A new modeling framework for fatigue damage of structural components under complex random spectrum

Available online at www.sciencedirect.com Available online at www.sciencedirect.com

ScienceDirect ScienceDirect

Structuralonline Integrity 00 (2019) 000–000 Available atProcedia www.sciencedirect.com Structural Integrity Procedia 00 (2019) 000–000

ScienceDirect

www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia

Procedia Structural Integrity 19 (2019) 528–537

Fatigue Design 2019 Fatigue Design 2019

A new modeling framework for fatigue damage of structural A new modeling framework for fatigue damage of structural components under complex random spectrum components under complex random spectrum Zhu Liaa, Ayhan Incea,b * Zhu Li , Ayhan Incea,b * 0F0F0F0F

0F0F0F0F

Purdue Polytechnic Institute, Purdue University, West Lafayette, Indiana, USA a Department of Mechanical, Industrial & Aerospace Engineering,Concordia University, Montreal, Purdue Polytechnic Institute, Purdue University, West Lafayette, Indiana, USA Quebec, Canada b Department of Mechanical, Industrial & Aerospace Engineering,Concordia University, Montreal, Quebec, Canada a

b

Abstract Abstract Time and frequency domains-based fatigue damage prediction approaches have been developed over past decades to predict Time frequency of domains-based fatigue damage prediction approaches have been developed over past to predict fatigueand performance mechanical structures subjected to random loads. Frequency domain approaches are decades increasingly being fatigue of mechanical structures subjected to random loads. Frequency approaches are increasingly adaptedperformance to provide fatigue assessment of mechanical components subjected to randomdomain loads due to computational efficiencybeing and adapted to provide fatigue assessment mechanical components to random loads due towhere computational efficiency and cost savings. Current frequency domainofdamage models only deal subjected with stationary random loadings Power Spectral Density cost savings. Current frequency damage only deal with loadings where Power Spectral (PSD) of random loadings doesdomain not change in models time. However, manystationary machine random components, such as jet engines and Density tracked (PSD) random loadings does not change in time. However, many machine components, such as jet engines and damage tracked vehiclesofare subjected to evolutionary PSD i.e. random-on-random loadings under real service loads. A new fatigue vehicles subjectedis to evolutionary PSDfatigue i.e. random-on-random loadings under realevolutionary service loads. new fatigue damage modelingare framework proposed to predict damage of structures under complex PSDAwhere the topology of modeling framework proposed to predict fatigue damageapproach of structures under evolutionary the topology of PSD function changesiswith time. The proposed modeling is based on complex the underlying conceptPSD that where the evolutionary PSD PSD function time. The proposed approach based PSDs. on the Each underlying concept thatinto the evolutionary PSD response of a changes structurewith can be decomposed intomodeling a finite number of is discrete PSD can be split narrow frequency response a structure can be decomposed a finitewith number of discrete PSDs. of Each PSDcycles. can beFatigue split into narrowcan frequency bands so of that each of narrowbands can be into associated Rayleigh distribution stress damage then be bands so that each of narrowbands associated with band Rayleigh of PSD stress function cycles. Fatigue then be predicted by summing up damagescan forbeeach individual and distribution each discrete on the damage basis ofcan a damage predicted by rule. summing up damages for each individual band and discrete PSD function the element basis ofmethod a damage accumulation The proposed modeling approach is numerically andeach experimentally validated by aonfinite and accumulation rule. The modeling approach experimentally validated by aapproach finite element method and experiments using threeproposed simplified structures made isofnumerically 5052-H32 and aluminum alloy. The proposed provides a more experiments using three simplified structures made offor5052-H32 aluminum alloy.ofThe proposed approach provides a more efficient and accurate modeling technique, and account complex random loadings structural components. efficient and accurate modeling technique, and account for complex random loadings of structural components. © 2019 The Authors. Published by Elsevier B.V. © 2019 The Authors. Published by Elsevier B.V. © 2019 Theunder Authors. Published by Elsevier B.V. Peer-review under responsibility of Fatigue the Fatigue Design 2019 Organizers. Peer-review responsibility of the Design 2019 Organizers. Peer-review under responsibility of the Fatigue Design 2019 Organizers. Keywords: Fatigue failure; accelerated test; power spectral density; frequency domain; random loading Keywords: Fatigue failure; accelerated test; power spectral density; frequency domain; random loading

