion without natural dynamics, the excitation profile is linearly translated to the fatigue loads. Therefore,we investigated three realistic industry spectra that are typical for the automotive industry, shown in Figure3. The spectral-width parameter α2 ranged from 0.40 to 0.48.

4.3 Results

The fatigue-life estimates for the frequency-domain methods were compared to the life estimate in the timedomain using a combination of the rainflow count and the Palmgren-Miner hypothesis, TRFC, which in thisstudy is assumed to be an exact reference value. The relative error is calculated using:

1In a real damage estimation the transfer functions of two material points are required and the damage estimation of these pointsis straightforward with respect to Equation (10).


Figure 2: The LDS V555 electro-dynamic shaker with the B&K accelerometer mounted on the vibratingplate and the B&K Nexus charge-conditioning amplifier.

200 400 600 800 1000

f [Hz]

10 1

100

101

102

2P

SD

[M

Pa/

Hz

]

Spectrum 1

Spectrum 2

Spectrum 3

Figure 3: Automotive spectra

Terr =TXX − TRFC

TRFC(48)

where TXX is a life estimate, as calculated using one of the frequency-domain methods. Because of thesignificant impact the S-N slope k has on the accuracy of the fatigue-life estimation, three different materialparameters, presented by Petrucci and Zuccarello [5], were used and the signal was normalized according toa suitable RMS value. The data is presented in Table 1.

For real-life applications it is very important that the damage-estimation method does not rely on the prop-erties of the response spectra (e.g., narrow band, close modes, etc.); therefore, in this research we are tryingto find the best methods in the sense that they perform well, regardless of the special features being presentin the analyzed spectrum. Table 2 shows the relative errors of estimates for each method.

Because they exhibited the best performance (other methods were not near as accurate), the Dirlik, Wirsching-Light, Zhao-Baker and improved Zhao-Baker methods were selected for a detailed analysis (see Figure 4).


Table 1: Material parameters [5] and RMS for the fatigue-life calculationC [MPak] k Su [MPa] RMS [MPa]

Steel 1.934 · 1012 3.324 725 5Aluminum 6.853 · 1019 7.300 446 10Spring steel 1.413 · 1037 11.760 1850 100

Table 2: Percentage of relative errors for different margins for k = 3.324rel. error

Spectrum A Spectrum B Spectrum CNarrow-band -0.68 -0.65 -0.74Wirsching-L. -0.61 -0.57 -0.68α0.75 method -0.51 -0.50 -0.58Gao-M. -0.28 -0.24 -0.44Dirlik +0.03 +0.03 +0.06Zhao-B. -0.08 -0.06 -0.07Imp. Zhao-B. -0.02 -0.02 -0.03Tovo-B. -0.24 -0.26 -0.22Imp. Tovo-B. +0.01 +0.02 +0.03Petrucci-Z. -0.73 -0.72 -0.74

0.2

0.1

0.0

0.1

0.2

Rel

ativ

eer

ror

Imp. Tovo-B.

Dirlik

Zhao-B.

Imp. Zhao-B.

k≈3

k 7≈

k 11≈

Figure 4: Comparison of the relative errors for the selected methods when k = 3.324, k = 7.3 and k =11.76, respectively.

5 Discussion

An ideal damage-estimation method for use in a design process should be consistent across different spectraand different slopes of the S-N curve. It would give results that are close or equal to those given by atime-domain approach and preferably be conservative when not accurate.

Zhao-Baker and improved Zhao-Baker methods stand out as the best-performing methods for the automotivespectra. Zhao-Baker only slightly overestimates the fatigue life, and achieves better accuracy than improvedZhao-Baker, which is always conservative. Only at the highest value of the slope k do these methods fallbehind the Dirlik and improved Tovo-Benasciutti in in terms of accuracy.


6 Conclusion

The focus of this research was on the applicability of a multitude of frequency-domain methods in theautomotive industry, specifically, on the accelerated vibration test profiles. After a concise theoretical back-ground, the frequency-domain methods along with their equations were listed. While some of these methodshave already been compared side by side, others (more recent) were not. Furthermore, this research focuseson special type of spectra and real data is measured with experiment.

The frequency-domain methods are judged based on the reference time-domain method, incorporating rain-flow counting and Palmgren-Miner linear damage accumulation hypothesis. The most accurate method forthe typical automotive-industry accelerated test profiles is found to be the improved Zhao-Baker method(giving a conservative fatigue-damage estimation). Following are the improved Tovo-Benasciutti and Dirlikmethods. Results are unlike those obtained by Benasciutti and Tovo [18, 11], where Dirlik was proven mostaccurate.

Regarding the impact of material parameter k, describing the S-N slope, the results were in agreement withstudies by Bouyssy et al. [15] and Benasciutti and Tovo [18]. Estimate consistency is high for a relativelylow k (k ≈ 3) but error and inconsistencies arise as the slope gets steeper.

The results clearly show that the improved Tovo-Benasciutti and the improved Zhao-Baker methods shouldalso be considered together with the Dirlik method for the purpose of vibration-fatigue estimation.

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