* Corresponding author. Tel.: + 1 (514) 848 2424 address:author. [email protected] * E-mail Corresponding Tel.: + 1 (514) 848 2424 E-mail address: [email protected] 2452-3216 © 2019 The Authors. Published by Elsevier B.V. Peer-review©under the Fatigue Design 2019 2452-3216 2019responsibility The Authors. of Published by Elsevier B.V. Organizers. Peer-review under responsibility of the Fatigue Design 2019 Organizers.

2452-3216 © 2019 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Fatigue Design 2019 Organizers. 10.1016/j.prostr.2019.12.057

2

Zhu Li et al. / Procedia Structural Integrity 19 (2019) 528–537 Author name / Structural Integrity Procedia 00 (2019) 000–000

529

1. Introduction Frequency-based fatigue damage prediction methods are now widely adapted for fatigue damage assessment of mechanical components subjected to random loadings. Those structural components are increasingly used in more complex and harsh environments. The components are subjected to complex random loadings that are both difficult to model and expensive to test using current fatigue assessment methods [1]. Therefore, more accurate and efficient modeling approaches for the fatigue assessments of structures operating under extreme operational conditions are needed. Experimental vibrational test methods have been increasingly used in mechanical designs of machine systems [2]. Despite their recent widely usage, vibration experiments are regarded to be excessively expensive and timeconsuming [3,4]. Certain components such as a jet engine usually experience very high loading cycles i.e. more than a billion cycles and may require many months of laboratory tests under normal service conditions [1]. Therefore, accelerated fatigue test methods has been developed to reduce long test time and corresponding high costs [5,6,7,8,9]. Accelerated test methods employ exaggerated random load levels so that the testing time can be reduced into reasonable timeframe to yield equivalent fatigue damage. The vibrational random loadings are generally transformed in the frequency domain by a power spectral density (PSD) to conduct accelerated tests. The PSD loadings are usually described as a random Gaussian process [5,10]. The PSD can characterize the spread of the mean square vibration loadings over a frequency range [10]. Frequency-domain based fatigue damage approaches have been developed to provide fatigue life assessment of structures by connecting the relationship between the response PSD and fatigue damage [11,12,13]. The most of those approaches/ methods are currently based on the stationary Gaussian process represented a stationary PSD function (time invariant PSD function). The PSD are generally classified into narrow-band and wideband processes depending on a frequency band width of the PSD function. A probability density function (PDF) of loadings (e.g. stress ranges) for narrowband and wideband PSD functions can be associated to Rayleigh and Dirlik’s distribution [8] respectively. The narrow-band allows for a direct derivation of the cycle distribution, as pointed out by Lutes and Sarkani [14]. As for a wide-band process the relation of the peak distribution and cycle amplitudes can be determined by empirical solutions (e.g., Dirlik [15]). Dirlik [15] has proposed an empirical solution for the probability density function of rainflow stress ranges. The Dirlik method is based on first four moments of the PSD. In the Dirlik method, the PSD versus frequency data is used to find the first four moments of the PSD function and these four moments are used to determine the PDF of stress ranges. Then, fatigue life is obtained by a damage accumulation method, e.g. Palmgren–Miner’s rule [16]. The PDF of stress distributions in Eq. (1) can be derived directly from first moments of the PSD function. The Narrowband method is a modeling approach based on the single moment (𝑚𝑚0 ) of the PSD function which is used to estimate the fatigue damage. In the Narrow-band process, it is assumed that every peak is coincident with a cycle, so the amplitudes of cycles can be associated with Rayleighdistribution. Braccesi et al. [13] proposed a damage modeling criterion which is called Bands method. According to the Bands method, a given PSD function regardless of its shape, can be divided into several bands. If each divided bandwidth is sufficiently narrow, then it can be related to the Rayleigh distribution of the stress cycles. Total fatigue damage for the given PSD function can be determined by summing of the damages of each individual narrowband [17,18]. Despite substantial progress, current frequency based damage assessment methods are based on stationary PSD function where shape and topology of PSD function does not change in time. However, many real systems, such as jet engine, tracked vehicles, helicopters experience complex random loadings where PSD functions are time varying [19]. These types of complex random loadings are defined as random-on-random loadings. Analysis and simulation of structural components need to account for real complex random loadings in order to provide more accurate fatigue predictions results. Therefore, Zhu and Ince [20,21] recently proposed a novel damage modeling approach to deal with complex random-on-random loadings (non-stationary PSD loadings). The modeling approach proposed by the present authors is further discussed to assess the fatigue damage associated with non-stationary PSD loading environments. The proposed modeling approach is numerically and experimentally validated by a finite element method and experiments using three simplified structures made of 5052-H32 aluminum alloy.

530

Zhu Li et al. / Procedia Structural Integrity 19 (2019) 528–537 Author name / Structural Integrity Procedia 00 (2019) 000–000

3

2. Computational Modeling Approach The proposed modeling approach is schematically shown in Fig. 1. The modeling approach is based on the discretization concept that time varying response PSDs for a given structure can be computed from a time-varying input PSD function on the basis of discretization of the input PSD. The time-varying input PSD is formed by three swept narrowbands superimposed on one stationary wideband. The time varying input PSD can be decomposed into u number of finite discrete PSD positions and corresponding response PSDs can be computed from all discrete PSD positions. Each response PSD position is further divided into v number of narrow frequency band of PSD segments; The PSD segments are divided sufficiently narrow frequency bandwidth to assume that the process is narrowband one. Therefore, the stress distribution of each narrowband PSD segment can be characterized by the probability density function of a Rayleigh distribution. Different PSD narrowband segments can be associated with different Rayleigh distributions. The fatigue damage for each PSD narrowband segment can be then determined by the stresslife curve (S-N curve) on the basis of Palmgren–Miner’s rule. Finally, the total fatigue damage for each PSD position can be determined by summation of damage values of each narrowband segment. In order to calculate the Fatigue Damage Index (FDI) for all PSD position, the modeling procedure is repeated for each PSD position of the complete evolutionary response PSD.

Fig. 1. Schematic representation of computational modeling approach.

2.1. Input Power Spectrum Density Many machine systems experience complex random vibrations due to service operational conditions. Several standards such as MIL-STD 810 and AECTP 400, were developed to represent actual operational loading conditions experienced by machine systems such as helicopters, tracked vehicles, and jet engines [19]. The AECTP 400 mechanical environmental tests are obtained from field test data of various vehicle types. A subset of AECTP 400 standard, the B1 tracked vehicle test spectrum, describes a wideband random vibration superimposed by three sweeping high-amplitude narrowbands [19]. The B1 test spectrum is used for the modeling approach proposed in this paper because it accurately represents the complex vibration environment that is experienced by tracked vehicles. The B1 spectrum consists of three swept narrowbands and one stationary wideband random PSD. The first narrowband sweeps from 20 Hz to 170 Hz, the second one sweeps from 40 Hz to 340 Hz, and the third one sweeps from 60 Hz to 510 Hz. The sweep rate is defined within the range of one-half to one octave per minute. Table 1 shows the simplified sweep rate values (1 Hz/sec, 2 Hz/sec, and 3 Hz/sec for first, second, and third narrowbands, respectively) that are assumed to reduce the computational intensity of the model.

4

Zhu Li et al. / Procedia Structural Integrity 19 (2019) 528–537 Author name / Structural Integrity Procedia 00 (2019) 000–000

531

Table 1. Heavy vehicle schedule breakpoints [19]. Wideband Random Spectrum

Harmonic Swept Narrowbands

Frequency (Hz)

Amplitude

Narrowband

f1

f2

f3

Bandwidth (Hz)

5

10

15

5

( 𝑔𝑔2 /𝐻𝐻𝐻𝐻) 0.001

Swept BW (Hz)

20170

40-340

60-510

0.15

0.15

0.15

energy per frequency

20

0.01

510

0.01

Moving rate (Hz/sec)

1

2

3

2000

0.001

# Narrowbands Sweep Cycles

2

2

2

(𝑔𝑔2 /𝐻𝐻𝐻𝐻)

The PSD data based on combination of the wideband and three swept narrowbands in Table 1 is used an input PSD base excitation in the modeling approach. Fig. 2 shows a schematic representation of the input PSD function as a function of the time.

Fig. 2. Schematic representation of time-varying input PSD.

2.2. Response PSD Dynamic response of a simple structure subjected to random loads can be mathematically represented as a singledegree-of-freedom (SDOF) system provided that the first mode shape is dominant mode. The assumption of a SDOF can be employed to simplify the modeling approach [22,23]. The system equation of a damped SDOF is the equation of the base-excited function shown in Eq. (1): mÿ + 𝑐𝑐(𝑦𝑦̇ − 𝑥𝑥̇ ) + 𝑘𝑘(𝑦𝑦 − 𝑥𝑥) = 0

(1)

1 + (2𝜁𝜁𝜁𝜁)2 𝑌𝑌 Tf = | | = √ (1 − 𝛽𝛽2 )2 + (2𝜁𝜁𝜁𝜁)2 𝑋𝑋

(2)

Where m is the mass of the system, and c and k are the damping and elastic constants of the SDOF system, respectively. The transfer function expressed in Eq. (2) can be defined as a gain function for the base excitation.

Where damping ratio ζ =

f/fn .

c

ccr

, critical damping ccr = 2𝑚𝑚𝑓𝑓𝑛𝑛 , natural frequency fn2 = 𝑘𝑘/𝑚𝑚 and frequency ratio β =

The response PSD for a given specific input PSD can be computed by the transfer function of the defined SDOF

532

Zhu Li et al. / Procedia Structural Integrity 19 (2019) 528–537 Author name / Structural Integrity Procedia 00 (2019) 000–000

5

system. The input excitation is represented as the power spectrum density, Gxx (𝑓𝑓), and the output response PSD by Gyy (𝑓𝑓), the relationship between Gxx (𝑓𝑓) and Gyy (𝑓𝑓) is defined by Eq. (3) considering that both input and response PSDs time depended [20,21]. The schematic representation of time varying response PSD, Gyy (𝑓𝑓) is illustrated in Fig. 3. Gyy (𝑓𝑓) = 𝑇𝑇𝑓𝑓2 𝐺𝐺𝑥𝑥𝑥𝑥 (𝑓𝑓)

(3)

Similar to the input PSD, the response PSD can also be decomposed into u number of discrete PSD positions as shown in Fig. 3 [21]. The response PSD can be formulated in Eq. (4). Gyyi (𝑓𝑓) =

1 + (2𝜁𝜁𝜁𝜁)2 𝐺𝐺 (𝑓𝑓) (1 − 𝛽𝛽2 )2 + (2𝜁𝜁𝜁𝜁)2 𝑥𝑥𝑥𝑥𝑥𝑥

Where Gyyi (𝑓𝑓) is the i-th number of the response PSD and i = 1… u (a number of PSD positions).

(4)

The input PSD is comprised of the wideband (in a range of 5 Hz to 2000 Hz) and three narrowbands that have been swept within the wideband range as given in Table 1. Each response PSD position 𝐺𝐺𝑦𝑦𝑦𝑦𝑖𝑖 (𝑓𝑓) can be divided into a finite number of narrow frequency bands of the PSD segment [21]. Each segment represents a narrowband PSD Gj (𝑓𝑓) with the central frequency 𝑓𝑓𝑗𝑗 located. The PSD division thus transforms the given discrete PSD, G(f) into a finite set of narrowband PSD segments , Gj (𝑓𝑓), where j= 1, 2, … v [24]. To simplify the computational modeling formulation and reduce solution time, each PSD position is further divided into a narrow frequency band of 1 Hz. Therefore, each of the u PSD decompositions described by Eq. (4) can be further split into the v number of PSD narrowband segments as illustrated in Fig. 3.

Fig. 3. Division of a given response PSD into finite number of narrowbands.

2.3. Rayleigh Probability Distribution of Stresses If each discrete response PSD function is split into v number of narrowband PSD segments, and each narrowband PSD segment can be associated with a different Rayleigh distribution of stress cycles [20,21]. Each j-th narrowband segment can be defined by a central frequency fj (where j is the number of the narrowband PSD segment), results in a different cycle as a function of fj. The Rayleigh distribution representing the narrowband PSD segment can thus be used to determine stress amplitudes and cycles. Each of the v narrowband PSD segments can thus be characterized by a distinct Rayleigh distribution which defines both the stress amplitudes and a number of stress cycles as shown in Fig. 4. The applied cyclic stresses associated with each Rayleigh distribution can be further split into k distinct stress amplitudes (e.g. five different stress amplitude and corresponding cycles as shown in Fig.4). Sk represent the expectation of the applied stress amplitude for each stress region.

6

Zhu Li et al. / Procedia Structural Integrity 19 (2019) 528–537 Author name / Structural Integrity Procedia 00 (2019) 000–000

533

Fig. 4. Schematic representation to determine associated stress amplitudes and cycles from each narrowband PSD segment.

2.4. Fatigue Damage Summation If stress amplitudes and cycles are obtained, the fatigue damage can be predicted by the fatigue stress-life curve of the given material on the basis of damage accumulation [25,26,27,28,29,30]. In order to calculate fatigue damage for each of the v PSD segments, k number of different applied stresses and associated stress cycles were calculated for each PSD segment by discretizing the corresponding Rayleigh distribution. A number of cycles, n𝑘𝑘 and applied stresses, Sk , provides the necessary information to compute fatigue damages for each narrowband PSD segment; the number of cycles to failure, N for each narrowband PSD segment can be calculated from the traditional S-N curve in Eq. (5) and (6) as shown in Fig. 4. −𝑏𝑏 Nj,k = 𝐶𝐶𝑆𝑆𝑗𝑗,𝑘𝑘

(5)

Where 𝑁𝑁𝑗𝑗,𝑘𝑘 is the number of cycles to failure, 𝑆𝑆𝑗𝑗,𝑘𝑘 is the stress amplitude, b is the slope of the S-N curve, and C is a constant corresponding to the material constant. Additionally, k = 1…5 and j = 1… v (the number of PSD segments):

Different stress regions in the PDF of Rayleigh distribution have different probabilities, as indicated in Figs. 4 and 5. Applied stress cycles for each of the v PSD segments can be obtained by Eq. (6). (6) n𝑗𝑗,𝑘𝑘 = 𝑓𝑓𝑗𝑗 𝑇𝑇𝑗𝑗 𝑃𝑃𝑗𝑗,𝑘𝑘 Where 𝑓𝑓𝑗𝑗 is the central frequency of j-th response PSD band, T the time duration, and 𝑃𝑃𝑗𝑗,𝑘𝑘 represents the probability of the Rayleigh stress distributions for the v-th PSD segments. According to the cumulative damage rule i.e. Miner’s rule, fatigue damage for all PSD segments can be calculated by summing up the fatigue damages of each PSD segment as expressed in Eq. (7): 𝑣𝑣

5

d = ∑∑

𝑗𝑗=1 𝑘𝑘=1

𝑛𝑛𝑗𝑗,𝑘𝑘 Nj,k

Where, v is the number of the PSD segments. Additionally, d is considered as damage per a PSD position.

(7)

Since d is damage per a single discrete PSD position, after considering u number of discrete PSD positions, n𝑖𝑖,𝑗𝑗,𝑘𝑘

Zhu Li et al. / Procedia Structural Integrity 19 (2019) 528–537 Author name / Structural Integrity Procedia 00 (2019) 000–000

534

7

and Ni,j,k can be determined by introducing i index such that i=1… u. Thus, total fatigue damage index for u PSD positions can be calculated on the basis of Miner’s rule in Eq. (8). The total (fatigue damage index) FDI can be formulated as the summation of the damages that can be induced by each of the v PSD segments and each of the u PSD positions. According to Miner’s rule, total fatigue damage i.e. FDI is obtained by Eq. (8). 𝑢𝑢

𝑢𝑢

𝑣𝑣

5

D = ∑ 𝑑𝑑𝑖𝑖 = ∑ ∑ ∑ 4. Discussion and Results

𝑖𝑖=1

𝑖𝑖=1 𝑗𝑗=1 𝑘𝑘=1

𝑓𝑓𝑖𝑖,𝑗𝑗 𝑇𝑇𝑖𝑖 𝑃𝑃𝑗𝑗,𝑘𝑘

(8)

−𝑏𝑏 𝐶𝐶𝑆𝑆𝑖𝑖,𝑗𝑗,𝑘𝑘

The test bracket shown in Fig. 5 is designed such that two different small blocks (527 g, 254 g and 0 g) were mounted on the bracket to achieve three different natural frequency (40 Hz, 60 Hz and 112 Hz) of the brackets. The bracket material made of 5052-H32 aluminum alloys is selected, and monotonic and cyclic properties of the 5052H32 aluminum alloy are given in Table 2.

Fig. 5. bracket configuration with added weights to obtain natural frequencies of 40,60 and 112 Hz for three different bracket samples; (b) a schematic representation of sample brackets. Table 2. Material Properties of Aluminum 5052-H32. Monotonic properties Yield strength

193 MPa

Ultimate strength

228 MPa

Modulus of elasticity

70.3 GPa

Elastic Poisson’s ratio

0.33

Density

2.68 g/mm3

Cyclic properties Material constant of S-N curve, C

1348 MPa

Slope of S-N curve, b

-0.17

Modal tests were conducted on test bracket to determine a natural frequency of the bracket components as a part of the model validation. The computed natural frequencies obtained from the FE modal analysis agree well with the experimental frequencies obtained from the modal test as provided in Table 3. The FE modal results are compared with experimental frequencies in Table 3 and error values between the FEA and the test is found to be in range of approximately 4-7%. The PSD spectrum tests could not be carried out because of missing appropriate software package for the shaker controller to conduct the time-varying PSD testing. Therefore, FEA spectrum analysis was

8

Zhu Li et al. / Procedia Structural Integrity 19 (2019) 528–537 Author name / Structural Integrity Procedia 00 (2019) 000–000

535

performed to validate the computational model. The input PSD function consists of a wideband and three sweeping narrowbands. Because of the dynamic nature of these narrowbands, the resonance behavior of the structure is expected to change in time., the bracket undergoes through three distinct dynamic response behaviors as shown in Fig.6 for 112 Hz bracket. Three distinct sweeping narrowbands have the potential to induce significant fatigue damages when any of narrowbands correspond the natural frequency of the given brackets. Three different brackets were designed and analyzed in order to assess the effects of the different natural frequencies on fatigue damage the structural brackets. Table 3. Natural frequency comparison between modal test and FE modal analysis. Modal results

40 Hz

60 Hz

112 Hz

Modal test (Hz)

40.5

58.5

113.5

FE modal analysis (Hz)

42.6

60.8

121.0

Error (%)

5.2

3.9

6.6

Fig. 6. PSD response of 112 Hz bracket.

Fig. 7 shows a three-dimensional surface of the PSD response estimated by the computational model, as a function of the frequency and PSD position in time. The comparison of response PSDs obtained from both FEA and computational models is shown at five different discrete PSD positions in Fig. 7. Based on the PSD responses for three different brackets, PSD responses of the computation model are in good agreement with ones of the FEA method even though that the model slightly overestimates the PSD responses in higher ends of frequency ranges. The FDI values of three different brackets are calculated from both the computational model and FEA method in Table 4. The prediction error values of FDI between the FEA method and computational model is listed in Table 4. The relatively small FDI values reported in Table 4 are contributed to very low 1G stresses incurred by the test structure. Since the weight of the brackets designed in the scope of this paper were relatively very small, they were consequently subject to very low 1G stresses of around 2 MPa, i.e. very low PSD stresses. The FDI error calculated was less than 44% for the bracket with 40 Hz natural frequency, less than 28% for the bracket with 60 Hz natural frequency, and less than 5% for the bracket with 112 Hz natural frequency. Table 4. FDI comparison between the model and FEA. Natural frequency of bracket

40 Hz

60 Hz

112 Hz

FEA method

1.76.10-6

1.67.10-7

6.56.10-9

Computational model model FDI

1.23.10-6

1.31.10-7

6.88.10-9

Error (%) for FDI

43.79%

27.13%

-4.65%

Time varying PSD loadings result in a significantly greater fatigue damage in comparison to stationary PSD loadings [35-36]. Therefore, the Dirlik and narrow band damage methods [14-19] are not considered to be appropriate damage modeling methods for accurate fatigue damage analysis of the structures subjected to non-

Zhu Li et al. / Procedia Structural Integrity 19 (2019) 528–537 Author name / Structural Integrity Procedia 00 (2019) 000–000

536

9

stationary PSD loadings. A novel computational modeling approach developed by the present authors is employed to estimate fatigue damage of structural components subject to complex non-stationary random loadings.

Fig. 7. PSD responses between the model and FEA for (a) 40 Hz bracket; (b) 60 Hz bracket; and (c) 112 Hz bracket.

Since the computational model showed good correlation with FEA and experiments, the model can be extended to capabilities of simulating multiple degree of freedom systems. Furthermore, complex structures can be represented multi-degree-of freedom systems and their dynamic characteristics can be easily determined by FEA. Therefore, the proposed modeling approach can be integrated with commercial FE packages or fully implemented with finite element codes 4. Conclusions A computational modelling approach was developed to estimate fatigue damage of simplified structural components under complex random-on-random loadings. The modeling approach is based on the underlying concept that the time varying response PSD is decomposed into a finite number of discrete PSD functions. Each PSD function is further divided into narrow frequency bands so that each of narrowbands can be related to Rayleigh distribution of stress cycles. Then, fatigue damage for each of the Rayleigh distributions associated with individual narrowband segments is calculated by Palmgren–Miner’s rule., the cumulative FDI can be obtained by summing up damages of each of all discrete PSD positions. Three structural bracket designs with different natural frequencies were used for numerical FEA model to validate the computational model. The predcited results from the computational model showed a good correlation with the FEA simulation. The computed PSD responses yielded that the swept narrowbands in the input PSD is of considerable significance in the fatigue damage of the excited

10

Zhu Li et al. / Procedia Structural Integrity 19 (2019) 528–537 Author name / Structural Integrity Procedia 00 (2019) 000–000

537

structure as evidenced by the PSD response. The higher applied stress and increased cycle times under resonance induce greater fatigue damage to the structure. Designing the natural frequency of the structure to be far from the frequencies of the swept narrowband inputs can reduce fatigue damage, thus extending lifetime. The modeling method developed here can be easily adapted to optimize the design of engineering structural components in the early design phase prior to building prototypes for laboratory vibration tests. References [1]. Hoksbergen, J., “Defining the Global Error of a Multi-Axis Vibration Test”, Sound and Vibration 48(9):8-13, 2014. [2]. Habtour, E., Connon, W.S., Pohland, M.F, Stanton, S.C., Paulus, M., and Dasgupta, A., “Review of response and damage of linear and nonlinear systems under multiaxial vibration”, Shock and Vibration, 2014: Article ID 294271, 21 pages, 2014. [3]. Steinwolf, A., “Shaker random testing with low kurtosis: Review of the methods and application for sigma limiting” Shock and Vibration, 17(3): 219-231, 2010. [4]. Rotem, A., "Accelerated fatigue testing method." International Journal of Fatigue, 3(4): 211-215, 1981. [5]. Allegri, G. and Zhang, X., "On the inverse power laws for accelerated random fatigue testing", International Journal of Fatigue, 30(6): 967977, 2008. [6]. Özsoy, S., Celik. M. and Kadıoğlu, F.S., "An accelerated life test approach for aerospace structural components", Engineering failure analysis, 15(7): 946-957, 2008. [7]. Shires, D., "On the time compression (test acceleration) of broadband random vibration tests", Packaging Technology and Science, 24(2): 75-87, 2011. [8]. Xu, K., Wu, Y., and Wu, Q., "Development of vibration loading profiles for accelerated durability tests of ground vehicles", PhD diss, University of Manitoba, 2011. [9]. Eldoǧan, Y., and Cigeroglu, E., "Vibration fatigue analysis of a cantilever beam using different fatigue theories", In Topics in Modal Analysis, 7:471-478, 2014. [10]. Wolfsteiner, P., and Sedlmair, S., "Deriving gaussian fatigue test spectra from measured non gaussian service spectra", Procedia Engineering, 101: 543-551, 2015. [11]. Lutes, L.D., and Larsen, C.E., “An improved spectral method for variable amplitude fatigue prediction”, J Struct Div. ASCE, 116:1149–64, 1990. [12]. Wirsching, P.H., Light, M.C., “Fatigue under wide band random stresses”, J Struct Div. ASCE, 106(ST7):1593–607, 1980. [13]. Braccesi, C., Cianetti, F., Lori, G., Pioli, D., “Fatigue behaviour analysis of mechanical components subject to random bimodal stress process frequency domain approach”, Int J Fatigue 27(4):335–45, 2005. [14]. Lutes, L.D., and Sarkani, S., “Random vibrations: analysis of structural and mechanical systems”, Butterworth-Heinemann, 2004. [15]. Dirlik, T., “Application of computers in fatigue analysis, Doctoral dissertation, University of Warwick, 1985. [16]. Miner, M.A., “Cumulative damage in fatigue”, J Appl Mech, 12:159–64, 1945. [17]. Benasciutti, D., Braccesi, C., Cianetti, F., Cristofori, A., Tovo, R., “A Fatigue damage assessment in wide-band uniaxial random loadings by PSD decomposition: outcomes from recent research”, International Journal of Fatigue, 91:248–250, 2016. [18]. Benasciutti, D., and Tovo, R., "Fatigue life assessment in non-Gaussian random loadings", International journal of fatigue, 28(7): 733-746, 2006. [19]. NATO/PFP UNCLASSIFIED, "AECTP 400 (Edition 3) – MECHANICAL ENVIRONMENTAL TESTS", 2006 [20]. Li, Z., Ince, A., and Lacombe, J., Fatigue Damage Modeling Approach Based on Evolutionary Power Spectrum Density, SAE Technical Paper, 2019-01-0524, 2019, https://doi.org/10.4271/2019-01-0524. [21]. Li, Z. and Ince, A., A unified frequency domain fatigue damage modeling approach for random-on-random spectrum, International Journal of Fatigue, 124:123-137, 2019, https://doi.org/10.1016/j.ijfatigue.2019.02.032. [22]. Gatti, P. L., “Applied Structural and Mechanical Vibrations: Theory and Methods”, CRC Press, 2014. [23]. Crandall, S. H., and William D. M., “Random vibration in mechanical systems”, Academic Press, 2014. [24]. Braccesi, C., Cianetti, F., Tomassini, L., “Random fatigue. A new frequency domain criterion for the damage evaluation of mechanical components”, Int J Fatigue, 70:417–427, 2015. [25]. Ince, A., “A mean stress correction model for tensile and compressive mean stress fatigue loadings”, Fatigue and Fracture of Engineering Materials and Structures, 40(6): 939-948, 2017. [26]. Ince, A., “A generalized mean stress correction model based on distortional strain energy”, International Journal of Fatigue, , 104: 273-282, 2017. [27]. Ince, A., “A novel technique for multiaxial fatigue modelling of ground vehicle notched components”, International Journal of Vehicle Design”, 67: 294-313, 2015. [28]. Ince, A., and Glinka, G. “A Numerical Method for Elasto-Plastic Notch-Root Stress-Strain Analysis”, Journal of Strain Analysis for Engineering Design, 48(4):229-224, 2013, doi:10.1177/0309324713477638. [29]. Ince A. and Glinka G. “A generalized damage parameter for multiaxial fatigue life prediction under proportional and non-proportional loadings”, International Journal of Fatigue 62:34-41, doi:10.1016/j.ijfatigue.2013.10.007. [30]. Ince, A. and Glinka, G. “Innovative Computational Modeling of Multiaxial Fatigue Analysis for Notched Components”, International Journal of Fatigue, 82(2):134-145, 2016, doi:10.1016/j.ijfatigue.2015.03.019.